Zengtao Chen Abdolhamid Akbarzadeh

Advanced Thermal Stress Analysis ofSmart Materials and Structures

Zengtao ChenAbdolhamid Akbarzadeh

Structural Integrity 10Series Editors: José A. F. O. Correia · Abílio M. P. De Jesus

Structural Integrity

Volume 10

Series Editors

José A. F. O. Correia, Faculty of Engineering, University of Porto, Porto, PortugalAbílio M. P. De Jesus, Faculty of Engineering, University of Porto, Porto, Portugal

Advisory Editors

Majid Reza Ayatollahi, School of Mechanical Engineering, Iran University ofScience and Technology, Tehran, IranFilippo Berto, Department of Mechanical and Industrial Engineering, Faculty ofEngineering, Norwegian University of Science and Technology, Trondheim,NorwayAlfonso Fernández-Canteli, Faculty of Engineering, University of Oviedo, Gijón,SpainMatthew Hebdon, Virginia State University, Virginia Tech, Blacksburg, VA, USAAndrei Kotousov, School of Mechanical Engineering, University of Adelaide,Adelaide, SA, AustraliaGrzegorz Lesiuk, Faculty of Mechanical Engineering, Wrocław University ofScience and Technology, Wrocław, PolandYukitaka Murakami, Faculty of Engineering, Kyushu University, Higashiku,Fukuoka, JapanHermes Carvalho, Department of Structural Engineering, Federal University ofMinas Gerais, Belo Horizonte, Minas Gerais, BrazilShun-Peng Zhu, School of Mechatronics Engineering, University of ElectronicScience and Technology of China, Chengdu, Sichuan, China

The Structural Integrity book series is a high level academic and professional seriespublishing research on all areas of Structural Integrity. It promotes and expeditesthe dissemination of new research results and tutorial views in the structuralintegrity field.

The Series publishes research monographs, professional books, handbooks,edited volumes and textbooks with worldwide distribution to engineers,researchers, educators, professionals and libraries.

Topics of interested include but are not limited to:

– Structural integrity– Structural durability– Degradation and conservation of materials and structures– Dynamic and seismic structural analysis– Fatigue and fracture of materials and structures– Risk analysis and safety of materials and structural mechanics– Fracture Mechanics– Damage mechanics– Analytical and numerical simulation of materials and structures– Computational mechanics– Structural design methodology– Experimental methods applied to structural integrity– Multiaxial fatigue and complex loading effects of materials and structures– Fatigue corrosion analysis– Scale effects in the fatigue analysis of materials and structures– Fatigue structural integrity– Structural integrity in railway and highway systems– Sustainable structural design– Structural loads characterization– Structural health monitoring– Adhesives connections integrity– Rock and soil structural integrity

Springer and the Series Editors welcome book ideas from authors. Potentialauthors who wish to submit a book proposal should contact Dr. Mayra Castro,Senior Editor, Springer (Heidelberg), e-mail: [email protected]

More information about this series at http://www.springer.com/series/15775

mailto:[email protected]

http://www.springer.com/series/15775

Zengtao Chen • Abdolhamid Akbarzadeh

Advanced Thermal StressAnalysis of Smart Materialsand Structures

123

Zengtao ChenDepartment of Mechanical EngineeringUniversity of AlbertaEdmonton, Canada

Abdolhamid AkbarzadehDepartment of Bioresource EngineeringMcGill UniversityIsland of Montreal, QC, Canada

Department of Mechanical EngineeringMcGill UniversityMontreal, QC, Canada

ISSN 2522-560X ISSN 2522-5618 (electronic)Structural IntegrityISBN 978-3-030-25200-7 ISBN 978-3-030-25201-4 (eBook)https://doi.org/10.1007/978-3-030-25201-4

© Springer Nature Switzerland AG 2020This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors, and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

https://doi.org/10.1007/978-3-030-25201-4

Preface

Advanced smart materials and structures are under rapid development to meetengineering challenges for multifunctionality, high reliability, and high efficiency inmodern technology. Advanced manufacturing technology calls for high-energy,non-contact, laser-forming materials in a sophisticated spatial and temporal envi-ronment. Thermal stress analysis of advanced materials offers a viable tool foroptimized design of advanced, multifunctional devices and for accurate modellingof advanced manufacturing processes.

In classical thermal analysis, thermal stress is caused by the constrained defor-mation when a temperature variation occurs in an elastic body. How the materialreaches the final temperature from the initial temperature will not affect the cal-culation of the steady-state or quasi-static thermal stresses. Classical Fourier heatconduction theory is widely used in the thermal stress analysis leading to perfectand trustworthy results. In transient and high-temperature gradient cases, whenmaterials experience sudden changes in temperature within an extremely short time,the reaction to this ultrafast, temporal temperature changes or heat flux would beexpected to have a time delay since heat propagation takes time to occur. This delaymight not be felt for most metallic materials as relaxation time of metals is in therange of 10–8–10–14 s, opposed to soft and organic materials with a relaxation timebetween 1 s and 10 s, where the delay is not negligible and the subsequent thermalwave propagation is evident. Non-Fourier, time-related heat conduction modelshave been proposed to compensate the effect of this delay in a heat transfer process.A natural outcome of the non-Fourier heat conduction models is the wave-like heatconduction equation, where a thermal wave is required to spread the heat. Thermalstress analysis in advanced materials based on the non-Fourier heat conductiontheories has become a popular topic in the past decades. The authors of this bookare among the researchers who initiated and continued working on promoting theresearch in the area of thermal stresses in advanced smart materials. Although theliterature in this area is booming in recent years, a single-volume monographsummarizing the recent progress in implementing non-Fourier heat conductiontheories to deal with the multiphysical behaviour of smart materials and structures isstill missing in the literature.

v

This monograph is a collection of the research on advanced thermal stressanalysis of advanced and smart materials and structures mainly written by theauthors of this book, and their students and colleagues. This book is organized intoseven chapters. Chapter 1 provides a brief introduction to the non-Fourier heatconduction theories, including the Cattaneo–Vernotte (C–V), dual-phase-lag(DPL), three-phase-lag (TPL) theory, fractional phase-lag, and non-local heatconduction theories. Chapter 2 introduces the fundamental of thermal wave char-acteristics by reviewing different methods for solving non-Fourier heat conductionproblems in representative homogenous and heterogeneous advanced materials.Chapter 3 provides the fundamentals of smart materials and structures, includingthe background, application, and governing equations. In particular, functionallygraded smart structures are introduced as they represent the recent development inthe industry; a series of uncoupled thermal stress analyses on one-dimensionalsmart structures are also presented. Chapter 4 presents coupled thermal stressanalyses in one-dimensional, homogenous and heterogeneous, smart piezoelectricstructures considering alternative coupled thermopiezoelectric theories. Chapter 5introduces a generalized method to deal with plane crack problems in smartmaterials and structures based on classical Fourier heat conduction. Thermal frac-ture analysis of cracked structures based on non-Fourier heat conduction theories ispresented in Chap. 6. Finally, Chap. 7 lists a few perspectives on the futuredevelopments in non-Fourier heat conduction and thermal stress analysis.

We sincerely thank our students, colleagues, and friends for their contributionsin preparation of this book. We are indebted to our families for their sacrifice,patience, and constant support during the composition of this book.

Edmonton, Canada Zengtao ChenMontreal, Canada Abdolhamid AkbarzadehApril 2019

vi Preface

Contents

1 Heat Conduction and Moisture Diffusion Theories . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Fourier Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Non-Fourier Heat Conduction . . . . . . . . . . . . . . . . . . . . . 7

1.3 Moisture Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 Fickian Moisture Diffusion . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Non-Fickian Moisture Diffusion . . . . . . . . . . . . . . . . . . . 18

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Basic Problems of Non-Fourier Heat Conduction . . . . . . . . . . . . . . . 232.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Laplace Transform and Laplace Inversion . . . . . . . . . . . . . . . . . . 23

2.2.1 Fast Laplace Inverse Transform . . . . . . . . . . . . . . . . . . . 242.2.2 Reimann Sum Approximation . . . . . . . . . . . . . . . . . . . . . 252.2.3 Laplace Inversion by Jacobi Polynomial . . . . . . . . . . . . . 25

2.3 Non-Fourier Heat Conduction in a Semi-infinite Strip . . . . . . . . . . 262.4 Nonlocal Phase-Lag Heat Conduction in a Finite Strip . . . . . . . . . 31

2.4.1 Molecular Dynamics to Determine CorrelatingNonlocal Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Three-Phase-Lag Heat Conduction in 1D Strips, Cylinders,and Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.1 Effect of Bonding Imperfection on Thermal Wave

Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.5.2 Effect of Material Heterogeneity on Thermal Wave

Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5.3 Thermal Response of a Lightweight Sandwich Circular

Panel with a Porous Core . . . . . . . . . . . . . . . . . . . . . . . . 502.6 Dual-Phase-Lag Heat Conduction in Multi-dimensional Media . . . 53

vii

2.6.1 DPL Heat Conduction in Multi-dimensional CylindricalPanels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.6.2 DPL Heat Conduction in Multi-dimensional SphericalVessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3 Multiphysics of Smart Materials and Structures . . . . . . . . . . . . . . . . 653.1 Smart Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.1.1 Piezoelectric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 673.1.2 Magnetoelectroelastic Materials . . . . . . . . . . . . . . . . . . . . 693.1.3 Advanced Smart Materials . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Thermal Stress Analysis in Homogenous Smart Materials . . . . . . . 743.2.1 Solution for the a Thermomagnetoelastic

FGM Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.2.2 Solution for Thermo-Magnetoelectroelastic

Homogeneous Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 823.2.3 Benchmark Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.3 Thermal Stress Analysis of Heterogeneous Smart Materials . . . . . . 903.3.1 Solution Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.2 Benchmark Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.4 Effect of Hygrothermal Excitation on One-Dimensional SmartStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.4.1 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.4.2 MEE Hollow Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.4.3 MEE Solid Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.4.4 Benchmark Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4 Coupled Thermal Stresses in Advanced Smart Materials . . . . . . . . . 1194.1 Functionally Graded Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.2 Hyperbolic Coupled Thermopiezoelectricity

in One-Dimensional Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.2.2 Homogeneous Rod Problem . . . . . . . . . . . . . . . . . . . . . . 1224.2.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.3 Hyperbolic Coupled Thermopiezoelectricity in CylindricalSmart Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.3.2 Hollow Cylinder Problem . . . . . . . . . . . . . . . . . . . . . . . . 1354.3.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.4 Coupled Thermopiezoelectricity in One-DimensionalFunctionally Graded Smart Materials . . . . . . . . . . . . . . . . . . . . . . 146

viii Contents

4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.4.2 The Functionally Graded Rod Problem . . . . . . . . . . . . . . 1474.4.3 Solution Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1564.4.5 Introduction of Dual Phase Lag Models . . . . . . . . . . . . . 1614.4.6 Results of Dual Phase Lag Model Analysis . . . . . . . . . . . 163

4.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5 Thermal Fracture of Advanced Materials Based on FourierHeat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.2 Extended Displacement Discontinuity Method and Fundamental

Solutions for Thermoelastic Crack Problems . . . . . . . . . . . . . . . . 1715.2.1 Fundamental Solutions for Unit Point Loading

on a Penny-Shaped Interface Crack . . . . . . . . . . . . . . . . . 1745.2.2 Boundary Integral-Differential Equations

for Interfacial Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.2.3 Stress Intensity Factor and Energy Release Rate . . . . . . . 194

5.3 Interface Crack Problems in Thermopiezoelectric Materials . . . . . . 1975.3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1985.3.2 Fundamental Solutions for Unit-Point Extended

Displacement Discontinuities . . . . . . . . . . . . . . . . . . . . . 2005.3.3 Boundary Integral-Differential Equations for an

Interfacial Crack in Piezothermoelastic Materials . . . . . . . 2045.3.4 Hyper-Singular Integral-Differential Equations . . . . . . . . . 2065.3.5 Solution Method of the Integral-Differential

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2105.3.6 Extended Stress Intensity Factors . . . . . . . . . . . . . . . . . . 219

5.4 Fundamental Solutions for MagnetoelectrothermoelasticBi-Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5.5 Fundamental Solutions for Interface Crack Problemsin Quasi-Crystalline Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255.5.1 Fundamental Solutions for Unit-Point EDDs . . . . . . . . . . 228

5.6 Application of General Solution in the Problemof an Interface Crack of Arbitrary Shape . . . . . . . . . . . . . . . . . . . 234

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Contents ix

6 Advanced Thermal Fracture Analysis Based on Non-FourierHeat Conduction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2436.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2436.2 Hyperbolic Heat Conduction in a Cracked Half-Plane

with a Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2436.2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.2.2 Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2476.2.3 Temperature Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . 2516.2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

6.3 Thermoelastic Analysis of a Partially Insulated Crackin a Strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2556.3.1 Definition of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 2566.3.2 Thermal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2596.3.3 Asymptotic Stress Field Near Crack Tip . . . . . . . . . . . . . 2646.3.4 Numerical Results and Discussions . . . . . . . . . . . . . . . . . 266

6.4 Thermal Stresses in a Circumferentially Cracked HollowCylinder Based on Memory-Dependent Heat Conduction . . . . . . . 2696.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2706.4.2 Thermal Axial Stress in an Un-cracked

Hollow Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.4.3 Thermal Stress in the Axial Direction . . . . . . . . . . . . . . . 2746.4.4 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 283

6.5 Transient Thermal Stress Analysis of a Cracked Half-Planeof Functionally Graded Materials . . . . . . . . . . . . . . . . . . . . . . . . . 2846.5.1 Formulation of the Problem and Basic Equations . . . . . . . 2876.5.2 Solution of the Temperature Field . . . . . . . . . . . . . . . . . . 2906.5.3 Solution of Thermal Stress Field . . . . . . . . . . . . . . . . . . . 2936.5.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . 296

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

7 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037.1 Heat Conduction Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037.2 Application in Advanced Manufacturing Technologies . . . . . . . . . 304

x Contents

Chapter 1Heat Conduction and MoistureDiffusion Theories

1.1 Introduction

The design of high performance micro/macro-scale composite structures working athigh temperature and humidity environmental conditions needs an accurate heatand moisture transfer analysis through the solid structure. The hygrothermaldeformation, developed by temperature and moisture distribution, under adverseoperating conditions degrades the structural integrity and results in lowering thestructural stiffness and strength. While increasing temperature could majorly inducethermal stresses, more fluid or moisture could be absorbed into voids and micro-scopic defects in the solids, specifically in composites with polymeric matrices,which results in additional moisture induced stresses [1, 2].

Heat is defined as the energy transport within a body from hotter regions tocooler regions according to the second law of thermodynamics. This energy couldbe provided by the constituent particles such as atoms, molecules, or free electrons.While the heat flow cannot be measured, there exists a measurable and macroscopicquantity known as temperature. Temperature can be interpreted as the combinedeffect of all kinetic energies of a large number of molecules in solid, liquid, orgaseous state. Conduction, convection, and radiation are three distinct modes ofheat transfer. In a conduction process, the heat passes through the materials bymicroscopic diffusion and collision of constituent particles. The process of trans-ferring heat by a moving fluid and a relative motion of a heated body is calledconvection [3, 4]. The energy could also be emitted directly between the distantportions of a body via the electromagnetic radiation. Since the convection andradiation are usually negligible in solids, the heat conduction is merely consideredin this book for thermal analysis except for the boundary conditions, which could beany type of heat transfer modes. Since the process of moisture transfer is basicallythe same as the heat transfer, the Fourier and non-Fourier heat conduction theoriesare first introduced in detail in Sect. 1.2 and then some of the moisture diffusionmodels are reviewed in Sect. 1.3.

© Springer Nature Switzerland AG 2020Z. T. Chen and A. H. Akbarzadeh, Advanced Thermal Stress Analysisof Smart Materials and Structures, Structural Integrity 10,https://doi.org/10.1007/978-3-030-25201-4_1

1

http://crossmark.crossref.org/dialog/?doi=10.1007/978-3-030-25201-4_1&domain=pdf



https://doi.org/10.1007/978-3-030-25201-4_1

1.2 Heat Conduction

Heat conduction is encountered in many engineering applications, such as elec-tronic packaging, casting, food processing machines, biomedical devices, andthermal shield design for aerospace vehicles [5]. The relation between the heat fluxvector q and temperature gradient rT is called the constitutive relation of heat flux.The heat flux, q [W/m2], is defined as the heat flow Q per unit time and per unitnormal vector of the area of an isothermal surface [6]. The heat conduction equationis then established by using the constitutive relation of heat flux and the energyconservation equation or first law of thermodynamics. The classical Fourier andnon-Fourier heat conduction theories are two main categories of thermal analysisdiscussed in this chapter.

1.2.1 Fourier Heat Conduction

The earliest constitutive relation of heat flux was proposed by Fourier in 1807,which based on experimental observations, assume that the heat flux and temper-ature gradient occur at the same instant of time. The Fourier heat conductionspecifies the proportionality between the heat flux and temperature gradient asfollows [7]:

qðx; tÞ ¼ �kTrTðx; tÞ ð1:1Þ

where kT [W/(m K)] is thermal conductivity coefficient and x and t, respectively,stand for the general coordinate of a material point and time. While kT is a positivescalar quantity for isotropic materials, kT should be replaced by the components of asecond-order tensor of thermal conductivity for anisotropic media [6]. The statetheorem and second law of thermodynamics prove that kT should be positivedefinite and a function of pressure and temperature (two independent properties) [8,9]. In a homogeneous medium, kT is constant through the body, while it is afunction of position in a heterogeneous material kTðxÞð Þ.

To derive the differential equation of heat conduction, the first and second lawsof thermodynamics should be utilized. As elucidated in [10, 11], the summation ofthe first variation of absorbed heat by a medium (Q) and the work done on themedium (W) is equal to the differential of internal energy change (U):

dQþ dW ¼ dU ð1:2Þ

where the first variation operator d is path dependent in contrast with the totalderivative operator d. The internal energy of a medium could be obtained by thesummation of the kinetic energy K and the intrinsic energy I as:

2 1 Heat Conduction and Moisture Diffusion Theories

U ¼ IþK ð1:3Þ

Whilst the first law of thermodynamics quantitatively describes the energytransport process, the second law of thermodynamics defines the direction of energytransfer. When a thermodynamic system accomplish a cycle, the Clausius inequalityreads according to the second law of thermodynamics [10, 12]:

ZdQT

� 0 ð1:4Þ

where T [K] stands for the absolute temperature. For an irreversible process, the lefthand side of the inequality is always negative, while it is zero for a reversibleprocess. Two independent properties of temperature and entropy define the ther-modynamic state of a system. Using Eq. (1.4), the entropy change of a reversiblecycle is defined as [13, 14]:

dS ¼ dQT

ð1:5Þ

The first and second law of thermodynamics are reduced to the following energyand entropy equations for a reversible thermodynamic process [10]:

�r:qðx; tÞþ qrTðx; tÞ ¼ qT0@sðx; tÞ

@tð1:6aÞ

qsðx; tÞ ¼ qcT0

Tðx; tÞ � T0ð Þ ð1:6bÞ

where rT ½W/kg� and s [J/kg K] are the heat generation per unit time per unit massand entropy per unit mass, respectively. The density, specific heat, initial temper-ature, and divergence operator are, respectively, represented by q ½kg/m3�, c [J/(kg K)], T0, and r. It is worth mentioning that the Fourier heat conductionEq. (1.1) and the first and second laws of thermodynamics in Eq. (1.6) needmodifications for the heat transport analysis of thermoelectrics. For thermoelec-tricity analysis, the heat loss from thermoelectric couples to interstitial gas isconsidered along with the Peltier effect, which is the contribution of electric currentdensity in heat flux and the first law of thermodynamics [15, 16].

Differential equation of transient Fourier heat conduction is achieved by sub-stituting Eqs. (1.1) and (1.6b) into Eq. (1.6a):

r: kTrTðx; tÞ� �þ qrTðx; tÞ ¼ qc@Tðx; tÞ

@tð1:7Þ

For steady-state analysis, the right-hand side of Eq. (1.7) is zero and the heatgeneration rate rT is omitted. Moreover, for a homogenous isotropic material, when

1.2 Heat Conduction 3

the thermal conductivity is assumed independent of temperature, Eq. (1.7) is sim-plified as [17, 18]:

r2Tðx; tÞþ qrTðx; tÞkT

¼ 1aTðx; tÞ ð1:8Þ

where a ¼ kTqc ½m2=s� is thermal diffusivity andr2 represents the Laplacian operator.

The typical mathematical operators used in the multiphysics analysis are given inTable 1.1 for different coordinate systems.

The boundary and initial conditions should be specified along with the differ-ential equation of heat conduction to obtain the temperature distribution within amedium. The number of boundary conditions on the spatial domain and number ofinitial condition in the time domain are equal to the order of the highest derivativeof a variable in the governing equation with respect to the space and time domain[10, 20].

1.2.1.1 Thermal Boundary Conditions

The typical thermal boundary conditions, encountered in thermal analysis, could becategorized as: (a) prescribed temperature, (b) prescribed heat flux, and (c) heattransfer by convection and irradiation [4, 10, 17]:

(a) The temperature along the boundary surface (S) of a medium is prescribed as:

Tðx; tÞjx¼S ¼ TðS; tÞ ð1:9Þ

where T is generally a known function of position and time.

(b) The heat flux across the boundary surface is known as a function of positionand time:

kTrTðx; tÞjx¼S ¼ �qSðS; tÞ ð1:10Þ

where qS is the prescribed heat flux transferred toward the boundary surface. Theplus or minus signs on the right-hand-side of Eq. (1.10) depends on the direction ofthe heat flux transferred from or to the surface. An insulated boundary condition is aspecial form of Eq. (1.10) where qS ¼ 0.


Tab

le1.1

Mathematical

operatorsin

differentcoordinates[19]

Coo

rdinate

Operator

Cartesian

Gradient:r

¼@ @xe xþ

@ @ye yþ

@ @ze z

Laplacian:r2

¼@2

@x2þ

@2

@y2þ

@2

@z2

Divergence:

r:A¼

@Ax

@xe xþ

@Ay

@ye yþ

@Az

@ze z

Curl:r

�A¼

@Az

@y�

@Ay

@z

�� e x

þ@Ax

@z�

@Az

@x

�� e y

þ@Ay

@x�

@Ax

@y

�� e z

Cylindrical

Gradient:r

¼@ @re rþ

1 r@ @he h

þ@ @ze z

Laplacian:r2

¼1 r@ @rr

@ @r�� þ

1 r2@2

@h2þ

@2

@z2

Divergence:

r:A¼

1 r@ @rArr

ðÞþ

1 r@Ah

@hþ

@Az

@z

Curl:r

�A¼

1 r@Az

@h�

@Ah

@z

�� e r

þ@Ar

@Az�

@Az

@r

�� e h

þ1 r@Ahr

ðÞ

@r

�1 r@Ar

@h

�� e z

Spherical

Gradient:r

¼@ @re rþ

1 r@ @he h

þ1

rsinh

@ @/e /

Laplacian:r2

¼1 r2

@ @rr2

@ @r

�� þ

1r2

sinh@ @hsinh

@ @h

�� þ

1r2

sinh

@2

@/2

Divergence:

r:A¼

1 r2@

r2Ar

ðÞ

@r

þ1

rsinh@Ahsinh

ðÞ

@h

þ1

rsinh@A/

@h

Curl:r

�A¼

1rsinh

@A/sinh

ðÞ

@h

�@Ah

@/

�� e r

þ1

rsinh

@Ar

@/�sinh@

rA/

ðÞ

@r

�� þ

1 r@rA

hð

Þ@r

�@Ar

@h

hi e z


(c) The convection heat transfer with ambient and radiation heat exchange with aradiator happens at the boundary surface:

kTrTðx; tÞjx¼S ¼ � h Tðx; tÞ � T1ð Þþ er Tðx; tÞ � T4r

� �� x¼S ð1:11Þ

where h, T1, and Tr are convection heat transfer coefficient, ambient temperature,and the temperature of radiative body, respectively. The Stefan-Boltzmann constantr has the value of r ¼ 5:67040� 0:00004ð Þ10�8 [W/m2 K4] and the emissivity ofe is always lower or equal to one e� 1ð Þ [20, 21].

1.2.1.2 Thermal Initial Conditions

Time derivatives of temperature appear in the differential equation of transient heatconduction. Therefore, the initial conditions are required to be specified fortime-dependent thermal analysis problems. Regarding the first-order derivative oftemperature in the Fourier heat conduction Eq. (1.7), the initial condition could bespecified as:

T x; t ¼ 0ð Þ ¼ T0ðxÞ ð1:12Þ

where T0ðxÞ is a known function of the spatial coordinate x. The initial conditionsfor higher-order time derivatives of temperature should also be specified fornon-Fourier heat conduction.

1.2.1.3 Thermal Interfacial Conditions

When a medium is composed of bonding the layers of dissimilar materials, thermalinterfacial conditions should be also specified along with the boundary and initialconditions. The bonding of two materials typically contains imperfection, such as

Fig. 1.1 Thermal boundaryconditions at the interface oftwo dissimilar materials


small voids and defects. The following thermal interfacial conditions need to beconsidered at the interface of two layers with thermally weak conduction (Fig. 1.1)[22, 23]:

T ð1Þðx; tÞ � T ð2Þðx; tÞ� ��

x¼S¼ RTqð1ÞðS; tÞ ð1:13aÞ

qð1ÞðS; tÞ ¼ qð2ÞðS; tÞ ð1:13bÞ

where RT is thermal compliance constant or thermal contact resistance for theimperfect interface. The perfectly bonded interfaces could achieved by substitutingRT ¼ 0 in Eq. (1.13a).

1.2.2 Non-Fourier Heat Conduction

The conventional heat conduction theory based on the classical Fourier law admitsan infinite speed for thermal wave propagation due to the parabolic-type partialdifferential equation of Fourier heat conduction [Eq. (1.7)]. Fourier law assumes aninstantaneous thermal response and a quasi-equilibrium thermodynamic condition,which implies that a thermal disturbance is felt instantaneously at all spatial pointswithin a medium. The classical diffusive-like theory has been widely used inmacroscopic heat transfer problems; however, the heat transmission has beenobserved to be a non-equilibrium phenomenon with a finite thermal wave speed forapplications involving very low temperature, extremely high temperature gradient,ultrafast laser heating, ballistic heat transfer, and micro temporal and spatial scales[16, 24, 25]. For example, it has been observed that the measured surface tem-perature of a slab immediately after an intense thermal shock is 300 °C higher thanthe temperature obtained by Fourier’s law [26, 27]. As a result, several non-Fourierheat conduction theories have been developed to eliminate the drawbacks of Fourierheat conduction while predicting the thermal wave behaviour, referred as secondsound, and incorporate the microstructural effects on the heat transport.

1.2.2.1 Cattaneo-Vernotte Heat Conduction

The simplest hyperbolic non-Fourier heat conduction theory was proposed byCattaneo [28] and Vernotte [29] (C-V model) to achieve a finite thermal wave speedfor heat propagation. The thermal relaxation time was introduced in Fourier’s lawof heat conduction as follows:

qðx; tÞþ sq@qðx; tÞ

@t¼ �kTrTðx; tÞ ð1:14Þ


where sq ½s�, a non-negative parameter, is the time delay called the thermal relax-ation time. The thermal relaxation time could be mathematically interpreted as thetime delay between heat flux vector and temperature gradient in the fast-transienttransport, or micro-structurally interpreted as the time when the intrinsic lengthscales in diffusion and the thermal wave becomes equal [16, 24]. It could becalculated in terms of thermal diffusivity and the speed of sound vsð Þ as [16]:

sq ¼ 3av2s

ð1:15Þ

The value of thermal relaxation time varies between 10−14 s and 103 s for metals,organic tissues, and materials with microstructural non-homogeneity, such aspolyethylene/graphite nanosheets [22, 24, 27, 30].

The C-V heat conduction differential equation is derived by omitting the heatflux q between the constitutive relation of heat flux (1.14) and the energy Eq. (1.6).For a homogenous isotropic material, when the thermal properties are assumedindependent of temperature, the C-V heat conduction could be simplified as:

r2Tðx; tÞ ¼ 1þ sq@

@t

1a@Tðx; tÞ

@t� qrTðx; tÞ

kT

ð1:16Þ

Equation (1.16) is a hyperbolic-type differential equation, in contrast to theparabolic-type differential equation of Fourier’s law [Eq. (1.7)], which depicts thepropagation of temperature disturbance as a wave with thermal damping.Comparing Eq. (1.16) with wave equation, the finite thermal wave speed of C-Vmodel CT

C�V

� �is obtained as [16, 31]:

CTC�V ¼

ffiffiffiffiffiasq

rð1:17Þ

In the absence of thermal relaxation time, CTC�V reaches infinity and Eqs. (1.14)

and (1.16) reduces to the classical Fourier heat conduction. The C-V model predictsa finite thermal wave speed based on the consideration of phonon collision inmicrostructural heat transport and results in the thermal shock formation andthermal resonance phenomenon, which cannot be observed by the conventionalFourier heat conduction. However, the heat transport by the dispersion of phononcollisions and phonon-electron interaction is overlooked. The C-V model does nottake into account the relaxation times among electrons and the atomic lattice due tothe macroscopic considerations. In addition, the C-V model presumes an instan-taneous heat flow due to the immediate response between the temperature gradientand the energy transport [16]. Some unusual physical solutions introduced by theC-V model have also been reported. Therefore, the applicability of C-V model insuperior conductors and fast-transient heat transport is debatable [16, 32–34].


1.2.2.2 Dual-Phase-Lag Heat Conduction

While the C-V model assumes an average macroscopic thermal behaviour overgrains of a medium, microstructural effects and delayed thermal responses becomepronounced in the fast transient process of heat transport as well as the induceddelayed responses by low-conducting pores in sand media and inert behaviour ofmolecules at low temperatures [16]. Dual-phase-lag (DPL) heat conduction wasintroduced by Tzou [35, 36] to remove the assumptions made in the C-V model andto take into account the effect of relaxation time between electrons and atomiclattices in the transient process of heat conduction. Since the heat transport processneeds a finite amount of time to take place on the macroscopic level due to theinteractions conducted on the microscopic level, Tzou presented the followingconstitutive equation for the delayed thermal responses and provided the experi-mental supports for the DPL formulation [16, 37]:

q x; tþ sq� � ¼ �kTrT x; tþ sTð Þ ð1:18Þ

where sq ½s�, like the C-V model, is the phase lag of the heat flux or thermalrelaxation time and sT ½s� represents the phase lag of temperature gradient.Furthermore, the phase lags sq and sT are positive and intrinsic properties ofmaterials [16]. It is worth mentioning that the three characteristic times should bedistinguished in the DPL heat conduction, which are: time t for the onset of the heattransport, time tþ sq at which heat flows through the material, and tþ sT for theoccurrence of temperature gradient within the medium.

While sq could be defined as the relaxation time due to the fast transient effectsof thermal inertia, sT could be interpreted as the time delay caused by themicrostructural interactions such as phonon-electron or phonon scattering [16, 38].In quantum mechanics, phonons determine the energy states of a metallic lattice.

Fig. 1.2 The delayed thermal responses caused by phonon-electron interaction: a electron gasheating by photons and b metal lattice heating by phonon-electron interactions. [Reproduced from[16] with permission from John Wiley and Sons]


The delayed thermal responses caused by phonon-electron interactions during theshort-pulse laser heating of metallic media is depicted in Fig. 1.2 [16]. The electrongases are first heated up by the photons of laser beams at time t when no tem-perature change can be detected in the metallic lattice (Fig. 1.2a). Thephonon-electron interactions then cause the energy transport from the heatedelectrons to phonons which leads to the temperature rise in the lattice at time tþ sT .However, as asserted by Tzou, a detailed understanding of the delayed thermalresponses caused by microstructural interactions needs a profound knowledge ofquantum mechanics and statistical thermodynamics [16, 39, 40].

As seen in Eq. (1.18), temperature gradient or heat flux vector could precede theother depending on the value of sq and sT . For instance, if sT [ sq, the temperaturegradient within a medium is established as a result of the heat flux vector, whichmeans that the heat flux vector is a cause and the temperature gradient is an effect [16].In order to omit the heat flux between Eq. (1.18) and energy equation [Eq. (1.6],where the thermo-physical phenomena occur at the same time, Taylor series expan-sion of Eq. (1.18) with respect to time t is used. To develop a model equivalent to themicroscopic hyperbolic two-step heat conduction model [35, 40], Tzou assumedsmall values for sq and sT so that the third- and higher-order terms of sq and second-and higher-order terms of sT are negligible. As a result, the second-order Taylor seriesexpansion of Eq. (1.18) for sq and the first-order expansion for sT lead to:

1þ sq@

@tþ s2q

2@2

@t2

!qðx; tÞ ¼ �kT rTðx; tÞþ sT

@

@trTðx; tÞ½ �

� ð1:19Þ

Eliminating the heat flux q between Eq. (1.19) and energy Eq. (1.6) results in theDPL heat conduction differential equations for isotropic materials withtemperature-independent material properties:

1þ sT@

@t

r2T x; tð Þ ¼ 1þ sq

@

@tþ s2q

2@2

@t2

!1a@Tðx; tÞ

@t� qrTðx; tÞ

kT

ð1:20Þ

The DPL heat conduction Eq. (1.20) has the same form as the phonon scatteringand phonon-electron interaction energy equations. Subsequently, the phase-lags ofDPL heat conduction could be determined directly in terms of microscopic thermalproperties [16]. Furthermore, Eq. (1.20) is a hyperbolic partial differential equation.As the characteristics of temperature distribution are governed by the highest-ordersof differentiation in the heat conduction equation [16], thermal wave behavior inheat propagation using the DPL model can be observed by isolating the twothird-order derivatives in Eq. (1.20) as follows:

@

@t@2Tðx; tÞ

@t2� CT

DPL

� �2r2Tðx; tÞ

þ lower � order terms ¼ 0 ð1:21Þ


where the finite thermal wave speed of DPL model ðCTDPLÞ is [41]:

CTDPL ¼

ffiffiffiffiffiffiffiffiffiffi2asT

psq

¼ CTC�V

ffiffiffiffiffiffiffi2sTsq

sð1:22Þ

Consequently, this type of DPL heat conduction could be named as wave-likeDPL or hyperbolic-type DPL model. Equation (1.22) shows that the DPL thermalwave speed depends on the phase-lag of heat flux and phase-lag of temperaturegradient as well as the thermal diffusivity. In addition, it is found that sq and sThave opposite effects on thermal wave speed. While increasing sT , enhances thethermal wave speed, increasing sq decreases the speed of thermal wave. Dependingon the ratio of sT

sq, Eq. (1.22) shows that CT

DPL could propagate faster than CTC�V , e.g.

in most metals ðsT\sqÞ, or slower, e.g. in superfluid liquid helium ðsT [ sqÞ [16].On the other hand, in the absence of Taylor series expansion term

s2q2 , Eq. (1.19)

leads to a parabolic differential Eq. (1.20), which reveals a diffusive thermalbehavior for heat flow. This type of DPL model is called diffusive-like DPL orparabolic-type DPL. It is also worthwhile mentioning that expanding both sq and sTin Eq. (1.18) up to the second-order leads to a parabolic differential equations andresults in the nonlinear thermal lagging behavior which could be employed todescribe the heat transport phenomena in biological systems with multiple energycarriers [42]. Moreover, in the absence of the phase-lag of temperature gradient,sT ¼ 0, and the second-order Taylor series expansion term, Eqs. (1.19) and (1.20)reduce to the C-V heat conduction equations [38].

1.2.2.3 Three-Phase-Lag Heat Conduction

The aforementioned Fourier and non-Fourier heat conduction theories have beenemployed to describe classical and generalized thermoelasticity models, respec-tively. Among generalized thermoelasticity models, Lord and Shulman (L-S) [43],Green and Lindsay (G-L) [44], Chandrasekhariah and Tzou (C-T) [24, 36], andGreen and Naghdi (G-N) [45] could be mentioned. However, the G-N model cannotbe obtained by the C-V and DPL heat conduction theories. As a result,three-phase-lag (TPL) heat conduction theory was proposed by Choudhuri [46] todescribe all generalized thermoelasticity models and encompass all previous the-ories for non-Fourier heat conduction at the same time. The utilization of phase-lagsof heat flux, temperature gradient, and thermal displacement gradient in the TPLtheory is significant for understanding several physical phenomena, such as bioheattransfer and laser heating in living tissues, exothermic catalytic reactions, andharmonic plane wave propagation [47, 48].

As an extension to the G-N thermoelastic model [45, 49], where thermal dis-placement gradient t was introduced along with the temperature gradient in the heat


flux constitutive equations, Choudhuri proposed the following generalized consti-tutive equation to describe the lagging responses:

q x; tþ sq� � ¼ � kTrT x; tþ sTð Þþ k�Trt x; tþ stð Þ� � ð1:23Þ

where t ¼ T and k�T � 0 is the TPL material constant (the rate of thermal con-ductivity). The third phase-lag st ½s� introduced in TPL could be interpreted as thephase-lag of thermal displacement gradient ð0� st\sT\sqÞ. For k�T ¼ 0,Eq. (1.23) reduces to the DPL constitutive equation given in Eq. (1.18).Furthermore, the TPL heat conduction theory is reduced to the C-V theory byassuming k�T ¼ sT ¼ 0 in Eq. (1.23). Finally, the TPL theory reduces to the con-ventional Fourier heat conduction by setting k�T ¼ sq ¼ sT ¼ 0. In the absence ofphase lags sq, sT , and st, Eq. (1.23) reduces to the one considered by Green andNaghdi, where the damped thermal wave behavior is admitted (G-N type III):

qðx; tÞ ¼ � kTrTðx; tÞþ k�Trtðx; tÞ� � ð1:24Þ

For kT k�T , Eq. (1.24) leads to the G-N model of type II without energydissipation. Several types of Taylor series expansion could be considered todevelop the TPL heat conduction equations. The second-order Taylor seriesexpansion for sq and the first-order Taylor series expansion for sT and st inEq. (1.23) result in:

1þ sq@

@tþ s2q

2@2

@t2

!qðx; tÞ

¼ � kT þ k�Tst� �rTðx; tÞþ kTsT

@

@trTðx; tÞ½ � þ k�Trtðx; tÞ

� ð1:25Þ

The TPL heat conduction differential equation for isotropic materials withtemperature-independent material properties is obtained by taking the divergenceand time derivative of Eq. (1.25) and using t ¼ T , taking the time derivative ofenergy Eq. (1.6), and then eliminating qðx; tÞ between the resulting equations:

k�T

kTþ 1þ k�Tst

kT

@

@tþ sT

@2

@t2

� �r2Tðx; tÞ

¼ 1þ sq@

@tþ s2q

2@2

@t2

!1a@2Tðx; tÞ

@t2� qkT

@rTðx; tÞ@t

ð1:26Þ

The TPL heat conduction Eq. (1.26) is a hyperbolic partial differential equation.As a result, this type of TPL is named wave-like TPL or hyperbolic-type TPLmodel. Thermal wave behavior in the TPL heat conduction equation could beobserved, similar to the DPL model, by isolating the two fourth-order derivatives inEq. (1.26) [50]:


@2

@t2@2Tðx; tÞ

@t2� CT

TPL

� �2r2Tðx; tÞ

þ lower � order terms ¼ 0 ð1:27Þ

where the finite thermal wave speed of TPL model ðCTTPLÞ is:

CTTPL ¼ CT

DPL ¼ffiffiffiffiffiffiffiffiffiffi2asT

psq

¼ CTC�V

ffiffiffiffiffiffiffi2sTsq

sð1:28Þ

Equation (1.28) reveals that the DPL and TPL predict the same thermal wavespeed; however, the characteristics of transient thermal responses of DPL and TPLare dissimilar due to the difference between the heat conduction equation of DPL[Eq. (1.20)] and TPL [Eq. (1.26)] [50]. The other types of TPL heat conduction canalso be developed by Taylor series expansion of Eq. (1.23). For instance, thefirst-order or second-order Taylor series expansion of all the phase-lags of sq, sT ,and st leads to a parabolic differential equation for heat conduction which does notshow the wave-like behavior for thermal response. These types of TPL models arecalled diffusive-like TPL or parabolic-type TPL models. Furthermore, another typeof wave-like TPL model could also be developed by second-order Taylor seriesexpansion of sq and st and the first-order Taylor series expansion of sT whichpredicts a higher thermal wave speed for TPL compared to Eq. (1.28):

CTTPL ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2asT þ a�s2t

psq

ð1:29Þ

where a� ¼ k�Tqcp.

1.2.2.4 Fractional Phase-Lag Heat Conduction

Another type of TPL model can be developed by taking the Taylor series expansionof time-fractional order aF on the both sides of Eq. (1.23) [47]. The fractionalcalculus differentiation and integration of arbitrary order has been employed tomodify the existing formulation for physical process in chemistry, biology, elec-tronics and signal processing, wave propagation, viscoelasticity, and chaos/fractals.Ezzat et al. [47, 54] retained terms up to 2aF-order for sq and up to aF-order for sTand st in fractional Taylor series expansion of Eq. (1.23) as follows:


1þ saFqaF !

@aF

@taFþ s2aFq

2aFð Þ!@2aF

@t2aF

!qðx; tÞ

¼ � kT þ k�TsaFtaF !

@aF�1

@taF�1

rT x; tð Þ

�

þ kTsaFTaF !

@aF

@taFrTðx; tÞ½ � þ k�Trtðx; tÞ

0� aF\1

ð1:30Þ

where the fractional derivative is defined as [54, 55]:

@aF

@taFf ðx; tÞ ¼

f ðx; tÞ � f ðx; 0Þ aF ! 0IaF�1 @f ðx;tÞ

@t 0\aF\1@f ðx;tÞ@t aF ¼ 1

8><>: ð1:31Þ

where the Riemann–Liouville fractional integral IaF could be written in aconvolution-type form as:

IaF f ðx; tÞ ¼Z t

0

t � fð ÞaF�1

C aFð Þ f ðx; fÞdf

I0f ðx; tÞ ¼ f ðx; tÞð1:32Þ

where C . . .ð Þ represents the Gamma function. Eliminating heat flux qðx; tÞ betweenEqs. (1.6) and (1.30) provides the following fractional TPL heat conductionequation:

k�T

kTþ 1þ k�TsaFt

kTaF !

@aF

@taFþ saFT

aF !@aF þ 1

@taF þ 1

� �r2Tðx; tÞ

¼ 1þ saFqaF !

@aF

@taFþ saFq

2aFð Þ!@2aF

@t2aF

1a@2Tðx; tÞ

@t2� qkT

@rTðx; tÞ@t

ð1:33Þ

For aF ¼ 1, Eq. (1.33) reduces to the TPL heat conduction given in Eq. (1.26).Other types of fractional TPL could be developed by considering different fractionalorder Taylor series expansion, for example, up to 2aF-order for phase lags of sq, sT ,and st. Another form of fractional TPL model could also be developed by thegeneralization of the fractional C-V heat conduction, proposed by Youssef [56], asfollows:

q x; tþ sq� � ¼ � kTIaF�1rT x; tþ sTð Þþ k�T IaF�1rt x; tþ stð Þ� �

0\aF � 2

ð1:34Þ


The associated fractional TPL heat conduction could be derived by Eqs. (1.6)and (1.34) as:

k�T

kTIaF�1 1þ st

@

@t

þ IaF�1 1þ sT

@

@t

@

@t

� �r2Tðx; tÞ

¼ 1þ sq@

@tþ s2q

2@2

@t2

!1a@2Tðx; tÞ

@t2� qkT

@rTðx; tÞ@t

ð1:35Þ

Implied by Kimmich [57] and Youssef [56], Eq. (1.35) could be called as ageneralized anomalous heat conduction which describers various heat transportphenomena in media with weak conductivity ð0\aF\1Þ, normal conductivityaF ¼ 1ð Þ, and superconductivity ð1\aF � 2Þ. Subdiffusive, e.g. dielectrics andsemiconductors, and superconductive, e.g. porous glasses and polymer chains,media are examples of materials that could exhibit anomalous diffusion and heatconduction [51–53, 58].

1.2.2.5 Nonlocal Phase-Lag Heat Conduction

To accommodate the effect of thermomass, the distinctive mass of heat, of dielectriclattices in the heat conduction, Tzou [59, 60] has also included the nonlocalbehavior, in space, in addition to the thermal lagging, in time. The nonlocal(NL) TPL constitutive equation could be derived as an expansion of the nonlocalDPL model proposed by Tzou in the following form:

q xþ kq; tþ sq� � ¼ � kTrT xþ kT ; tþ sTð Þþ k�Trt xþ kt; tþ stð Þ� � ð1:36Þ

where kq, kT , and kt are correlating nonlocal lengths of heat flux, temperaturegradient, and thermal displacement gradient in the heat transport constitutiveequation [60]. The Taylor series expansion of Eq. (1.36) with respect to eithernonlocal lengths and/or phase-lags could lead to a number of local or nonlocal heatconduction constitutive equations. However, only the constitutive equations withcoordinate independent property must be considered. Eliminating the heat fluxqðx; tÞ between Eqs. (1.6) and (1.35) provides the nonlocal TPL heat conductionequation. Since the nonlocal behavior in the heat flux is only related to existingmicroscale heat transfer models, the nonlocal length kq is involved hereafter.Recently, Akbarzadeh et al. [61] has also introduced a heat conduction model,called nonlocal fractional three-phase-lag (NL FTPL), to take into account thesize-dependency of thermophysical properties, subdiffusion or superdiffusion ofheat transport, and phonon-electron interaction in ultrafast heat transport.

q xþ kq; tþ sq� � ¼ � kTIaF�1rT x; tþ sTð Þ�

þ k�T IaF�1rt x; tþ stð Þ� 0\aF � 2ð1:37Þ


Among the abovementioned non-Fourier heat conduction theories, C-V, DPL,NL C-V, TPL, and fractional TPL models are considered in this book.

It is worth mentioning that the application of interfacial thermal boundaryconditions [Eq. (1.13)] is straightforward for Fourier heat conduction. However,mathematical implementations are required to correctly apply the interfacial andboundary conditions for non-Fourier models. For instance, the interfacial conditionsshould be applied in the Laplace domain for local non-Fourier models and/or we

need to assume @qðx;tÞ@x

��x¼S

¼ 0, due to the continuity of heat flux in the interface, for

nonlocal non-Fourier models.

1.3 Moisture Diffusion

Moisture, similar to temperature, could induce significant strains and stresses withina solid material. Moisture and temperature could also cause the reduction ofstrength and stiffness. For instance, the failure mode caused by moisture diffusionduring the manufacturing and operation of fiber-reinforced composites and plasticencapsulated microcircuits, is a major reliability concern [1, 62, 63]. As a result, thedetermination of moisture distribution within solids could be as significant astemperature for design of composite structures.

Following the heat conduction study by Fourier [64], Fick [65] recognized thatthe diffusion of moisture in solids is similar to the heat conduction process.Therefore, the rate of moisture flux p is related to the moisture concentrationgradient rm with the constitutive equation of moisture flux. The moisture flux,p [kg/m2s], is the moisture transfer per unit time and per unit normal vector of thearea, while moisture concentration, m [kg/m3], is defined as the mass of moistureper unit volume of the dry solid [66, 67]. The moisture diffusion equation isobtained by employing the constitutive equation of moisture flux and the conser-vation equation of mass of moisture. In view of the similarity between the moisturediffusion and heat conduction, the Fickian and non-Fickian moisture diffusionequations are briefly reviewed in this Section.

1.3.1 Fickian Moisture Diffusion

The Fickian moisture diffusion equation, similar to Fourier heat conduction, assumesthe proportionality between the moisture flux and moisture concentration as [1]:

pðx; tÞ ¼ �kmrmðx; tÞ ð1:38Þ

where km ½m2=s� is the moisture diffusion coefficient, which is a positive scalarquantity or a second-order tensor for isotropic or anisotropic materials, respectively.


The negative sign in Eq. (1.38) implies that the moisture diffuses in the direction ofthe decreasing moisture concentration. Furthermore, the moisture coefficient hasbeen shown to be dependent on temperature as [1]:

km ¼ km0 exp � Em

RgT

ð1:39Þ

where km0 is a constant, Em and Rg are, respectively, the activation energy and gasconstant, and T is temperature in Kelvin. Some typical values of km0 and Em havealready been reported in [1, 68] for composite laminates. It has also been observedthat moisture content has negligible effect on moisture diffusion coefficient km [69];however, this observation may not hold for polymers, molding compounds, andorganic substrates [63, 70, 71].

Furthermore, the conservation law for the mass of moisture, equivalent to energyconservation for temperature, is given by [66, 67]:

�r:pðx; tÞþ qrmðx; tÞ ¼ @mðx; tÞ@t

ð1:40Þ

where rm is the moisture source per unit time per unit mass of dry solid. Thedifferential equation of transient Fickian moisture diffusion is derived by substi-tuting Eq. (1.38) into Eq. (1.40) as follows:

r: kmrmðx; tÞð Þþ qrmðx; tÞ ¼ @mðx; tÞ@t

ð1:41Þ

For a homogenous isotropic material with moisture independent moisture dif-fusivity, Eq. (1.7) is further simplified as:

r2mðx; tÞþ qrmðx; tÞkm

¼ 1km

@mðx; tÞ@t

ð1:42Þ

For steady-state moisture diffusion analysis, the right-hand side of Eqs. (1.41)and (1.42) are set zero and the moisture generation rate rm is overlooked.Appropriate boundary and initial conditions for moisture or hygroscopic fieldshould be specified for the moisture diffusion differential equation to a uniquesolution for the system of differential equations.

Similar to thermal boundary conditions, hygroscopic boundary conditions couldbe:

(a) Prescribed moisture concentration along the boundary surface (S):

mðx; tÞjx¼S ¼ mðS; tÞ ð1:43Þ

where m is a prescribed function of position and time, or:

1.3 Moisture Diffusion 17

(b) Prescribed moisture flux across the boundary surface:

kmrmðx; tÞjx¼S ¼ �pSðS; tÞ ð1:44Þ

where pS is the prescribed moisture flux. Hygroscopic insulation boundaryconditions could also be reached by substituting pS ¼ 0 in Eq. (1.44).

Due to the time derivatives of moisture concentration in the differential equationof moisture diffusion, initial conditions should also be specified. For the Fickianmoisture diffusion, the initial condition for moisture concentration could be spec-ified as:

m x; t ¼ 0ð Þ ¼ m0ðxÞ ð1:45Þ

where m0ðxÞ is a specified function of the spatial coordinate x.When a multilayered medium is considered, hygroscopic interfacial conditions

need to also be specified. The following hygroscopic interfacial conditions could beconsidered at the interface of two layers with a hygroscopic weak moisture diffusion[72]:

mð1Þ x; tð Þ � mð1Þðx; tÞ� ��

x¼S¼ Rmpð1ÞðS; tÞ ð1:46aÞ

pð1ÞðS; tÞ ¼ pð2ÞðS; tÞ ð1:46bÞ

where Rm is the hygroscopic compliance constant or hygroscopic contact resistancefor the imperfect interface. Moreover, Rm ¼ 0, in Eq. (1.46), represents the per-fectly bonded interfaces.

1.3.2 Non-Fickian Moisture Diffusion

Fickian moisture diffusion provides a good approximation for most of the engi-neering applications [73, 74]. However, it is well known that Fickian moisturediffusion exhibits an unrealistic infinite speed for propagation of mass of moisturedue to the parabolic-type partial differential equation of Fickian moisture diffusion[Eq. (1.41)]. Moreover, the validity of traditional Fickian moisture diffusionequation breaks down for short-time inertial motion of mass and very high fre-quency of mass flux density [73]. Thus, non-Fickian moisture diffusion theorieswith a hyperbolic diffusion equation, which describe the moisture diffusion with afinite speed, have been developed. Non-Fickian moisture diffusion equations haverecently found many practical application in superionic conductors, molten salts,laser drying, laser melting, and rapid solidification [75].

The non-Fickian moisture diffusion theories in the absence of a potential field arethe same as non-Fourier heat conduction theories. Nonetheless, the potential field,


which does not appear in non-Fourier heat conduction, plays an important role innon-Fickian moisture diffusion. One of the non-Fickian moisture diffusion theorieswas developed by Das [75] to accommodate the assumption of a local Maxwellianequilibrium. The constitutive non-Fourier moisture flux could be written as [73, 75]:

1þ sp@

@t

pðx; tÞ ¼ �kmrmðx; tÞ � sprVðxÞ

pmmðx; tÞ ð1:47Þ

where sp ½s� is the relaxation time of the mass flux, pm ½kg� is the particle mass, andV is potential field. In the absence of VðxÞ, Eq. (1.47) is equivalent to the C-Vmodel of non-Fourier heat conduction. The non-Fickian moisture diffusion equationis derived by omitting the moisture flux pðx; tÞ between Eqs. (1.40) and (1.47). Fora homogenous isotropic material with moisture independent moisture diffusivity,the non-Fickian moisture diffusion equation is written as:

r2mðx; tÞþ spkmpm

r: rVðxÞmðx; tÞð Þ ¼ 1þ sp@

@t

1km

@mðx; tÞ@t

� qrmðx; tÞkm

ð1:48Þ

Equation (1.48) is a hyperbolic-type differential equation and reveals awave-like behaviour for the propagation of moisture concentration. The finitehygroscopic wave speed Cm

D

� �predicted by Das model in Eq. (1.48) is obtained as:

CmD ¼

ffiffiffiffiffikm

sp

sð1:49Þ

If the relaxation time of the mass flux sp and potential field VðxÞ are neglected,Eq. (1.48) reduces to the Fickian moisture diffusion and Cm

D reaches infinity.Moreover, another effort for developing a non-Fickian moisture diffusion was madeby Akbarzadeh [25] where the following dual-phase lag moisture diffusion equationwas proposed analogous to the DPL heat conduction:

1þ sp@

@tþ s2p

@2

@t2

pðx; tÞ ¼ �km 1þ sm

@

@t

rmðx; tÞ � sprVðxÞ

pmmðx; tÞ

ð1:50Þ

where sp and sm are the phase lags of moisture flux (the relaxation time of the massflux) and moisture gradient, respectively. Although some of the non-Fickianmoisture diffusion theories have been introduced in this section, the non-Fourierheat conduction has received more attention in the literature compared tonon-Fickian moisture diffusion. As a result, we only focus in this book on theinfluence of Fourier/non-Fourier heat conduction and Fickian moisture diffusion onstructural responses of smart materials.

1.3 Moisture Diffusion 19

References

1. Sih GC, Michopoulos J, Chou S-C (1986) Hygrothermoelasticity. Springer, Berlin2. Brischetto S (2013) Hygrothermoelastic analysis of multilayered composite and sandwich

shells. J Sandwich Struct Mater 15(2):168–2023. Carslaw HS, Jaeger JC (1959) Conduction of heat in solids. Oxord at the Clarendon Press,

Great Britain4. Ozisik MN (1980) Heat conduction. Wiley, New York5. Jiji LM (2009) Heat conduction. Springer, New York6. Wang L, Zhou X, Wei X (2008) Heat conduction: mathematical models and analytical

solutions. Springer, Berlin, Heidelberg7. Lienhard JH (2008) A heat transfer text book. Phlogiston Press, Cambridge, MA8. Wang L (1994) Generalized Fourier law. Int J Heat Mass Transf 37(17):2627–26349. Cengel YA, Boles MA (2006) Thermodynamics: an engineering approach, 5th edn.

McGraw-Hill, Boston10. Hetnarski RB, Eslami MR (2008) Thermal stresses-advanced theory and applications:

advanced theory and applications, vol 158. Springer, Berlin11. Parkus H (1976) Thermoelasticity. Springer, New York12. Bejan A (1988) Advanced engineering thermodynamics. Wiley, New York13. Clausius R (1865) On different forms of the fundamental equations of the mechanical theory

of heat and their convenience for application. Physics (Leipzig) 125:31314. Rankine WJM (1869) On the thermal energy of molecular vortices. Trans R Soc Edinburgh

25:557–56615. Tian Z, Lee S, Chen G (2014) A comprehensive review of heat transfer in thermoelectric

materials and devices. Ann Rev Heat Transf 17:425–48316. Tzou DY (1996) Macro-to micro-scale heat transfer: the lagging behavior. CRC Press, Boca

Raton17. Eslami MR et al (2013) Theory of elasticity and thermal stresses. Springer, Berlin18. Theodore L (2011) Heat transfer application for the practicing engineer. Wiley, Hoboken, NJ19. Corral M (2008) Vector calculus. Michael Corral20. Arpaci VS (1966) Conduction heat transfer. Addison-Wesley, Reading21. Baher HD, Stephan K (2006) Heat and mass transfer, 2nd edn. Springer, Berlin22. Akbarzadeh AH, Pasini D (2014) Phase-lag heat conduction in multilayered cellular media

with imperfect bonds. Int J Heat Mass Transf23. Chen T (2001) Thermal conduction of a circular inclusion with variable interface parameter.

Int J Solids Struct 38(17):3081–309724. Chandrasekharaiah D (1998) Hyperbolic thermoelasticity: a review of recent literature. Appl

Mech Rev 51(12):705–72925. Akbarzadeh AH (2013) Multiphysical behaviour of functionally graded smart structures.

Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB,Canada

26. Antaki P (1995) Key features of analytical solutions for hyperbolic heat conduction27. Babaei MH (2009) Multiphysics analysis of functionally graded piezoelectrics. University of

New Brunswick, Canada28. Cattaneo C (2011) Sulla conduzione del calore. In: Some aspects of diffusion theory.

Springer, Berlin, pp 485–48529. Vernotte P (1958) Les paradoxes de la theorie continue de l’equation de la chaleur. Comptes

rendus de l’academie bulgare des sciences: sciences mathematiques et naturelles 246:3154–3155

30. Zhou L et al (2008) Electrical-thermal switching effect in high-density polyethylene/graphitenanosheets conducting composites. J Mater Sci 43(14):4886–4891

31. Chester M (1963) Second sound in solids. Phys Rev 131:2013–2015


32. Godoy S, García-Colín L (1997) Nonvalidity of the telegrapher’s diffusion equation in twoand three dimensions for crystalline solids. Phys Rev E 55(3):2127

33. Körner C, Bergmann H (1998) The physical defects of the hyperbolic heat conductionequation. Appl Phys A 67(4):397–401

34. Xu F, Seffen K, Lu T (2008) Non-Fourier analysis of skin biothermomechanics. Int J HeatMass Transf 51(9):2237–2259

35. Tzou DY (1995) The generalized lagging response in small-scale and high-rate heating. Int JHeat Mass Transf 38(17):3231–3240

36. Tzou D (1995) A unified field approach for heat conduction from macro-to micro-scales.J Heat Transfer 117(1):8–16

37. Tzou DY (1995) Experimental support for the lagging behavior in heat propagation.J Thermophys Heat Transfer 9(4):686–693

38. Akbarzadeh A, Chen Z (2012) Heat conduction in one-dimensional functionally graded mediabased on the dual-phase-lag theory. Proc Inst Mech Eng, Part C: J Mech Eng Sci 227(4):744–759

39. Qiu T, Tien C (1992) Short-pulse laser heating on metals. Int J Heat Mass Transf 35(3):719–726

40. Qiu T, Tien C (1993) Heat transfer mechanisms during short-pulse laser heating of metals.J Heat Transfer 115(4):835–841

41. Akbarzadeh A, Pasini D (2014) Phase-lag heat conduction in multilayered cellular media withimperfect bonds. Int J Heat Mass Transf 75:656–667

42. Tzou DY, Dai W (2009) Thermal lagging in multi-carrier systems. Int J Heat Mass Transf 52(5):1206–1213

43. Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J MechPhys Solids 15(5):299–309

44. Green A, Lindsay K (1972) Thermoelasticity. J Elast 2(1):1–745. Green A, Naghdi P (1993) Thermoelasticity without energy dissipation. J Elast 31(3):189–

20846. Choudhuri SR (2007) On a thermoelastic three-phase-lag model. J Therm Stresses 30(3):231–

23847. Ezzat MA, El-Bary AA, Fayik MA (2013) Fractional fourier law with three-phase lag of

thermoelasticity. Mech Adv Mater Struct 20(8):593–60248. Miranville A, Quintanilla R (2011) A phase-field model based on a three-phase-lag heat

conduction. Appl Math Optim 63(1):133–15049. Green A, Naghdi P (1885) A re-examination of the basic postulates of thermomechanics.

Proc R Soc London. Ser A: Math Phys Sci 432:171–19450. Akbarzadeh A, Fu J, Chen Z (2014) Three-phase-lag heat conduction in a functionally graded

hollow cylinder. Trans Can Soc Mech. Eng 38(1):155–17151. Dzieliński A, Sierociuk D, Sarwas G (2010) Some applications of fractional order calculus.

Bull Polish Acad Sci: Tech Sci 58(4):583–59252. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives,

fractional differential equations, to methods of their solution and some of their applications,vol 198. Academic press, New York

53. Stiassnie M (1979) On the application of fractional calculus for the formulation of viscoelasticmodels. Appl Math Model 3(4):300–302

54. Ezzat MA, El Karamany AS, Fayik MA (2012) Fractional order theory in thermoelastic solidwith three-phase lag heat transfer. Arch Appl Mech 82(4):557–572

55. Jumarie G (2010) Derivation and solutions of some fractional Black–Scholes equations incoarse-grained space and time. Application to Merton’s optimal portfolio. Comput Math Appl59(3):1142–1164

56. Youssef HM (2010) Theory of fractional order generalized thermoelasticity. J Heat Transf 132(6):061301

57. Kimmich R (2002) Strange kinetics, porous media, and NMR. Chem Phys 284(1):253–285

References 21

58. Povstenko YZ (2004) Fractional heat conduction equation and associated thermal stress.J Therm Stresses 28(1):83–102

59. Tzou D (2011) Nonlocal behavior in phonon transport. Int J Heat Mass Transf 54(1):475–48160. Tzou D, Guo Z-Y (2010) Nonlocal behavior in thermal lagging. Int J Therm Sci 49(7):

1133–113761. Akbarzadeh AH, Cui Y, Chen ZT (2015) Non-local fractional phase-lag heat conduction:

continuum to molecular dynamic simulation. Appl Phys Lett62. Reddy JN (2004) Mechanics of laminated composite plates and shells: theory and analysis.

CRC Press, New York63. Shirangi M, Michel B (2010) Mechanism of moisture diffusion, hygroscopic swelling, and

adhesion degradation in epoxy molding compounds, in moisture sensitivity of plasticpackages of IC devices. Springer, Berlin, pp 29–69

64. Fourier J (1822) Theorie analytique de la chaleur, par M. Fourier. Chez Firmin Didot, père etfils

65. Fick A (1855) Ueber diffusion. Ann Phys 170(1):59–8666. Smittakorn W (2001) A theoretical and experimental study of adaptive wood composites.

Colorado State University67. Heyliger WSPR (2000) A discrete-layer model of laminated hygrothermopiezoelectric plates.

Mech Compos Mater Struct 7(1):79–10468. Shen C-H, Springer GS (1976) Moisture absorption and desorption of composite materials.

J Compos Mater 10(1):2–2069. Augl J, Trabocco R (1975) Environmental degradation studies on carbon fiber reinforced

epoxies. In: Workshop on durability of composite materials (held on September)70. Chen X, Zhao S, Zhai L (2005) Moisture absorption and diffusion characterization of molding

compound. J Electron Packag 127(4):460–46571. Celik E, Guven I, Madenci E (2009) Experimental and numerical characterization of

non-Fickian moisture diffusion in electronic packages. IEEE Trans Adv Packag 32(3):666–674

72. Akbarzadeh A, Pasini D (2014) Multiphysics of multilayered and functionally gradedcylinders under prescribed hygrothermomagnetoelectromechanical loading. J Appl Mech 81(4):041018

73. Chen H-T, Liu K-C (2001) Numerical analysis of non-Fickian diffusion problems in apotential field. Numer Heat Transf: Part B: Fundam 40(3):265–282

74. Ding S, Petuskey WT (1998) Solutions to Fick’s second law of diffusion with a sinusoidalexcitation. Solid State Ionics 109(1):101–110

75. Das AK (1991) A non-Fickian diffusion equation. J Appl Phys 70(3):1355–1358


Chapter 2Basic Problems of Non-Fourier HeatConduction

2.1 Introduction

In this chapter, the non-Fourier heat conduction equations along with the boundaryand initial conditions are solved for one-dimensional (1D)media with semi-infinite orfinite dimensions in Cartesian, cylindrical, and spherical coordinate systems. Inparticular, semi-analytical solutions for C-V, DPL, TPL, and nonlocal C-V heatconduction models are provided. The influence of non-Fourier heat conduction the-ories on thermal responses of heterogeneous multilayered/functionally graded solid/cellular materials is also presented. Since applying Laplace transform to transientproblems is a well-known methodology for dealing with the time-dependency ofsolutions, Laplace transform and numerical Laplace inversion techniques are firstintroduced in Sect. 2.2. Non-Fourier heat conduction in a homogeneous semi-infinitemedium is considered in Sect. 2.3. Then non-Fourier heat conduction problem infinite homogenous and functionally graded media is studied in Sect. 2.4, whileSect. 2.5 studies the non-Fouirer heat conduction inmultilayeredmedia and disucss itsapplication in porous or cellular media. Finally, a set of solutions is given in Sect. 2.6for non-Fourier heat conduction problems in 1D media with finite dimensions.

2.2 Laplace Transform and Laplace Inversion

Laplace transform along with the method of separation of variables are frequentlyused to solve the time-dependent differential equations. Particularly, Laplacetransform has been used in several studies on non-Fourier heat conduction. TheLaplace transform of a function f x; tð Þ, denoted by L f x; tð Þf g or f x; sð Þ, is defined as:

f x; tð Þ ¼ L�1 f x; sð Þf g ¼ 12pi

Zaþ i1

a�i1f x; sð Þest ds ð2:1Þ


23




https://doi.org/10.1007/978-3-030-25201-4_2

where s represents the Laplace variable. The time-dependent function f x; tð Þ,defined for t� 0, should be a piecewise continuous and of exponential order.A function f x; tð Þ is said to be of exponential order if there exist constants c, 0�M,0� T , such that:

f x; tð Þj j �Mect for T � t ð2:2Þ

The inverse Laplace transform of a function f x; sð Þ, denoted by L�1 f x; tð Þf g, isdefined by the Bromwich integral formula as:

f x; tð Þ ¼ L�1 f x; sð Þf g ¼ 12pi

Zaþ i1

a�i1f x; sð Þest ds ð2:3Þ

where a is an arbitrary real number larger than all real parts of singularities of f x; sð Þand i ¼ ffiffiffiffiffiffiffi�1

pis the imaginary unit. Using Cauchy’s residue theorem, the contour

integration Eq. (2.3) could be analytically reduced to:

12pi

Zaþ i1

a�i1f x; sð Þest ds ¼

Xj¼1

rj ð2:4Þ

where rj are the residues of f x; sð Þest at the singularities of f x; sð Þ. However, thisanalytical methodology could be cumbersome for complicated f x; sð Þ. As a result,numerical Laplace inversion techniques are commonly employed. Among variousnumerical approaches for inverse Laplace transform, the three major techniquesusing fast Fourier transform, Jacobi polynomial, and Reimann sum approximationare briefly reviewed here.

2.2.1 Fast Laplace Inverse Transform

Durbin used the fast Fourier transform (FFT) to speed up the computation time forLaplace inversion. According to the fast Laplace inverse transform (FLIT), pro-posed by Durbin, the Laplace inversion of f x; sð Þ at time tk is obtained by:

f x; tj� � � C jð Þ � 1

2Re f x; að Þf gþRe

XN�1

k¼0

A x; kð Þþ iB x; kð Þð ÞWjk

( )" #ð2:5Þ

24 2 Basic Problems of Non-Fourier Heat Conduction

where:

A x; kð Þ ¼XLl¼0

Re f x; aþ i kþ lNð Þ 2pTperiod

� �� ;

B x; kð Þ ¼XLl¼0

Im f x; aþ i kþ lNð Þ 2pTperiod

� ��

C jð Þ ¼ 2Tperiod

eajDt;Dt ¼ TperiodN ;W ¼ ei

2pN ; i ¼ ffiffiffiffiffiffiffi�1

p

ð2:6Þ

where Tperiod is the time period for performing the Laplace inversion and D t standsfor the time increment; Re and Im represent the real and imaginary parts of theirarguments, respectively. Moreover, a is an arbitrary real number larger than all thereal parts of the singularities present in the function f x; sð Þ and L and N are twoarbitrary parameters that affect the accuracy of the solutions. To minimize thenumerical discretization and truncation errors, it is recommended to consider thefollowing constraints for the arbitrary parameters:

5� aTperiod � 10; 50�NL� 5000 ð2:7Þ

2.2.2 Reimann Sum Approximation

The Laplace inversion of f x; sð Þ at time t could also be obtained by Reimann sumapproximation of the Fourier integral transformed from Laplace inversion integral as:

f x; tð Þ ¼ ect

t12F x; s ¼ cð ÞþRe

XNn¼1

F x; s ¼ cþ inpt

� ��1ð Þn

" #ð2:8Þ

where c and N are two real constants. The value of c and truncation error deter-mined by N dictate the accuracy of the Reimann-sum approximation. Furthermore,c should satisfy the following relation to achieve a faster convergence:

ct ffi 4:72 ð2:14Þ

As employed by Tzou for non-Fourier heat conduction analysis, the summationin Eq. (2.8) could be performed till the Cauchy norm is smaller than 10�15.

2.2.3 Laplace Inversion by Jacobi Polynomial

Jacobi polynomial could also be employed to obtain the Laplace inversion of f x; sð Þas follows:

2.2 Laplace Transform and Laplace Inversion 25

f x; tð Þ ¼XNn¼0

CnP0;bð Þn 2e�dt � 1� � ð2:10Þ

where d is a real positive number and b[ � 1. Furthermore, P a;bð Þn represents the

Jacobi polynomial of degree n, defined as:

P a;bð Þn zð Þ ¼ 1� zð Þ�a 1þ zð Þ�b �1ð Þn

2nn!dn

dzn1� zð Þnþ a 1þ zð Þnþ b

h ið2:11aÞ

or:

P a;bð Þn xð Þ ¼ C aþ nþ 1ð Þ

n!C aþ bþ nþ 1ð ÞXnk¼0

nk

� �C aþ bþ nþ kþ 1ð Þ

C aþ kþ 1ð Þx� 12

� �k

ð2:11bÞ

The unknown coefficients Cn in Eq. (2.10) are obtained by the followingrecurrence relation:

df x; bþ 1þ kð Þdð Þ ¼Xkm¼0

k k � 1ð Þ � � � k � m� 1ð Þð Þkþ bþ 1ð Þ kþ bþ 2ð Þ � � � kþ bþ 1þmð ÞCm ð2:12Þ

For an accurate approximation of Laplace inversion, it is recommended tochoose the b and d as:

�0:5� b� 5 ð2:13aÞ

0:05� d� 2 ð2:13bÞ

2.3 Non-Fourier Heat Conduction in a Semi-infinite Strip

To consider the effect of non-Fourier heat conduction on thermal responses, weconsider a semi-infinite, isotropic, homogeneous 1D medium, as seen in Fig. 2.1.The solid medium is initially t ¼ 0ð Þ kept at a constant temperature T0 and ther-

mally disturbed from a stationary state, @T x;tð Þ@t ¼ 0 at t ¼ 0. The surface temperature

at x ¼ 0 is abruptly raised to TWL which leads to the propagation of thermal dis-turbance through the medium. However, temperature at a distance far from theheated zone keeps its initial value. The semi-infinite medium facilitates theexamination of the way temperature distributes and thermal affected zone evolves,with no concern about the thermal wave separation/reflection from boundaries.

To accommodate comparing the heat conduction in a semi-infinite medium usingFourier, C-V, nonlocal C-V, and DPL theories, the heat conduction constitutiveequation iswritten in the following nonlocalDPL form, in accordancewithEq. (1.35):


1þ kq@

@xþ sq

@

@tþ s2q

2@2

@t2

!q x; tð Þ ¼ �kT 1þ sT

@

@t

� �@T x; tð Þ

@xð2:14Þ

where kq is the correlating length parameter for nonlocal analysis. The correlatinglength is equivalent, by two times, to the length parameters in the thermomass modelof heat transfer in dielectric lattices. Specifically, kq and sq could be correlated to themean free time and themean free path inmicroscale heat transport [1]. Eliminating theheat flux q x; tð Þ between Eq. (2.14) and 1D form of the energy Eq. (1.6) leads to:

1þ sT@

@t

� �@2T x; tð Þ

@x2¼ 1þ kq

@

@xþ sq

@

@tþ s2q

2@2

@t2

!1a@T x; tð Þ

@tð2:15Þ

The following non-dimensional parameters are also introduced for analysis:

h ¼ T � T0Tw � T0

; b ¼ tsq

; n ¼ xffiffiffiffiffiffiffiasq

p ; Z ¼ sTsq

; L ¼ kqffiffiffiffiffiffiffiasq

p ð2:16Þ

The heat conduction Eq. (2.15) and the initial and boundary conditions could bewritten in terms of the non-dimensional parameters. To identify different heatconduction models, two artificial coefficients A and B are included in thenon-dimensional heat conduction equation as follows:

@2h n; bð Þ@n2

þ Z@3h n; bð Þ@n2@b

¼ @h n; bð Þ@b

þ L@2h n; bð Þ@n@b

þA@2h n; bð Þ

@b2þB

@3h n; bð Þ@b3

ð2:17aÞ

h n; 0ð Þ ¼ @h n; 0ð Þ@b

¼ 0 Initial conditionsð Þ ð2:17bÞ

h 0; bð Þ ¼ 1; limn!1

h n; bð Þ ¼ 0 Boundary conditionsð Þ ð2:17cÞ

As seen in Eq. (2.17), the thermal responses are characterized by Z; L; A; and Bparameters. For the case of L ¼ 0, A ¼ 1, B ¼ 1

2 and Z 6¼ 0, Eq. (2.17) reduces to

Fig. 2.1 A semi-infinite,homogeneous 1D mediumwith a suddenly raised surfacetemperature TW

2.3 Non-Fourier Heat Conduction in a Semi-infinite Strip 27

hyperbolic-type DPL model, while L ¼ 0, A ¼ 1, B ¼ 0, and Z 6¼ 0 lead toparabolic-type DPL model. Furthermore, for L ¼ 0, A ¼ 1, and B ¼ 0, Eq. (2.17)reduces to C-V and classical Fourier models when Z is set to 0 sT ¼ 0ð Þ and 1sq ¼ sT� �

, respectively. The nonlocal C-V (NL C-V) model could also be derivedfrom Eq. (2.17), by setting L 6¼ 0, A ¼ 1, and B ¼ Z ¼ 0:

Due to the time dependency of transient thermal responses in Eq. (2.17), solu-tion for temperature is found in the Laplace transform as:

~h n; sð Þ ¼ 1sexp

Ls�ffiffiffiffiffiffiffiffiffiffiffiLsð Þ2

qþ 4s 1þAsþBs2ð Þ 1þ Zsð Þ2 1þ Zsð Þ n

24

35 ð2:18Þ

shown by Tzou [1–3], using the partial expansion technique and the limiting the-orem in Laplace transform, Eq. (2.18) presents the thermal wave behavior fornonlocal C-V CT

NLC-V� �

model with the following thermal wave speed:

CTNLC-V ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiasq

þ kq2sq

� �2s

þ kq2sq

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCTC-V

� �2 þ kq2sq

� �2s

þ kq2sq

ð2:19Þ

which reveals that the CTNLC-V [CT

C-V . Reimann sum approximation is thenemployed to numerically transform temperature in Laplace domain, given inEq. (2.18), to time domain.

A code in MATLAB could be developed to numerically conduct the Laplaceinversion of Eq. (2.18). Reimann sum approximation has been used here for theLaplace inversion. Figure 2.2 compares temperature distribution at non-dimensionaltime b ¼ 1 for classical Fourier, C-V, diffusive-like DPL, wave-like DPL, and NLC-Vheat conductionmodels. As shown in Fig. 2.2, C-V,wave-likeDPL, andNLC-Vmodels result in awave-like behavior for temperature and reveals a sharpwavefront inthermal wave which divides the thermal response domain into the heat affected andunaffected zones. In accordance with Eq. (2.19), NL C-V model predicts a higherthermal wave speed compared to the C-Vmodel. Furthermore, as derived in Eq. (1.22), thermal wave speed of the wave-like DPL model is related to the C-V model as:CTDPL ¼ CT

C-Vffiffiffiffiffiffi2Z

p. As a result:

CTDPL\CT

C-V for Z\12

ð2:20aÞ

CTDPL �CT

C-V for Z� 12

ð2:20bÞ

For the assumed Z ¼ 10 in Fig. 2.2, the wave front of the wave-like DPL modelis ahead of the C-V and NL C-V models. As observed in Figs. 2.2 and 2.3, the

mixed derivative term Z @3h n;bð Þ@n2@b

in Eq. (2.17a) removes the singularity at the thermal

wavefront, compared to the wave-like DPL model. As Z increases in thediffusive-like DPL model the thermal wavefront is completely destroyed and


temperature responses show a diffusive behavior same as classical Fourier heatconduction, Fig. 2.3a. Although both of the diffusive-like DPL and classical Fourierheat conduction models do not show a finite thermal wave speed, the thermallyaffected zones are not the same for these two models. While for Z[ 1, thediffusive-like DPL model reveals a wider affected zone compared to the Fourierheat conduction; the affected zone is narrower for Z\1. For diffusive-like DPLmodel, thermal wave with discontinue temperature distribution around the wave-front is detected in Fig. 2.3b. Thermal wave speed increases by increasing Z. Asopposed to diffusive-like DPL, decreasing Z could result in a diverged and noisytemperature distribution for very low values of Z� 0ð Þ.

The effect of non-dimensional correlation length L on temperature distribution isillustrated in Fig. 2.4, at non-dimensional time b ¼ 1 using the NL C-V heatconduction model. As seen in this figure, the NL C-V model reduces to the C-Vmodel for = 0 kq ¼ 0

� �. As L increases, the wavefront of the NL C-V thermal wave

advances and temperature in the heat affected zone is raised. The NL model forL\2 has been shown by Tzou to be identical to the thermomass heat transfer modelin phonon gas.

Fig. 2.2 Non-dimensional temperature distribution predicted by Fourier, C-V, diffusive-like DPL,hyperbolic-type DPL, and NL C-V heat conduction models at non-dimensional time b ¼ 1

2.3 Non-Fourier Heat Conduction in a Semi-infinite Strip 29

Fig. 2.3 Effect of temperature and heat flux phase lag ratio Zð Þ on non-dimensional temperaturedistribution predicted by a diffusive-like DPL and b wave-like DPL at non-dimensional timeb ¼ 1

Fig. 2.4 Effect of non-dimensional correlation length Lð Þ on non-dimensional temperaturedistribution at non-dimensional time b ¼ 1 using NL C-V heat conduction model. [Reproducedfrom [1] with permission from Elsevier Masson SAS]


2.4 Nonlocal Phase-Lag Heat Conduction in a Finite Strip

As shown in Fig. 2.2, C-V, NL C-V, and wave-like DPL heat conduction modelsresult in a sharp wavefront of temperature with an infinite temperature gradientacross the thermal wavefront. To remove the singularity of thermal wavefront inC-V and NL-CV model, the introduction of sT has been shown to be effective [1].However, this approach can not remove the singularity of thermal wave observed inthe wave-like DPL/TPL models unless a fractional-order is used for the heat con-duction. As a result, we use in this section the wave-like NL FTPL, introduced inEq. (1.36), for a heat conduction analysis in a finite, isotropic, homogeneous 1Dstrip. As shown in Fig. 2.5, the strip is initially at T0 and a stationary state@T x;tð Þ

@t ¼ 0. The surface temperature of the left side x ¼ 0ð Þ is suddenly raised toTWL, while the surface temperature of the right side of the strip is kept at initialtemperature.

According to Eq. (1.36), the heat conduction constitutive equation is written inthe wave-like NL FTPL form in the one-dimensional Cartesian coordinate as:

1þ kq@

@xþ sq

@

@tþ s2q

2@2

@t2

!q x; tð Þ

¼ �kTIaF�1 1þ sT@

@t

� �@T x; tð Þ

@x� kT IaF�1 1þ st

@

@t

� �@t x; tð Þ@x

ð2:21Þ

Equation (2.21) is one of the forms of the wave-like NL FTPL heat conductionderived by the first-order Taylor series expansion of kq in space and thesecond-order Taylor series expansion of sq and the first-order Taylor seriesexpansion of sT and st in time. The other types of wave-like NL FTPL heatconduction could be achieved with a similar approach given in Ref. [4] for the TPLheat conduction. The current NL FTPL is not only able to remove the singularity ofthermal wavefront, but also is capable to take into account size-dependency, sub-diffusion or superdiffusion, and phonon-electron interaction in heat transport. InEq. (2.21), aF is the order of Riemann-Liouville fractional integral [5, 6]. The heatconduction equation of NL FTPL could be obtained by eliminating the heat fluxq x; tð Þ between Eq. (2.21) and energy Eq. (1.6). Taking the divergence andtime-derivative of Eq. (2.21) and eliminating r:q via the time-derivation of energyEq. (2.21) leads to the following heat conduction in 1D Cartesian coordinate:

Fig. 2.5 Thermal boundaryand initial conditions of a 1Dfinite strip

2.4 Nonlocal Phase-Lag Heat Conduction in a Finite Strip 31

IaF�1 1þ sT@

@t

� �@3T x; tð Þ@x2@t

þ kT

kTIaF�1 1þ st

@

@t

� �@2T x; tð Þ

@x2

¼ 1þ kq@

@xþ sq

@

@tþ s2q

2@2

@t2

!1a@2T x; tð Þ

@t2

ð2:22Þ

Following the non-dimensional parameters defined in Eq. (2.16), Eq. (2.22) andinitial and boundary conditions can be rewritten as:

@2h f; bð Þ@b2

þ L@3h f; bð Þ@b2@f

þ @3h f; bð Þ@b3

þ 12@4h f; bð Þ

@b4

¼ IaF�1CK@2h f; bð Þ

@f2þ IaF�1 1þCKZ

ð Þ @3h f; bð Þ@f2@b

þ IaF�1Z@4h f; bð Þ@f2b2

ð2:23aÞ

h n; 0ð Þ ¼ @h n; 0ð Þ@b

¼ 0 Initial conditionsð Þ ð2:23bÞ

h 0; bð Þ ¼ 1; h fR; bð Þ ¼ 0 Boundary conditionsð Þ ð2:23cÞ

where CK ¼ sqkT

kT and Z ¼ stsq; fR represents the non-dimensional length of the

medium, which is assumed fR ¼ 10 for the numerical results. Solution ofEq. (2.23a) in Laplace domain could be found as:

~h f; sð Þ ¼ er2fR

er2fR � er1fRer1fR � er1fR

er2fR � er1fRer2fR ð2:24Þ

in which r1 and r2 are characteristic roots of Eq. (2.23a) in the Laplace domain:

r1;2 ¼Ls2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2s4 þ 4s�aF þ 3 CK þ 1þCKZð Þsþ Zs2ð Þ 1þ sþ s2

2

� �q2s�aF þ 1 CK þ 1þCKZð Þsþ Zs2ð Þ ð2:25Þ

It is worth reminding the following identity for the Laplace transform ofRiemann-Liouville fractional integral as [5]:

L IaF f tð Þf g ¼ 1saF

L f tð Þf g aF [ 0 ð2:26Þ

To retrieve the temperature in the time domain, the Reimann sum approximationhas been used here.

Figure 2.6 presents the effect of fractional order aF on temperature distribution atno-dimensional time b ¼ 1 for the wave-like NL FTPL heat conduction. The tem-perature contours of the 1D slab are also given in the inset of Fig. 2.6. The fractionalorder varies in the range of 0\aF � 1. The nonlocal and phase-lag rations areassumed as L ¼ 1, Z ¼ 10, and Z ¼ 10, respectively. For aF ¼ 1, the wave-like


NL FTPL reduces to a wave-like NL TPL with a sharp wavefront with a transitionfrom a thermally affected zone f� 4:48ð Þ to an unaffected one f[ 4:48ð Þ, whichleads to an infinite temperature gradient across the thermal wavefront. As seen inFig. 2.6, fractional order aF is an alternative for removing the singularity of thermalwave in NL TPL heat conduction. A slight variation of aF from 1 to 0.95 effectivelysmoothens the thermal wavefront and the transition between the heat affected andunaffected zones. The effectiveness of aF on smoothening the thermal wave front ismore evident for lower values of aF , e.g. aF ¼ 0:85. Further decrease of aF couldcompletely destroyed the thermal wave front, as seen for aF ¼ 0:5. Temperaturelevel of the heat affected zone decreases while the temperature level of thermallyunaffected zone increases by a decrease in the value of aF . The effect of aF onthermal wave propagation in wave-like NL FTPL heat conduction is similar to theeffect of Z on diffusive-like Nonlocal DPL (diffusive-like NL DPL) heat conduction.As found by Zou and Guo [1], an increase in the value of Z in the NL DPL heatconduction could effectively remove the discontinuity of thermal wavefront whileresults in an increase in temperature level.

Figure 2.7 illustrates temperature distribution and temperature contours in the1D medium at non-dimensional time b ¼ 1 for a wave-like NL FTPL withaF ¼ 0:8, Z ¼ 10, and Z ¼ 10. Effect of non-dimensional TPL parameter CKð Þ

Fig. 2.6 Effect of fractional order aFð Þ on non-dimensional temperature distribution atnon-dimensional time b ¼ 1 using NL FTPL heat conduction model L ¼ 1; Z ¼ 10; Z ¼ 10ð Þ.Reproduced from [7] with permission from the Royal Society of Chemistry


and non-dimensional correlation length Lð Þ on the thermal behavior is specificallyinvestigated. Smooth thermal wavefront is detected in the thermal responses due tothe fractional heat conduction of the order aF ¼ 0:8. As seen in Fig. 2.7, CK and Lhave similar effect on thermal responses; temperature is increased by an increase ineither CK or L. Moreover, an increase in CK or L leads to an enhance in temperaturegradient around the thermal wavefront. While the correlation length L alters thethermal wave speed in NL C-V heat conduction, the effect of L on the thermal wavespeed of wave-like NL FTPL is not considerable.

To reveal the effect of phase lag ratios on thermal responses of wave-likeNL FTPL heat conduction, Fig. 2.8 shows temperature distribution atnon-dimensional time b ¼ 1 for alternative values of Z and Z. The fractional orderof heat conduction and TPL parameter are assumed as: CK ¼ 0:3 and aF ¼ 0:8. Asseen in Fig. 2.8, increasing the phase-lag ratio Z, which means either an increase inthe phase-lag of temperature gradient sT or a decrease in the phase-lag of heat fluxsq, increases the temperature level throughout the heat affected zone of the mediumand increases the thermal wave speed. Enhancing the phase-lag ratio Z, equiva-lently an increase in the phase-lag of thermal displacement st or a decrease in thephase-lag of heat flux sq, leads to an increase in the temperature level. Thermalwave speed, however, does not change by st in the wave-like NL FTPL model.

Fig. 2.7 Effect of non-dimensional TPL parameter CKð Þ and non-dimensional correlation lengthLð Þ on non-dimensional temperature distribution at non-dimensional time b ¼ 1 using NL FTPLheat conduction model ðaF ¼ 0:8, Z ¼ 10, and Z ¼ 10Þ [7]


Comparing the current observations for the thermal wave speed of wave-likeNL FTPL heat conduction with those found for wave-like TPL model in Eq. (1.27)and Refs. [4, 8] shows that all phase-lags have the same effect on the thermal wavespeed of both aforementioned heat conduction models.

To summarize the characteristics of alternative heat conduction in a continuumscale, temperature distribution developed within a semi-infinite slab subjected to anincreased temperature on the left side is illustrated in Fig. 2.9. The thermal boundaryconditions are the same as those assumed in Fig. 2.9 for a semi-infiniteone-dimensional strip. Deduced from Eq. (2.23), temperature distribution atnon-dimensional time b ¼ 1 is compared in Fig. 2.9 for the Fourier, C-V, nonlocalC-V (NL C-V) L ¼ 1ð Þ, diffusive-like and wave-like DPL Z ¼ 1ð Þ, wave-likefractional DPL (FDPL) Z ¼ 10; aF ¼ 0:95ð Þ, wave-like TPL Z ¼ Z ¼ 10;ðCK ¼ 0:3Þ, wave-like FTPL ðaF ¼ 0:95; Z ¼ Z ¼ 10; CK ¼ 0:3Þ, wave-likeNL TPL L ¼ 1; Z ¼ Z ¼ 10; CK ¼ 0:3ð Þ, and wave-like NL FTPL ðL ¼ 1; aF ¼0:95; Z ¼ Z ¼ 10; CK ¼ 0:3Þ continuum heat conduction models. The propaga-tion of thermal disturbance in the form of thermal wave can be observed in all theaforementioned heat conduction models, except the Fourier and diffusive-like DPLmodels. Temperature profile of C-V, NL C-V, wave-like DPL, and wave-like TPL,wave-like NL TPL heat conduction models reveals unrealistic sharp thermal

Fig. 2.8 Effect of non-dimensional phase-lag ratios Z and Z on non-dimensional temperaturedistribution at non-dimensional time b ¼ 1 using NL FTPL heat conduction model: ðCK ¼ 0:3;aF ¼ 0:8Þ [7]


wavefront which divides the thermal domain intro heat affected and heat unaffectedzones causing the unrealistic discontinuity of temperature and resulting thermalstrain [7]. Reported by Tzou [1], thermal wave speed of NL C-V model CNLC-Vð Þ ishigher than the C-V model CC-Vð Þ: CNLC-Vð Þ ¼

ffiffiffiffiffiffiffiffiffi4þ L2

p þ L2 CC-V where the dimen-

sional thermal wave speed of C-V is: CC-V ¼ffiffiffiasq

qx ¼ CC-V t ¼

ffiffiffiasq

qt

� �. Thermal

Fig. 2.9 Temperature profile of alternative continuum heat conduction models atnon-dimensional time b ¼ 1 in an infinite homogenous medium subjected to an abrupttemperature increase on its left side [7]


wave speed of wave-like DPL Cwave-like DPLð Þ, wave-like TPL Cwave-like TPLð Þ, andwave-like NL TPL Cwave-like NL TPLð Þ are equal and their thermal wave speed can bemathematically expressed as: Cwave-like DPL ¼ Cwave-like TPL ¼ Cwave-like NL TPL ¼ffiffiffiffiffiffi2Z

pCC-V . Thermal wave speed of different continuum non-Fourier heat conduction

models depends on the phase-lags of temperature gradient and heat flux, thermaldiffusivity, and correlating length and can be compared as: CNLC-V [CC-V [Cwave-like DPL ¼ Cwave-like TPL ¼ Cwave-like NL TPL for Z\ 2þ L2 þL

ffiffiffiffiffiffiffiffiffi4þ L2

p4 .

Cwave-like DPL ¼ Cwave-like TPL ¼ Cwave-like NL TPL �CNLC-V �CC-V

for Z� 2þ L2 þ Lffiffiffiffiffiffiffiffiffiffiffiffiffi4þ L2

p

4

ð2:27Þ

As shown in Fig. 2.9, while both classical Fourier heat conduction anddiffusive-like DPL models do not show finite thermal wave speed, the thermalaffected zones are not the same for these models. For Z[ 1, the diffusive-like DPLmodel leads to a wider affected zone compared to the classical Fourier heat con-duction, while the thermal affected zone is narrower for Z\1. Introducingtime-fractional derivatives in non-Fourier heat conduction models adds

IaF�1Z @4h f;bð Þ@f2@b2

term in the heat conduction Eq. (2.23a) and effectively destroys the

singularity of temperature field around the thermal wavefront. Smooth variation oftemperature is observed in the temperature profile of all fractional non-Fourier heatconduction. In addition, thermal wave speed is the same for all wave-like FDPL,FTPL, and NL FTPL models; however, NL FTPL model leads to a higher tem-perature range for the thermal affected zone. The NL FTPL heat conduction, as arecently introduced non-Fourier nonlocal continuum heat conduction model [7],simultaneously detects thermal wave propagation, removes the discontinuity oftemperature at thermal wavefront, and enables taking into account the effect oflength scale and microstructural heat transport on the conductive heat transport.

2.4.1 Molecular Dynamics to Determine CorrelatingNonlocal Length

Determining the value of correlating nonlocal length is one of the most intricatechallenges for application of recently developed nonlocal non-Fourier heat con-duction models to nanoscale materials. Since the correlating nonlocal length is anintrinsic property of material, it is required to be determined for each material. Theexperimental testing is a cumbersome task for measuring the thermal nonlocallength. While the experimental testing is inevitable for accurate determination ofnonlocal length, molecular dynamics (MD) and atomistic simulation are feasiblemethods for determining nonlocal length by comparing the characteristics ofthermal wave in nonlocal non-Fourier heat conduction and MD thermal results.Herein, we introduce the MD approach for measuring thermal nonlocal length of


copper. In specific, we present the results of MD simulation for a relatively-longnano-slab subjected to thermal excitation on its left side, an example whichresembles thermal wave propagation in one-dimensional heat transport.

The MD simulation is conducted by LAMMPS software [9] for a coppersingle-crystalline nano-slab of 361.5 nm length, 7.23 nm width, and 7.23 nmheight as shown schematically in Fig. 2.10. The steps required to be taken for theMD simulation are [7]:

(1) Creating the copper nano-slab by face-centred-cubic lattices.(2) Initializing the atoms with random velocities.(3) Equilibrating the nano-slab at room temperature 300 K for 20 picosecond

(ps) under Noose-Hoover thermostat (NVT) ensembles [10]. We fix the tem-perature of both hot and cool zones of the equilibrated nano-slab by rescalingtheir atoms at each time step.

(4) Increasing the temperature of the fixed hot zone of the nano-slab to 1000 Ktemperature, a condition that replicates the thermal boundary condition ofone-dimensional continuum NL FTPL heat conduction.

We apply MD simulation for time steps of 1 femtosecond (fs) and for a total timeperiod of 10 ps before thermal wavefront reaches the right end of nano-slab. Heattransport in solids is carried out by electrons and phonons. The atomistic interac-tions between atoms and electrons are introduced into the MD simulation throughthe embedded atom method (EAM) potential defined as [11]:

E ¼Xi

FiXi 6¼j

qi rij� � !

þ 12

Xij;i6¼j

/ij rij� � ð2:28Þ

where rij represents the distance between atoms i and j, qi is the contribution of theelectron charge density, Fi is the summation of individual embedding function ofatom i, and /ij represents a pairwise potential function between atoms. To deter-mine temperature distribution along the slab length, we divide the slab to finitenumbers of segments (here 100 segments) and obtain the average temperature ofeach segment as:

Fig. 2.10 A nano-slab considered for MD simulation of thermal wave propagation [7]


Tseg ¼P

i mm2i

2NsegkBð2:29Þ

In this equation, m, mi, Nseg, and kB are, respectively, atomic mass, velocity ofatom i, number of atoms in each segment, and Boltzmann constant and Tseg presentthe average value at each segment. As seen in Eq. (2.29), average thermodynamictemperature is related to the mean square velocity of atoms.

Figure 2.11 shows temperature profile in the copper nano-slab at different time.Similar to the temperature profile observed in a nano argon film [12], temperatureevolves in the nano-slab in the form of thermal wave. Thermal wave is observed totravel from the hot surface on the left side of the nano-slab towards the cold surfaceon the right with estimated thermal wave speed of CMD ¼ 23� 25� 103 m/s. If wecorrelate the thermal wave speed estimated by MD simulation with the speed ofsharp thermal wavefront in NL C-V model, the correlating length kq defined inEq. (2.14) can be estimated as:

kq ¼ sq CNLC-V � C2C-V

CNLC-V

� �� sq CMD � C2

C-VCMD

� �¼ 5:79� 7:11 nm ð2:30Þ

whereCC-V andCNLC-V represent the thermalwave speed predicted by theC-Vmodeland NL C-V models, respectively. While MD results in Fig. 2.11 show the propa-gation of temperature disturbance in the form of thermal wave, slight temperature risein atoms (locations) ahead of thermal wavefront is observed in temperature distri-bution. This observation is compatible with the characteristics of thermal wave pre-dicted by the NL FTPL heat conduction and those reported in Ref. [8] for phase-lagheat conduction in homogenous and heterogeneous porous materials.

Fig. 2.11 Temperaturedistribution in a nano-slab atdifferent time obtained by MDsimulation [7]


2.4.1.1 Nonlocal Heat Conduction in Functionally Graded Materials

Most of biological materials with extreme mechanical and thermochemical prop-erties, e.g. Moso culm bamboo [13], dento-enamel-junction of natural teeth [14],and the Humboldt squid beak [15], reveal a multi-scale hierarchical and function-ally graded (FG) microstructure. Examples of the extreme properties of functionallygraded materials (FGMs) are resistant to contact damage, cracking, deformation,thermal stresses, and heat flow due to the gradation of microstructural morphology,porosity, and chemical/material ingredients in FGMs [16–18]. FGMs enable theengineering of advanced materials with tuned multiphysics properties to satisfymechanical, hygrothermal, electrical, and biological requirements for structuraldesign in a wide range of applications as thermal barriers, bone tissues andimplants, thermoelectric generators, and energy harvesters. Advances in powdermetallurgy [19], laser cutting [20], and additive manufacturing/3D printing [21]have also facilitated fabrication and the arbitrary variation of material compositionand micro-architecture of FGMs.

Due to the importance of FGMs, we present here the temperature evolution andthermal wave propagation in an FGM nano-slab. The material properties of FGMscan be arbitrarily tailored within FGMs through the variation of volume fraction ofconstituent solid components (Two-phase solid FGMs) or relative density of porousmaterials (Single-phase porous FGMs). To be able to obtain closed-form solutionsfor transient temperature in the Laplace domain, we adopt here an exponentialfunction for variation of thermal conductivity kð Þ, material constant of the TPLtheory kð Þ, and specific heat per volume qcð Þ through the length of FGMnano-slab:

k ¼ k0enGx; k ¼ k0e

nGx; qcp� � ¼ C0e

nGx ð2:31Þ

where nG represents the FGM exponential index for the variation of materialproperties [22]. We assume that other thermosphysical properties are constantthroughout the nano-slab. The wave-like NL FTPL heat conduction equation for theexponentially graded medium in the absence of heat source can be obtained byusing the NL FTPL heat conduction Eq. (2.21) along with the energy Eq. (1.6) andnon-dimensional parameters (2.16):

1þ n0Lð Þ @2h f; bð Þ@b2

þ L@3h f; bð Þ@b2@f

þ @3h f; bð Þ@b3

þ 12@4h f; bð Þ

@b4

¼ IaF�1CK@2h f; bð Þ

@f2þ n0IaF�1 @

2h f; bð Þ@b@f

þ IaF�1 1þCKZð Þ @

3h f; bð Þ@f2@b

þ IaF�1Z@4h f; bð Þ@f2@b2

ð2:32Þ


where n0 ¼ nGffiffiffiffiffiffiffiasq

p. The closed-form solution of Eq. (2.32) in the Laplace domain

is the same as the one provided in Eq. (2.24) with the characteristics roots modifiedas follows:

r1;2 ¼

Ls2 � n0S�aF þ 2ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLs2 � n0S�aF þ 2ð Þ

þ 4s�aF þ 3 CK þ 1þCKZð Þsþ Zs2ð Þ1þ n0Lþ sþ s2

2

vuuuut

2s�aF þ 1 CK þ 1þCKZð Þsþ Zs2ð Þ ð2:33Þ

Temperature can then be retrieved in the time domain by using a numericalLaplace inversion technique.

Figure 2.12 shows the effect of FGM exponential index nG [presented inEq. (2.31)] on the characteristics of NL FTPL thermal wave at the non-dimensionaltime b ¼ 1. As seen in this figure, the NL FTPL thermal wave speed is constant andindependent of non-homogeneity index nG for exponential type of FGM materials;a phenomenon caused by the absence of non-homogeneity index parameter n0 ¼nG

ffiffiffiffiffiffiffiasq

pin terms of the highest order of temperature derivatives in NL FTPL

Fig. 2.12 Effect of material non-homogeneity index of FGM nano-slab on non-dimensionaltemperature distribution at non-dimensional time b ¼ 1ð Þ using NL FTPL heat conduction modelL ¼ 1;CK ¼ 0:3; aF ¼ 0:95;Z ¼ 10;Z ¼ 10ð Þ [7]


differential equation of heat conduction [Eq. (2.32)]. It is important mentioning thatwhile the exponential material gradation does not alter the thermal wave speedbased on the NL FTPL model for this specific FGM medium, thermal wave speedcan vary within an FGM medium with the FGM non-homogeneity index for FGMmaterials with an arbitrary variation of material properties [8, 23, 24]. Figure 2.12shows that material gradation can effectively tailor temperature within the thermalaffected zone of FGM medium. In addition, increasing the value of FGM expo-nential index nG from −0.5 to 0.5 can remarkably reduce temperature within thethermal affected zone of the FGM medium. Interestingly, decreasing the FGMexponential index nG can also magnify temperature at the thermal wavefrontcausing that the temperature within the FGM medium exceeds the temperature atthe boundaries; for example, the maximum temperature within an FGM mediumwith nG ¼ �0:5 is about 30% higher than the maximum temperature occurredwithin a homogenous medium nG ¼ 0. Consequently, material gradation canpotentially improve the performance of advanced materials used in extreme envi-ronmental conditions if the material gradation index is optimized.

2.5 Three-Phase-Lag Heat Conduction in 1D Strips,Cylinders, and Spheres

Till now, the one-dimensional heat conduction problems discussed in this chapterwere limited to planar medium in Cartesian coordinate system. In this section, wepresent a framework for investigating non-Fourier heat conduction in 1D media ingeneral coordinate system. In specific, a methodology is introduced for solving TPLheat conduction equation in a multilayered 1D (solid or porous) medium in ageneral coordinate system, which can present thermal wave propagation in 1D rod,1D infinitely-long axisymmetric cylinder, and 1D axisymmetric sphere. Each layerof medium is assumed to be homogenous, for which the TPL heat conductionequation is written. The TPL heat conduction differential equations for the multi-layered medium can be solved analytically in Laplace domain by applyingappropriate boundary and interfacial equations. Temperature is then can beretrieved in the time domain by a numerical Laplace Inversion to investigate thecharacteristics of thermal wave in general 1D coordinate systems.

Figure 2.13 illustrates a heterogeneous N-layered multilayered medium in ageneral 1D coordinate system~x. The position of the inner and outer surfaces of themedium is represented by xi and xo. In addition, xn n ¼ 1; 2; . . .; Nð Þ is the innersurface of the nth layer with x1 ¼ xi and xNþ 1 ¼ xo. The heterogeneous medium isinitially at ambient temperature T0 and the material properties in each layer isassumed to be constant.

To deal with the heat conduction problem in the multilayered medium, weintroduce the TPL heat conduction equation in the following form [25]:


~q nð Þ ~x; tþ s nð Þq

¼ � K nð Þr!T ~x; tþ s nð Þ

T

þK nð Þr!t ~x; tþ s nð Þ

t

h ið2:34Þ

In this equation, superscript n represents the layer number of the multilayeredmedium and associated material properties in that layer. As mentioned earlier in thischapter,~q nð Þ, T nð Þ, t nð Þ, K nð Þ, and K nð Þ n ¼ 1; 2; . . .; Nð Þ are, respectively, heat fluxvector, absolute temperature, thermal displacement _t nð Þ ¼ T nð Þ� �

, thermal conduc-

tivity, and material constant characteristics of the TPL theory; s nð Þq , s nð Þ

T , and s nð Þt also

represent, respectively, the phase-lag of heat flux, temperature gradient, and thermaldisplacement gradient in each layer. To develop the wave-like and diffusive-likeTPL heat conduction models, we expand the heat conduction equation of (2.34) by

using the second-order Taylor series expansion for s nð Þq and the first-order for s nð Þ

T

and s nð Þt as follows [8]:

1þ s nð Þq

@

@tþ s nð Þ2

q

2@2

@t2

!~q nð Þ

¼ � K nð Þ þK nð Þs nð Þt

r!T nð Þ þK nð Þs nð Þ

T@

@tr!T nð Þ þK nð Þr!t nð Þ

� � ð2:35Þ

where @@t t

nð Þ ¼ T nð Þ. The energy conservation equation can also be written as:

�r!:~q nð Þ þR nð Þ ¼ q nð Þc nð Þp

@T@t

ð2:36Þ

Fig. 2.13 A Multilayered medium in a general 1D coordinate system.[Reproduced from [8] with permission from Elsevier]

2.5 Three-Phase-Lag Heat Conduction in 1D Strips, Cylinders, and Spheres 43

where R nð Þ, q nð Þ, and c nð Þp are internal heat generation density, material density, and

specific heat in layer n. Heat conduction and energy equations introduced inEqs. (2.35) and (2.36) can be applied to any coordinate systems fromone-dimensional (1D) to three-dimensional (3D) media. If Eqs. (2.35) and (2.36)are combined together in one-dimensional media in Cartesian, cylindrical, andspherical coordinate systems, the differential equation of TPL heat conduction isobtained in a general 1D coordinate system~x shown in Fig. 2.13:

1þ s nð Þq

@

@tþ s nð Þ2

q

2@2

@t2

!q nð Þc nð Þ

p@2T nð Þ

@t2� @R nð Þ

@t

� �

¼ @

@xþ m

x

� �K nð Þ þK nð Þs nð Þ

t

@2T nð Þ

@x@tþK nð Þs nð Þ

T@3T nð Þ

@x@t2þK nð Þ @T

nð Þ

@x

� � ð2:37Þ

where m ¼ 0, m ¼ 1, and m ¼ 2 are respectively used for 1D Cartesian, cylindrical,and spherical coordinate systems. Similar to the non-dimensional parametersintroduced in Eq. (2.16), we use the following non-dimensional parameters for theheat conduction analysis in multilayered media:

f ¼ K 0 1ð Þtx2o

; g ¼ xxo

; h nð Þ ¼ T nð Þ � T0T0

;Q nð Þx ¼ xoq nð Þ

x

K 1ð ÞT0; e nð Þ

0 ¼ K 0 1ð Þs nð Þq

x2o

d nð Þ0 ¼ K 0 1ð Þs nð Þ

T

x2o; a nð Þ

0 ¼ K 0 1ð Þs nð Þt

x2o;K 0 1ð Þ ¼ K 1ð Þ

q 1ð Þc 1ð Þp

;K 0 nð Þ ¼ K nð Þ

q nð Þc nð Þp

gc ¼xixo

; hwi ¼ Twi � T0T0

; hwo ¼ Two � T0T0

ð2:38Þ

where Twi and Two are the temperature on the inner and outer surfaces. In theabsence of internal heat generation, Eq. (2.37) can be rewritten in the followingnon-dimensional form:

1þ e nð Þ0

@

@fþ e nð Þ2

0

2@2

@f2

!@2h nð Þ

@f2

¼ C nð Þ2T þ K 0 nð Þ

K 0 1ð Þ þC nð Þ2T a nð Þ

0

� �@

@fþ K 0 nð Þ

K 0 1ð Þ dnð Þ0

@2

@f2

� �mg@h nð Þ

@gþ @2h nð Þ

@g2

" # ð2:39Þ

where C nð Þ2T ¼ K nð Þx20

q nð Þc nð Þp K 0 1ð Þ2 . The thermal wave speed in each layer for C-V, wave-like

DPL, and wave-like TPL models can deduced from Eq. (2.39) as follows:

C nð ÞC-V ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK 0 nð Þ

K 0 1ð Þe nð Þ0

sC-Vð Þ ð2:40aÞ


C nð ÞDPL ¼ C nð Þ

TPL ¼ 1

e nð Þo

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2K 0 nð Þd nð Þ

0

K 0 1ð Þ

sWave-like DPL and TPLð Þ ð2:40bÞ

It is found that thenon-dimensional thermalwave speed in1Dmediadependson thephase-lag of the heat flux, phase-lag of temperature gradient, and the material prop-erties of each layer of a multilayered composite. The difference of material propertiesand interfacial imperfection between neighboring layers of multilayered media cancause the separation of the thermal wave into transmitted and reflected parts [8, 26].

Applying Laplace transform to Eq. (2.39) and considering zero initial conditionslead to:

g2@~h nð Þ

@g2þmg

@~h nð Þ

@g� D nð Þ

TPLg2~h nð Þ ¼ 0 ð2:41Þ

where

D nð ÞTPL ¼

1þ e nð Þ0 sþ e nð Þ2

02 s2

� �s2

C nð Þ2T þ K0 nð Þ

K0 1ð Þ þC nð Þ2T a nð Þ2

0

sþ K 0 nð Þ

K 0 1ð Þ dnð Þ0 s2

ð2:42Þ

The differential Eq. (2.41) can be solved in terms of Bessel functions as:

~h nð Þ g; sð Þ ¼ g1�m2 A nð Þ

1 JG nð Þ I nð Þg

þA nð Þ2 YG nð Þ I nð Þg

ð2:43Þ

where

G nð Þ ¼ m� 12

; I nð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�D nð Þ

TPL

qð2:44Þ

and A nð Þ1 and A nð Þ

2 n ¼ 1; 2; . . .;Nð Þ are integration constants, and JG nð Þ and YG nð Þ areG nð Þth-order Bessel functions of the first and second kind. The heat flux can also bewritten in the Laplace domain using Eqs. (2.35) and (2.43):

~Q nð Þx g; sð Þ ¼ P nð Þ

TPL

2g�m�1

2

A nð Þ1 M nð ÞJG nð Þ I nð Þg

� 2I nð ÞgJG nð Þ þ 1 I nð Þg

þA nð Þ

2 M nð ÞYG nð Þ I nð Þg

� 2I nð ÞgYG nð Þ þ 1 I nð Þg

0B@

1CA ð2:45Þ

where

M nð Þ ¼ 2G nð Þ � m� 1ð Þ;P nð ÞTPL ¼ �

C nð Þ2T þ K nð Þ

K 1ð Þ þC nð Þ2T a nð Þ

0

sþ K nð Þ

K 1ð Þ dnð Þ0 s2

s 1þ e nð Þ0 sþ e nð Þ2

02 s2

� � ð2:46Þ


The integration constants in Eqs. (2.43) and (2.45) are obtained by satisfying thethermal boundary and interfacial conditions:

h 1ð Þ g; fð Þjg¼gc¼ hwifi fð Þ ð2:47aÞ

v jð ÞT K 1ð Þ

xoQ jð Þ

x g; fð Þjg¼gjþ 1¼ h jð Þ g; fð Þ � h jþ 1ð Þ g; fð Þ

g¼gjþ 1

ð2:47bÞ

Q jð Þx g; fð Þjg¼gjþ 1

¼ Q jþ 1ð Þx g; fð Þjg¼gjþ 1

ð2:47cÞ

h Nð Þ g; fð Þjg¼1¼ hwofo fð Þ j ¼ 1; 2; . . .;N � 1ð Þ ð2:47dÞ

where v jð ÞT stands for the thermal compliance constant or thermal contact resistance for

the imperfect interface between layers j and jþ 1; fi and fo are temporal functions forthe applied transient thermal boundary conditions on the inner and/or outer surfaces ofthe multilayered medium. The bonding imperfection with thermally weak conductionhas taken into account in Eq. (2.47) to conduct a reliable thermal analysis for mul-tilayered composites [27, 28]. The perfectly bonded interfaces are associated with

v jð ÞT ¼ 0. Although a Heaviside step function, fi fð Þ ¼ fo fð Þ ¼ H fð Þ, is considered for

the time-dependent functions in this case study, similar procedure can be followed forother types of transient thermal disturbances [23]. Using Eqs. (2.43) and (2.45) andthe thermal boundary and interfacial conditions (2.47) in the Laplace domain results inthe following algebraic equation that allows obtaining the integration constants:

KTPL½ �2N�2N XTPLf g2N�1¼ FTPLf g2N�1 ð2:48Þ

where KTPL½ � is a 2N � 2N matrix, XTPLf g is a 2N � 1 vector of integration con-

stants XTPLf gT¼ A 1ð Þ1 A 1ð Þ

2 � � � A Nð Þ1 A Nð Þ

2

n o, and FTPLf g is a 2N � 1 vector.

By solving Eq. (2.48), the transient temperature change and heat flux in the Laplacedomain are obtained in the Laplace domain. Finally, temperature change and heatflux can be retrieved in the time domain by implementing a numerical Laplaceinversion technique introduced in Sect. 2.2. If this methodology is applied, we caninvestigate the effect of bonding interface, material heterogeneity, and continuousvariation of material properties on thermal responses of advanced materials.

2.5.1 Effect of Bonding Imperfection on Thermal WavePropagation

To examine the effect of bonding interface on transient thermal responses, Fig. 2.14illustrates the effect of imperfectly bonded interface on the temperature and heatflux distribution of a bilayered cylinder m ¼ 1ð Þ with inner radius xi ¼ 0:6 and


Fig. 2.14 Effect of thermal compliance on the distribution of a temperature and b heat-flux atdimensionless time f ¼ 0:126 m ¼ 1; e0 ¼ 0:35; d0 ¼ 0:25; Hyperbolic DPLð Þ [8]

outer radius xi ¼ 1. Both bonded layers are made of the same materials, i.e. copper(Cu). The outer surface of the bilayered cylinder is subjected to a sudden tem-perature rise Two ¼ 600 K and the hyperbolic DPL heat conduction model is usedfor the thermal analysis. Figure 2.14a, b depict temperature and heat flux at thenon-dimensional time f ¼ 0:126. Thermal excitation causes the thermal wave topropagate towards the inner surface of the cylinder. The bonding imperfectioncauses the separation of the initial thermal wave into the transmitted and a reflectedparts, each travelling, at a given thermal wave speed of CDPL, towards the inner andouter surfaces of cylinder, respectively. While temperature is discontinuous at theinterface of the bilayered cylinder, the radial heat flux is continuous at the interfacewhich is compatible with the thermal boundary conditions considered in Eq. (2.47).

Figure 2.15 reveals that the absolute value of the transient and steady-statetemperature and heat flux in the middle of the inner layer g ¼ 0:7ð Þ decreases by


Fig. 2.15 Effect of thermal compliance on: a temperature and b heat flux time-histories at g ¼ 0:7m ¼ 1; e0 ¼ 0:35; d0 ¼ 0:25;Hyperbolic DPLð Þ [8]

stiffening the thermal compliance of the bonding interface. Figures 2.14 and 2.15show that thermal compliance of imperfectly bonded interface amplifies tempera-ture difference at the interface of cylinder and reduces the heat flux transmittedthrough the interface.

2.5.2 Effect of Material Heterogeneity on Thermal WavePropagation

Although bonding imperfections often trigger thermal wave separation in a multi-layered system, material heterogeneity is generally the main culprit for the thermalwave separation. Figure 2.16 shows the influence of heterogeneity of the middle


layer in a sandwich slab m ¼ 0ð Þ on thermal responses predicted by the hyperbolicTPL model. The position of the inner and outer surfaces are assumed to be xi ¼ 0:7and xo ¼ 1. The sandwich slab is perfectly bonded vT ¼ 0ð Þ and is made by theinner and outer layers of Cu while the middle layer of the slab is of Cu, Ag, or Au.We track the thermal waves propagation towards the inner layer of the slab for threedifferent layer arrangements: Cu/Cu/Cu, Cu/Ag/Cu, and Cu/Au/Cu. Thermal waveseparation occurs only for the heterogeneous material arrangements of Cu/Ag/Cu,and Cu/Au/Cu. This is caused by the meeting of thermal wave front at the interfacebetween layers with dissimilar material properties, thereby generating waves withreflected and transmitted portions. This phenomenon is depicted in Fig. 2.16a, b at

Fig. 2.16 Effect of material heterogeneity in the middle layer of a sandwich slab on temperaturedistribution at a f ¼ 0:084 and b f ¼ 0:126 non-dimensional time. m ¼ 0; n ¼ 3; vT ¼ 0;ð e0 ¼0:35; d0 ¼ 0:25; a0 ¼ 0:15;C2

T ¼ 2; Hyperbolic TPLÞ [8]


two different non-dimensional time. Figure 2.16a shows that the thermal wavespeed at the middle layer of Ag is higher than that of Au due to its thermalproperties; however, the transient temperature at the middle layer of Ag is lowerthan Au. If time increases, thermal wave will transmit and reflects from all inter-faces and it will reflect back from the inner and outer surfaces of the slab to finallyreach the steady-state temperature.

2.5.3 Thermal Response of a Lightweight SandwichCircular Panel with a Porous Core

We present here the thermal response of a lightweight sandwich panel with a porouscore, which is commonly used for aerospace, automotive, electronics, biomedicineapplications. These lightweight porous materials are recently of great significancedue to their capabilities for satisfying multiple functionalities if their microarchi-tecture is optimized. The advances in 3D printing and additive manufacturing havealso enabled engineers to tune the morphology of porous materials and fabricatethis new type of advanced porous materials. Herein, we focus on the application ofporous materials as a core of sandwich panels.

Since a fully detailed microscale analysis of porous materials is computationallyexpensive, a multiscale model based on homogenization theory and microme-chanics is commonly used. In homogenization, a representative volume element ofthe porous materials is selected and effective material properties are obtained byapplying the periodic boundary conditions [29–32]. Following this approach, theeffective specific heat of porous foams can be obtained by the classical rule ofmixture which includes the contribution of the solid and gas as follows [33]:

qcp ¼ qscps� �

qr þ qgcpg� �

1� qrð Þ ð2:49Þ

where subscripts “s” and “g” represent respectively thematerial properties of solid andgas, and the overbar parameters specify the effectivematerial properties offoams;qr isthe relative density definedas:qr ¼ q

qs. If theheat convectionof thegas is neglecteddue

to the small size of pores, we can approximate its thermal conductivity [33]:

K ¼ 13

qr þ 2q32r

Ks þ 1� qrð ÞKg ð2:50Þ

We can assume that the specific heat and thermal conductivity of the gas to beequal to those of dry air: qgcpg ¼ 1:006� 103 J/kgK and Kg ¼ 0:025 W/mK. Weuse the effective material properties presented in Eqs. (2.49) and (2.50) to inves-tigate heat conduction in sandwich panels with a porous core.

When thermal disturbance hits a sandwich panel, the relative density of thecellular foam core controls the thermal response of the lightweight sandwich


structure. We examine a three-layer perfectly bonded sandwich cylinder m ¼ 1ð Þwith inner and outer solid layers of Cu and a porous middle layer of Cu with therelative density of qr. The inner and outer radius of the sandwich cylinder arexi ¼ 0:7 and xo ¼ 1, respectively. The thermal compliance constants for all inter-

faces are expressed by the index: vT ¼ v jð ÞT K 1ð Þ

xoj ¼ 1; 2; . . .;N � 1ð Þ.

Temperature and heat flux distribution obtained by the hyperbolic DPL model isshown in Figs. 2.17 and 2.18. The figures confirm the impact of relative density; inparticular in porous middle layer, the thermal wave speed decreases with the rel-ative density. The reduction of relative density in the middle layer increases thetransient temperature in the middle and outer layers of the cylinder, while it

Fig. 2.17 Effect of relative density of the porous middle layer in a sandwich cylinder on thetemperature distribution at non-dimensional time: a f ¼ 0:112 and b steady-statem ¼ 1; n ¼ 3; vT ¼ 0; e0 ¼ 0:35; d0 ¼ 0:25; Hyperbolic DPLð Þ [8]


decreases the transient temperature in the inner layer. A decrease in relative densityalso lessens the absolute value of the heat flux passing through the inner and middlelayer, whereas it amplifies the thermal insulation of the inner surface from thethermal shock applied on the outer surface of the lightweight sandwich cylinder.Further investigations on the transient responses reveal that from one hand areduced relative density of the middle layer decreases the steady-state temperatureof the first half of the cylinder, far from the thermal excitation; from the other, itincreases the steady-state temperature of the second half of the cylinder, close to thethermal disturbance. The heat flux within the cylinder, however, decreases with therelative density of the middle layer [8].

Fig. 2.18 Effect of relative density of porous middle layer in a sandwich cylinder on the heat fluxdistribution at non-dimensional time: a f ¼ 0:112 and b steady-state m ¼ 1; n ¼ 3; vT ¼ 0;ðe0 ¼ 0:35; d0 ¼ 0:25; Hyperbolic DPLÞ [8]


It is worth mentioning that the multilayered methodology presented in thissection can also be used for investigating the thermal response of FG solid andporous materials. In this case, FG materials are divided into a finite number ofhomogenous layers in which the material properties of each layer are obtainedaccording to the function associated with the variation of material properties in FGmaterials. This method is called “piecewise homogenous layer” for the simulationof FGMs, which is consistent with the methods which are used by 3D printing oradditive manufacturing techniques for fabrication of FGMs.

2.6 Dual-Phase-Lag Heat Conductionin Multi-dimensional Media

All heat conduction problems yet discussed in this chapter have been limited to 1Dmedia. However, the heat conduction differential equations introduced earlier in thischapter can be applied to 2D and 3D problems in all coordinate systems. Thesolution procedure for 2D and 3D problems are usually more complex than 1Dproblems. For general types of thermal boundary conditions, numerical method,e.g. finite element method or boundary element method, are efficient for solving theheat conduction problems in spatial coordinate systems. Closed-form solutions for2D and 3D problems may also be developed for specific thermal boundary con-ditions. In this section, we introduce a semi-analytical methodology for the heatconduction analysis of 2D and 3D problems in the form of cylindrical/sphericalpanel based on the DPL model.

2.6.1 DPL Heat Conduction in Multi-dimensionalCylindrical Panels

We consider a radially graded FG cylindrical panel of the inner and outer radii riand ro, azimuthal angle u0, and length L. As shown in Fig. 2.19, we consider thecylindrical coordinate system r; u; zð Þ. The initial temperature of the FG cylin-drical panel is assumed to be ambient temperature T1.

The DPL heat conduction equation in heterogeneous materials can be written as[34]:

1þ sq@

@tþ s2q

2@2

@t2

!qcp

@T@t

� R

� �¼ r!: K 1þ sT

@

@t

� �r!T

� �ð2:51Þ

To simplify the solution procedure, we assume that phase-lags to be constant.Quintanilla [35] proved the stability of Eq. (2.51) for sh

sq[ 1

2. To investigate the


temperature field in the heterogeneous cylindrical structures, heat conductionEq. (2.51) can be written in 3D cylindrical coordinate system r; u; zð Þ:

1þ sq@

@tþ s2q

2@2

@t2

!qcp

@T@t

¼ 1r@

@rrK

@T@r

þ sT@2T@t@r

� ��

þ 1r2

@

@uK

@T@u

þ sT@2T@t@u

� �� þ @

@zK

@T@z

þ sT@2T@t@z

� �� ð2:52Þ

Considering the thermal boundary conditions and the geometry of the cylindricalpanel, Eq. (2.52) can be used to solve the DPL heat conduction problem in anaxisymmetric cylinder with an infinite length (r space), a cylinder with finite length[ r; zð Þ space], an infinitely long cylindrical panel [ r;uð Þ space], and a cylindricalpanel with finite length [ r;u; zð Þ space]. The FG cylindrical panels are assumed tohave a continuous transition of material properties, except for the phase lags,according to a power law formulation:

K gð Þ ¼ K0gn1 ; q gð Þ ¼ q0g

n2 ; cp gð Þ ¼ cp0gn3 ð2:53Þ

where K0, q0, and cp0 are constants; nj j ¼ 1; 2; 3ð Þ are non-homogeneity indicesand g ¼ r

r0. To simplify our analysis, the following non-dimensional parameters are

used:

f ¼ K 00tr2o

; e0 ¼ K 00sqr2o

; d0 ¼ K 00sTr2o

; g ¼ rro;u ¼ u

uo; z ¼ z

L; rc ¼ ri

ro

h ¼ T � T1Two � T1

; Tc ¼ Twi � T1Two � T1

;Qr ¼ roqrK0T1

;Qu ¼ rou0quK0T1

;Qz ¼ LqzK0T1

ð2:54Þ

where Twi and Two are temperature on the inner and outer surfaces of the cylindricalpanel and K 0

0 ¼ K0q0c0

; qr, qu, and qz are the radial, azimuthal, and longitudinal heat

fluxes, respectively.We consider an FG cylindrical panel with a length L and azimuthal angle u0.

The non-dimensional form of the DPL heat conduction equation for the FG panel iswritten as:

Fig. 2.19 Radially gradedFG cylindrical panel.[Reproduced from [34] withpermission from Springer]


gn2 þ n3 1þ e0@

@fþ e20

2@2

@f2

� �@h@f

¼ d0@

@fþ 1

� �gn1

@2h@g2

þ n1 þ 1ð Þgn1�1 @h@g

þ 1u20gn1�2 @

2h

@u2 þ roL

2gn1

@2h

@z2

� �ð2:55Þ

To develop a semi-analytical solution for the heat conduction Eq. (2.55), weassume the thermal boundary and initial conditions in the following form:

h g;u; z; fð Þjg¼rc¼ Tc sin mpuð Þ sin ppzð Þh g;u; z; fð Þjg¼1¼ sin mpuð Þ sin ppzð Þ ð2:56aÞ

h g;u; z; fð Þjf¼0¼ 0;@

@fh g;u; z; fð Þjf¼0¼ 0 ð2:56bÞ

To satisfy the thermal boundary conditions of Eq. (2.56a), temperature change isassumed as follows:

h g;u; z; fð Þ ¼ hmp g; fð Þ sin mpuð Þ sin ppzð Þ ð2:57Þ

where m and p stand for the mode number and hmp g; fð Þ is an unknown function fortemperature. Using Eqs. (2.55) and (2.57) and implementing the Laplace transformlead to:

g2@2~hmp@g2

þ n1 þ 1ð Þg @~hmp@g

� Egn2 þ n3�n1 þ 2 þ nproL

2g2 þ pp

u0

� �2 !

~hmp ¼ 0

ð2:58Þ

where E is defined as: E ¼s

e202 s

2 þ e0sþ 1

d0sþ 1 . For n1 ¼ n2 þ n3, the solution of

Eq. (2.58) can be found as:

~hmp g; sð Þ ¼ g�n12 A1JG Igð ÞþA2YG Igð Þð Þ ð2:59Þ

where

G ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin12

2þ mp

u0

� �2s

; I ¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEþ ppro

L

2rð2:60Þ

2.6 Dual-Phase-Lag Heat Conduction in Multi-dimensional Media 55

and the integration constants A1 and A2 are obtained as:

A1 ¼ � YG Irc� �þ r

n12c TcYG Ið Þ

s �YG Irc� �

JG Ið Þþ JG Irc� �

YG Ið Þ� �A2 ¼

JG Irc� �� r

n12c TcJG Ið Þ

s �YG Irc� �

JG Ið Þþ JG Irc� �

YG Ið Þ� �ð2:61Þ

The radial, azimuthal, and longitudinal heat fluxes in the Laplace domain can beobtained as:

~Qr g;u; z; sð Þ ¼ P2g

n12�1 A1 MJG Igð Þ � 2HIgHJGþ 1 Igð Þ� ��

þA2 MYG Igð Þ � 2HIgHYGþ 1 Igð Þ� ��sin mpuð Þ sin ppzð Þ

~Qu g;u; z; sð Þ ¼ mpPð Þgn12�1 A1JG Igð ÞþA2YG Igð Þ½ � cos mpuð Þ sin ppzð Þ

~Qz g;u; z; sð Þ ¼ ppPð Þgn12 A1JG Igð ÞþA2YG Igð Þ½ � sin mpuð Þ cos ppzð Þ

ð2:62Þ

where P and M are defined as:

P ¼ � Two � T1T1

� �d0sþ 1

e202 s

3 þ e0s2 þ s

!;M ¼ 2G� n1 ð2:63Þ

The steady-state temperature change can also be defined as:

hs g;u; zð Þ ¼ hmps gð Þ sin mpuð Þ sin ppzð Þ ð2:64Þ

For n1 ¼ n2 þ n3, Eq. (2.64) is substituted into the steady-state form ofEq. (2.55):

g2@2hmps@g2

þ n1 þ 1ð Þg @hmps@g

� mpu0

� �2

þ pproL

2 !hmps ¼ 0 ð2:65Þ

The solution of Eq. (2.65) can be obtained as follows

hmps gð Þ ¼ g�n12 B1JGs Isgð ÞþB2YGs Isgð Þð Þ ð2:66Þ

where Gs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin12

� �2 þ mpu0

2rand Is ¼ ppro

L

� �i. The integration constants B1 and B2

in Eq. (2.66) are obtained as:


B1 ¼�YGs Isrc

� �þ rn12c TcYGs Isð Þ

�YGs Isrc� �

JGs Isð Þþ JGs Isrc� �

YGs Isð Þ

B2 ¼JGs Isrc� �� r

n12c TcJGs Isð Þ

�YGs Isrc� �

JGs Isð Þþ JGs Isrc� �

YGs Isð Þ

ð2:67Þ

Finally, temperature and heat flux are retrieved in the time domain by imple-menting a numerical Laplace inversion technique. It is worth mentioning thatEq. (2.55) shows the thermal wave speed in the radial direction based on the DPLmodel CDPL gð Þ½ � within axisymmetric infinitely-long hollow FG cylinder dependson the location of thermal wavefront since CDPL gð Þ½ � can be expressed as:

CDPL gð Þ ¼ffiffiffiffiffiffiffi2d0e0

sg

n1�n2�n32 ð2:68Þ

According to Eq. (2.68), when the non-homogeneity indices follow an specificrelation, i.e. n1 ¼ n2 þ n3, the radial thermal wave speed is independent of both theradial coordinate and the non-homogeneity indices. To numerically confirm that themathematical conclusion is also valid for cylindrical panels with a finite length, thetemperature distribution along the radial direction is depicted in Fig. 2.20 for an FGcylindrical panel with the azimuthal angle /0 ¼ p

2 and length L ¼ 1 for differentcombinations of non-homogeneity indices, in which n1 ¼ n2 þ n3. The radial tem-perature distribution has been shown at the mid-section of the panel with / ¼ 0:5and z ¼ 0:5. As shown in Fig. 2.20, all the wavefronts are in the same position at thenon-dimensional time f ¼ 0:126 independent of the value of non-homogeneity

Fig. 2.20 In an FG cylindrical panel with a finite length ð/0 ¼ p2 and L ¼ 1Þ, thermal wavefront

location is independent of the non-homogeneity indices when n1 ¼ n2 þ n3 [34]


indices. This observation corroborate the DPL thermal wave speed is the same atdifferent locations of FG cylindrical panels when n1 ¼ n2 þ n3.

2.6.2 DPL Heat Conduction in Multi-dimensional SphericalVessels

To obtain temperature field in heterogeneous spherical vessels, we should writeDPL heat conduction Eq. (2.51) in the spherical coordinate system r;u;wð Þ pre-sented in Table 1.1. The thermal boundary conditions for the spherical vessel canbe assumed to be: (1) Spherically symmetric one-dimensional or (2) Axisymmetrictwo-dimensional. As a result, the heat conduction equation is simplified in the formof the following partial differential equations in the spherical coordinate system:

1þ sq@

@tþ s2q

2@2

@t2

!qcp

@T@t

¼ 1r2

@

@rKr2 1þ sT

@

@t

� �@T@r

� �þ 1

r sinu@

@usinur

K 1þ sT@

@t

� �@T@u

� � ð2:69Þ

In this section, we focus on the heat conduction in spherically axisymmetrictwo-dimensional problem. The material properties of the spherical vessel isassumed to vary radially according to the power law formulation, similar to thoseintroduced in Sect. 2.6.1. Except for phase lags, which are assumed constant, allother thermal properties varies according to Eq. (2.53). To simplify the solutionprocedure, we employ the non-dimensional parameters of Eq. (2.54).

Using the assumed material properties for the FG spherical vessel, we canrewrite Eq. (2.69) in the following non-dimensional form:

gn2 þ n3 1þ e0@

@fþ e20

2@2

@f2

� �@h@f

¼ d0@

@fþ 1

� �gn1

@2h@g2

þ n1 þ 2ð Þgn1�1 @h@g

þ gn1�2 1sinu

@

@usinu

@h@u

� �� ð2:70Þ

The axisymmetric thermal boundary and initial conditions are assumed for theFG spherical vessel to enable us obtaining a semi-analytical solution for the 2D heatconduction problem in the spherical coordinate system [36]:

h g;u; fð Þjg¼rc¼ Tc cosu; h g;u; fð Þjg¼1¼ cosu ð2:71aÞ

h g;u; fð Þjf¼0¼ 0;@

@fh g;u; fð Þjf¼0¼ 0 ð2:71bÞ


Considering the above-mentioned thermal boundary conditions, temperature canbe written as:

h g;u; fð Þ ¼ h1 g; fð Þ cosu ð2:72Þ

where h1 g; fð Þ is an unknown temperature that is needed to be determined using theinitial and boundary thermal conditions. Substituting Eq. (2.72) into Eq. (2.70) andperforming the Laplace transform with regard to the initial conditions (2.71b), leadto:

g2@2~h1@g2

þ n1 þ 2ð Þg @~h1@g

� Egn2 þ n3�n1 þ 2 þ 2� �

~h1 ¼ 0 ð2:73Þ

where E is defined as: E ¼s

e202 s

2 þ e0sþ 1

d0sþ 1 . The differential Eq. (2.73) can be solved

as:

~h g; sð Þ ¼ g�n1 þ 1

2 A1JG IgH� �þA2YG IgH

� �� for n1 � n2 � n3 6¼ 2ð Þ

~h g; sð Þ ¼ A1gk1 þA2gk2 for n1 � n2 � n3 ¼ 2ð Þð2:74Þ

where

G ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1

2

� �2 þ 2q

n2 þ n3 � n1 þ 2; k1;2 ¼

� n1 þ 1ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ 1ð Þ2 þ 4 Eþ 2ð Þ

q2

H ¼ 1þ n2 þ n3 � n12

; I ¼ 2ffiffiffiffiffiffiffi�E

p

n2 þ n3 � n1 þ 2

ð2:75Þ

The integration constants A1 and A2 can be also obtained in the Laplace domainas:

For n1 � n2 � n3 6¼ 2:

A1 ¼�YG IrHc

þ r

n1 þ 12

c TcYG Ið Þs �YG IrHc

JG Ið Þþ JG IrHc

YG Ið Þ

A2 ¼JG IrHc

� rn1 þ 1

2c TcJG Ið Þ

s �YG IrHc

JG Ið Þþ JG IrHc

YG Ið Þ ð2:76aÞ

For n1 � n2 � n3 ¼ 2:


A1 ¼Tc � rk2c

s rk1c � rk2c

A2 ¼rk1c � Tc

s rk1c � rk2c

ð2:76bÞ

Non-dimensional radial and polar heat fluxes in the Laplace domain are alsoobtained as:

For n1 � n2 � n3 6¼ 2:

~Qr g;u; sð Þ ¼ P2g

n1�32 A1 MJG IgH

� �� 2IHgHJGþ 1 IgH� ��

þA2 MYG IgH� �� 2IHgHYGþ 1 IgH

� �� cosu

~Qu g;u; sð Þ ¼ �Pgn1�32 A1JG IgH

� �þA2YG IgH� ��

sinu

ð2:77aÞ

and for n1 � n2 � n3 ¼ 2:

~Qr g;/; s� � ¼ P A1k1g

n1 þ k1�1 þA2k2gn1 þ k2�1� �

cosu

~Q/ g;/; s� � ¼ �P A1gn1 þ k1�1 þA2gn1 þ k2�1

� �sinu

ð2:77bÞ

where M and P are defined as:

M ¼ 2GH � n1 þ 1ð Þ; P ¼ � Two � T1T1

� �d0sþ 1

e202 s

3 þ e0s2 þ s

!ð2:78Þ

The temperature in time domain can then be obtained by implementing anumerical Laplace inversion technique. It is worth mentioning that while thermalwave can propagate in multiple directions in 2D or 3D spherical vessels, the radialthermal wave speed in all 1D, 2D, and 3D structures are the same.

To clarify the effects of each non-homogeneity indices on the thermal responsesof an axisymmetric hollow sphere (2D) based on the DPL heat conduction theory,among the three different non-homogeneity indices n1 (thermal conductivity index),n2 (density index), and n3 (specific heat index), two of them are kept constant andonly one varies. Figure 2.21a–c show the temperature history of the mid-plane inthree different cases: (a) n2 ¼ n3 ¼ 1; (b) n1 ¼ n3 ¼ 1; and (c) n1 ¼ n2 ¼ 1.Figure 2.21a reveals that increasing the non-homogeneity index of thermal con-ductivity n1 leads to higher transient and steady-state temperature. Althoughincreasing the non-homogeneity indices of density n2 and specific heat n3 increasesthe amplitudes of transient temperature, the steady-state temperature does notchange by tailoring n2 and n3 as shown in Fig. 2.21b, c.


Fig. 2.21 Effect of non-homogeneity indices on temperature time-history of the mid-plane of theaxisymmetric (2D) hollow sphere: a Effect of thermal conductivity, b Effect of density, andc Effect of specific heat. [Reproduced from [36] with permission from World Scientific PublishingCo., Inc]


References

1. Tzou D, Guo Z-Y (2010) Nonlocal behavior in thermal lagging. Int J Therm Sci 49(7):1133–1137

2. Tzou D (2011) Nonlocal behavior in phonon transport. Int J Heat Mass Transf 54(1):475–4813. Tzou DY (2014) Macro-to microscale heat transfer: the lagging behavior. Wiley4. Akbarzadeh A, Fu J, Chen Z (2014) Three-phase-lag heat conduction in a functionally graded

hollow cylinder. Trans Can Soc Mech Eng 38(1):1555. Youssef HM (2010) Theory of fractional order generalized thermoelasticity. J Heat Transf

132(6):0613016. Ezzat MA, El Karamany AS, Fayik MA (2012) Fractional order theory in thermoelastic solid

with three-phase lag heat transfer. Arch Appl Mech 82(4):557–5727. Akbarzadeh A, Cui Y, Chen Z (2017) Thermal wave: from nonlocal continuum to molecular

dynamics. RSC Adv 7(22):13623–136368. Akbarzadeh A, Pasini D (2014) Phase-lag heat conduction in multilayered cellular media with

imperfect bonds. Int J Heat Mass Transf 75:656–6679. Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput

Phys 117(1):1–1910. Nosé S (1984) A unified formulation of the constant temperature molecular dynamics

methods. J Chem Phys 81(1):511–51911. Foiles S, Baskes M, Daw M (1986) Embedded-atom-method functions for the fcc metals Cu,

Ag, Au, Ni, Pd, Pt, and their alloys. Phys Rev B 33(12):798312. Liu Q, Jiang P, Xiang H (2008) Molecular dynamics simulations of non-Fourier heat

conduction. Progr Nat Sci 18(8):999–100713. Tan T et al (2011) Mechanical properties of functionally graded hierarchical bamboo

structures. Acta Biomater 7(10):3796–380314. Rahbar N, Soboyejo W (2011) Design of functionally graded dental multilayers. Fatigue Fract

Eng Mater Struct 34(11):887–89715. Miserez A et al (2008) The transition from stiff to compliant materials in squid beaks. Science

319(5871):1816–181916. Reddy J, Chin C (1998) Thermomechanical analysis of functionally graded cylinders and

plates. J Therm Stresses 21(6):593–62617. Vel SS, Batra R (2003) Three-dimensional analysis of transient thermal stresses in

functionally graded plates. Int J Solids Struct 40(25):7181–719618. Guo L, Wang Z, Noda N (2012) A fracture mechanics model for a crack problem of

functionally graded materials with stochastic mechanical properties. Proc R Soc A 468(2146),2939–2961. The Royal Society

19. Kieback B, Neubrand A, Riedel H (2003) Processing techniques for functionally gradedmaterials. Mater Sci Eng, A 362(1):81–106

20. Overvelde JT et al (2016) A three-dimensional actuated origami-inspired transformablemetamaterial with multiple degrees of freedom. Nat Commun 7:10929

21. Bartlett NW et al (2015) A 3D-printed, functionally graded soft robot powered bycombustion. Science 349(6244):161–165

22. Akbarzadeh A, Abedini A, Chen Z (2015) Effect of micromechanical models on structuralresponses of functionally graded plates. Compos Struct 119:598–609

23. Akbarzadeh A, Chen Z (2013) Heat conduction in one-dimensional functionally graded mediabased on the dual-phase-lag theory. Proc Inst Mech Eng Part C J Mech Eng Sci 227(4):744–759

24. Babaei M, Chen Z (2008) Hyperbolic heat conduction in a functionally graded hollow sphere.Int J Thermophys 29(4):1457–1469

25. Choudhuri SR (2007) On a thermoelastic three-phase-lag model. J Therm Stresses 30(3):231–238


26. Ramadan K (2009) Semi-analytical solutions for the dual phase lag heat conduction inmultilayered media. Int J Therm Sci 48(1):14–25

27. Akbarzadeh A, Chen Z (2013) On the harmonic magnetoelastic behavior of a compositecylinder with an embedded polynomial eigenstrain. Compos Struct 106:296–305

28. Chen T (2001) Thermal conduction of a circular inclusion with variable interface parameter.Int J Solids Struct 38(17):3081–3097

29. Wang M, Pan N (2008) Modeling and prediction of the effective thermal conductivity ofrandom open-cell porous foams. Int J Heat Mass Transf 51(5):1325–1331

30. Arabnejad S, Pasini D (2013) Mechanical properties of lattice materials via asymptotichomogenization and comparison with alternative homogenization methods. Int J Mech Sci77:249–262

31. Akbarzadeh A et al (2016) Electrically conducting sandwich cylinder with a planar latticecore under prescribed eigenstrain and magnetic field. Compos Struct 153:632–644

32. Mirabolghasemi A, Akbarzadeh AH, Rodrigue D, Therriault D (2019) Thermal conductivityof architected cellular metamaterials. Acta Materialia 174:61–80

33. Ashby MF, Cebon D (1993) Materials selection in mechanical design. Le Journal dePhysique IV 3(C7):C7-1–C7-9

34. Akbarzadeh A, Chen Z (2012) Transient heat conduction in a functionally graded cylindricalpanel based on the dual phase lag theory. Int J Therm 33(6):1100–1125

35. Quintanilla R, Racke R (2006) A note on stability in dual-phase-lag heat conduction. Int JHeat Mass Transf 49(7):1209–1213

36. Akbarzadeh AH, Chen ZT (2014) Dual phase lag heat conduction in functionally gradedhollow spheres. Int J Appl Mech 6(1):1450002

References 63

Chapter 3Multiphysics of Smart Materialsand Structures

3.1 Smart Materials

In this section, the definition of smart materials are presented. The concept ofmultiphysics is introduced and different types of coupled multiphysical fields areelucidated. Moreover, the piezoelectric and piezomagnetic materials as the twocommonly used smart materials are introduced. Finally, some potential applicationsof these advanced smart materials are mentioned.

Multiphysics involves the investigation of the interaction among differentphysical fields in multiple simultaneous physical phenomena; multiphysical simu-lation typically leads to a set of coupled systems of partial differential equations [1,2]. As an example, the coupled physical fields can be displacement, electricpotential, magnetic potential, temperature, and moisture concentration in a hy-grothermomagnetoelectroelastic medium. The interaction of multiple physicalfields may be observed in natural (wood, bone, and liquid crystals) or synthetic(piezoelectric, piezomagnetic, magnetoelectroelastic, magnetostrictive, and poly-electrolyte gel) smart materials [3]. Piezoelectric materials exhibit interestingphenomena; as seen in Fig. 3.1, an electric field is generated when piezoelectricmaterials are mechanically deformed and vice versa. The intrinsic property makesthe mechanical displacement and electric potential coupled. Common piezoelectricmaterials are made of ceramics subjected to a strong DC electric field so that apermanent dipole moment is aligned during the poling process. The process inducesthe piezoelectricity and anisotropy property in piezoelectric materials [4, 5].

Piezomagnetic materials possess a magnetoelastic coupling similar to the elec-troelastic coupling in piezoelectric materials; they are mechanically strained whensubjected to a magnetic field and vice versa. In the same way, magnetoelectriccoupling is identified as the effect of magnetic (or electric) field on the dielectricpolarization (or magnetization) of smart materials. This phenomenon can beobserved directly in single-phase multiferroics, or indirectly by stress or strain inmagnetoelectroelastic (MEE) composites. As seen in Fig. 3.2, the in-plane


65




https://doi.org/10.1007/978-3-030-25201-4_3

Fig. 3.1 Effect of mechanical stresses on the generation of electric potential in a verticallypolarized piezoelectric material

Fig. 3.2 Magnetoelectric coupling in magnetoelectroelastic (MEE) composites: a Direct effect;b Converse effect. [Reproduced from [7] with permission from Springer Nature]

66 3 Multiphysics of Smart Materials and Structures

magnetic field H leads to strain in the magnetic component due to the magne-tostrictive effect, which is mechanically transferred to the ferroelectric componentinducing a dielectric polarization because of the piezoelectric effect. Conversely, themagnetoelectric coupling is observed when the MEE composite is subjected to anelectric field E which results in the magnetization change DM [6, 7].

Multiphysical materials are frequently called smart, intelligent, active, or adap-tive materials. Due to their multifunctional capabilities, these materials may befound in the following physical fields: electromagnetic, hygrothermal, thermoe-lastic, magnetoelectroelastic, optothermoelectromagnetoelastic, and hygrother-mopiezoelectric fields. For instance, piezoelectric, piezomagnetic, electrostrictive,magnetostrictive, magnetoelectroelastic, and photovoltaic materials as well aselectro/magnetorheological fluids are some of the common synthetic smart mate-rials that are being used in different applications in science and technology [8].

Some of the synthetic multiphysical or smart materials exhibit a full couplingamong different physical fields; however, natural ones rarely do so. Smart materialshave been used in vibration and damping suppression, noise reduction, controlledactive deformation, health monitoring, and improved fatigue and corrosion resis-tance. They are reportedly being used in transportation and aerospace industries [3,9]. Smart structures with piezoelectric and/or piezomagnetic patches to control thestructural vibration are of great interest. These types of active control have beenemployed in axisymmetric shells [10, 11] and laminated beams [12]. Smart mate-rials could also be employed in active noise control. A numerical approach for thedesign of smart lightweight structures was presented in reference [13] for activereduction of noise and vibration. Furthermore, smart materials can be used foractive shape control, health monitoring, and damage detection of structural ele-ments [14]. A new development is a wear detection system for train wheels bydetecting the vibration behaviour of the entire wheel caused by surface change onthe rolling contact area. Another application of smart materials is in energy har-vesting, which is the process of changing parasitic mechanical energy into electricalenergy. This energy can be used for driving electrical circuits or storage in a battery[15, 16]. In the following sections the piezoelectric and magnetoelectroelastic fieldsare formulated. Then, the equations of motion for analyzing smart hollow cylinders,subjected to different physical fields, are derived. Sample results are then presentedto describe the behavior of various types of smart materials.

3.1.1 Piezoelectric Materials

The linear constitutive equations for a multiphysical piezo electric medium arewritten as [17, 18]:

3.1 Smart Materials 67

rij ¼ Cijklekl � ekijEk � bij#� nijm; i; j; k; l ¼ 1; 2; 3ð ÞDi ¼ eijkejk þ eijEj þ ci#þ vim; i; j; k; l ¼ 1; 2; 3ð Þ ð3:1Þ

in which rij and Di are, respectively, stress and electric displacement; eij, Ek, # andm are strain, electric field, temperature change, and moisture concentration changewith respect to the reference state; Cijkl, ekij and �ij are elastic, piezoelectric,dielectric, and coefficients, respectively; bij, nij, ci and vi are thermal stress,hygroscopic stress, pyroelectric and hygroelectric, coefficients. Furthermore, # ¼T � T0 and m ¼ M �M0, in which T and M are the absolute temperature andmoisture concentration while T0 and M0 represent the stress-free temperature andmoisture concentration.

The stress and strain tensors for the considered problem are symmetric whichresult in the following symmetrical properties:

rij ¼ rji; eij ¼ eji; Cijkl ¼ Cjikl ¼ Cijlk ¼ Cklij; ekij ¼ ekji;

�ij ¼ �ji; bij ¼ bji; fij ¼ fjið3:2Þ

The symmetric properties lower the total number of independent coefficients inEq. (3.1). For a general case of triclinic system with the least symmetry, there exist21 elastic, 18 piezoelectric, 6 dielectric 6 thermal stress, 6 hygroscopic stress, 3pyroelectric and 3 hygroelectric constants. Nonetheless, the number of constantsdepends on the symmetry of crystal structure of multiphysical materials. Thenonzero matrix elements for some piezoelectric materials such as quartz, lithiumniobate LiNbO3, cadmium sulphide CdS, polarized ceramic, and gallium arsenideGaAs could be found in [19]. It is more convenient to utilize a system withabbreviated subscripts for material properties to simplify the problems in elasticity.Since stress and strain tensors are symmetric, each component can be specified byone subscript rather than two as follows [19]:

r11 ¼ r1; r22 ¼ r2; r33 ¼ r3;r23 ¼ r4; r13 ¼ r5; r12 ¼ r6

e11 ¼ e1; e22 ¼ e2; e33 ¼ e3; e23 ¼ 12e4; e13 ¼ 1

2e5; e12 ¼ 1

2e6

ð3:3Þ

The same logic is used for the following material properties:

cad ¼ Cijkl; eka ¼ ekij; ba ¼ bij; na ¼ nij i; j; k; l ¼ 1; 2; 3; a; d ¼ 1; 2; . . .; 6ð Þð3:4Þ

It should be mentioned that the contracted notations for stresses and strains arenot applied in this book.


3.1.1.1 Potential Field Equations

The strain and electric fields are related to their potentials by the following gradientequations. The relation between strain and displacement components for smallstrain is:

eij ¼ 12

ui;j þ uj;i� � ð3:5Þ

where ui is the displacement component and a comma denotes partial differentiationwith respect to the space variables. The quasi-stationary electric field equations inthe absence of free conducting electromagnetic current are expressed as:

Ei ¼ �/;i ð3:6Þ

where / is the scalar electric potential.

3.1.2 Magnetoelectroelastic Materials

In order to consider the effect of magnetic field in the constitutive equations,Eq. (3.1) must be modified to:

rij ¼ Cijklekl � ekijEk � dkijHk � bij#� nijm

Di ¼ eijkejk þ �ijEj þ gijHj þ ci#þ vim

Bi ¼ dijkejk þ gijEj þ lijHj þ si#þ tim

i; j; k; l ¼ 1; 2; 3ð Þ ð3:7Þ

in which Bi and Hk are respectively magnetic induction and magnetic field. dkij, gkij,and lij are piezomagnetic, magnetoelectric, and magnetic permeability coefficients,respectively; si and ti are pyromagnetic, and hygromagnetic coefficients. Similar topiezoelectric materials, the symmetry of stress and strain tensors leads to the fol-lowing symmetrical properties:

dkij ¼ dkji; gij ¼ gji; lij ¼ lji ð3:8Þ

These symmetric properties result in 18 piezomagnetic, 6 magnetoelectric and 6magnetic permeability constants for the most general case of triclinic system. As anexample, the constitutive equations (3.7) for orthotropic and radially polarized andmagnetized materials in a cylindrical coordinate system r; h; zð Þ can be written as:


rrrrhhrzzrzhrrzrrh

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

¼

c33 c13 c23 0 0 0

c13 c11 c12 0 0 0

c23 c12 c22 0 0 0

0 0 0 2c66 0 0

0 0 0 0 2c44 0

0 0 0 0 0 2c55

2666666664

3777777775

errehhezzezherzerh

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

�

e33 0 0

e31 0 0

e32 0 0

0 0 0

0 0 e240 e15 0

2666666664

3777777775

Er

Eh

Ez

8><>:

9>=>;

�

d33 0 0

d31 0 0

d32 0 0

0 0 0

0 0 d240 d15 0

2666666664

3777777775

Hr

Hh

Hz

8><>:

9>=>;�

b1b2b30

0

0

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;#�

n1n2n30

0

0

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;#

Dr

Dh

Dz

8><>:

9>=>; ¼

e33 e31 e32 0 0 0

0 0 0 0 0 2e150 0 0 0 2e24 0

264

375

errehhezzezherzerh

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

þ�33 0 0

0 �11 0

0 0 �22

264

375

Er

Eh

Ez

8><>:

9>=>;

þg33 0 0

0 g11 0

0 0 g22

264

375

Hr

Hh

Hz

8><>:

9>=>;þ

c1c2c3

8><>:

9>=>;#þ

v1v2v3

8><>:

9>=>;m

Br

Bh

Bz

8><>:

9>=>; ¼

d33 d31 d32 0 0 0

0 0 0 0 0 2d150 0 0 0 2d24 0

264

375

errehhezzezherzerh

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

þg33 0 0

0 g11 0

0 0 g22

264

375

Er

Eh

Ez

8><>:

9>=>;

þl33 0 0

0 l11 0

0 0 l22

264

375

Hr

Hh

Hz

8><>:

9>=>;þ

s1s2s3

8><>:

9>=>;#þ

t1t2t3

8><>:

9>=>;m

ð3:9Þ

3.1.2.1 Potential Field Equations

The quasi-stationary magnetic field equations in the absence of free conductingelectromagnetic current are expressed as:


Hi ¼ �u;i ð3:10Þ

where u is the scalar magnetic potential.

3.1.2.2 Conservation Equations

The conservation or divergence equations for a hygrothermomagnetoelectroelasticmedium are provided in this section. The equation of motion is:

rij;j þ fi ¼ qui;tt ð3:11Þ

where fi, q, and t, respectively, stand for body force, density, and time. Inmagneto-hygrothermoelectroelastic analysis, an electrically conducting elastic solidsubjected to an external magnetic field experiences the Lorentz force via theelectromagnetic-elastic interaction which works as a body force in Eq. (3.11) asfollows [20]:

~J ¼ r�~h; r�~e ¼ �l~h;t; r:~h ¼ 0; ~e ¼ �l ~u;t � ~H� �

~h ¼ r� ~u� ~H� �

; ~f ¼ l ~J � ~H� � ð3:12Þ

in which, ~J, ~h, ~e, ~u, ~H, and ~f are, respectively, the electric current density, per-turbation of magnetic field, perturbation of electric field, displacement, magneticintensity, and the Lorenz force vectors; l represents the magnetic permeability.Maxwell’s electromagnetic equations or equations of charge and current conser-vation are written as [21]:

Di;i ¼ qe; Bi;i ¼ 0 ð3:13Þ

in which qe is the charge density.Furthermore, the classical energy conservation equation is [22]:

qi;i þ q cm#;t � R� � ¼ 0 ð3:14Þ

where qi, cm, and R are heat flux component, specific heat at constant volume, andinternal heat source per unit mass, respectively. However, Biot [23] introduced theeffects of elastic term in the energy equation to obtain more accurate results forthermoelastic analysis. The energy equation (3.14) was modified for the classical,coupled thermoelasticity as follows:

qi;i þ q S;tT0 � R� � ¼ 0 ð3:15Þ

where S denotes the entropy per unit mass and is defined as:


qS ¼ bijeij þqcmT0

# ð3:16Þ

Through Eqs. (3.15) and (3.16), the energy equation is coupled with the strainrate. Considering the advent of smart materials with coupled multiphysical inter-actions, the classical coupled thermoelasticity equations could be modified to alsoconsider the coupling effects of electric, magnetic, and hygroscopic fields on theenergy equations [24]. Accordingly, Eq. (3.16) could be written in the followingform for classical, coupled hygrothermomagnetoelectroelasticity:

qS ¼ bijeij þ ciEi þ siHi þ qcmT0

#þ dtm ð3:17Þ

where dt is the specific heat-moisture coefficient. On the other hand, the conservationlaw for the mass of moisture in the absence of a moisture source is given by [25]:

pi;i þm;t ¼ 0 ð3:18Þ

in which, pi represents the moisture flux component that is the rate of moisturetransfer per unit area.

3.1.2.3 Fourier Heat Conduction and Fickian Moisture Diffusion

The following Fourier heat conduction theory which relates the heat flux qi to thetemperature gradient is the most widely used theory in the literature:

qi ¼ �kTij#;j ð3:19Þ

Furthermore, the diffusion of moisture in a solid is basically the same as that oftemperature. As a result, the Fickian moisture diffusion equation for moisture flux pican be defined similar to Fourier heat conduction equation as follows:

pi ¼ �fHij m;j ð3:20Þ

In the above equations, kTij and fHij are the thermal conductivity and moisturediffusivity coefficients, respectively. Substituting Eq. (3.19) into (3.14) andEq. (3.20) into (3.18) lead to a diffusion-like equations with parabolic-type gov-erning differential equations for temperature and moisture concentration. To con-sider the possible effect of other physical fields on the heat and mass flux,Eqs. (3.19) and (3.20) could be modified as [3, 18]:

qi ¼ kMijklekl;j � kEijkEk;j � kBijkHk;j � kTij#;j � kHij m;j

pi ¼ fMijklekl;j � fEijkEk;j � fBijkHk;j � fTij#;j � fHij m;jð3:21Þ


where kMijkl, kEijk, kBijk, kHij , fMijkl, fEijk, fBijk, fTij i; j; k; l ¼ 1; 2; 3ð Þ are, respectively,strain-thermal conductivity, electric-thermal conductivity, magnetic-thermal con-ductivity, moisture-thermal conductivity (Dufour effect), strain-moisture diffusivity,electric-moisture diffusivity, magnetic-moisture diffusivity, heat-moisture diffusiv-ity (Soret effect) coefficients. These coefficients represent the degree of thermal andmechanical, thermal and electrical, thermal and magnetic, thermal and hygroscopic,hygroscopic and mechanical, hygroscopic and electrical, hygroscopic and magnetic,and hygroscopic and thermal field interactions.

The conventional heat conduction and moisture diffusion theories based on theclassical Fourier and Fickian laws lead to an infinite speed of thermal and moisturewave propagation due to the parabolic-type heat and mass transport equations.Fourier and Fickian laws assume instantaneous hygrothermal responses and aquasi-equilibrium thermodynamic condition. The classical diffusion theories havebeen widely used in heat and mass transfer problems; however, the heat and masstransmission is observed to be a non-equilibrium phenomenon, and they propagatewith a finite speed for applications involving very low temperature, high temper-ature gradients, short-pulse heating, laser drying, laser melting and welding, rapidsolidification, very high frequencies of heat and mass flux densities, and microtemporal and spatial scales [26]. Consequently, different non-Fourier andnon-Fickian heat and mass transfer theories have been developed to remove thesedrawbacks.

3.1.3 Advanced Smart Materials

Functionally graded materials (FGMs) have become considerably important inextremely high temperature environments such as rocket nozzles and chemicalplants. In 1984, the concept of FGMs was proposed in Japan as thermal barriermaterials [27]. As shown in Table 3.1, FGMs are composite materials, micro-scopically non-homogeneous, in which material properties vary continuously withrespect to spatial coordinates. FGMs are typically made from a mixture of ceramicand metal or a combination of different metals. The advantage of using FGMs canbe expressed as their sustainability in high temperature environments while main-taining their structural integrity. The ceramic constituents of FGMs provide the hightemperature resistance due to their low thermal conductivity. On the other hand, theductile metal constituent of FGMs impedes fracture due to high temperature gra-dient in a very short period of time as seen in laser impulse applications [28]. Thesmooth and continuous changes of material properties and thermomechanicalstresses in FGMs distinguish them from the conventional laminated compositeswith a mismatch of material properties across the laminate interfaces. The laminatedcomposites are prone to debonding, crack initiation, and the presence of residualstresses due to the difference in thermal expansion coefficients of different layers.The continuous transition of volume fraction in FGMs eliminates the deficiency.


This gradual variation in material properties reduces the likelihood of delaminationcaused by stress concentration, in-plane and transverse thermal stresses, and thestress intensity factors [2, 28].

FGMs were first introduced as thermal barriers to withstand high temperaturechanges; nonetheless, they have lots of applications in modern industry. For sizereduction and enhancement of the reliability of electric power equipment, FGMswith spatial distribution of dielectric permittivity have been used recently [29].Some of the applications of FGMs in biomedical engineering such as implants forbone and knee joint replacement are mentioned in [30]. The normal and shearstresses in a double-layered pressure vessel due to internal pressure and thermalloadings were reduced by using FGM materials [31]. Moreover, there are variousapplications of FGMs in aerospace structures, fusion reactors, turbine rotors, fly-wheels, gears, wear resistant linings, thermoelectric generators, prostheses, etc.

3.2 Thermal Stress Analysis in Homogenous SmartMaterials

In this section, the constitutive relations and governing equations for solving thetwo thermomagnetoelastic and thermo-magnetoelectroelastic problems are pre-sented. Consider an infinitely long, hollow cylinder rotating at a constant angularvelocity x as shown in Fig. 3.3. The cylinder is magnetized and polarized in theradial direction. The inner and outer radii of the cylinder are a and b, respectively.The cylinder experiences the magnetic scalar potential, u, electric scalar potential,/, and pressure, P, at the inner and outer surfaces. The inner surface of the cylinderis subjected to the temperature change, #a ¼ Ta � T1, where Ta is the absolutetemperature at the inner surface and T1 is the ambient temperature. The outer

Table 3.1 Characteristics of FGMs [27]

FGM Non-FGM

Property:1. Mechanical

strength2. Thermal

conductivity

Constituentelements:1. Ceramic ●2. Metal ○3. Fiber □


surface is under convection boundary condition with the heat convectivity coeffi-cient, h1. Subscripts “a” and “b” are employed to indicate the load on the inner andouter surfaces, respectively.

The non-zero components of strain, electric, and magnetic fields for theaxisymmetric, plane strain problem are written as:

�err ¼ ur; �ehh ¼ ur; Er ¼ �/;r; Hr ¼ �u;r ð3:22Þ

where u ¼ ur is the radial displacement; r and h are the radial and circumferentialcoordinates. Using constitutive Eqs. (3.9) and (3.22), one can obtain:

rrr ¼ c33ur þ c13urþ e33/r þ d33u;r � b1# ð3:23aÞ

rhh ¼ c13ur þ c11urþ e31/r þ d31u;r þ d31u;r � b3# ð3:23bÞ

Dr ¼ e33ur þ e31ur� e33/;r � g33u;r þ c1# ð3:23cÞ

Br ¼ d33ur þ d31ur� g33/;r � l33u;r þ s1# ð3:23dÞ

in which, cmn ¼ Cijkl, emk ¼ epij, dmk ¼ dpij i; j; k; l; p ¼ 1; 2; 3; m; n ¼ 1; 2; . . .; 6ð Þ,b1 ¼ b11, and b3 ¼ b33. The governing equations for a rotating magnetoelectroe-lastic cylinder under axisymmetric loading, when the body force, free chargedensity, and current density are absent, are expressed as:

Fig. 3.3 Rotating hollowcylinder and its boundaryconditions [Reproduced from[35] with permission fromTaylor & Francis Ltd.]

3.2 Thermal Stress Analysis in Homogenous Smart Materials 75

rrr;r þ 1r

rrr � rhhð Þ ¼ qdu;tt ð3:24aÞ

Dr;r þ 1rDr ¼ 0 ð3:24bÞ

Br;r þ 1rBr ¼ 0 ð3:24cÞ

where qd is the mass density, and t stands for time. Furthermore, the inertial effectfor the rotating cylinder with angular velocity x can be written as:

u;tt ¼ �rx2 ð3:25Þ

3.2.1 Solution for the a Thermomagnetoelastic FGMCylinder

The solution for a thermomagnetoelastic FGM rotating hollow cylinder is obtainedin this section. It is assumed that the material properties of the FGM cylinder varyaccording to a power law along the radial direction as follows:

v rð Þ ¼ v0rb

� �2Nð3:26Þ

where v rð Þ, v0, and N represent, respectively, the general material properties of thecylinder, their values at the outer surface, and the non-homogeneity parameter.Substituting Eqs. (3.23), (3.25), and (3.26) into Eq. (3.24) leads to the followingcoupled governing differential equations in terms of displacement and magneticpotential:

r2c330u;rr þ r 2N þ 1ð Þc330u;r þ 2Nc130 � c110ð Þuþ r2d330/;rr

þ r 2Nþ 1ð Þd330 � d310ð Þ/;r � r2b10#;r � r 2N þ 1ð Þb10 � b30ð Þ#þ qd0x2 ¼ 0

ð3:27aÞ

r2d330u;rr þ r 2Nþ 1ð Þd330 þ d310ð Þu;r þ 2Nd310u� r2u330/;rr

�r 2Nþ 1ð Þl330/;r þ r2s10#;r þ r 2N þ 1ð Þs10# ¼ 0ð3:27bÞ

Using the following non-dimensional parameters:


a ¼ c110c330

; b ¼ e310e330

; d ¼ c130c330

; m ¼ d310d330

; g ¼ b330b110

; c ¼ e330c330e2330

f ¼ g330c330d330e330

; k ¼ l330c330d2330

; X ¼ c10c330e330b10

; Y ¼ s10c330d330b10

ð3:28Þ

as well as a new electric potential, a new magnetic potential, and a new temperaturechange:

U ¼ e330c330

/; W ¼ d330c330

u; H ¼ b10c330

# ð3:29Þ

Equation (3.27) can be rewritten in the following form:

r2u;rr þ r 2Nþ 1ð Þu;r þ 2Nd� að Þuþ r2W;rr

þ r 2Nþ 1� mð ÞW;r � r2H;r � r 2Nþ 1� gð ÞHþ qd0x2r3

c330¼ 0

ð3:30aÞ

r2u;rr þ r 2Nþ 1þ mð Þu;r þ 2Nmu� r2kW;rr

�r 2N þ 1ð ÞkW;r þ r2YH;r þ r 2Nþ 1ð ÞYH ¼ 0ð3:30bÞ

Employing the normalized radial coordinate, q ¼ ra, Eq. (3.30) is rearranged as

follows:

q2u;qq þ q 2N þ 1ð Þu;q þ 2Nd� að Þuþ q2W;qq

þ q 2Nþ 1� mð ÞW;q � q2aH;q � q 2Nþ 1� gð ÞaHþ qd0x2q3a3

c330¼ 0

ð3:31aÞ

q2u;qq þ q 2Nþ 1þ mð Þu;q þ 2Nmu� kq2W;qq

�q 2Nþ 1ð ÞkW;q þ q2aYH;q þ qa 2N þ 1ð ÞYH ¼ 0ð3:31bÞ

Solving Eq. (3.31) requires us to determine the temperature distribution alongthe radial direction. The axisymmetric, steady state heat conduction equation for aninfinitely long hollow cylinder can be written as:

1r

rk#;r� �

;r¼ 0 a� r� b ð3:32Þ

where k is the thermal conductivity varying according to Eq. (3.26) for the FGMcylinder. Rearranging Eq. (3.32), using the normalized radial coordinate andemploying Eqs. (3.26) and (3.29), leads to:


1q

q2Nþ 1H;q

� �;q¼ 0 1� q� i ð3:33Þ

where i ¼ ba is the aspect ratio of the hollow cylinder. The general solution is

obtained as:

H ¼ C1q�2N þC2; N 6¼ 0 ð3:34aÞ

H ¼ C1 ln qð ÞþC2; N ¼ 0 ð3:34bÞ

where C1 and C2 are integration constants to be determined by thermal boundaryconditions. The following general thermal boundary conditions are considered forEq. (3.33) [32]:

A11H 1ð ÞþA12H0 1ð Þ ¼ f1

A21H ið ÞþA22H0 ið Þ ¼ f2

ð3:35Þ

in which, Aij is the Robin-type thermal boundary condition coefficients, and f1 andf2 are known functions on the inner and outer radii. Equations (3.34) are substitutedinto Eq. (3.35) and are solved for the integration constants C1 and C2. When N 6¼ 0:

C1 ¼ A11f2 � A21f1A11 A21i� 2NA22ð Þi�2N�1 � A21 A11 � 2NA12ð Þ

C2 ¼ f1A11

� A11 � 2NA12ð Þ A11f2 � A21f1ð ÞA211 A21i� 2NA12ð Þi�2N�1 � A21 A2

11 � 2NA12A11� � ð3:36Þ

and when N ¼ 0:

C1 ¼ A11f2 � A21f1A11 A21 ln ið ÞþA22i�1ð Þ � A21A12

C2 ¼ f1A11

� A12 A11f2 � A21f1ð ÞA211 A21 ln ið ÞþA22i�1ð Þ � A21A12A11

ð3:37Þ

Using the aforementioned temperature distribution, the coupled governing dif-ferential equations (3.31) can be solved. First, Eq. (3.31) is converted to the fol-lowing new differential equations about 1 with constant coefficients by introducinga variable substitution, q ¼ e1:

€uþ 2N _uþ 2Nd� að Þuþ €Wþ 2N � mð Þ _W ¼ ae1 _Hþ 2N þ 1� gð Þae1H� xnae31

ð3:38aÞ

€uþ 2Nþ mð Þ _uþ 2Nmu� k €W� 2Nk _W ¼ �aYe1 _H� 2Nþ 1ð ÞaYe1H ð3:38bÞ


where the overdot stands for differentiation with respect to 1 and

xn ¼qd0x

2a2

c330ð3:39Þ

Eliminating €W between the two equations of (3.38) and solving for _W leads to:

W:

¼ 1þ kmk

€uþ 2N 1þ kð Þþ mmk

_uþ 2N dkþ mð Þ � akmk

uþ a Y � kð Þmk

e1 H:

þ 2N þ 1ð ÞY � 2Nþ 1� gð Þkmk

ae1Hþ axn

me31

ð3:40Þ

Equation (3.40) and its derivative are substituted into the first equation of (3.38)to give us the following decoupled differential equation for u:

a3vuþ a2€uþ a1 _uþ a0u ¼ d3e31 þ d2e

1 þ d1e�2N þ 1ð Þ1 ð3:41Þ

in which,

a3 ¼ 1þ kmk

; a2 ¼ 4N 1þ kð Þmk

; a1 ¼ 2N dkþ mð Þþ 4N2 kþ 1ð Þ � ak� m2

mk;

a0 ¼ 4N2 mþ dcð Þ � 2N akþ m2ð Þmk

; d3 ¼ � axn 3þ 2Nð Þm

;

d2 ¼ � aY 2Nþ 1ð Þ 2Nþ 1� mð Þ � ak 2Nþ 1� gð Þ 1þ 2Nð Þmk

C2;

d1 ¼ � aY 1� mð Þþ ak g� 2Nm� 1ð Þmk

C1

ð3:42Þ

The solution of Eq. (3.41) can be exactly obtained as follows

u ¼ Aem11 þBem21 þCem31 þK1e�2Nþ 1ð Þ1 þK2e

1 þK3e31 ð3:43Þ

where mi i ¼ 1; 2; 3ð Þ are the roots of the characteristic equation of Eq. (3.41); A, B,and C are the constants of integration determined by the boundary conditions; andKi i ¼ 1; 2; 3ð Þ are obtained as:

K1 ¼ d1�2Nþ 1ð Þ3a3 þ �2Nþ 1ð Þ2a2 þ �2Nþ 1ð Þa1 þ a0

K2 ¼ d2a3 þ a2 þ a1 þ a0

; K3 ¼ d327a3 þ 9a2 þ 3a1 þ a0

ð3:44Þ

The final solution for u in terms of q is:


u ¼ Aqm1 þBqm2 þCqm3 þK1q�2Nþ 1 þK2qþK3q

3 ð3:45Þ

Substituting Eq. (3.43) into (3.40) and doing the integration leads to the fol-lowing expression for W:

W ¼ b1Aqm1 þ b2Bq

m2 þ b3Cqm3 þDþK5q

�2N þK6q�2Nþ 1 þK7qþK8q

3

ð3:46Þ

where D is a new integration constant and

bi ¼ 1mk

1þ kð Þmi þ 2N 1þ kð Þþ mþ 1mi

2N mþ dkð Þ � akð Þ� �

K4 ¼ 1mk

K1 1þ kþ mð Þþ K1

�2Nþ 12N dkþ mð Þ � akð Þþ C1a Y � k 1� gð Þð Þ

�2Nþ 1

� �

K5 ¼ 1mk

K2 2N þ 1ð Þ 1þ kð Þþ mð ÞþK2 2N dkþ mð Þ � akð ÞðþC2a Y 2Nþ 1ð Þ � k 2N þ 1� gð Þð Þ

K6 ¼ 1mk

K3 2Nþ 3ð Þ kþ 1ð Þþ mð Þþ K3

32N dkþ mð Þ � akð Þþ xnak

3

� �ð3:47Þ

The following non-dimensional stresses, displacement, electric potential, mag-netic potential and magnetic induction are used for convenience in this paper:

Rrr ¼ rrrc330

;Rhh ¼ rhhc330

; u1 ¼ ua;U1 ¼ U

a;W1 ¼ W

a;Br1 ¼

Br

d330ð3:48Þ

Using Eqs. (3.23), (3.45), (3.46), and (3.48), we achieve:

Rrr ¼ i�2N q2N�1

a

Aqm1 m1 þ dþ b1m1ð ÞþBqm2 m2 þ dþ b2m2ð ÞþCqm3 m3 þ dþ b3m3ð Þ

þ q�2Nþ 1 K1 �2Nþ 1ð ÞþK1dþK4 �2Nþ 1ð Þ � aC1ð Þþ q K2 1þ dð ÞþK5 � aC2ð Þþ q3 3K3 þK3dþ 3K6ð Þ

0BB@

1CCA

ð3:49aÞ

Rhh ¼ i�2N q2N�1

a

Aqm1 m1dþ aþ mb1m1ð ÞþBqm2 m2dþ aþ mb2m2ð ÞþCqm3 m3dþ aþ mb3m3ð Þ

þ q�2N þ 1 K1d �2Nþ 1ð ÞþK1aþK4m �2Nþ 1ð Þ � gaC1ð Þþ q K2 dþ að Þþ mK5 � gaC2ð Þþ q3 K3 3dþ að Þþ 3K6mð Þ

0BB@

1CCA

ð3:49bÞ


W1 ¼ b1aAqm1 þ b2

aBqm2 þ b3

aCqm3 þ D

aþ K4

aq�2Nþ 1 þ K5

aqþ K6

aq3 ð3:49cÞ

Br1 ¼ i�2N q2N�1

a

Aqr1 r1 þ m� kb1r1ð ÞþBqr2 r2 þ m� kb2r2ð ÞþCqr3 r3 þ m� kb3r3ð Þþ q�2Nþ 1 K1 �2N þ 1ð Þþ mK1 þC1aYð Þ

þ q K2 1þ mð Þ � kK5 þC2Yað Þþ q3 K3 3þ mð Þ � 3kK6ð Þ

0@

1A

ð3:49dÞ

u1 ¼ Aaqm1 þ B

aqm2 þ C

aqm3 þ K1

aq�2Nþ 1 þ K2

aqþ K3

aq3 ð3:49eÞ

The following boundary conditions are assumed for thermomagnetoelasticanalysis of the rotating FGM cylinder:

Rrr 1ð Þ ¼ Rrri;W1 1ð Þ ¼ W1i ð3:50aÞ

Rrr ið Þ ¼ Rrro;W1 ið Þ ¼ W1o ð3:50bÞ

Using Eqs. (3.49) and (3.50), the four unknown A, B, C, and D are obtained bysolving the following linear algebraic system of equations:

I

ABCD

8>><>>:

9>>=>>; ¼ J ð3:51Þ

where I is a 4� 4 nontrivial matrix with the following arrays:

I11 ¼ i�2N m1 1þ b1ð Þþ da

; I12 ¼ i�2N m2 1þ b2ð Þþ da


; I14 ¼ 0

I21 ¼ b1a; I22 ¼ b2

a; I23 ¼ b3

a; I24 ¼ 1

a

I31 ¼ im1�1 m1 1þ b1ð Þþ da

; I32 ¼ im2�1 m2 1þ b2ð Þþ da


; I34 ¼ 0

I41 ¼ im1b1a; I42 ¼ im2

b2a; I43 ¼ im3

b3a; I44 ¼ 1

a

ð3:52Þ

and the vector J has the following components:


J1 ¼ Rrri � i�2N

aK1 �2N þ 1þ dð ÞþK4 �2Nþ 1ð Þ � aC1 þK2 1þ dð ÞþK5 � aC2ð

þK3 3þ dð Þþ 3K6ÞJ2 ¼ W1i � K4

a� K5

a� K6

a

J3 ¼ Rrro � 1a

K1 �2Nþ 1þ dð ÞþK4 �2N þ 1ð Þ � aC1ð Þi�2N þK2 1þ dð ÞþK5 � aC2�

þK3 3þ dð Þþ 3K6Þi2

J4 ¼ W1o � K4

ai�2Nþ 1 � K5

ai� K6

ai3

ð3:53Þ

For brevity the expressions for A, B, C, and D are not presented here.

3.2.2 Solution for Thermo-MagnetoelectroelasticHomogeneous Cylinder

An analytical solution for thermal analysis of a magnetoelectroelastic hollowcylinder is obtained in this part. The cylinder is assumed to be orthotropic andhomogeneous. The governing differential equations are obtained using Eqs. (3.23),(3.25), (3.26), (3.28), (3.29) and (3.39) into (3.24) and setting N ¼ 0 as follows:

r2u;rr þ ru;r � auþ r2U;rr þ r 1� bð ÞU;r þ r2W;rr þ r 1� mð ÞW;r�r2H;r � r 1� gð ÞHþxnr3

a2¼ 0

ð3:54aÞ

r2u;rr þ r 1þ bð Þu;r � r2cU;rr � rcU;r � r2fW;rr � rfW;r þ r2XH;r þ rXH ¼ 0

ð3:54bÞ

r2u;rr þ r 1þ mð Þu;r � r2fU;rr � rfU;r � r2kW;rr � rkW;r þ r2YH;r þ rYH ¼ 0

ð3:54cÞ

Using the normalized radial coordinate q ¼ ra and then changing the variable of

q ¼ e1, the following differential equations with constant coefficients are obtained:

€u� auþ €U� b _Uþ €W� m _W ¼ ae1 _Hþ 1� gð Þae1H� xnae31 ð3:55aÞ

€uþ b _u� c€U� f€U ¼ �Xae1 _H� Xae1H ð3:55bÞ


€uþ m _u� f€U� k€U ¼ �Yae1 _H� Yae1H ð3:55cÞ

We eliminate U between Eqs. (3.55) and reach the two differential equationsabout u and W as:

1þ cð Þu� acuþ bu� bcUþ c� fð ÞW� mcW ¼ ae1H c� Xð Þþ ae1H c 1� gð Þ � Xð Þ � Xace31

ð3:56aÞ

c� fð Þ€uþ mc� bfð Þ _uþ f2 � kc� �

€W ¼ ae1 _H Xf� Ycð Þþ ae1H Xf� Ycð Þð3:56bÞ

Eliminating W between Eqs. (3.56) and considering the temperature distributionaccording to Eq. (3.34b) leads to the following fourth-order ordinary differentialequation:

a4 €u::

þ a2€u ¼ b1C2 þ b2C1ð Þe1 þ b1C1te1 þ b3e

31 ð3:57Þ

in which,

a4 ¼ 1mc� bf

1þ cþ c� fð Þ2kc� f2

!; a2 ¼ � acþ b2

mc� bfþ mc� bf

kc� f2

� �

b1 ¼ aXf� Yc

f2 � kcþ c 1� gð Þþ b� 1ð ÞX

mc� bf� c� fmc� bf

Xf� Yc

f2 � kc

� �

b2 ¼ a 2Xf� Yc

f2 � kcþ c 3� 2gð Þþ 2b� 3ð ÞX

mc� bf� 3

c� fmc� bf

Xf� Yc

f2 � kc

� �

b3 ¼ � 9acxn

mc� bf

ð3:58Þ

The solution of Eq. (3.57) can be written as:

u ¼ AþB ln qð ÞþCqm þDq�m þ K1 ln qð ÞþK2ð ÞqþK3q3 ð3:59Þ

where

m ¼ffiffiffiffiffiffiffiffiffi� a2a4

rm 2 Rð Þ; K1 ¼ b1C1

a4 þ a2; K2 ¼ b1C2 þ b2C1

a4 þ a2� b1C1 4a4 þ 2a2ð Þ

a4 þ a2ð Þ2 ;

K3 ¼ b581a4 þ 9a2

ð3:60Þ

In Eq. (3.59), A, B, C, and D are integration constants. Using the secondequation of (3.56), W can be found as:


W ¼ FþE ln qð ÞþBc0 ln qð Þð Þ2 þCc1qm þDc2q

�m þ c3 ln qð Þþ c4ð ÞqþK3c5q3

ð3:61Þ

where E and F are new integration constants and

c0 ¼ mc� bf

2 kc� f2� � ; c1 ¼ 1

m kc� f2� � m c� fð Þþ mc� bfð Þ;

c2 ¼ 1

m kc� f2� � m c� fð Þ � mcþ bfð Þ;

c3 ¼ 1

kc� f2K1 c 1þ mð Þ � f 1þ bð Þð Þ � aC1 Xf� Ycð Þð Þ;

c4 ¼ 1

kc� f22K1 þK2ð Þ c� fð Þþ K2 þK1ð Þ mc� bfð Þ

�aC2 Xf� Ycð Þ � K1 c 1þ mð Þ � f 1þ bð Þð Þ

� �;

c5 ¼ 1

3 kc� f2� � 3 c� fð Þþ mc� bfð Þ

ð3:62Þ

Considering the expressions for u and W, one can obtain the followingexpression for U using Eq. (3.55c):

U ¼ HþG ln qð ÞþBl0 ln qð Þð Þ2 þCl1qm þDl2q

�m þ l3 ln qð Þþ l4ð ÞqþK3l5q3

ð3:63Þ

where G and H are integration constants and

l0 ¼ 12c

b� 2fc0ð Þ; l1 ¼ 1mc

mþ b� mfc1ð Þ; l2 ¼ 1mc

mþ b� mfc2ð Þ;

l3 ¼ 1c

K1 1þ bð Þ � fc3 þXaC1ð Þ;

l4 ¼ 1c

K1 þK2 1þ bð Þ � f c3 þ c4ð ÞþXaC2ð Þ; l5 ¼ 13c

3þ b� 3fc5ð Þ

ð3:64Þ

Since we have six boundary conditions while there are eight integration con-stants, we need two more complimentary equations. These equations are acquiredby inserting Eqs. (3.59), (3.61), and (3.63) into Eq. (3.55a) as follows:

B ¼ 0; aAþ mEþ bG ¼ 0 ð3:65Þ

Using the non-dimensional parameters (3.48), we have:


Rrr ¼ q�1

aAdþGþEð Þþ qm�1

am 1þ l1 þ c1ð Þþ dð ÞC

þ q�m�1

a�m 1þ l2 þ c2ð Þð ÞDþ ln qð Þ

aK1 1þ dð Þþ l3 þ c3 � C1að Þ

þ 1a

K1 þ 1þ dð ÞK2 þ l3 þ l4 þ c3 þ c4 � C2að Þþ q2

a3 1þ l5 þ c5ð Þþ dð ÞK3

ð3:66aÞ

Rhh ¼ q�1

aAaþGbþEmð Þþ qm�1

am dþ bl1 þ mc1ð Þþ að ÞC

þ q�m�1

a�m dþ bl2 þ mc2ð Þþ að ÞDþ ln qð Þ

aK1 dþ að Þþ bl3 þ mc3 � gC1að Þ

þ 1a

dK1 þ dþ að ÞK2 þ b l3 þ l4ð Þþ m c3 þ c4ð Þ � gqC2ð Þ

þ q2

a3 dþ bl5 þ mc5ð Þþ að ÞK3

ð3:66bÞ

U1 ¼ Haþ G

aln qð Þþ C

al1q

�m þ Dal2q

�m þ l3 ln qð Þþ l4ð Þ qaþ K3l5

aq3 ð3:66cÞ

W1 ¼ Faþ E

aln qð Þþ C

ac1q

m þ Dac2q

�m þ c3 ln qð Þþ c4ð Þ qaþ K3c5

aq3 ð3:66dÞ

The following boundary conditions in this thermo-magnetoelectroelastic analy-sis are:

Rrr 1ð Þ ¼ Rrri; U1 1ð Þ ¼ U1i; W1 1ð Þ ¼ W1i ð3:67aÞ

Rrr ið Þ ¼ Rrro; U1 ið Þ ¼ U1o; W1 ið Þ ¼ W1o ð3:67bÞ

Employing Eqs. (3.65) through (3.67), we can obtain a linear algebraic equationfor the integration constants as:

I

ACDEFGH

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

¼ J ð3:68Þ

where I is a 7� 7 nontrivial matrix with the following arrays:


I11 ¼ da; I12 ¼ 1

am 1þ l1 þ c1ð Þþ dð Þ; I13 ¼ 1

a�m 1þ l2 þ c2ð Þþ dð Þ; I14 ¼ 1

a;

I15 ¼ 0; I16 ¼ 1a; I17 ¼ 0

I21 ¼ 0; I22 ¼ l1a; I23 ¼ l2

a; I24 ¼ 0; I25 ¼ 0; I26 ¼ 0; I27 ¼ 1

a

I31 ¼ 0; I32 ¼ c1a; I33 ¼ c2

a; I34 ¼ 0; I35 ¼ 1

a; I36 ¼ 0; I37 ¼ 0

I41 ¼ a; I42 ¼ 0; I43 ¼ 0; I44 ¼ m; I45 ¼ 0; I46 ¼ b; I47 ¼ 0

I51 ¼ dai�1; I52 ¼ im�1

am 1þ l1 þ c1ð Þþ dð Þ; I53 ¼ i�m�1

a�m 1þ l2 þ c2ð Þþ dð Þ;

I54 ¼ i�1

a; I55 ¼ 0; I56 ¼ i�1

a; I57 ¼ 0

I61 ¼ 0; I62 ¼ l1aim; I63 ¼ l2

ai�m; I64 ¼ 0; I65 ¼ 0; I66 ¼ ln ið Þ

a; I67 ¼ 1

a

I71 ¼ 0; I72 ¼ c1aim; I73 ¼ c2

ai�m; I74 ¼ ln ið Þ

a; I75 ¼ 1

a; I76 ¼ 0; I77 ¼ 0

ð3:69Þ

and the components of vector J are:

J1 ¼ Ri � 1a

K1 þ 1þ dð ÞK2 þ l3 þ l4 þ c3 þ c4 � C2að Þ � 1a

3 1þ l5 þ c5ð Þþ dð ÞK3

J2 ¼ U1i � l4a� K3l5

a; J3 ¼ W1i � c4

a� K3c5

a; J4 ¼ 0

J5 ¼ Ro � ln ið Þa

K1 1þ dð Þþ l3 þ c3 � C1að Þ � 1a

K1 þ 1þ dð ÞK2 þ l3 þ l4 þ c3 þ c4 � C2að Þ

� i2


J6 ¼ U1o � ia

l3 ln ið Þþ l4ð Þ � K3l5a

i3; J7 ¼ W1o � ia

c3 ln ið Þþ c4ð Þ � K3c5a

i3

ð3:70Þ

Solving Eq. (3.68) renders the integration constants and completes the analyticalsolution for the thermo-magnetoelectroelastic hollow cylinder.

3.2.3 Benchmark Results

In this section, the numerical results are presented to show the multiphysicalbehavior of the hollow thermomagnetoeleastic functionally graded cylinder and thehomogeneous thermo-magnetoelectroelastic orthotropic cylinder. The outer surfaceof the FGM cylinder is assumed to be BaTiO3/CoFe2O4 with material properties


given in Table 3.2, and these material properties are taken for the homogenousorthotropic cylinder as well. The following mechanical and magnetic boundaryconditions are assumed for the thermomagnetoelastic FGM hollow cylinder:

Rrr 1ð Þ ¼ �1;W1 1ð Þ ¼ 1

Rrr ið Þ ¼ 0;W1 ið Þ ¼ 0

The non-dimensional temperature on the inner and outer surfaces of the FGMhollow cylinder are assigned to be Ha and Hb, respectively. For the thermomag-netoelastic analysis, the radial and hoop stresses, magnetic potential, magneticinduction, and temperature distribution are given for a variety of non-dimensionalvalues of inner temperature, inertial effects, aspect ratios and non-homogeneityparameters.

The influence of the thermal boundary condition on the distribution of stresses,magnetic induction, and magnetic potential along the radial direction for an FGMthermomagnetoelastic hollow cylinder are shown in Fig. 3.4a–e. Thenon-homogeneity parameter, angular velocity, and aspect ratio are N ¼ 1, xn ¼ 1,and i ¼ 4, respectively. The electromagnetic boundary conditions in above

Table 3.2 Materialproperties of BaTiO3/CoFe2O4 [33, 34]

Properties BaTiO3/CoFe2O4

c330 Nm2

� �2:96� 1011

c110 Nm2

� �2:86� 1011

c130 Nm2

� �1:70� 1011

e310 Cm2

� � �4:4

e330 Cm2

� �18:6

d310 NAm

� �580:3

d330 NAm

� �699:7

e330 C2

Nm2

� �9:3� 1011

g330 Ns2C2

� �3:0� 10�12

l330Ns2VC

� �1:57� 10�4

b110Nkm2

� �4:395� 106

b330Nkm2

� �4:560� 106

c10Ckm2

� � �13:0� 10�5

s10 NAmK

� �6:0� 10�3


Fig. 3.4 a Radial stress, b Hoop stress, c Magnetic induction, d Total magnetic potentialdistribution, e Zoomed-in magnetic potential distribution for different thermal boundaryconditions, xn ¼ 1, N ¼ 1, i ¼ 4. [Reproduced from [35] with permission from Taylor &Francis Ltd.]


equations are used. The outer non-dimensional temperature of the FGM is assumedto be zero, while the non-dimensional temperature on the inner surface is increasedgradually.

Increasing the temperature on the inner surface changes the radial stress distri-bution monotonically as shown in Fig. 3.4a and shifts the transition point of theradial stress, in which the radial stress changes from negative to positive, toward theouter surface of the cylinder. As depicted in Fig. 3.4b, the value of hoop stressdecreases before q ¼ 2:4958 and increases after this point when the inner tem-perature is increased. Furthermore, the non-dimensional magnetic induction inFig. 3.4c increases when the inner temperature is augmented. Although the mag-netic potential does not change significantly when altering the inner temperature inFig. 3.4d, the zoomed-in examination clarifies that it decreases slightly when theinner temperature increases as shown in Fig. 3.4e. It should be mentioned that theresults are similar to those reported in [36] when the inner and outer temperaturesare set to be zero.

Fig. 3.4 (continued)


3.3 Thermal Stress Analysis of Heterogeneous SmartMaterials

A similar approach is used in this section to analyze heterogeneous smart materials.An infinitely long, hollow, FGPM cylinder rotating at a constant angular velocity xis considered as shown in Fig. 3.5. The cylinder is poled and graded in the radialdirection. As depicted, the inner and outer radii of the cylinder are a and b,respectively. The inner and outer surfaces of the cylinder are subjected to tem-perature change h, electric potential /, and pressure P. Subscripts “a” and “b” areused to indicate the load on the inner and outer surfaces, respectively. The cylinderis placed in a constant magnetic field H0 acting in the z direction of cylindricalcoordinate system r; h; zð Þ.

The material properties vary along the radial direction according to a power lawas follows:

v rð Þ ¼ v0rb

� �2Nð3:71Þ

where v rð Þ, v0, and N are the general material property of the cylinder, its value atthe outer surface, and the non-homogeneity parameter, respectively. The constitu-

Fig. 3.5 FGPM rotating hollow cylinder and its boundary conditions [Reproduced from [37] withpermission from IOP Publishing]


tive equations for a linear piezoelectric material are given in Eq. (3.1). Furthermore,the quasi-stationary electric field and the linear strain-displacement are in accor-dance with Eqs. (3.5) and (3.6).

The non-zero components of strains and electric fields for the current axisym-metric, plane strain problem are given as follows:

�err ¼ u;r; �ehh ¼ ur

Er ¼ �/;r

ð3:72Þ

where u is the radial displacement; r and h are the radial and circumferentialcoordinates. For the cylindrically orthotropic piezoelectric material polarized in theradial direction, substituting Eq. (3.72) into Eq. (3.1) leads to:

rrr ¼ c33u;r þ c13urþ e33/;r � b1h ð3:73aÞ

rhh ¼ c13u;r þ c11urþ e31/;r � b3h ð3:73bÞ

Dr ¼ e33u;r þ e31ur� e33/;r þ c1h ð3:73cÞ

where cmn ¼ Cijkl, emk ¼ epij i; j; k; l; p ¼ 1; 2; 3;m; n ¼ 1; 2; . . .; 6ð Þ, b1 ¼ b11, andb3 ¼ b33.

The equation of motion in the rotating electromagnetic medium and Maxwellequation in the absence of electric charge under axisymmetric loading are expressedas:

rrr;r þ 1r

rrr � rhhð Þþ fz ¼ qdu;tt ð3:74aÞ

Dr;r þ 1rDr ¼ 0 ð3:74bÞ

where qd is the mass density and t stands for time; fz and hz are defined as Lorentz`sforce and perturbation of magnetic field, respectively, which for a constant mag-netic field H0 can be written as follows [20]:

fz ¼ H20 l u;r þ 1

ru

� �� ;r

hz ¼ �H0 u;r þ 1ru

� � ð3:75Þ

in which, l is magnetic permeability.

3.3 Thermal Stress Analysis of Heterogeneous Smart Materials 91

3.3.1 Solution Procedures

Substituting Eqs. (3.71), (3.73), and (3.75) into Eq. (3.74) and consideringutt ¼ �rx2, the two coupled governing differential equations for the problem areobtained:

r2 c330 þ l0H20

� �u;rr þ r 2Nþ 1ð Þ c330 þ l0H

20

� �u;r

þ 2Nc130 � c110 þ l0H20 2N � 1ð Þ� �

uþ r2e330u;rr

þ r 2Nþ 1ð Þe330 � e310ð Þu;r � r2b10h;r

�r 2Nþ 1ð Þb10 � b30ð Þhþ qd0x2r3 ¼ 0

ð3:76aÞ

r2e330u;rr þ r 2Nþ 1ð Þe330 þ e310ð Þu;r þ 2Ne310u� r2e330u;rr

�r 2N þ 1ð Þe330ur þ r2c10h;r þ r 2Nþ 1ð Þc10h ¼ 0ð3:76bÞ

in which, x is the constant angular velocity of the cylinder; c110; c130; c330,e130; e330; e330, qd0; b10; b30; c10, and l0 represent the corresponding values ofc11; c13; c33; e13; e33; e33, qd; b1; b3; c1, and l at the outer surface of the cylinder,respectively. To simplify the analysis, we introduce the following non-dimensionalparameters:

a ¼ c110c330

; b ¼ e310e330

; c ¼ e330c330e2330

; d ¼ c130c330

; g ¼ b30b10

;X ¼ c10c330e330b10

;X ¼ l0H20

c330ð3:77Þ

as well as a new electric potential and temperature change as follows:

U ¼ e330c330

/; H ¼ b110c330

h ð3:78Þ

Therefore, Eq. (8) can be rewritten in the following form:

r2 1þXð Þu;rr þ r 2Nþ 1ð Þ 1þXð Þu;rþ 2Nd� aþX 2N � 1ð Þð Þuþ r2U;rr þ r 2Nþ 1� bð ÞU;r

�r2H;r � r 2Nþ 1� gð ÞHþ qd0x2r3

c330¼ 0

ð3:79aÞ

r2u;rr þ r 2Nþ 1þ bð Þu;r þ 2Nbu� r2cU;rr

�r 2N þ 1ð ÞcU;r þ r2XH;r þ r 2N þ 1ð ÞXH ¼ 0ð3:79bÞ

Using the normalized radial coordinate q ¼ ra and employing the chain rule for

differentiation, Eq. (3.79) is reduced to:


q2 1þXð Þu;qq þ q 2N þ 1ð Þ 1þXð Þu;qþ 2Nd� aþX 2N � 1ð Þð Þuþ q2U;qq þ q 2Nþ 1� bð ÞU;q

�q2aH;q � q 2Nþ 1� gð ÞHþ qd0x2q3a3

c330¼ 0

ð3:80aÞ

q2u;qq þ 2N þ 1þ bð Þqu;q þ 2Nbu� q2cU;qq

�q 2Nþ 1ð ÞcU;q þ q2aXH;q þ qa 2N þ 1ð ÞXH ¼ 0ð3:80bÞ

To solve Eq. (3.80), the temperature distribution along the radial direction of thehollow cylinder must be obtained. The axisymmetric, steady state heat conductionequation for an infinitely long hollow cylinder can be represented as:

1r

rkh;r� �

;r¼ 0 a� r� b ð3:81Þ

in which, k is the thermal conductivity varying according to Eq. (3.70).Rearranging Eq. (3.81), using the normalized radial coordinate and employingEqs. (3.70) and (3.78) leads to:

1q

q2Nþ 1H;q� �

;q¼ 0 1� q� i ð3:82Þ

where i ¼ ba is the aspect ratio of the hollow cylinder. The general solution for

N 6¼ 0 is obtained as follows:

H ¼ C1q�2N þC2 ð3:83Þ

in which, C1 and C2 are integration constants to be determined by thermal boundaryconditions. The general boundary conditions for Eq. (3.82) are [32]:

A11H 1ð ÞþA12H0 1ð Þ ¼ f1 ð3:84aÞ

A21H ið ÞþA22H0 ið Þ ¼ f2 ð3:84bÞ

where Aij are the Robin-type boundary condition coefficients, and f1 and f2 areknown functions on the inner and outer radii. Substituting Eq. (3.83) intoEq. (3.84) gives us the following integration constant C1 and C2:

C1 ¼ A11f2 � A21f1A11 A21#� 2NA22ð Þ#�2N�1 � A21 A11 � 2NA12ð Þ

C2 ¼ f1A11

� A11 � 2NA12ð Þ A11f2 � A21f1ð ÞA211 A21#� 2NA12ð Þ#�2N�1 � A21 A2

11 � 2NA12A11� � ð3:85Þ


Having the temperature distribution, we can solve the coupled governingequations (3.80). Making a variable change of q ¼ e1, Eq. (3.80) is converted to thenew differential equations with constant coefficients in the following form:

1þXð Þuþ 2N 1þXð Þuþ 2Nd� aþX 2N � 1ð Þð ÞuþUþ 2N � bð ÞU¼ ae1Hþ 2N þ 1� gð Þe1aH� xnae

31 ð3:86aÞ

€uþ 2Nþ bð Þ _uþ 2Nbu� c€U� 2cN _U ¼ �ae1X _H� 2Nþ 1ð ÞYae1H ð3:86bÞ

in which, the superposed dot represents differentiation with respect to 1 and

xn ¼ qd0x2a2

c330ð3:87Þ

By eliminating €U between the two equations in Eq. (3.87) and solving for _U, weobtain:

_U ¼ 1þXð Þcþ 1bc

€uþ 2N 1þ 1þXð Þcð Þþ bbc

_u

þ 2N dcþ bð Þ � acþXc 2N � 1ð Þbc

uþ a X � cð Þbc

e1 _H

þ 2Nþ 1ð ÞX � 2Nþ 1� gð Þcbc

ae1Hþ axn

be31 ð3:88Þ

Substituting Eq. (3.88) and its derivative into (3.86a), we obtain the followingdecoupled differential equation about u:

a3vuþ a2€uþ a1 _uþ a0u ¼ d3e31 þ d2e

1 þ d1e�2N þ 1ð Þ1 ð3:89Þ

where

a3 ¼ 1þXð Þcþ 1bc

; a2 ¼ 4N 1þ c 1þXð Þð Þbc

;

a1 ¼ 2N cdþ bð Þþ 4N2 1þXð Þcþ 1ð Þþ c X 2N � 1ð Þ � að Þ � b2

bc;

a0 ¼4N2 cdþ bð Þþ 2N Xc 2N � 1ð Þ � ac� b2

� �bc

;

d3 ¼ � axn 3þ 2Nð Þb

; d2 ¼ aX 2Nþ 1ð Þ 2Nþ 1� bð Þ � ac 2Nþ 1� gð Þ 1þ 2Nð Þbc

C2;

d1 ¼ aX 1� bð Þþ ac g� 2Nb� 1ð Þbc

C1

ð3:90Þ


The solution of the differential equation (3.89) includes two parts, the generalsolution and the particular solution. The general solution is:

ug ¼ Aem11 þBem21 þCem31 ð3:91Þ

where, mi i ¼ 1; 2; 3ð Þ are the roots of the characteristic equation (3.89). Also, A;B;and C are constants of integration determined by the boundary conditions. Theparticular solutions can be written in the following form:

up ¼ K1e�2Nþ 1ð Þ1 þK2e

1 þK3e31 ð3:92Þ

Substituting Eq. (3.92) into Eq. (3.89), we obtain:

K1 ¼ d1�2N þ 1ð Þ3a3 þ �2N þ 1ð Þ2a2 þ �2N þ 1ð Þa1 þ a0

;

K2 ¼ d2a3 þ a2 þ a1 þ a0

; K3 ¼ d327a3 þ 9a2 þ 3a1 þ a0

ð3:93Þ

Consequently, the solution for u in terms of q is:

u ¼ ug þ up ¼ Aqm1 þBqm2 þCqm3 þK1q�2Nþ 1 þK2qþK3q

3 ð3:94Þ

Substituting Eq. (3.94) into Eq. (3.88) and integrating, U is obtained as:

U ¼ b1Aqm1 þ b2Bq

m2 þ b3Cqm3 þDþK4q

�2Nþ 1 þK5qþK6q3 ð3:95Þ

in which D is a new integration constant and

bi ¼ 1bc

1þXð Þcþ 1ð Þmi þ 2N 1þXð Þcþ 1ð Þþ bþ 2N bþ dcð Þ � acþXc 2N � 1ð Þmi

� �

K4 ¼ 1bc

K1 1þXð Þcþ 1þ bð Þþ K1

�2N þ 12N dcþ bð Þþ c X 2N � 1ð Þ � að Þð Þ

�

þ C1a X � c 1� gð Þð Þ�2N þ 1

�

K5 ¼ 1bc

K2 2Nþ 1ð Þ 1þXð Þcþ 1ð Þþ bð ÞþK2 2N dcþ bð Þþ c X 2N � 1ð Þ � að Þð ÞðþC2a X 2Nþ 1ð Þ � c 2N þ 1� gð Þð ÞÞ

K6 ¼ 1bc

K3 2N þ 3ð Þ 1þXð Þcþ 1ð Þþ bð Þþ K3

3 2N dcþ bð Þþ c X 2N � 1ð Þ � að Þð Þþ xnac3

!

ð3:96Þ

For convenience, we define the following non-dimensional stresses, displace-ment, electric potential, electric displacement, and perturbation of magnetic field asfollows:


Rrr ¼ rrrc330

;Rhh ¼ rhhc330

; u1 ¼ ua;U1 ¼ U

a;Dr1 ¼

Dr

e330; h�z ¼

hzH0

ð3:97Þ

Using Eqs. (3.73), (3.94), (3.95), and (3.97), we obtain:

Rrr ¼ i�2N q2N�1

a

Aqm1 m1 þ dþ b1m1ð ÞþBqm2 m2 þ dþ b2m2ð ÞþCqm3 m3 þ dþ b3m3ð Þ

þ q�2Nþ 1 K1 �2Nþ 1ð ÞþK1dþK4 �2Nþ 1ð Þ � aC1ð Þþ q K2 þK2dþK5 � aC2ð Þþ q3 3K3 þK3dþ 3K6ð Þ

0BB@

1CCA

ð3:98aÞ

Rhh ¼ i�2N q2N�1

a

Aqm1 m1dþ aþ bb1m1ð ÞþBqm2 m2dþ aþ bb2m2ð ÞþCqm3 m3dþ aþ bb3m3ð Þ

þ q�2N þ 1 K1d �2Nþ 1ð ÞþK1aþK4b �2Nþ 1ð Þ � gaC1ð Þþ q K2dþK2aþ bK5 � gaC2ð Þþ q3 3K3dþK3aþ 3K6bð Þ

0BB@

1CCA

ð3:98bÞ

Dr1 ¼ i�2N q2N�1

a

Aqm1 m1 þ b� cb1m1ð ÞþBqm2 m2 þ b� cb2m2ð ÞþCqm3 m3 þ b� cb3m3ð Þ

þ q�2Nþ 1 K1 �2Nþ 1ð ÞþK1b� K4c �2N þ 1ð Þ � gaC1ð Þþ q K2 þ bK2 � cK5 þXaC2ð Þþ q3 3K3 þK3b� 3K6cð Þ

0BB@

1CCA

ð3:98cÞ

h�z ¼ � 1a

Aqm1�1 m1 þ 1ð ÞþBqm2�1 m2 þ 1ð ÞþCqm3�1 m3 þ 1ð Þþ 2K1q�2N �N þ 1ð Þþ 2K2 þ 4K3q2

� �ð3:98dÞ

u1 ¼ 1aAqm1 þ 1

aBqm2 þ 1

aCqm3 þ K1

aq�2Nþ 1 þ K2

aqþ K3

aq3 ð3:98eÞ

U1 ¼ b1aAqm1 þ b2

aBqm2 þ b3

aCqm3 þ 1

aDþ K4

aq�2Nþ 1 þ K5

aqþ K6

aq3 ð3:98fÞ

The boundary conditions according to Fig. 3.5 can be expressed in the followingform:

Rrr 1ð Þ ¼ Rrri; U1 1ð Þ ¼ U1i ð3:99aÞ

Rrr ið Þ ¼ Rrro; U1 ið Þ ¼ U1o ð3:99bÞ

Applying Eqs. (3.98) and (3.99), we reach a set of four linear algebraic equationsfor the four unknowns, A, B, C, and D, as follows:


I

ABCD

8>><>>:

9>>=>>; ¼ J ð3:100Þ

in which, I is a 4� 4 nontrivial matrix with the following arrays:


; I12 ¼ i�2N m2 1þ b2ð Þþ da

;


; I14 ¼ 0;

I21 ¼ b1a; I22 ¼ b2

a; I23 ¼ b3

a; I24 ¼ 1

a;


; I32 ¼ im2�1 m2 1þ b2ð Þþ da

;


; I34 ¼ 0;

I41 ¼ im1b1a; I42 ¼ im2

b2a; I43 ¼ im3

b3a; I44 ¼ 1

a

ð3:101Þ

and the vector J has the following components:

J11 ¼ Rrri � 1ai�2N K1 �2N þ 1ð ÞþK1dþK4 �2Nþ 1ð Þ

�aC1 þK2 þK2dþK5 � aC2 þ 3K3 þK3dþ 3K6

� �ð3:102Þ

Finally, solving Eq. (3.100) completes our analysis. For brevity the expressionsfor A, B, C, and D have been omitted here.


Numerical examples of the analytical results for the multiphysical response of theFGPM hollow cylinder are presented graphically in this section. The outer surfaceof the cylinder is taken to be PZT-4 with its material properties listed in Table 3.3.Two mechanical and electrical boundary conditions are assumed here to illustratethe behavior of the FGPM cylinder:

Case 1: Case 2:

Rrr 1ð Þ ¼ �1;U1 1ð Þ ¼ 1 Rrr 1ð Þ ¼ �1;U1 1ð Þ ¼ 0

Rrr ið Þ ¼ 0;U1 ið Þ ¼ 0 Rrr ið Þ ¼ 0;U1 ið Þ ¼ 0


Although the formulation for thermal analysis includes a general form of thermalboundary conditions, the numerical results are obtained for the FGPM hollowcylinder with assigned non-dimensional temperature Ha and Hb on the inner andouter surfaces, respectively. The stresses, electric potential, electric displacement,perturbation of magnetic field and temperature distribution are given for variousnon-dimensional magnetic fields, inertial effects, aspect ratios andnon-homogeneity parameters as well as different non-dimensional values of innerand outer temperature.

Figure 3.6a–d show the effects of non-dimensional magnetic field X on thedistribution of stresses, electric displacement, and electric potential along the radialdirection of the hollow FGPM cylinder with N ¼ �1, angular velocity xn ¼ 1, andaspect ratio i ¼ 4 under Case 1 boundary conditions in the equations above withoutany thermal disturbance. The results for X ¼ 0 are completely similar to thosereported in (Babaei and Chen 2008), which validates our solution procedure.

3.4 Effect of Hygrothermal Excitationon One-Dimensional Smart Structures

The constitutive, potential field, and conservation equations for solving theuncoupled hygrothermomagnetoelectroelastic problems are presented in this sec-tion. The geometry of an infinitely long MEE cylinder is depicted in Fig. 3.7. Thecylinder is radially polarized and magnetized and rotating at a constant angularvelocity x about the z-axis of cylindrical coordinate system r; h; zð Þ. The inner andouter radii of the cylinder are a and b, respectively. The MEE cylinder rests on the

Table 3.3 Materialproperties of the outer surfaceof the FGPM cylinder

Properties PZT-4 (outer surface)

c330 Nm2

� �115�109

c110 Nm2

� �139� 109

c130 Nm2

� �74:3� 109

e310 Cm2

� � �5:20

e330 Cm2

� �15:1

e330 C2 Nm2

� �5:62� 10�9

qd0kgm3

� �7:5� 103

b10NKm2

� �1:0089� 106

b30NKm2

� �0:8439� 106

c10CKm2

� � �2:5� 10�5


Fig. 3.6 a Radial stress for different X, xn ¼ 1, N ¼ �1, i ¼ 4, case 1, b Hoop stress fordifferent X, xn ¼ 1, N ¼ �1, i ¼ 4, case 1, c Electric displacement for different X, xn ¼ 1,N ¼ �1, i ¼ 4, case 1, d Electric potential for different X, xn ¼ 1, N ¼ �1, i ¼ 4, case 1.[Reproduced from [37] with permission from IOP Publishing]

3.4 Effect of Hygrothermal Excitation on One-Dimensional Smart Structures 99

elastic foundation with Winkler-type foundation stiffness kw at the inner and/orouter surfaces or exposed to internal and/or external pressure pa and pb. The innerand outer surfaces of the cylinder are subjected to moisture concentration changema and mb, temperature change #a and #b, magnetic potential ua and ub, andelectric potential /a and /b. Subscripts “a” and “b” are used, respectively, toindicate loads on the inner and outer surfaces.

For transversely isotropic and radially polarized and magnetized materials,substituting Eq. (3.22) into (3.7) results in [39]:

rrr ¼ c33u;r þ c13urþ e33/;r þ d33u;r � b1#� f1m ð3:103aÞ

rhh ¼ c13u;r þ c11urþ e31/;r þ d31u;r � b3#� f3m ð3:103bÞ

Fig. 3.7 Rotating hollow MEE cylinder resting on elastic foundation and its boundary conditions.[Reproduced from [38] with permission from IOP Publishing]



rzz ¼ c13u;r þ c12urþ e31/;r þ d31u;r � b3#� f3m ð3:103cÞ

Dr ¼ e33u;r þ e31ur� 233 /;r � g33u;r þ c1#þ v1m ð3:103dÞ

Br ¼ d33u;r þ d31ur� g33/;r � l33u;r þ s1#þ t1m ð3:103eÞ

where, cmn ¼ Cijkl, eml ¼ ekij, dml ¼ dkij, bm ¼ bij, and nm ¼ nij i; j; k; l ¼ 1; 2; 3;ðm; n ¼ 1; 2; . . .; 6Þ. In the absence of body force, free charge density, and currentdensity, the equation of motion and Maxwell’s electromagnetic equations for theaxisymmetric, infinitely long cylinder are written as Eq. (3.24).

3.4.1 Solution Procedure

The solution procedure for hollow and solid MEE cylinders is given in this sectionsimilar to those reported in [40] for magnetoelectroelastic cylinders. SubstitutingEqs. (3.103) and (3.25) into Eq. (3.24) results in the following coupled ordinarydifferential equations in terms of moisture concentration, temperature change,magnetic potential, electrical potential, and displacement:

c33r2u;rr þ c33ru;r � c11uþ e33r

2u;rr þ e33 � e31ð Þru;r

þ d33r2/;rr þ d33 � d31ð Þr/;r � b1r

2#;r

� b1 � b3ð Þr#� n1r2m;r � n1 � n3ð Þrmþ qdr

3x2 ¼ 0 ð3:104aÞ

e33r2u;rr þ e31 þ e33ð Þru;r� 233 r

2u;rr� 233 r/;r

�g33r2u;rr � g33ru;r þ c1r

2#;r þ c1r#þ v1r2m;r þ v1rm ¼ 0 ð3:104bÞ

d33r2u;rr þ d31 þ d33ð Þru;r � g33r

2/;rr � g33r/;r � l33r2u;rr

�l33ru;r þ s1r2#;r þ s1r#þ t1r

2m;r þ t1rm ¼ 0 ð3:104cÞ

The non-dimensional parameters are introduced in Eq. (3.28) and new electricpotential, magnetic potential, temperature change, and moisture concentrationchange as:

U ¼ e33c33

/; W ¼ d33c33

u; H ¼ b1c33

#; M ¼ n1c33

m ð3:105Þ

Using Eqs. (3.28) and (3.105), Eq. (3.104) can be rewritten in the followingform:


r2u;rr þ ru;r � auþ r2U;rr þ 1� bð ÞrU;r þ r2W;rr þ 1� mð ÞrW;r

�r2H;r � 1� gð ÞrH� r2M;r � 1� 1ð ÞrMþXr3

a2¼ 0

ð3:106aÞ

r2u;rr þ 1þ mð Þru;r � fr2U;rr � frU;r � kr2W;rr � krW;r

þ Yr2H;r þ YrHþWr2M;r þWrM ¼ 0ð3:106bÞ

r2u;rr þ 1þ bð Þru;r � cr2U;rr � crU;r � fr2W;rr � frW;r

þXr2H;r þXrHþVr2M;r þVrM ¼ 0ð3:106cÞ

in which,

X ¼ qdx2a2

c33ð3:107Þ

The following set of second-order coupled ordinary differential equations withconstant coefficients is obtained by using the non-dimensional radial coordinateq ¼ r

a and then changing variable q with s by q ¼ es as follows:

€u� auþ €U� b _Uþ €W� m _W ¼ aes _Hþ 1� gð ÞaesHþ aes _Mþ 1� 1ð ÞaesM � Xae3s

ð3:108aÞ

€uþ b _u� c€U� f €W ¼ �Xaes _H� XaesH� Vaes _M � VaesM ð3:108bÞ

€uþ m _u� f€U� k €W ¼ �Yaes _H� YaesH�Waes _M �WaesM ð3:108cÞ

in which, the overdot stands for differentiation with respect to s. For uncoupledhygrothermomagnetoelectroelastic problems, temperature and moisture concentra-tion distributions are obtained separately by solving heat conduction and moisturediffusion equations. The axisymmetric and steady state Fourier heat conduction andFickian moisture diffusion equations for an infinitely long hollow cylinder arewritten as [17]:

1r

rkTh;r� �

;r¼ 0 Heat conduction equationð Þ ð3:109aÞ

1r

rkCm;r� �

;r¼ 0 Moisture diffusion equationð Þ ð3:109bÞ

where, kT and kC are thermal conductivity and moisture diffusivity coefficients,respectively. Solving Eq. (3.109) and using the non-dimensional radial coordinateas well as the new parameters defined in Eq. (3.105) result in the following tem-perature and moisture concentration distribution along the radial direction for ahomogenous cylinder:


H ¼ C1 ln qð ÞþC2 ð3:110aÞ

M ¼ C3 ln qð ÞþC4 ð3:110bÞ

where C1, C2, C3, and C4 are integration constants which are determined by satis-fying the hygrothermal boundary conditions. The non-dimensional temperaturechange on the inner and outer surfaces of the hollow cylinder are, respectively,assumed to be Ha and Hb. Similarly, non-dimensional moisture concentrationchange are assigned to beMa andMb on the inner and outer surfaces. The integrationconstants according to the hygrothermal boundary conditions can be obtained as:

C1 ¼ Hb �Ha

ln ið Þ ; C2 ¼ Ha; C3 ¼ Mb �Ma

ln ið Þ ; C4 ¼ Ma ð3:111Þ

in which, i ¼ ba is the aspect ratio of the hollow cylinder. It should be mentioned that

for solid cylinders as well as hollow cylinders with uniform temperature andmoisture concentration rise, we have C1 ¼ C3 ¼ 0. For more general hygrothermalboundary conditions, one may refer to the Robin-type boundary conditions con-sidered in [17, 32]. It is worth noting that temperature and moisture concentrationhave similar effects on magnetoelectroelastic responses in uncoupled hygrother-momagnetoelectroelasticity according to Eqs. (3.108) through (3.111). The solutionof Eq. (3.108) can be found analytically by successive decoupling method [40].Eliminating U between equations in (3.108) leads to the following two ordinarydifferential equations about u and W:

1þ cð Þvu� b2 þ ac� �

_uþ c� fð ÞvWþ bf� mcð Þ €W¼ c� Xð Þaes €Hþ c 2� gð ÞþX b� 2ð Þð Þaes _Hþ c 1� gð ÞþX b� 1ð Þð ÞaesH

þ c� Vð Þaes €Mþ c 2� 1ð ÞþV b� 2ð Þð Þaes _Mþ c 1� 1ð ÞþV b� 1ð Þð ÞaesM � 3aXce3s

ð3:112aÞ

c� fð Þ€uþ mc� bfð Þ _uþ f2 � kc� �

€W ¼ Xf� Ycð Þaes _Hþ Xf� Ycð ÞaesH� Vf�Wcð Þaes _Mþ Vf�Wcð ÞaesM ð3:112bÞ

By eliminating W between Eqs. (3.112) and considering the temperature changeand moisture concentration change distributions according to Eq. (3.110), the fol-lowing ordinary differential equation with constant coefficients for radial dis-placement u is achieved:

a2vuþ a1 _u ¼ b1C2 þ b2C1 þ b4C4 þ b5C3ð Þes þ b1C1 þ b4c3ð Þses þ b5e3s ð3:113Þ


where,

a1 ¼ mc� bfð Þ2f2 � kc

� b2 þ ac� �

; a2 ¼ 1þ c� c� fð Þ2f2 � kc

b1 ¼ a c 1� gð ÞþX b� 1ð Þ � Xf� Ycð Þ bf� mcþ cfð Þf2 � kc

� �

b2 ¼ a c 2� gð ÞþX b� 2ð Þ � Xf� Ycð Þ bf� mcþ 2c� 2fð Þf2 � kc

� �

b3 ¼ a c 1� 1ð ÞþV b� 1ð Þ � Vf�Wcð Þ bf� mcþ c� fð Þf2 � kc

� �

b4 ¼ a c 2� 1ð ÞþV b� 2ð Þ � Vf�Wcð Þ bf� mcþ 2c� 2fð Þf2 � kc

� �b5 ¼ �3acX

ð3:114Þ

The solution of Eq. (3.113) in terms of variable q can be expressed as:

u ¼ AþCqm þDq�m þ K1 ln qð ÞþK2ð ÞqþK3q3 ð3:115Þ

in which,

m ¼ iffiffiffiffiffia1a2

rm 2 R and i ¼

ffiffiffiffiffiffiffi�1

p� �ð3:116Þ

and A, C, and D are integration constants. Employing Eqs. (3.113) and (3.115), thefollowing expression for W can be obtained:

W ¼ FþE ln qð ÞþCc1qm þDc2q

�m þ c3 ln qð Þþ c4ð ÞqþK3c5q3 ð3:117Þ

where, E and F are new integration constants and:

c1 ¼ 1

m kc� f2� � m c� fð Þþ mc� bfð Þ; c2 ¼ 1

m kc� f2� � m c� fð Þ � mcþ bfð Þ

c3 ¼ 1

kc� f2K1 c 1þ mð Þ � f 1þ bð Þð Þ � aC1 Xf� Ycð Þ � aC3 Vf�Wcð Þð Þ

c4 ¼ 1

kc� f2K2 c� fð Þþ K2 � K1ð Þ mc� bfð Þþ a C1 � C2ð Þ Xf� Ycð Þð

þ a C3 � C4ð Þ Vf�Wcð ÞÞc5 ¼ 1

3 kc� f2� � 3 c� fð Þþ mc� bfð Þ

ð3:118Þ


Using Eq. (3.108b) and considering Eqs. (3.115) and (3.117), U can beexpressed by:

U ¼ HþG ln qð ÞþCl1qm þDl2q

�m þ l3 ln qð Þþ l4ð ÞqþK3l5q3 ð3:119Þ

in which, G and H are integration constants and:

l1 ¼ 1mc

mþ b� mfc1ð Þ; l2 ¼ 1mc

m� b� mfc2ð Þ

l3 ¼ 1c

K1 1þ bð Þ � fc3 þ aXC1 þ aVC3ð Þ

l4 ¼ 1c

K2 þ K2 � K1ð Þb� c4fþ aX C2 � C1ð Þþ aV C4 � C3ð Þð Þ

l5 ¼ 13c

3þ b� 3fc5ð Þ

ð3:120Þ

For convenience, the following non-dimensional stresses, displacement, electricpotential, and magnetic potential are used:

Rrr ¼ rrrc33

;Rhh ¼ rhhc33

;Rzz ¼ rzzc33

;U ¼ ua;U1 ¼ U

a;W1 ¼ W

að3:121Þ

Then, we reach:

Rrr ¼ q�1

adAþGþEð Þþ qm�1

am 1þ l1 þ c1ð Þþ dð ÞC

þ qm�1

a�m 1þ l2 þ c2ð Þþ dð ÞDþ ln qð Þ

aK1 1þ dð Þþ l3 þ c3 � a C1 þC3ð Þð Þ

þ 1a

K1 þ 1þ dð ÞK2 þ l3 þ l4 þ c3 þ c4 � a C2 þC4ð Þð Þþ q2


ð3:122aÞ

Rhh ¼ q�1

adAþbGþ mEð Þþ qm�1

am dþ bl1 þ mc1ð Þþ dð ÞC

þ qm�1

a�m dþbl2 þ mc2ð Þþ dð ÞDþ ln qð Þ

aK1 dþ að Þþbl3 þ mc3 � a gC1 þ 1C3ð Þð Þ

þ 1a

dK1 þ dþ að ÞK2 þ b l3 þ l4ð Þþ m c3 þ c4ð Þ � a gC2 � 1C4ð Þð Þ

þ q2

a3 dþ bl5 þ mc5ð Þþ að ÞK3

ð3:122bÞ


Rzz ¼ q�1

ad�Aþ bGþ mEð Þþ qm�1

am dþ bl1 þ mc1ð Þþ d�ð ÞC

þ qm�1

a�m dþ bl2 þ mc2ð Þþ d�ð ÞDþ ln qð Þ

aK1 dþ d�ð Þþ bl3 þ mc3 � a gC1 þ 1C3ð Þð Þ

þ 1a

dK1 þ dþ d�ð ÞK2 þ b l3 þ l4ð Þþ m c3 þ c4ð Þ � a gC2 þ 1C4ð Þð Þ

þ q2

a3 dþ bl5 þ mc5ð Þþ d�ð ÞK3

ð3:122cÞ

W1 ¼ Faþ E

aln qð Þþ C

ac1q

m þ DaC2q

�m þ c3 ln qð Þþ c4ð Þ qaþ K3c5

aq3 ð3:122dÞ

U1 ¼ Haþ G

aln qð Þþ C

al1q

m þ Dal2q

�m þ l3 ln qð Þþ l4ð Þ qaþ K3l5

aq3 ð3:122eÞ

U ¼ Aaþ C

aqm þ D

aq�m þ K1

aln qð Þþ K2

a

� �qþ K3

aq3 ð3:122fÞ

We have seven integration constants in the aforementioned equations; however,there exist only six boundary conditions in the magnetoelectroelastic medium.Therefore, one complimentary equation is needed which is obtained by substitutingEqs. (3.115), (3.117), and (3.119) into Eq. (3.108):

aAþ mEþ bG ¼ 0 ð3:123Þ

The integration constants are obtained in the following subsections for hollowand solid cylinders. Although the solution procedure for this hygrothermomagne-toelectroelastic analysis under steady-state condition is the same as [40], the work isa pioneer in such emerging multiphysical analysis. The analytical solutions given inEq. (3.122) could be employed for the design of MEE structures as well as abenchmark solution for verification of the other analytical and numerical resultswhich will be used later for the multiphysical problem.

To consider the effect of temperature and moisture dependency of elastic coef-ficients on the magnetoelectroelastic response, the elastic coefficients are expressedin the following form [41, 42]:

Cij ¼ Cij0 1þ a�#þ b�Mð Þ ð3:124Þ

in which, Cij0 is an elastic coefficient at stress-free temperature and moisture con-centration; a� and b� are empirical material constants for temperature and moisturedependency. In the current work, the temperature and moisture dependency is onlyconsidered for uniform temperature and moisture concentration rise to avoiddealing with non-linear–problems [43].


3.4.2 MEE Hollow Cylinder

The MEE hollow cylinder may be exposed to different hygrothermomagnetoelec-troelastic boundary conditions. The hygrothermal boundary conditions were con-sidered earlier; here we specify other magnetoelectroelastic boundary conditions.Since the hollow cylinder may be simulated with or without Winkler-type elasticfoundation on the inner and outer surfaces, different non-dimensional elasticboundary conditions could be considered as follows [44, 45]:

• Internal and external pressure:

Rrr 1ð Þ ¼ Rrra and Rrr ið Þ ¼ Rrrb ð3:125aÞ

• Internal pressure and outer surface elastic foundation:

Rrr 1ð Þ ¼ Rrra and Rrr ið Þ ¼ �KWUb ð3:125bÞ

• Inner surface elastic foundation and external pressure:

Rrr 1ð Þ ¼ KWUa and Rrr ið Þ ¼ Rrrb ð3:125cÞ

• Elastic foundation on both the inner and outer surfaces:

Rrr 1ð Þ ¼ KWUa and Rrr ið Þ ¼ �KWUb ð3:125dÞ

Moreover, the magnetoelectric boundary conditions are specified as:

U1 1ð Þ ¼ U1a; W1 1ð Þ ¼ W1a ð3:126aÞ

U1 ið Þ ¼ U1b; W1 ið Þ ¼ W1b ð3:126bÞ

in which, KW ¼ akwc33

and subscripts “a” and “b” are associated with quantities on theinner and outer surfaces. It is worth to note that the Winkler-type elastic foundationchanges the type of mechanical boundary conditions and does not affect the gov-erning differential equations. Employing Eqs. (3.122), (3.124), (3.125), and (3.126)lead to the following linear algebraic equation for integration constants as follows:

I

ACDEFGH

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

¼ J ð3:127Þ


where, I is a 7� 7 matrix and J is a 7� 1 vector and their components are given in[38]. Once the integration constants are obtained by solving Eq. (3.127), the ana-lytical solution for hollow MEE cylinder under hygrothermal loading is eventuallyobtained.

3.4.3 MEE Solid Cylinder

The solution procedure for an MEE solid cylinder with outer radius b is analogousto an MEE hollow cylinder; however, the new non-dimensional radial coordinateq ¼ r

b 0� q� 1ð Þ is needed to enable us to use previously obtained expressions forstresses, displacement, electric potential, and magnetic potential. The followingnew non-dimensional displacement, electric potential, and magnetic potential aredefined for the MEE solid cylinder:

U ¼ ub; U1 ¼ U

b; W1 ¼ W

bð3:128Þ

As the displacement, electric potential, and magnetic potential should be finite atthe axis of symmetry of solid cylinders, we can deduce from Eqs. (3.115), (3.117),(3.119), and (3.123):

A ¼ C ¼ E ¼ G ¼ 0 ð3:129Þ

Therefore, one can obtain the following results for the MEE solid cylinder fromEqs. (3.115), (3.117), (3.119), and (3.122):

Rrr ¼ q�m�1

b�m 1þ l2 þ c2ð Þþ að ÞDþ 1

b1þ dð ÞK2 þ l4 þ c4 � b C2 þC4ð Þð Þ

þ q2

b3 1þ l5 þ c5ð Þþ dð ÞK3

ð3:130aÞ

Rhh ¼ q�m�1

b�m dþ bl2 þ mc2ð Þþ að ÞD

þ 1b

dþ að ÞK2 þ bl4 þ mc4 � b gC2 � 1C4ð Þð Þþ q2

b3 dþ bl5 þ mc5ð Þþ að ÞK3

ð3:130bÞ

Rzz ¼ q�m�1

b�m dþ bl2 þ mc2ð Þþ d�ð ÞD

þ 1b

dþ d�ð ÞK2 þ bl4 þ mc4 � b gC2 þ 1C4ð Þð Þþ q2

b3 dþ bl5 þ mc5ð Þþ d�ð ÞK3

ð3:130cÞ


W1 ¼ Fbþ D

bc2q

�m þ c4bqþ K3c5

bq3 ð3:130dÞ

U1 ¼ Hbþ D

bl2q

�m þ l4bqþ K3l5

bq3 ð3:130eÞ

U ¼ Dbq�m þ K2

bqþ K3

bq3 ð3:130fÞ

where all constants are similar to those defined for the hollow cylinder except thefollowing parameters:

C1 ¼ C3 ¼ 0

K1 ¼ 0;K2 ¼ b1C2 þ b3C4

a2 þ a1

c3 ¼ 0; c4 ¼ 1

kc� f2K2 c� fð ÞþK2 mc� bfð Þ � bC2 Xf� Ycð Þ � bC4 Vf�Wcð Þð Þ

l3 ¼ 0; l4 ¼ 1c

K2 1þ bð Þ � c4fþ bXC2 þ bVC4ð Þð3:131Þ

Furthermore, it is worth recalling that the temperature and moisture concentra-tion remain constant through the radial direction of axisymmetric solid cylinders inthe steady-state condition according to the heat conduction and moisture diffusionequations. The constant values can be determined by the non-dimensional tem-perature and moisture concentration at the outer surface of cylinders. To obtain theintegration constants in Eq. (3.130), the magnetoelectroelastic boundary conditionsare required. The outer surface of the solid cylinder could be exposed to externalpressure or rest on a Winkler-type elastic foundation; the following elastic boundaryconditions can be expressed accordingly:

• External pressure:

Rrr 1ð Þ ¼ Rrrb ð3:132aÞ

• Elastic foundation on the outer surface:

Rrr 1ð Þ ¼ �KWUb ð3:132bÞ

The electromagnetic boundary conditions are:

U1 1ð Þ ¼ U1b and W1 1ð Þ ¼ W1b ð3:133Þ

Employing Eqs. (3.130) and (3.132a, 3.132b), the following linear algebraicequation can be established to obtain the integration constants:


PDFH

8<:

9=; ¼ Q ð3:134Þ

in which, P is a 3� 3 matrix and Q is a 3� 1 vector with components defined in[38]. Solving Eq. (3.134) gives the integration constants and complete the analyt-ical solution. According to Eq. (3.130), the value of displacement, electric potential,and magnetic potential are always finite along the central axis of solid cylinder;

however, the stress components are singular for m ¼ a1a2

[ � 1 depending on the

material properties of the MEE cylinder [40].


The numerical results for uncoupled hygrothermomagnetoelectroelastic behavior oftransversely isotropic hollow and solid cylinders are presented graphically in thissection. The numerical results include radial, hoop, and axial stresses as well aselectrical and magnetic potentials. There does not exist any experimental results forsuch multifield analysis; however, due to the application of smart wood structureswith piezoelectric/piezomagnetic sensors and actuators in different environmentalconditions, such multiphysical experiment is feasible. These theoretical resultsreveal the possible interaction of different physical fields when studying thestructural behavior of smart materials.

The material properties of MEE cylinders are given in Table 3.4 according to thematerial properties of an adaptive wood made of BaTiO3/CoFe2O4 [34, 46, 47].

The effect of hygrothermal boundary conditions on the multiphysical responsesof an MEE hollow cylinder is depicted in Fig. 3.8. The cylinder is assumed to beunder internal pressure and the traction-free boundary condition exists on the outersurface. The following magnetoelectroelastic boundary conditions are considered:

Rrr 1ð Þ ¼ �1; U1 1ð Þ ¼ 1; W1 1ð Þ ¼ 1

Rrr ið Þ ¼ 0; U1 ið Þ ¼ 0; W1 ið Þ ¼ 0

The aspect ratio, inner radius, and non-dimensional angular velocity of therotating MEE hollow cylinder are i ¼ 4, a ¼ 1, and X ¼ 1, respectively. Thenon-dimensional moisture concentration and temperature on the inner surface arekept at be zero, while the moisture concentration and temperature rise on the outersurface are Hb and Mb. Since the effects of moisture concentration and temperatureon the multiphysical responses are similar, the same values are taken for Hb andMb. As depicted in Fig. 3.11, the results are quite close to those reported in [36] forthe magnetoelectroelastic response of rotating MEE hollow cylinders in the absenceof elastic foundation and hygrothermal loading Ha ¼ Hb ¼ 0;Ma ¼ Mb ¼ 0ð Þ.


Figure 3.8a–c illustrate the effect of hygrothermal loading Hb and Mb on thedistribution of stresses through the thickness. The greater hygrothemal loadings onthe outer surface result in greater absolute values of maximum radial stress.Increasing the outer hygrothermal loading amplifies the hoop stress on the innersurface and lessens the hoop stress on the outer surface. Since electroelasticcylinders have been observed to fail at a critical hoop stress [48], the effect ofhygrothermal loading on the hoop stress is noteworthy for design and manufac-turing of magnetoelectroelastic cylinders. Furthermore, the hygrothermal loadinggenerally decreases the axial stresses through the thickness of the MEE cylinder.The effect of hygrothermal loading on the distribution of electric and magneticpotentials is shown in Fig. 3.8d and e. As depicted in Fig. 3.8d, the electricpotential distribution shows double concavities in the absence of temperature andmoisture concentration; however, applying the hygrothermal loading on the outersurface leads to the disappearance of one concavity and increases the maximum

Table 3.4 Materialproperties of an adapativewood made of BaTiO3/CoFe2O4

c33 Nm2

� �2:695� 1011

c11 Nm2

� �2:86� 1011

c13 Nm2

� �1:705� 1011

c12 Nm2

� �1:73� 1011

e31 Cm2

� � �4:4

e33 Cm2

� �18:6

d31 NAm

� �580:3

d31 NAm

� �699:7

233C2

Nm2

� �9:3� 10�11

g33 Ns2C2

� �3:0� 10�12

l33Ns2C2

� �1:57� 10�4

aT1 ¼ aT2 ¼ aT31K

� �1� 10�5

bC1m3

kg

� �0

bC2 ¼ bC3m3

kg

� �1:1� 10�4

c1Ckm2

� � �13:0� 10�5

v1Cmkg

� �0

s1 NAmK

� �6:0� 10�3

t1 Nm2

Akg

� �0


Fig. 3.8 Effect of hygrothermal loading on the distribution of: a radial stress, b hoop stress,c axial stress, d electric potential, e magnetic potential, f temperature, and g moisture concentration[38]




absolute value of electric potential in the MEE cylinder. Moreover, increasing thetemperature and moisture concentration on the outer surface decreases the magneticpotential through the thickness of the cylinder as shown in Fig. 3.8e. As mentionedearlier, in uncoupled hygrothermomagnetoelectroelasticity, temperature and mois-ture distributions are independent of other multiphysical fields. As seen in Fig. 3.8fand g, the non-dimensional temperature and moisture concentration distributionincrease through the thickness in the same manner as the applied temperature andmoisture concentration on the outer surface increases.

3.5 Remarks

Analytical solutions are obtained for the uncoupled hygrothermomagnetoelectroe-lastic response of rotating MEE hollow and solid cylinders on a Winkler-type elasticfoundation. The cylinders are exposed to hygrothermal loading and assumed to beinfinitely long. The combined hygroscopic, thermal, magnetic, electric, andmechanical loads is considered. For a uniform temperature and moisture concentra-tion rise, the effect of temperature and moisture dependency of elastic coefficients onthe magnetoelectroelastic response is investigated. Using the axisymmetric,steady-state Fourier heat conduction and Fickian moisture diffusion equations, theradial temperature and moisture concentration distributions through the thickness aredetermined. The coupled governing ordinary differential equations in terms of mag-netic potential, electric potential, and displacement, including the effects ofhygrothermal loading, are solved analytically by a successive decoupling method.Numerical results are calculated to reveal the effect of hygrothermal boundary con-ditions, elastic foundation, and temperature and moisture dependency of elasticcoefficients on the multiphysical responses of the hollow and solid cylinders. The



investigation reveals that the coupling effects of magneto-electro-elastic fields cannotbe ignored when the material properties exhibit piezomagnetic/piezoelectric effectssimultaneously. Although the governing and constitutive equations of the magneticand electric potentials are similar to each other, their distributions are not the same dueto the different coupling coefficients. Furthermore, it is seen that imposing a propermagnetic field can reduce the hoop stress in a rotating FGPM cylinder, and as a resultcan make the smart structures more reliable. Finally, the investigation shows thatmoisture concentration and temperature have similar effects on the multiphysicalresponses of an MEE cylinder in uncoupled hygrothermomagnetoelectroelasticity. Itis observed that hygrothermal loading can change the radial stress, hoop stress, axialstress, and electric potential significantly for both hollow and solid MEE cylinders. Itis worth mentioning that a theoretical micromechanical model or a computationalhomogenization technique can be used to obtain the effective properties of smartmaterials to be used in the closed-form solutions obtained in this chapter for multi-physical analysis of smart materials and structures [49, 50].

References

1. Krzhizhanovskaya VV, Sun S (2007) Simulation of multiphysics multiscale systems:introduction to the ICCS’2007 workshop. In: Proceedings of the 7th international conferenceon computational science, Part I: ICCS 2007. Springer, Beijing, China, pp 755–761

2. Babaei MH (2009) Multiphysics analysis of functionally graded piezoelectrics. MechanicalEngineering Department, University of New Brunswick, Ph.D. thesis

3. Altay G, Dökmeci MC (2008) Certain hygrothermopiezoelectric multi-field variationalprinciples for smart elastic laminae. Mech Adv Mater Struct 15(1):21–32

4. Qin QH (2001) Fracture mechanics of piezoelectric materials5. Ballato A (2001) Modeling piezoelectric and piezomagnetic devices and structures via

equivalent networks. IEEE Trans Ultrason Ferroelectr Freq Control 48(5):1189–12406. Nan C-W (1994) Magnetoelectric effect in composites of piezoelectric and piezomagnetic

phases. Phys Rev B 50(9):6082–60887. Wang Y et al (2010) Multiferroic magnetoelectric composite nanostructures. NPG Asia Mater

2:61–688. Tzou HS, Lee HJ, Arnold SM (2004) Smart materials, precision sensors/actuators, smart

structures, and structronic systems. Mech Adv Mater Struct 11(4–5):367–3939. Robert GL (1997) Recent developments in smart structures with aeronautical applications.

Smart Mater Struct 6(5):R1110. Javanbakht M, Shakeri M, Sadeghi SN (2009) Dynamic analysis of functionally graded shell

with piezoelectric layers based on elasticity. Proc Inst Mech Eng Part C: J Mech Eng Sci 223(9):2039–2047

11. Yas MH, Shakeri M, Khanjani M (2011) Layer-wise finite-element analysis of a functionallygraded hollow thick cylinder with a piezoelectric ring. Proc Inst Mech Eng Part C: J MechEng Sci 225(5):1045–1060

12. Kapuria S, Yasin MY (2010) Active vibration control of piezoelectric laminated beams withelectroded actuators and sensors using an efficient finite element involving an electric node.Smart Mater Struct 19(4):045019

13. Gabbert U, Ringwelski S (2014) Active vibration and noise control of a car engine: modelingand experimental validation. In: Belyaev AK, Irschik H, Krommer M (eds) Mechanics andmodel-based control of advanced engineering systems. Springer Vienna, Vienna, pp 123–135

3.5 Remarks 115

14. Roederer AG, Jensen NE, Crone GAE (1996) Some European satellite-antenna developmentsand trends. IEEE Antennas Propag Mag 38(2):9–21

15. Barman S.L.a.G.P., (2003) Damage detection of structures based on spectral methods usingpiezoelectric materials. In: Proceeding of 4th international workshop on structural healthmonitoring, Stanford University

16. Bein T, Nuffer J (2006) Application of piezoelectric materials in transportation industry. In:Global symposium on innovative solution for the advancement of the transport industry

17. Sih GC, Michopoulos JG, Chou SC (1986) Coupled diffusion of temperature and moisture. In:Sih GC, Michopoulos JG, Chou SC (eds) Hygrothermoelasticity. Springer Netherlands:Dordrecht, pp 17–45

18. Smittakorn W (2001) A theoretical and experimental study of adaptive wood composites. Ph.D. thesis

19. Parton VZ, Kudryavtsev BA (1988) Electromagnetoelasticity: piezoelectrics and electricallyconductive solids. Gordon and Breach Science Publishers

20. Kraus JD (1984) Electromagnetics. McGraw Hill21. Chen P, Shen Y (2007) Propagation of axial shear magneto–electro-elastic waves in

piezoelectric–piezomagnetic composites with randomly distributed cylindrical inhomo-geneities. Int J Solids Struct 44(5):1511–1532

22. Eslami MR, Hetnarski RB (2009) Heat Conduction problems. In: Thermal stresses—advanced theory and applications. Springer Netherlands, Dordrecht, pp 132–218

23. Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27(3):240–253

24. Heyliger WSPR (2000) A discrete-layer model of laminated hygrothermopiezoelectric plates.Mech Compos Mater Struct 7(1):79–104

25. Aboudi J, Williams TO (2000) A coupled micro–macromechanical analysis of hygrother-moelastic composites. Int J Solids Struct 37(30):4149–4179

26. Chandrasekharaiah DS (1998) Hyperbolic thermoelasticity: a review of recent literature. ApplMech Rev 51(12):705–729

27. Reddy JN, Chin CD (1998) thermomechanical analysis of functionally graded cylinders andplates. J Therm Stresses 21(6):593–626

28. Reddy J (2011) A general nonlinear third-order theory of functionally graded plates. Int JAerosp Lightweight Struct (IJALS) 1(1)

29. Kurimoto M et al (2010) Application of functionally graded material for reducing electric fieldon electrode and spacer interface. IEEE Trans Dielectr Electr Insul 17(1):256–263

30. Pompe W et al (2003) Functionally graded materials for biomedical applications. Mater SciEng, A 362(1):40–60

31. Carrera E, Soave M (2011) Use of functionally graded material layers in a two-layeredpressure vessel. J Press Vessel Technol 133(5):051202

32. Poultangari R, Jabbari M, Eslami M (2008) Functionally graded hollow spheres undernon-axisymmetric thermo-mechanical loads. Int J Press Vessels Pip 85(5):295–305

33. Tang T, Yu W (2009) Micromechanical modeling of the multiphysical behavior of smartmaterials using the variational asymptotic method. Smart Mater Struct 18(12):125026

34. Challagulla K, Georgiades A (2011) Micromechanical analysis ofmagneto-electro-thermo-elastic composite materials with applications to multilayered struc-tures. Int J Eng Sci 49(1):85–104

35. Akbarzadeh AH, Chen ZT (2014) Thermo-magneto-electro-elastic responses of rotatinghollow cylinders. Mech Adv Mater Struct 21(1):67–80

36. Babaei M, Chen Z (2008) Analytical solution for the electromechanical behavior of a rotatingfunctionally graded piezoelectric hollow shaft. Arch Appl Mech 78(7):489–500

37. Akbarzadeh AH, Babaei MH, Chen ZT (2011) The thermo-electromagnetoelastic behavior ofa rotating functionally graded piezoelectric cylinder. Smart Mater Struct 20(6):065008

38. Akbarzadeh AH, Chen ZT (2012) Magnetoelectroelastic behavior of rotating cylinders restingon an elastic foundation under hygrothermal loading. Smart Mater Struct 21(12):125013


39. Yu J, Ma Q, Su S (2008) Wave propagation in non-homogeneous magneto-electro-elastichollow cylinders. Ultrasonics 48(8):664–677

40. Babaei M, Chen Z (2008) Exact solutions for radially polarized and magnetizedmagnetoelectroelastic rotating cylinders. Smart Mater Struct 17(2):025035

41. Youssef HM (2005) Generalized thermoelasticity of an infinite body with a cylindrical cavityand variable material properties. J Therm Stresses 28(5):521–532

42. Adams DF, Miller AK (1977) Hygrothermal microstresses in a unidirectional compositeexhibiting inelastic material behavior. J Compos Mater 11(3):285–299

43. Babaei M, Akhras G (2011) Temperature-dependent response of radially polarizedpiezoceramic cylinders to harmonic loadings. J Intell Mater Syst Struct 22(7):645–654

44. Ying J, Lü CF, Chen WQ (2008) Two-dimensional elasticity solutions for functionally gradedbeams resting on elastic foundations. Compos Struct 84(3):209–219

45. Kiani Y, Bagherizadeh E, Eslami M (2011) Thermal buckling of clamped thin rectangularFGM plates resting on Pasternak elastic foundation (Three approximate analytical solutions).ZAMM-J Appl Math Mech (Z Angew Math Mech) 91(7):581–593

46. Smittakorn W, Heyliger PR (2007) An adaptive wood composite: theory. Wood Fiber Sci 33(4):595–608

47. Hou P-F, Leung AY (2004) The transient responses of magneto-electro-elastic hollowcylinders. Smart Mater Struct 13(4):762

48. Galic D, Horgan C (2003) The stress response of radially polarized rotating piezoelectriccylinders. Trans Am Soc Mech Eng J Appl Mech 70(3):426–435

49. Shi J, Akbarzadeh AH (2019) Architected cellular piezoelectric metamaterials:Thermo-electro-mechanical properties. Acta Materialia 163:91–121

50. Kalamkarov AL, Andrianov IV, Danishevs'Kyy VV (2009) Asymptotic homogenization ofcomposite materials and structures. Appl. Mech. Rev. 62(3):669–676

References 117

Chapter 4Coupled Thermal Stresses in AdvancedSmart Materials

4.1 Functionally Graded Materials

Besides being the snack of choice of the Chinese Giant panda, the bamboo plant alsorepresents a near-perfect natural example of a functionally graded material [1]. Thistype of materials can be defined as a composite characterized by a spatially varyingmicrostructure [2]. The properties of the functionally graded material (FGM) varygradually along a given spatial axis throughout the material. This is usuallyaccomplished by continuously and gradually alternating the presence of the rein-forcement and matrix materials in creating composites, essentially using biomimicryinspired by naturally occurring examples such as bamboo [2]. Most species ofbamboo plant have hollow culms with varying structural characteristics between theinner and outer peripheries. The fibres at the outermost layer are more numerous andhave a compact, circular cross-section compared to the larger and elliptically shapedfibres at the inner layer. This variation in the microstructure leads to a tensilestrength of 160 and 45 kg/mm2 at the outer and inner peripheries, respectively [1].

The principle advantage of FGMs is the possibility of “combining” advantagesand desired properties based on the constituent materials used. For example, using ametal and a ceramic correctly in an FGM would incorporate the heat and corrosionresistance of the ceramic as well as the mechanical strength of the metal [2].Moreover, FGMs have smoothly varying material layers instead of the abrupt layerchanges of typical composites. This gradual variation leads to a reduction of stressconcentration at layer interfaces, reduces creep and failure, and increases the life ofthe material [2, 3].

In certain materials such as quartz and tourmaline, the piezoelectric effect occursnaturally. However, even when the piezoelectric effect is not naturally present, itcan be induced through an electric polarization process. Barium titanate (BaTiO3),polyvinylidene fluoride (PVDF) and other polycrystalline materials can be providedwith piezoelectric properties by excessive heating and exposure to a strong DC field(higher than 2000 V/mm). This process aligns the molecular dipoles of the material


119




https://doi.org/10.1007/978-3-030-25201-4_4

according to the direction of the applied field [2]. FGMs can be used in this contextas more efficient piezoelectric materials with less mechanical stress developedthroughout the material [2]. Typical applications of piezoelectric materials that canbe enhanced through the use of FGMs include: microelectromechanical systems,accelerometers, acoustic, pressure, and monitoring sensors, as well as precisionposition control among others [2, 3]. Furthermore, FGMs were found to haveinherent thermal characteristics due to their unique nature. By correctly choosingthe degree of variation of the composite materials, it is possible to essentiallycontrol (increase or decrease) the temperature of the material when exposed toexcessive thermal stress [4]. For example, a spacecraft re-entering earth’s atmo-sphere is subjected to a temperature gradient of approximately 1000 °C from theouter surface to the inside of the vessel. The design of an FGM with outer ceramicproperties and inner thermally conductive properties, as well as a properly selecteddegree of gradation could greatly reduce the temperature within the walls of thespacecraft and reduce the risk of failure [2, 4].

Knowing this, the intelligent design of smart graded materials can greatlyimprove performance in extreme thermal conditions and in electrical applications,benefitting the fields and industries that could potentially rely on these materials.The very nature of functionally graded materials make them inherently useful sincethey offer the property-blending benefits of typical composites without the dangersof stress concentration at layer interfaces. Their potential as smart materials tran-scends any one scientific discipline, instead blurring the lines between mechanical,electrical, and thermal applications for use in structural design, electrical machineryand aerospace technologies. Manufacturing advanced materials like FGMs usingsmart, responsive, piezoelectric materials generates a powerful tool that is not onlymore efficient, but also more useful than traditional materials.

This chapter seeks to further this potential by examining the response of FGPMsto thermal stresses. Within the context of various governing thermoelasticity the-ories, analyses are carried out in order to graphically compare the results with thosefrom previous authors. In Sect. 4.2, a generalized theory is used to quantify thebehaviour of a homogeneous piezoelectric rod under thermal stress. Section 4.3employs another generalized theory to analyze a functionally graded piezoelectriccylinder undergoing thermal shock. Finally, a functionally graded piezoelectricrod’s response to a heat source is examined through coupled, uncoupled andgeneralized theories of thermoelasticity in Sect. 4.4.

4.2 Hyperbolic Coupled Thermopiezoelectricityin One-Dimensional Rod

In this section, the dynamic response of a thermospiezoelectric rod is analyzed onthe basis of the Lord and Shulman theory of generalized thermoelasticity [5]. Therod, which is assumed to be made of a homogeneous material, is subject to a

120 4 Coupled Thermal Stresses in Advanced Smart Materials

moving heat source travelling along its length. Solutions for the displacement,temperature and electric potential are analytically obtained from three coupled,dynamic, governing different equations for the given problem [5]. The equations arefirstly solved in the Laplace domain through successive decoupling, then thetime-dependent dynamic solutions are attained through a numerical inversion in theLaplace domain. Finally, through numerical examples, the present results are jux-taposed with those previously reported in the literature to substantiate theconclusions.

In many applications, piezoelectric materials are used under conditions of hightemperature [2, 6]. For this reason, pyroelectricity, which plays a crucial role inthermopiezoelectric media and showcases the potential of piezoelectric materials togenerate electricity out of temperature changes, is incorporated in the presentanalysis.

4.2.1 Introduction

In analysis, the classical theory of thermoelasticity yields unrealistic results; thermalwave speeds are found to be infinite, and the thermal and elastic fields are inde-pendent of each other [5]. Vernotte [7] and Cattaneo [8] introduced a hyperbolicnon-Fourier heat conduction theory which included a “relaxation time” to addressthe first issue of infinite speed of heat propagation. The concept of this addition willbe explored in Sect. 4.3. Biot [9] investigated the idea of coupling the thermal andelastic fields using the first law of thermodynamics and successfully proposed theaptly named coupled thermoelasticity theory. Following these two breakthroughs,more generalized thermoelasticity theories have been developed which amend boththe thermal wave velocity and the coupling effect problems. One such theory arosefrom Lord and Shulman (L-S) [10] who used the hyperbolic heat conduction theoryto define a more generalized classical thermoelasticity theory. Using the samehyperbolic theory, Chandrasekharaiah [11] proposed the generalized ther-mopiezoelasticity, essentially extending the work of LS to a thermopiezoelectrictheory. This theory will be seen in Sect. 4.3 as well.

Using the generalized Lord-Shulman theory of thermoelasticity, the investigationof a finite thermopiezoelectric rod was performed by He et al. [12]. However, in thiscase the electric displacement in the rod was considered to be independent of time inorder to simplify the analysis. As such, the time histories of both electric displace-ment and electric potential were omitted from the research. Furthermore, theequation for entropy was stated in terms of strain, temperature, and electric fieldinstead of electric displacement. This leads to an inconsistency whereby the vari-ables used in the entropy equation differ from those used in the equations for stressand electric field [5]. When the pyroelectric constant of the material becomes large,the effect of this discrepancy is magnified, and there is also no evidence of thermalwavefronts present in the temperature distributions found. Finally, the solutionsproposed by He et al. for the displacement and temperature of the rod only contain

4.2 Hyperbolic Coupled Thermopiezoelectricity in One-Dimensional Rod 121

four integration constants, arising from two boundary conditions for temperature andtwo for displacement. This system implies that there is no mathematical option tohave the rod electroded at its ends in order to create specified voltage conditions [5].

We now investigate the problem of a thermopiezoelectric rod of finite lengthsubjected to a moving heat source. In the following analysis, the simplifications in[12] will not be included. As such, six integration constants are utilized, allowingthe ends of the rod to experience preset conditions of voltage. Results, nowincluding stress and electric displacement distributions, and thermal wavefronts, areobtained and compared graphically to those based on the simplified problem.

4.2.2 Homogeneous Rod Problem

Consider a thermopiezoelectric rod of length L, lying on the horizontal z-axissituated in a one-dimensional coordinate system, as shown in Fig. 4.1. The left endof the rod is located at the origin of the coordinate system. The boundary conditionsof the ends of the rod are as follows:

w z ¼ 0; tð Þ ¼ w0; w z ¼ L; tð Þ ¼ wL ð4:1a; bÞ

/ z ¼ 0; tð Þ ¼ /0; / z ¼ L; tð Þ ¼ /L ð4:1c; dÞ

d10T z ¼ 0; tð Þþ d20@T@z

z ¼ 0; tð Þ ¼ H0;

d1LT z ¼ L; tð Þþ d2L@T@z

z ¼ L; tð Þ ¼ HL

ð4:1e; fÞ

where w, /, T, and t are, respectively, the displacement, electric potential, absolutetemperature and time. d10, d1L, d20, d2L, w0, wL, /0, /L, H0 and HL are arbitraryconstants and @

@z stands for partial differentiation with respect to z.

L

Thermally insulated, fixed ends

The moving heat sourcez

Fig. 4.1 A thermopiezoelectric rod subjected to a moving heat source


The initial conditions are given below:

w z; t ¼ 0ð Þ ¼ 0;@w@t

z; t ¼ 0ð Þ ¼ 0 ð4:2a; bÞ

/ z; t ¼ 0ð Þ ¼ 0;@/@t

z; t ¼ 0ð Þ ¼ 0 ð4:2c; dÞ

T z; t ¼ 0ð Þ ¼ T0;@T@t

z; t ¼ 0ð Þ ¼ 0 ð4:2e; fÞ

where T0 is the initial temperature of the rod.

4.2.2.1 Fundamental and Governing Equations

The constitutive equations for linear thermopiezoelectric media are [11]:

rij ¼ Cijklekl � eijkEk � bijh ð4:3aÞ

qS ¼ qCE

T0hþ bijeij þ piEi ð4:3bÞ

Di ¼ eijkejk þ 2ij Ej þ pih ð4:3cÞ

1þ s@

@t

� �qi ¼ �Kijh;j ð4:3dÞ

where rij, S, Di and qi are stress, entropy, electric displacement and heat flux,respectively. Cijkl, eijk, bij, pi, 2ij, s, CE, Kij and q are, respectively, elastic andpiezoelectric constants, thermal moduli, pyroelectric and dielectric constants,thermal relaxation, specific heat, coefficient of thermal conductivity, and density.Here, a subscript comma denotes the partial differentiation with respect to thevariable that follows it and h is the temperature change ðh ¼ T � T0Þ.

The equations of energy and motion, as well as the Coulomb equation and linearstrain-displacement, in the absence of electric current and free charge, are:

q S;tT0 � R� �þ qi;i ¼ 0 ð4:4aÞ

rij;j þ qFi ¼ qui;tt ð4:4bÞ

Di;i ¼ 0; Ei ¼ �/;i i; j ¼ 1; 2; 3ð Þ ð4:4c; dÞ


eij ¼ 12

ui;j þ uj;i� � ð4:4eÞ

where Fi, R and ui are the external body force, heat source intensity and dis-placement components. We can see that for a problem in which the only dimensionis z, the remaining displacement and electric field components are u3 ¼ w andE3 ¼ Ez. Likewise, the only components of heat flux, stress and electric displace-ment that are non-zero are in the z direction, and any spatial differentiations are zeroexcept those with respect to z. Thus, Eqs. (4.3) and (4.4) are reduced to:

rzz ¼ c33ezz � e33E3 � b3h ð4:5aÞ

qS ¼ qCE

T0hþ b3ezz þ p3Ez ð4:5bÞ

Dz ¼ e33ezz þ 23 Ez þ p3h ð4:5cÞ

1þ s@

@t

� �qz ¼ �K3h;z ð4:5dÞ

and

q S;tT0 � R� �þ qz;z ¼ 0 ð4:6aÞ

rzz;z ¼ qwtt ð4:6bÞ

Dz;z ¼ 0; Ez ¼ �/;z ð4:6c; dÞ

ezz ¼ w;z ð4:6eÞ

in which, C3333, e333, b33, 233 and k33 are replaced by c33, e33, b3, 23 and k3 forconvenience. By using Eqs. (4.5) and (4.6d, e), Eq. (4.6a–c) can be reduced to thefollowing three governing differential equations of the stated problem:

c33w;zz þ e33/;zz � b3h;z ¼ qw;tt ð4:7aÞ

e33w;zz� 233 /;zz þ p3h;z ¼ 0 ð4:7bÞ

K3h;ij þ 1þ s@

@t

� �qR� qCEh;t � b3T0w;zt þ p3T0/;zt

� � ¼ 0 ð4:7cÞ



4.2.3.1 Solution in Laplace Domain

Using the following relations, we can normalize the governing equations in order tomake the solution procedure more convenient:

z ¼ c0g0z; w ¼ c0g0w; / ¼ 23

e33L/; R ¼ R

K3T0c20g20

ð4:8a�dÞ

t ¼ c20g0t; s ¼ c20g0s ð4:8e; fÞ

h ¼ hT0

; rzz ¼ rzzc33

; Dz ¼ Dz

e33ð4:8g�iÞ

where c0 ¼ffiffiffiffiffic33q

qand g0 ¼ qCE

K3are the propagation speed of an elastic wave in a

homogeneous, purely elastic, linearly isotropic solid, and the reciprocal of thermaldiffusivity, respectively. Using the above normalized values and after dropping theoverbars for convenience, the non-dimensional form of Eq. (4.7) is:

w;zz þ c0g0Le233

23 c33/;zz �

b3T0c33

h;z � w;tt ¼ 0 ð4:9aÞ

w;zz � c0g0L/;zz þp3T0e33

h;z ¼ 0 ð4:9bÞ

h;zz þ 1þ s @@t

� �qR� h;t � b3

qCEw;zt þ p3c0Le33

K323/;zt

� �¼ 0 ð4:9cÞ

In this problem, it is assumed that the intensity of the heat source has thefollowing form on a non-dimensional basis:

R ¼ R0d z� vtð Þ ð4:10Þ

where R0, d, and v are the non-dimensional magnitude of the moving heat source,the Dirac delta function, and the non-dimensional velocity of the heat source,respectively. In order to simplify the solution procedure, we first find the solution ofEq. (4.9) in the Laplace domain. The Laplace transform of a function f ðtÞ, in termsof its argument t, is defined by f ðsÞ as follows:

L f ðtÞ½ � ¼ f ðsÞ ¼Z10

e�stf ðtÞdt ReðsÞ[ 0ð Þ ð4:11Þ


where s is the Laplace parameter and Re is the real part of its argument. Using theintegral transform above and incorporating the initial conditions stated in Eq. (4.2),after dropping overbars for simplicity, the governing, non-dimensional equations ofthe rod in the Laplace domain are:

w;zz � a1s2wþ a2/;zz � a3h;z ¼ 0 ð4:12aÞ

w;zz � b1/;zz þ b2h;z ¼ 0 ð4:12bÞ

h;zz � s 1þ ssð Þh� d1s 1þ ssð Þw;z þ d2s 1þ ssð Þ/;z ¼ � 1þ ssð Þde�svz ð4:12cÞ

where

a1 ¼ c20qc33

; a2 ¼ c0g0Le233

233 c33; a3 ¼ b3T0

c33

b1 ¼ c0g0L; b2 ¼ p3T0e33

d1 ¼ b3qCE

; d2 ¼ p3c0e33LK3 23

; d ¼ qR0

v

ð4:13Þ

Typically, a system of ordinary differential equations with constant coefficientsis solved using its eigenvalues and eigenvectors [13]. For current problem, how-ever, one unknown can be consecutively eliminated from the equations. Thismethod is more straightforward, as seen below:

By eliminating h;z between Eqs. (4.12a) and (4.12b), we obtain the followingequations for w and /:

1þ a3b2

� �w;zz � a1s

2wþ a2 � b1a3b2

� �/;zz ¼ 0 ð4:14Þ

After solving for h;z using Eq. (4.12a) and using its derivative h;zz� �

andEq. (4.12c), we can find h in terms of other unknowns:

h ¼ 1a3s 1þ ssð Þw;zzz � d1 þ a1s

a3 1þ ssð Þ� �

w;z þ a2ass 1þ ssð Þ/;zzz þ d2/;z þ

dse�

svz

ð4:15Þ

Using the above equations, we can eliminate h in Eq. (4.12b) and obtain thesecond equation of w and /:

b2a3s 1þ ssð Þw;zzzz þ 1� b2 d1 þ a1s

a3 1þ ssð Þ� ��

w;zz

þ a2b2a3s 1þ ssð Þ/;zzzz þ b2d2 � b1ð Þ/;zz ¼

db2v

e�svz

ð4:16Þ


Similarly, / can be eliminated between Eqs. (4.14) and (4.16). Then we canobtain the final differential equation of w:

f4w;zzzz þ f2w;zz þ f0w ¼ fe�svz ð4:17Þ

where

f4 ¼ b2a3s 1þ ssð Þ 1� a2

b2 þ a3a2b2 � a3b1

� �

f2 ¼ 1� b2 d1 þ a1sa3 1þ ssð Þ

� �þ a2a1b22s

a3 a2b2 � a3b1ð Þ 1þ ssð Þ� b2d2 � b1ð Þ b2 þ a3

a2b2 � a3b1

f0 ¼ b2d2 � d1ð Þ a1b2s2

a2b2 � a3b1; f ¼ db2

v

ð4:18Þ

The solution of Eq. (4.17) can be expressed by the sum of its general andparticular solutions, as:

w ¼ Ciekiz þKe�

svz i ¼ 1; . . .; 4ð Þ ð4:19Þ

where Ci are integration constants to be found using the boundary conditions and kiare corresponding characteristic roots of Eq. (4.17):

k1 ¼ �k2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�f2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 22 � 4f4f0

p2f4

s

k3 ¼ �k4 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�f2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 22 � 4f4f0

p2f4

s ð4:20Þ

Also, we can define K as:

K ¼ f

f4 sv

� �4 þ f2 sv

� �2 þ f0ð4:21Þ

Substituting Eq. (4.19) into Eqs. (4.14) and (4.12), we can find / and h,respectively, as follows:

/ ¼ jiCiekiz þ jKe�

svz þC5zþC6 i ¼ 1; . . .; 4ð Þ ð4:22aÞ

h ¼ l1Ciekiz þ lKe�svz þ b1

b2C5 þC7 i ¼ 1; . . .; 4ð Þ ð4:22bÞ


where C5, C6, and C7 are three new constants of integration and:

j1 ¼ j2 ¼ 1a2b2 � a3b1

a1b2s2

k21� b2 � a3

!

j3 ¼ j4 ¼ 1a2b2 � a3b1

a1b2s2

k23� b2 � a3

!; j ¼ 1

a2b2 � a3b1a1b2v

2 � b2 � a3� �

l1 ¼ �l2 ¼ k1b2

b1j1 � 1ð Þ; l3 ¼ �l4 ¼ k3b2

b1j3 � 1ð Þ;

¼ svb2

1� b1jð Þð4:23Þ

Substituting Eqs. (4.19) and (4.22) into Eq. (4.12c), we can obtain anotherauxiliary equation:

d2 � b1b2

� �C5 � C7 ¼ 0 ð4:24Þ

Having solved the governing different equations for displacement, electricpotential and temperature of the rod, we can obtain equations for stress and electricdisplacement using Eqs. (4.5a, c):

rzz ¼ Ci ki 1þ jie233c0g0L23 c33

� �� b3T0

c33li

� �ekiz

þK � sv

1þ je233c0g0L23 c33

� �� b3T0

c33l

� �e�

svz

þC5e233c0g0L23 c33

� b1b2

b3T0c33

� �� b3T0

c33C7

ð4:25aÞ

Dz ¼ Ci ki 1� jic0g0Lð Þþ p3T0e33

li

� �ekiz

þK � sv1� jc0g0Lð Þþ p3T0

e33l

� �e�

svz

þC5p3T0e33

b1b2

� c0g0L

� �þ p3T0

e33C7 i ¼ 1; . . .; 4ð Þ

ð4:25bÞ

After normalizing and transforming the boundary conditions in Eq. (1) into theLaplace domain, we encounter a linear system of equations with seven unknowns,Ci i ¼ 1; . . .; 7ð Þ, as follows:


A7�7C7�1 ¼ B7�1 ð4:26Þ

The array of coefficients and the matrices A7�7 ¼ aij and B7�1 ¼bj i; j ¼ 1; . . .; 7ð Þ are given below:

a11 ¼ a12 ¼ a13 ¼ a14 ¼ 1; a15 ¼ a16 ¼ a17 ¼ 0

b1 ¼ c0g0w0

s� K

a21 ¼ j1; a22 ¼ j2; a23 ¼ j3; a24 ¼ j4; a25 ¼ 0; a26 ¼ 1; a27 ¼ 0

b2 ¼ 23

e33L/0

s� jK

a31 ¼ d10 þ d20k1c0g0ð Þl1; a32 ¼ d10 þ d20k2c0g0ð Þl2; a33 ¼ d10 þ d20k3c0g0ð Þl3;a34 ¼ d10 þ d20k4c0g0ð Þl4; a35 ¼ d10

b1b2

; a36 ¼ 0; a37 ¼ d10

b3 ¼ d20svc0g0 � d10

� �lKþ 1

sH0

T0� d10

� �a41 ¼ ek1c0g0L; a42 ¼ ek2c0g0L; a43 ¼ ek3c0g0L; a44 ¼ ek4c0g0L; a45 ¼ a46 ¼ a47 ¼ 0

b4 ¼ c0g0wL

s� Ke�

svc0g0L

a51 ¼ a52 ¼ a53 ¼ a54 ¼ 0; a55 ¼ d2 � b1b2

; a56 ¼ 0; a57 ¼ �1

b5 ¼ 0

a61 ¼ j1ek1c0g0L; a62 ¼ j2e

k2c0g0L; a63 ¼ j3ek3c0g0L; a64 ¼ j4e

k4c0g0L;

a65 ¼ c0g0L; a66 ¼ 1; a67 ¼ 0

b6 ¼ 23

e33L/L

s� jKe�

svc0g0L

a71 ¼ d10 þ d20k1c0g0ð Þl1ek1c0g0L; a72 ¼ d10 þ d20k2c0g0ð Þl2ek2c0g0L;a73 ¼ d10 þ d20k3c0g0ð Þl3ek3c0g0L; a74 ¼ d10 þ d20k4c0g0ð Þl4ek4c0g0L;a75 ¼ d1L

b1b2

; a76 ¼ 0; a77 ¼ d1L

b7 ¼ d2Lsvc0g0 � d1L

� �lKe�

svc0g0L þ 1

sHL

T0� d1L

� �

Finally, we can solve Eq. (4.26), and find the unknowns. The next step in ourprocedure is the inversion of the results from the Laplace domain to the timedomain, which is performed numerically.


4.2.3.2 Numerical Inversion of Laplace Transform

The analytical inversion of the solutions to the time domain is not straightforward.Therefore, a numerical technique is employed [14]. In this method, a function f ðsÞð Þin the Laplace domain can be inverted at discrete time points tj using the followingformula:

f ðtjÞ ¼ CðjÞ � 12Re f ðaÞf gþRe

XN�1

k¼0

AðkÞþ iBðkÞð ÞWjk

( )" #; j ¼ 0; 1; 2; . . .;N � 1

ð4:27Þ

where

AðkÞ ¼XLl¼0

Re f aþ i kþ lNð Þ 2pT

� �

BðkÞ ¼XLl¼0

Im f aþ i kþ lNð Þ 2pT

� �

CðjÞ ¼ 2TeajDt;Dt ¼ T

N

W ¼ ei2pN ð4:28Þ

In the above equations, T, N and Dt are the total time over which the numericalinversion is performed, the number of time points to which the total time is divided,and the time increment, respectively; a is an arbitrary real number larger than thereal parts of all singularities of f ðsÞ; and Im denotes the imaginary part of a complexnumber. To minimize both discretization and truncation errors, the followingconstraints are defined [14]:

5� aT � 10

50�NL� 5000ð4:29Þ

For our current solution, the above-mentioned parameters are chosen as follows:

aT ¼ 7:5

L ¼ 10

N ¼ 450

ð4:29Þ


4.2.4 Results and Discussion

Let us now use an example to quantify the results found thus far. We consider aquartz thermopiezoelectric rod with the properties listed as follows [15, 16]:

c33 ¼ 8:674� 1010 Nm2 b3 ¼ 1:16� 106 N

m2K

q ¼ 2:65� 103 kgm3CE ¼ 782 J

kg K

e33 ¼ 0:2 Cm2 23¼ 0:392� 10�10 F

m

p3 ¼ 4� 10�4 Cm2K K3 ¼ 1:4 W

mK

Other numerical values used include: R0 ¼ 10q , T0 ¼ 293, m ¼ 0:5, and s ¼ 0:05.

The non-dimensional length of the rod ðLNÞ is taken to be 1 [5]. Now, let us assumethat the rod’s ends are fixed and thermally insulated, such that no electric potentialexists at either end. Under these conditions, the parameters in Eq. (4.1) reduce tothe following:

w0 ¼ wL ¼ 0 ð4:31aÞ

d10 ¼ d1L ¼ 0; d20 ¼ d2L ¼ 1 ð4:31b; cÞ

H0 ¼ HL ¼ 0 ð4:31dÞ

/0 ¼ /L ¼ 0 ð4:31eÞ

Using these restrictions, the current solutions for displacement, temperaturechange, and stress are compared with those based on simplified assumptions in thefigures that follow. When comparing the solution in this section with the simplifiedsolution, one can note a few new and interesting phenomena.

In Fig. 4.2, the displacements along the length of the rod at non-dimensionaltime t ¼ 0:1848 are plotted using both the solution found in this section and thesimplified one. As shown, the current plot is negative and the peak displacement isslightly lower than that of the simplified solution. As can be expected, the extremaof the two solutions occur at the same location on the rod ðz ¼ 0:19Þ. Finally, it canbe seen that in the simplified solution, a significant portion of the rod (roughly0:8� z� 1) has yet to respond to the thermal disturbance while in thenon-simplified solution, the entire rod has responded.

Figure 4.3 depicts the temperature change at non-dimensional instant t ¼ 0:1848along the length of the rod. As shown, the thermal wavefront of the simplifiedsolution is still travelling towards the right end of the rod. For the solution solved inthis section, the wavefront is travelling in the opposite direction back towards theleft end, implying a higher thermal wave speed for the current solution.

In Fig. 4.4, the time history of stress at a specific location on the rod ðz ¼ 0:25Þis displayed. The main difference between the two solutions arises from the


difference in displacement, as seen in Fig. 4.2. Also shown in Fig. 4.4 is the pointin time at which the heat source exits the rod, after which both stress fields vibratefreely.

As seen in Fig. 4.5, the history of electric potential can be analyzed based on twoperiods of time; before and after the heat source finished travelling along the rod.The former is dominated by the thermal effects of the disturbance, while the latter isdominated by the coupling between electric potential and displacement. Thiscoupling leads to the fluctuation seen after the heat source exits the rod.Additionally, the positivity of the electric potential is changed along its lengthbefore the heat source exits the rod.

Fig. 4.2 Comparison of thedisplacement distributions att ¼ 0:1848 of the currentsolution [5] and simplifiedsolution using theassumptions of [12].[Reproduced from [5] withpermission from IOPPublishing]

Fig. 4.3 Comparison of thetemperature distributions att ¼ 0:1848 of the currentsolution [5] and that using theassumptions of [12].[Reproduced from [5] withpermission from IOPPublishing]


The plot of Fig. 4.6 shows that electric displacement is time dependent beforethe heat source exits the material, as was implied in the non-simplified solution [5].Afterwards, the plot remains at a constant value, since for a one-dimensionalproblem the electric displacement is expected to remain spatially uniform along thelength of the rod from Eq. (4.4c).

Given its non-simplified nature, the solution procedure in this section shows amore consistent and discrepancy-free approach to the hyperbolic theory of gener-alized thermoelasticity.

Finally, it should be noted that when the pyroelectric constant ðp3Þ decreases, theplots for both displacement and temperature change become equivalent to thesimplified approach’s plots. This relationship is demonstrated in Fig. 4.7 for dis-placement. Moreover, even when ðp3 ! 0Þ, electric potential and electric dis-placement remain non-zero and time dependent along the rod, respectively.

Fig. 4.4 Comparison of thestress history at z ¼ 0:25 ofthe current solution [5] andthat using the assumptions of[12]. [Reproduced from [5]with permission from IOPPublishing]

Fig. 4.5 Time history ofelectric potential at z ¼ 0:25.[Reproduced from [5] withpermission from IOPPublishing]


4.3 Hyperbolic Coupled Thermopiezoelectricityin Cylindrical Smart Materials

In the following section, we will analyze the transient response of a functionallygraded, radially polarized hollow cylinder under dynamic axisymmetric loadings[17]. In this case, the Chandrasekharaiah hyperbolic theory of generalizedthermo-piezoelectricity is used to simultaneously couple the displacement, tem-perature and electric fields under non-Fourier heat conduction. Moreover, allmaterial properties (except thermal relaxation time) vary gradually throughout thematerial according to a volume fraction-based rule with varying degrees ofnon-homogeneity. The Galerkin finite element method is employed in the Laplace

Fig. 4.6 Time history of theelectric displacement atz ¼ 0:25. [Reproduced from[5] with permission from IOPPublishing]

Fig. 4.7 Effect of thepyroelectric constant on thedisplacement verifying thatthe current results [5] willreduce to the simplified ones[12] when p3 ! 0.[Reproduced from [5] withpermission from IOPPublishing]


domain in order to solve the three coupled partial differential equations, after whicha numerical method of the Laplace inversion restores the time variable. This sectionemphasizes the effects of the non-homogeneity index and the thermal relaxationtime on the results.

4.3.1 Introduction

As previously stated at the beginning of this chapter, the benefits of functionallygraded materials (FGMs) include lower stress concentrations at layer interfaces andimproved residual stress distribution, among others [17]. When combined with thedesign of piezoelectric components, FGMs can improve the performance ofadvanced structures such as surface acoustic wave (SAW) sensors [18] and bimorphactuators [19]. This being said, piezoelectric devices operated at high temperaturesrequire the coupling of temperature with elastic and electric fields. In order to modelthe response of piezoelectric media when exposed to different loading situations, itis imperative to choose a proper thermo-piezoelectricity theory. In the past, classicuncoupled theories were unable to quantify certain physical phenomena such as thewave-like behaviour of temperature and the effect of strain and electric field on thetemperature distribution [20]. To account for these observations, generalized the-ories of thermo-piezoelectricity were developed such as Chandrasekharaiah’s [11]work in extending the generalized Lord and Shulman theory of thermoelasticity intoa thermo-piezoelectricity theory. Previous works in cylindrical piezoelectricstructures include Babaei and Akhras [21] modeling the response of a radiallypolarized piezoceramic cylinder to harmonic loadings. Temperature dependencewas incorporated in the analysis through material properties that fluctuated withtemperature, pyroelectricity, and thermally-dependent dimensions.

Let us now introduce the problem solved in this section. The Chandrasekharaiahtheory is used to analyze the transient response of a functionally graded piezo-electric hollow cylinder to an axisymmetric thermal shock [17]. The cylinder isassumed to be radially polarized and its materials vary in the radial direction, henceits functionally graded nature. The degree to which the properties vary is givenaccording to a volume fraction rule and different non-homogeneity indices. Thesolutions for the displacement, electric potential and temperature, and the effects ofthe non-homogeneity indices and thermal relaxation time on the solutions, aredisplayed graphically.

4.3.2 Hollow Cylinder Problem

For this problem, we consider a long, hollow cylinder whose central axis is alignedwith the z-direction of the coordinate system, as shown in Fig. 4.8 [17]. Thecylinder, with inner and outer radii a and b respectively, is polarized and graded in

4.3 Hyperbolic Coupled Thermopiezoelectricity … 135

the radial direction. On each surface, the temperature change h, electric potential /,and pressure P are specified.

The variation of the material properties follows the volume fraction-based gra-dient rule as follows:

vðrÞ ¼ va þ vbar � ab� a

� �nv ð4:32aÞ

vba ¼ vb � va ð4:32bÞ

where v is any property of the cylinder (except the thermal relaxation time, which isconstant), the subscripts a and b denote the surface at which the property is con-sidered, and nv is the non-homogeneity index of the corresponding materialproperty, v. Figure 4.9 shows how an arbitrary material property v varies radiallyaccording to Eq. (4.32) with different non-homogeneity indices. When nv ¼ 0, theproperty is constant at the vb value, as shown in Fig. 4.9.


The constitutive relations for a linear piezoelectric material and their variables,previously stated in Sect. 4.1, are repeated here for consistency [11]:

rij ¼ Cijklekl � eijkEk � bijh ð4:33aÞ

qS ¼ qcT0

hþ bijeij þ ciEi ð4:33bÞ

Fig. 4.8 An FGPM hollowcylinder and boundarycondition locations.[Reproduced from [17] withpermission from The RoyalSociety]


Di ¼ eijkejk þ 2ij Ej þ cih ð4:33cÞ

qi þ sqi;t ¼ �Kijh;j i; j; k ¼ 1; 2; 3 ð4:33dÞ

where rij, S, Di, and qi are the stress, entropy, electric displacement and heat fluxrespectively. Cijkl, eijk, bij, q, c, ci, are, respectively, the elastic and piezoelectriccoefficients, thermal moduli, density, specific heat, and pyroelectric coefficient. 2ij,s, and Kij are the dielectric coefficient, thermal relaxation time, and coefficient ofthermal conductivity. Here, a comma subscript denotes partial differentiation withrespect to what follows it. h is the temperature change with respect to the initialtemperature T0, i.e. h ¼ T � T0, while ejk and Ek represent the strain and electricfields. In the absence of electric current, free charge and body force, the equationsof energy and motion, as well as the Coulomb equation for the conservation ofelectric charge are [17]:

qS;tT0 þ qi;i ¼ 0 ð4:34aÞ

rij;j ¼ qui;tt ð4:34bÞ

Di;i ¼ 0 ð4:34cÞ

where ui are the displacement components. The linear strain-displacementrelations are:

eij ¼ 12

ui;j þ uj;i� � ð4:35Þ

Fig. 4.9 Profiles of amaterial property, v, in theradial direction; variationswith differentnon-homogeneity indices.Solid line: nv ¼ 0, dashedline: nv ¼ 1, dotted line:nv ¼ 5, dash-dotted line:nv ¼ 100. [Reproduced from[17] with permission fromThe Royal Society]


and for a quasi-stationary electric field the following relation holds:

Ei ¼ �/;i ð4:36Þ

For the current axisymmetric plane strain problem:

uz ¼ uh ¼ 0

ð�Þ;z ¼ ð�Þ;h ¼ 0ð4:37Þ

where h is the azimuthal direction and ð�Þ is an arbitrary variable. It should be statedthat a symbol h as a subscript corresponds to the azimuthal direction, while aregular lowercase h represents temperature change. Considering Eq. (4.37), thenon-zero components of strain and electric field, Eqs. (4.35) and (4.36), are:

err ¼ u;r; ehh ¼ ur

ð4:38a; bÞ

Er ¼ �/;r ð4:38cÞ

Here, ur has been replaced by u for convenience. Therefore, the entropy andnon-zero components of the stress, electric displacement and heat flux [Eq. (4.33)]are:

rrr ¼ c33u;r þ c13ruþ e33/;r � b1h; rhh ¼ c13u;r þ c11

ruþ e31/;r � b3h

ð4:39a; bÞ

rzz ¼ c13u;r þ c12ruþ e31/;r � b3h; qS ¼ b1u;r þ

b3ru� c1/;r þ

qcT0

h ð4:39c; dÞ

Dr ¼ e33u;r þ e31ru� 233 /;r þ c1h; qr þ sqr;t ¼ �K11h;r ð4:39e; fÞ

where cad ¼ cijkl i; j; k; l ¼ 1; 2; 3; a; d ¼ 1; 2; . . .; 6ð Þ, ema ¼ emij m ¼ 1; 2; . . .; 6ð Þand b1 ¼ b11, b3 ¼ b33. The notations are changed here for brevity and conve-nience. Similarly, Eq. (4.34) is reduced to the following set of equations:

rrr;r þ rrr � rhhr

¼ qu;tt ð4:40aÞ

1r

rDrð Þ;r¼ 0 ð4:40bÞ

qS;tT0 þ 1r

rqrð Þ;r¼ 0 ð4:40cÞ


We now introduce the normalized parameters as follows:

r ¼ c�g�r; u ¼ c�g�u; t ¼ c� 2g�t; s ¼ c� 2g�s

h ¼ hT0

;/ ¼ 233a

e33ab/; rij ¼ rij

c33a;Di ¼ Di

e33a

ð4:41Þ

where an overbar denotes a normalized value, c� ¼ffiffiffiffiffiffic33aqa

qrepresents the propaga-

tion speed of the elastic wave in a homogeneous, linearly elastic, isotropic medium,and g� ¼ qaca

K11ais the reciprocal of the thermal diffusivity of the inner surface of the

cylinder. By substituting Eq. (4.39) into Eq. (4.40), the normalized governingequations of the current problem are defined as follows:

c33u;rr þ c33;r þ c33r

� �u;r þ 1

rc13;r � c11

r

� �uþ te33/;rr

þ t e33;r þ 1r

e33 � e31ð Þ� �

/;r � b1T0h;r

þ �b1;r þ1r

b3 � b1ð Þ� �

T0h ¼ qc� 2u;tt

ð4:42aÞ

e33u;rr þ e33;r þ 1r

e31 þ e33ð Þ� �

u;r þ e31;rr

u� t 233 /;rr

� t 233;r þ 233

r

� �/;r þ c1T0h;r þ c1;r þ

c1r

� �T0h ¼ 0

ð4:42bÞ

� 1þ s@

@t

� �qch;t þ b1u;rt þ

b3ru;t � tc1/;rt

� �

þ g� K11;r þ K11

r

� �h;r þ g�K11h;rr ¼ 0

ð4:42cÞ

where t ¼ e33abc�g�

233aand overbars are once again omitted for convenience.


Equation (4.42) is a system of linear partial differential equations with variablecoefficients. To solve this system, we will use the steps previously mentioned in theintroduction to this section. Details of the solution procedure are presented in thefollowing sections.



The Laplace transform and its inversion are defined as follows:

f ðsÞ ¼ L f ðtÞ½ � ¼Z10

e�stf ðtÞdt

f ðtÞ ¼ L�1 f ðsÞ½ � ¼ 12pi

Zaþ i1

a�i1estf ðsÞds

ð4:43Þ

where a is an arbitrary real number greater than all real parts of the singularities off ðsÞ, s is the Laplace parameter, a tilde denotes a transformed function and i rep-resents the imaginary unit. After applying the Laplace the transform to Eq. (4.42),and considering zero initial conditions, the governing equations in the Laplacedomain become:

c33u;rr þ c33;r þ c33r

� �u;r þ 1

rc13;r � c11

r� rqc� 2s2

� �uþ te33/;rr

þ t e33;r þ 1r

e33 � e31ð Þ� �

/;r � b1T0h;r þ �b1;r þ1r

b3 � b1ð Þ� �

T0h ¼ 0

ð4:44aÞ

e33u;rr þ e33;r þ 1r

e31 þ e33ð Þ� �

u;r þ e31;rr

u� t 233 /;rr

� t 233;r þ 233

r

� �/;r þ c1T0h;r þ c1;r þ

c1r

� �T0h ¼ 0

ð4:44bÞ

�ib1u;r �ib3r

uþ itc1/;r þ g�K11h;rr þ g� K11;r þ K11

r

� �h;r � iqch ¼ 0 ð4:44cÞ

where i ¼ ð1þ ssÞs. In Eq. (4.44), tildes are omitted for convenience.

4.3.3.2 Galerkin Finite Element Method

We now make use of a finite element method to solve Eq. (4.44). In order todiscretize the governing equations, linear elements and shape functions are incor-porated in the analysis method. The shape of the elements, the local coordinates andthe shape functions are shown in Fig. 4.10 [17].


The unknowns of the problem (displacement, electric potential, and temperaturechange) can then be approximated over the defined linear elements as follows:

u;/; hð ÞðeÞ¼ Nk uk;/;k; h;k� �

; k ¼ i; j ð4:45Þ

Now, we apply the Galerkin method [22] to Eq. (4.44):

ZL0

c33u;nn þ c33;n þ c33ri þ n

� �u;n

þ 1ri þ n c13;n � c11

ri þ n � ri þ nð Þqc� 2s2� �

uþ te33/;nn

þ t e33;n þ 1ri þ n e33 � e31ð Þ

� �/;n � b1T0h;n

þ �b1;n þ 1ri þ n b3 � b1ð Þ

� �T0h

0BBBBBB@

1CCCCCCANkdn ¼ 0 ð4:46aÞ

ZL0

e33u;nn þ e33;n þ 1ri þ n e31 þ e33ð Þ

� �u;n þ e31;n

ri þ n u� t 233 /;nn

�t 233;n þ 233ri þ n

� �/;n þ c1T0h;n þ c1;n þ c1

ri þ n

� �T0h

0@

1ANkdn ¼ 0

ð4:46bÞ

ZL0

�ib1u;n � ib3

ri þ nuþ itc1/;n þ g� K11;n þ K11

ri þ n

� �h;n � iqch

� �Nkdn ¼ 0; k ¼ i; j

ð4:46cÞ

To decrease the order of the second order derivatives and obtain boundaryconditions (the stresses on the boundaries), we now perform integration by parts onall of the second order derivatives in Eq. (4.46). After the integrations and sufficientmanipulation, element matrices are found. These matrices are then assembled into asystem of linear algebraic equations which contain the variables at each node.Finally, all the unknowns can be obtained.

Fig. 4.10 The element, localcoordinate and shapefunctions used in the currentsection [17]


4.3.3.3 Numerical Inversion of the Laplace Transform

The procedure for numerical inversion used in this section is identical to thatperformed in Sect. 4.2.3 and is omitted here for brevity. The same discretizationand truncation error constraints are used as well.


In this section, we analyze the effects of both the non-homogeneity index andthermal relaxation time on the response of the cylinder to thermal shock. Allboundary conditions shown in Fig. 4.8 are assumed to be zero except thenon-dimensional temperature at the inner surface of the cylinder, which is definedas ha ¼ 104te�1000t (t is non-dimensional). The non-dimensional inner and outerradii of the cylinder are taken to be 0:01 and 0:02, respectively [17]. The innersurface of the cylinder is made of Lead Zirconate Titanate while the outer surface isCadmium Selenide. Their properties are listed in Table 4.1.

Figure 4.11a–c depicts varying non-homogeneity indices and the effects of thisvariation on the distribution of the displacement, electric potential, and temperaturechange respectively, with t ¼ 0:0011 and s ¼ 0:05. In this section, thenon-homogeneity indices are assumed to be identical for every varying material

Table 4.1 Material properties of the inner and outer surfaces of the cylinder [17, 23–25]

Properties Lead Zirconate Titanate(inner surface)

Cadmium Selenide(outer surface)

c33 (GPa) 117 83.6

c13 (GPa) 53 39.3

c11 (GPa) 126 74.1

e33 Cm2

� �23.3 0.347

e31 Cm2

� �−6.5 −0.16

b1 � 106 NKm2

� �1.41 0.551

b3 � 106 NKm2

� �1.97 0.621

233 �10�11 C2

Nm2

� �1300 9.03

c1 � 10�6 CKm2

� �−5.48 −2.94

q� 103 kgm3

� �7.87 5.68

c� 103 Jkg K

� �0.33 0.46

K11WmK

� �50 12.9


property, i.e. nv ¼ n. Three different non-homogeneity indices were used in thecalculations, namely, n ¼ 0 (Cadmium Selenide), n ¼ 1 (linear variation), and n �1 (Lead Zirconate Titanate).

The non-zero thermal relaxation time in Fig. 4.11 leads to non-Fourier heatconduction and consequently the presence of thermal wavefronts in the distribu-tions. These wavefronts travel along the thickness of the cylinder from inner surfaceto outer surface, acting as messengers of the thermal shock. Intuitively, there existundisturbed portions of the rod, located ahead of the wavefront, where the solutionsremain at their initial values. As can be seen in Fig. 4.11a, c, an increase inn shortens the length of the unresponsive portion of the cylinder, although inFig. 4.11b no wavefront is present. These contractions can be attributed to the factthat the velocity of a thermal wave, according to hyperbolic heat conduction, is

equal toffiffiffiffiffiffiffiKqcs

r. When n is increased, the average of this term increases as well.

It can also be noted that the change in the undisturbed portion of the displacementsolution between the two different material indices in Fig. 4.11a is almostinsignificant.

In Fig. 4.12a–c, the time history of the solutions at the midpoint of the cylinderare plotted for the same non-homogeneity indices seen in Fig. 4.11. It can be seenin Fig. 4.12a that the smallest amplitude of displacement occurs when n ¼ 0, while

Fig. 4.11 The effect of different non-homogeneity index on the distribution of: a the displace-ment, b the electric potential, c temperature change, t ¼ 0:0011, s ¼ 0:05. Dashed line: n ¼ 0(cadmium selenide), continuous line: n ¼ 1, dotted line: n � 1 (lead zirconate titanate).[Reproduced from [17] with permission from The Royal Society]


this same index results in the largest amplitude of electric potential as seen inFig. 4.12b. Furthermore, both the displacement and temperature change exhibit thelargest fluctuations when n ¼ 1, as depicted in Fig. 4.12a, c, respectively.

In the following two sets of figures, a non-homogeneity index of 0 is selected foranalysis. Figure 4.13a–c depicts the effects of thermal relaxation time, s, on theresults. By definition, when s ¼ 0 there exists no wavefront in the distributions ofthe solutions. As such, the thermal disturbance spreads throughout the cylinder’sthickness immediately after the inner side is subjected to the thermal shock and noportion of the cylinder is undisturbed, as shown in Fig. 4.13a, c. In these samefigures, it can also be noted that a larger thermal relaxation time leads to anexpanded unaffected portion of the cylinder where the variables remain at theirinitial values.

Figure 4.14a–d displays the time histories of the solutions under conditions ofvarying thermal relaxation time. It can be seen that the fluctuations of all thevariables are smallest when s ¼ 0. In Fig. 4.14a, the amplitude of the displacementincreases in a monotonic fashion as the thermal relaxation time becomes larger.Moreover, by increasing the thermal relaxation time, the fluctuations of temperatureare maintained for a longer time period such that the time needed for the temper-ature difference of the cylinder to return to zero is extended. The time history ofradial stress is shown in Fig. 4.14d for interest.

Fig. 4.12 The effect of the non-homogeneity index on the history of: a displacement, b electricpotential, c temperature change, Middle point, s ¼ 0:05. Dashed line: n ¼ 0 (cadmium selenide),continuous line: n ¼ 1, dotted line: n � 1 (lead zirconate titanate). [Reproduced from [17] withpermission from The Royal Society]


Fig. 4.13 The effect of the thermal relaxation time on the distribution of: a displacement,b electric potential, c temperature change, t ¼ 0:0011, n ¼ 0. Dashed line: s ¼ 0, continuous line:s ¼ 0:05, dotted line: s ¼ 0:5. [Reproduced from [17] with permission from The Royal Society]

Fig. 4.14 The effect of the thermal relaxation time on the history of: a displacement, b electricpotential, c temperature change, d radial stress, Middle point, n ¼ 0. Dashed line: s ¼ 0,continuous line: s ¼ 0:05, dotted line: s ¼ 0:5. [Reproduced from [17] with permission from TheRoyal Society]


The figures presented in this section all provide insight into the effects of varyingboth non-homogeneity and thermal relaxation time. In general, it is vital to note thefollowing [17]:

1. Of the three values of the non-homogeneity index studied, the amplitude of thedisplacement reaches its smallest value at n ¼ 0. In addition, the fluctuations ofthe displacement and temperature change reach their largest values at n ¼ 1.

2. As thermal relaxation time grows larger, the undisturbed regions of the cylindergrow in the radial direction and when s ¼ 0, all distributions experience min-imal fluctuations.

4.4 Coupled Thermopiezoelectricity in One-DimensionalFunctionally Graded Smart Materials

In this section, we once again analyze the behaviour of a rod under the effects of amoving heat source, similar to what was seen in Sect. 4.2, only now the rod isassumed to be functionally graded [3]. As such, its material properties, exceptspecific and thermal relaxation time, are assumed to vary exponentially throughoutits length. In this particular section, the governing equations of displacement,temperature and electric potential are stated in a general form than incorporates bothcoupled and uncoupled thermoelasticity theories. Within the coupled formulation,we can find that both classic and generalized thermoelasticity are considered. Onceagain using the Laplace transform, solutions are obtained in the Laplace domainand subsequently inverted into the time domain. Finally, a numerical example helpsto illustrate the analyzed problem.

4.4.1 Introduction

Introduced by Biot in 1956, classic coupled thermoelasticity helped to consider theeffects of the elastic terms in the heat equation [9]. However, since the heat equationused in this theory is parabolic in nature, the results predict an unrealistic infinitespeed for heat propagation. Consequently, generalized theories were developed toaccount for the second sound effect of the wave [26, 27]. For instance, Lord andShulman presented a new heat conduction law including a so-called relaxation time,accounting for the time required for acceleration of heat flow [10]. Following this,Green and Lindsay introduced another interpretation of generalized thermoelasticityusing two relaxation times in the constitutive equations of stress and entropy [28].

In the past, many authors have produced works on the different theories ofthermoelasticity. Tzou investigated the thermodynamic and mechanic nature ofrelaxation time in 1993 [29]. He found that this phenomenon can be interpreted as


the phase-lag between the heat flux vector and the temperature gradient. In 2001,the thermally induced displacement of a rod was analyzed by Al-Huniti et al. basedon the hyperbolic heat conduction model [30]. Using the Lord-Shulman theory ofgeneralized thermoelasticity, a one-dimensional piezoelectric rod exposed to asudden heat source was studied by He et al. The Laplace transform and thestate-space approach allowed the solving of the governing partial differentialequations [31]. In addition, Aouadi solved the problem of a coupled,two-dimensional, thermopiezoelectric, thick infinite plate by making use of thehybrid Laplace transform-finite element method. By considering both generalizedand classical coupled thermoelasticity, the author was able to quantify the wave-likeheat propagation in the plate [32].

Although previously investigated by Babaei and Chen [33], this section seeks asolution to the problem of a one-dimensional FGPM based on the Lord-Shulmantheory of thermoelasticity. In particular, a more general formulation is sought outfor understanding the coupled and uncoupled behaviour of the functionally gradedpiezoelectric medium at hand [3]. In the following section, the governing equationsof the problem are given in general forms which include the generalized coupled(Lord-Shulman and Green-Lindsay), classical coupled, and classical uncoupledthermoelasticity theories. Solutions for three unknown fields, displacement, tem-perature and electric potential, are found in the Laplace domain for coupled ther-moelasticity through the successive decoupling method. Then, a morestraightforward approach is used for the analysis of the uncoupled theory. Finally, anumerical example provides results that are plotted for different non-homogeneityindices and for the different theories.

4.4.2 The Functionally Graded Rod Problem

Let us now consider a FGPM rod of length L that is aligned with the x-axis of aone-dimensional coordinate system as shown in Fig. 4.15. The left end of the rod islocated at the origin. The rod’s ends are fixed in space, thermally insulated and havezero voltage. A mobile heat source is located at the rod’s left end at t ¼ 0 andtravels towards the right end at a constant speed t.

Fig. 4.15 FGP rod subjected to a moving heat source. [Reproduced from [3] with permissionfrom World Scientific Publishing Co., Inc.]

4.4 Coupled Thermopiezoelectricity in One-Dimensional … 147


The governing equations are written below in a general form for the analysisaccording to the coupled and uncoupled thermoelastic theories of a piezoelectricrod [26, 34, 35].

rij ¼ cijklekl � bij hþ vhð Þ � eijkEk ð4:47aÞ

Di ¼ eijkejk þ pi hþ vhð Þþ 2ij Ej ð4:47bÞ

1þ n0s@

@t

� �qi ¼ �Kijh;j ð4:47cÞ

The variables and constants in these equations have previously been defined inSect. 4.2.2 and are omitted here. New additions include a constant and tworelaxation times, represented by n0, s, and v, respectively. Given Eq. (4.47), it ispossible to obtain equations for the following thermoelasticity theories:

1. Classical coupled thermoelasticity, i.e. when s ¼ v ¼ n0 ¼ 0.2. Lord-Shulman (L-S) generalized theory with one relaxation time, i.e. when

v ¼ 0, n0 ¼ 1, s[ 0.3. Green-Lindsay (G-L) generalized theory with two relaxation times, i.e. when

n0 ¼ 0, v[ s[ 0.

For the classical coupled and L-S theories we can write:

qS ¼ bijeij þqCh0

hþ piEi ð4:48aÞ

while, for the G-L theory, the entropy equation is given as:

qS ¼ beij þ qCh0

hþ s _h� �

þ piEi ð4:48bÞ

where S, q, and C are, respectively, entropy per unit mass, density, and the specificheat. For thermopiezoelectricity in the absence of body force and volume charges,the set of governing equations, i.e. the equation of motion (a), Gaussian law (b), andenergy equation (c), is as follows:

rij;j ¼ q€ui ð4:49aÞ

Di;i ¼ 0 ð4:49bÞ

q _Sh0 � R� �þ qi;i ¼ 0 ð4:49cÞ

in which, a superposed dot represents a time derivative, while a comma subscriptdenotes partial differentiation with respect to the space variable x. In addition, the


quasi-stationary electric field equation (a) and the linear strain-displacement rela-tionship (b) are:

Ei ¼ �/;i ð4:50aÞ

eij ¼ 12

ui;j þ uj;i� � ð4:50bÞ

where / is the electric potential. For the current problem, the constitutive andgoverning equations depend purely on x and t. Knowing this, using Eq. (4.50) wecan simplify Eqs. (4.47) and (4.49):

rxx ¼ cu;x � b hþ vhð Þ � eE1 ð4:51aÞ

Dx ¼ eu;x þ p hþ vhð Þþ 2 E1 ð4:51bÞ

1þ n0s@

@t

� �qx ¼ �Kh;x ð4:51cÞ

rxx;x ¼ q€u ð4:51dÞ

Dx;x ¼ 0 ð4:51eÞ

q _Sh0 � R� �þ qx;x ¼ 0 ð4:51fÞ

where u is the displacement in the x direction and the subscripts of materialproperties have been omitted for convenience. Once again, the rod isnon-homogenous and all material properties are assumed to be functions of x,varying along the rod’s length. The specific heat and the relaxation times areassumed constant for simplicity. Using Eqs. (4.48a), (4.48b) and (4.51), the gov-erning differential equations for the aforementioned theories can be written asfollows:

c;xu;x þ cu;xx � qu� b;xh� bh;x � b;xvh� bvh;x þ e;x/;x þ e/;xx ¼ 0 ð4:52aÞ

e;xu;x þ eu;xx þ p;xhþ ph;x þ vp;xhþ vph;x � e;x/;x � e/;xx ¼ 0 ð4:52bÞ

K;xh;x þKh;xx þ q Rþ n0sRð Þ � qC hþ n0shð Þ� h0b u;x þ n0su;x

� �þ h0p /;x þ n0s/;x

� � ¼ 0ð4:52cÞ

K;xh;x þKh;xx þ q Rþ n0s _R� �� qC _hþ s€h

� �� n0sqC €hþ svh

� �� h0b _u;x þ n0s€u;x

� �þ h0p _/;x þ n0s€/;x

� �¼ 0

ð4:52dÞ


The following non-dimensional parameters are introduced to streamline thesolution procedure:

g� ¼ q0CK0

; c� ¼ffiffiffiffiffic0q0

r; x; uð Þ ¼ c�g� x; uð Þ; t; s; vð Þ ¼ c� 2g� t; s; vð Þ

h ¼ hh0

;/ ¼ 20

e0L/;R ¼ R

K0h0c� 2g� 2; rxx ¼ rxx

c0;Dx ¼ Dx

e0; l ¼ Lc�g�

ð4:53Þ

in which, K0, c0, q0, e0 and 20 are the thermal conductivity, elastic coefficients,density, piezoelectric and dielectric coefficients at x ¼ 0, respectively. The materialproperties of the rod vary exponentially along the x-axis, following the relationshown below:

v ¼ v0ekx ð4:54Þ

where v is an arbitrary material property, and k is an arbitrary non-homogeneityindex. Finally, the moving thermal disturbance is defined in the following form:

R ¼ R0d x� ttð Þ ð4:55Þ

where R0 and t are the intensity and velocity of the heat source while d is the Diracdelta function.

4.4.3 Solution Procedures


Anytime a solution is obtained in the Laplace domain, a numerical Laplaceinversion must be used to obtain the final solution in the time domain. The Laplacetransform, defined in Sect. 4.3.3, is also used in this section. The initial conditionsfor the displacement, electric potential, and temperature change are assumed to bezero [3]. The non-dimensional governing equations in the Laplace domain arewritten below:

u;xx þ a1u;x � s2uþ a2c�g�/;xx þ a1a2c

�g�/;x � a3a8h;x � a1a3a8h ¼ 0 ð4:56aÞ

u;xx þ a1u;x � Lc�g�/;xx � a1Lc�g�/;x þ a4a8h;x þ a1a4a8h ¼ 0 ð4:56bÞ

a9h;xx þ a1a9h;x � a10h� a6a10u;x þ a7a10/;x ¼ � a5a10s

e�stx ð4:56cÞ


a9h;xx þ a1a9h;x � a11h� a6a10u;x þ a7a10/;x ¼ � a5a10s

e�stx ð4:56dÞ

where

a1 ¼ kc�g�

; a2 ¼ e20L20 c0

; a3 ¼ b0h0c0

; a4 ¼ p0h0e0

; a5 ¼ q0R0

t; a6 ¼ b0

qC

a7 ¼ p0c�e0L20 K0

; a8 ¼ 1þ svð Þ; a9 ¼ 1; a10 ¼ s 1þ n0ssð Þa11 ¼ s 1þ ss 1þ n0ð Þþ s2s2n0

� �ð4:57Þ

For convenience, overbar and tilde signs have been omitted in the aboveequations.

4.4.3.2 Coupled Thermopiezoelectricity Analysis

In this section, the coupled thermopiezoelectrical response of the FGPM rod isanalyzed based on the classical and generalized L-S theories. As such, inEq. (4.56), only the energy equation based on the L-S theory is considered. Thesolution of the linear ordinary differential equations of Eq. (4.56) contains twocomponents; the particular solution and the general solutions. The former can bewritten in the following format:

up;/p; hp� � ¼ Pu;P/;Ph

� �e�

stx ð4:58Þ

Here, subscript p is used to denote the particular solution. By substitutingEq. (4.58) into Eq. (4.56), we can solve the algebraic equation and obtain:

st

� �2�a1 st � s2 I s

t

� �2�a1 st

� ��a3a8 � s

t þ a1� �

st

� �2�a1 st �J s

t

� �2�a1 st

� �a4 �a8 s

t þ a1a8� �

a6a10 st �a7a10 s

t a9 st

� �2�a1 st

� �� a10

26664

37775

Pu

P/

Ph

8<:

9=; ¼

00

� a5a10s

8<:

9=;

ð4:59Þ

where, I ¼ a2c�g� and J ¼ Lc�g�. To obtain the general solution, we successivelyeliminate / and u in the governing equations, which results in an ordinary differ-ential equation containing h. Then we find hg;x from the third equation of thehomogenous form of Eq. (4.56):

/g;x ¼ � a9a7a10

hg;xx � a1a9a7a10

hg;x þ 1a7

hg þ a6a7

ug;x ð4:60Þ


where subscript g is used to indicate the general solution. Using Eq. (4.60), thegoverning equations are reduced to:

A1ug;xx þ a1A1ug;x þA2hg;xxx þ 2a1A2hg;xx þA3hg;x þ a1A4hg ¼ 0 ð4:61aÞ

B1ug;xx þ a1B1ug;x � s2ug � B2hg;xxx � 2a1B2hg;xx þB3hg;x þ a1B4hg ¼ 0 ð4:61bÞ

where

A1 ¼ 1� Ja6a7

;A2 ¼ Ja9a7a10

;A3 ¼ Ja7

a21a9a10

� 1� �

þ a4a8;A4 ¼ � Ja7

þ a4a8

B1 ¼ 1þ Ia6a7

;B2 ¼ Ia9a7a10

;B3 ¼ Ia7

1� a21a9a7a10

� �� a3a8;B4 ¼ I

a7� a3a8

ð4:62Þ

To obtain the following equation for ug, we can multiply Eq. (4.61b) by � A1B1and

add this result with (4.61a):

ug ¼ D1hg;xxx þ 2a1D1hg;x þD2hg;x þ a1D3hg ð4:63Þ

in which,

D1 ¼ � 1s2

B2 þ B1A2

A1

� �;D2 ¼ 1

s2B3 � B1A3

A1

� �;D3 ¼ 1

s2B4 � B1A4

A1

� �ð4:64Þ

Substituting Eq. (4.63) and its derivatives into Eq. (4.61a) leads to the ordinarydifferential equation with constant coefficients below:

E1hg;xxxxx þ 3a1E1hg;xxxx þE2hg;xxx þ a1E3hg;xx þE4hg;x þ a1A4hg ¼ 0 ð4:65Þ

where

E1 ¼ A1D1;E2 ¼ A1 D2 þ 2a21D1� �þA2;E3 ¼ A1 D2 þD3ð Þþ 2A2;E4

¼ a21A1D3 þA3 ð4:66Þ

Solving Eq. (4.65) allows us to obtain the following characteristic equation:

E1f5 þ 3a1E1f

4 þE2f3 þ a1E3f

2 þE4fþ a1A4 ¼ 0 ð4:67Þ

Analytical methods have been proposed for solving quintic equations, such asusing the Hermit-Kronecker method and the Mellin method [36, 37]. However, itcan be noted that one of the characteristic roots of Eq. (4.67) is f1 ¼ �a1, and assuch we can factor this term from Eq. (4.67). Therefore, it is possible to analytically


solve the resulting fourth order algebraic equation [38] and the solution for thetemperature can be written as follows:

h x; sð Þ ¼ hg þ hp ¼ Chi efix þPhe

�stx ð4:68Þ

Substituting Eq. (4.68) into Eqs. (4.63) and (4.60) leads to the followingequations for displacement and electric potential:

u x; sð Þ ¼ CuiChi efix þPue�

stx ð4:69aÞ

/ x; sð Þ ¼ C/iChi e

fix þC0 þP/e�s

tx ð4:69bÞ

where Chi and C0 are integration constants. Cui and C/iare defined in the following

forms:

Cui ¼ D1f3i þ 2a1D1f

2i þD2fi þ a1D3 ð4:70aÞ

C/i¼ 1

a7� a9a10

fi �a1a9a10

þ 1fi

þ a6Cui

� �ð4:70bÞ

Applying the boundary condition u;/; @h@x� ��

x¼0;l¼ 0; 0; 0ð Þ on the system results

in the following system of equations which can be solved for the integrationconstants:

Cu1 Cu2 Cu3 Cu4 Cu5 Cu6Cu1e

f1l Cu2ef2l Cu3e

f3l Cu4ef4l Cu5e

f5l Cu6ef6l

C/1C/2

C/3C/4

C/5C/6

C/1ef1l C/2

ef2l C/3ef3l C/4

ef4l C/5ef5l C/6

ef6l

f1 f2 f3 f4 f5 f6f1ef1l f2ef2l f3ef3l f4ef4l f5ef5l f6ef6l

0BBBBBB@

1CCCCCCA

Ch1Ch2Ch3Ch4Ch5C0

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

�Pu

�Pue�svl

�P/

�P/e�svl

Phsv

Phsv e

�svl

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð4:71Þ

Moreover, the normalized stress and electric displacement can be found in theLaplace domain as shown below:

rx ¼ Chi fi Cui þ a2JL C/i

� �� a3 1þ vsð Þ� efix

� st Pu þ a2J

L P/� �þ a3 1þ vsð ÞPh

� ea1x ð4:72aÞ

D1 ¼ Chi fi Cui � JC/i

� �þ a4 1þ vsð Þ� efix � s

t Pu � JP/� �� a4 1þ vsð ÞPh

� � �ea1x

ð4:72bÞ


In the case where the strain rate or the time rate of change of thermal sources isrelatively low, the displacement effects are ignored in the energy equation. Then wecan solve the governing equations with a similar approach by using the followingvalues:

a6 ¼ 0; a8 ¼ 1; a9 ¼ 1; a10 ¼ s; a11 ¼ s ð4:73Þ

4.4.3.3 Uncoupled Thermopiezoelectricity Analysis

In this section, the FGPM rod is studied on the basis of classical uncoupled ther-mopiezoelectricity. The classical uncoupled theory does not take into account thecoupling effect of strain and electric potential on temperature. Despite this flaw, itremains accurate enough to successfully model many engineering applications,especially if the rate of strain and electric field and relatively small [39]. When thisoccurs, the governing equations from Eq. (4.56) are simplified as follows:

u;xx þ a1u;x � s2uþ a2c�g�/;xx þ a1a2c

�g�/;x � a3h;x � a1a3h ¼ 0 ð4:74aÞ

u;xx þ a1u;x � Lc�g�/;xx � a1Lc�g�/;x þ a4h;x þ a1a4h ¼ 0 ð4:74bÞ

h;xx þ a1h;x � sh ¼ �a5e�s

tx ð4:74cÞ

Multiplying the second equation of Eq. (4.74) by a2L and summing it to the first

equation:

F1u;xx þ a1F1u;x � s2uþF2h;x þ a1F2h ¼ 0 ð4:75aÞ

h;xx þ a1h;x � sh ¼ �a5e�s

tx ð4:75bÞ

where

F1 ¼ 1þ a2L

F2 ¼ a2a4L

� a3ð4:76Þ

It can be noted that the second equation of Eq. (4.75) is an ordinary differentialequation. As such, the solution can be obtained as:

h ¼ C1er1x þC2er2x þ hp ð4:77aÞ

hp ¼ � a5sv

� �2�a1 sv

� �� se�

svx ð4:77bÞ


r1;2 ¼ �a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 þ 4s

p2

ð4:77cÞ

Substituting the temperature relation from Eq. (4.77) into Eq. (4.75a) leads to:

F1u;xx þ a1F1u;x � s2u ¼ �F2C1 r1 þ a1ð Þer1x � F2C2 r2 þ a1ð Þer2x� F2hp � s

vþ a1

� �ð4:78Þ

The solution of Eq. (4.78) contains two parts; the general solution and theparticular solution:

u ¼ C3en1x þC4e

n2x þ up1 þ up2 þ up3 ð4:79aÞ

n1;2 ¼�a1F1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21F

21 þ 4F1s2

p2F1

ð4:79bÞ

up1 ¼ � F2C1 r1 þ a1ð ÞF1r21 þ a1F1r1 � s2

er1x ð4:79cÞ

up2 ¼ � F2C1 r2 þ a1ð ÞF1r22 þ a1F1r2 � s2

er2x ð4:79dÞ

up3 ¼ � F2Ch1 � sv þ a1

� �F1

sv

� �2�a1F1sv

� �� s2e�

svx ð4:79eÞ

Inserting the displacement and the temperature equations into Eq. (4.74b) resultsin the ordinary differential equation for electric potential written below:

Lc�g�/;xx þ a1Lc�g�/;x ¼ C1a4 r1 þ a1ð Þer1x þC2a4 r2 þ a1ð Þer2x

þ hpa4 � sv

� �þ a1

� �þC3 n21 þ a1n1

� �en1x þC4 n22 þ a1n2

� �en2x

þ up1 r21 þ a1r1� �þ up2 r22 þ a1r2

� �þ up3sv

� �2�a1

sv

� �� ð4:80Þ

By solving Eq. (4.80), we complete the solution procedure for the uncoupledthermoelasticity analysis:

/ ¼ C5e�a1x þC6 þ/p1 þ/p2 þ/p3 þ/p4 þ/p5 ð4:81aÞ

/p1 ¼C3 n21 þ a1n1� �

Lc�g�n21 þ a1Lc�g�n1en1x ð4:81bÞ


/p2 ¼C4 n22 þ a1n2� �

Lc�g�n22 þ a1Lc�g�n2en2x ð4:81cÞ

/p3 ¼up1 r21 þ a1r1� �þC1a4 r1 þ a1ð Þer1x

Lc�g�r21 þ a1Lc�g�r1ð4:81dÞ

/p4 ¼up2 r22 þ a1r2� �þC2a4 r2 þ a1ð Þer2x

Lc�g�r22 þ a1Lc�g�r2ð4:81eÞ

/p5 ¼up3

sv

� �2�a1 sv

� �� þ hpa4 � s

v þ a1� �

Lc�g� sv

� �2�a1Lc�g� sv

� � ð4:81fÞ

4.4.3.4 Numerical Inversion of the Laplace Transform

Having obtained the coupled and uncoupled solutions in the Laplace domain, wenow employ the so-called fast Laplace inverse transform [14]. Performing thisnumerical inversion will transform the results into the time domain for later anal-ysis. The form of this inversion process have been previously stated in Sect. 4.2.3and is omitted here. The only change concerns the values of the constraintparameters, which are stated below:

aT ¼ 10015

; Ln ¼ 5;N ¼ 900


The numerical example presented in this section will help to quantify the results ofthe solution procedure. Consider an FGPM rod with the properties listed inTable 4.2 [31], in which the left side is composed of Cadmium Selenide. The rod’sinitial temperature is taken to be the ambient temperature h0 ¼ 293K and theintensity of the mobile heat source is R0 ¼ 10

q0. The relaxation time incorporated in

the L-S theory has a non-dimensional value of 0.05, and the velocity of the thermaldisturbance is 0.5 [3].

For the first set of figures, the rod is assumed to be homogeneous and is analyzedbased on the generalized coupled (L-S), classical coupled, and classical uncoupledtheories. The distributions of temperature change, displacement, stress, and electricpotential at non-dimensional time t ¼ 0:2 in response to a moving heat source aredisplayed in Figs. 4.16, 4.17, 4.18 and 4.19 respectively. It is immediately noticedthat the results for classical coupled and classical uncoupled are practically identical


and are therefore indiscernible in the figures. Intuitively, the maximum temperatureoccurs at the location of the heat source at this time x ¼ tt ¼ 0:1ð Þ. On the otherhand, the maxima of displacement, stress and electric potential all occur ahead ofthis point as shown in Figs. 4.17, 4.18 and 4.19. These results are in agreement withthose found by Babei and Chen [33] analytically.

The effects of non-Fourier heat conduction can be seen in Figs. 4.16 and 4.18.For the classical coupled and classical uncoupled solutions, the temperature dis-tribution is diffusive, and as such thermal wave characteristics are not observed dueto the parabolic nature of Fourier heat conduction. Contrarily, in the generalizeddistributions, the presence of thermal wavefronts is obvious due to the finite thermalwave speed and hyperbolic heat conduction. Therefore, there are distinguishableundisturbed portions of the rod in the distributions of temperature and stress.

Table 4.2 Materialproperties of the left end ofthe rod [3, 31]

Properties Cadmium Selenide

c0 (GPa) 74.1

e0 Cm2

� �0.347

b0 � 106 NKm2

� �0.621

20 �10�11 C2

Nm2

� �9.03

p0 � 10�6 CKm2

� �–2.94

q0 � 103 kgm3

� �7.60

CE � 103 Jkg K

� �0.42

K0WmK

� �12.9

Fig. 4.16 Comparison of thetemperature distribution basedon different thermoelasticitytheories at non-dimensionaltime t ¼ 0:2. [Reproducedfrom [3] with permission fromWorld Scientific PublishingCo., Inc.]


The time history of the difference between the temperature solutions of thecoupled and uncoupled theories is depicted in Fig. 4.20. The amplitude of fluctu-ation remains constant after the heat source leaves the rod at texit ¼ 2.

Figures 4.21, 4.22 and 4.23 display the effect of the non-homogeneity index k onthe histories of displacement, temperature and electric displacement, respectively.This analysis is performed at a non-dimensional location of x ¼ 0:5 based on theclassical coupled thermoelasticity theory. The absolute mean value of fluctuation forthe displacement distribution increases as the value of k increases. It is shown inFig. 4.22 that before the heat source exits the rod, the temperature at each locationincreases monotonically. After this point, the temperature will reach its constantvalue while exhibiting small fluctuations.

Fig. 4.17 Comparison of thedisplacement distributionbased on differentthermoelasticity theories atnon-dimensional time t ¼ 0:2.[Reproduced from [3] withpermission from WorldScientific Publishing Co.,Inc.]

Fig. 4.18 Comparison of thestress distribution based ondifferent thermoelasticitytheories at non-dimensionaltime t ¼ 0:2. [Reproducedfrom [3] with permission fromWorld Scientific PublishingCo., Inc.]


In Fig. 4.23 the history of electric displacement with different non-homogeneityindices is depicted. It is clear to see that the absolute value of electric displacementincreases when k becomes larger. The distribution smoothly increases until thethermal disturbance reaches the end of the rod, at which point it remains constant.This phenomenon is consistent with the results obtained by Babei and Chen for ahomogeneous rod under L-S theory [33].

The effect of non-homogeneity index on the distribution of stress is depicted inFig. 4.24. The results are again analyzed for classical coupled thermoelasticity atnon-dimensional time t ¼ 0:2. Before it reaches its maximum, the absolute value ofstress decreases when k increases. This relationship is completely reversed after themaximum of stress occurs. Once again, these findings can also be observed for thecoupled thermoelasticity analysis based on L-S theory.

Fig. 4.19 Comparison of theelectric potential distributionbased on differentthermoelasticity theories atnon-dimensional time t ¼ 0:2.[Reproduced from [3] withpermission from WorldScientific Publishing Co.,Inc.]

Fig. 4.20 Time history of thedifference of the temperaturedistribution for coupled anduncoupled thermoelasticity atnon-dimensional time t ¼ 0:2.[Reproduced from [3] withpermission from WorldScientific Publishing Co.,Inc.]


In general, certain phenomena can be noted based on the results found in thissection. In classical coupled thermoelasticity, there are no wave fronts in the dis-tributions of temperature or stress, but they exist for the generalized L-S solutions.Additionally, the extrema of temperature and stress based on classical coupled andclassical uncoupled thermoelasticity are lower compared to those based on thegeneralized theory. For any thermoelasticity theory discussed, an increase in kresults in an increase of the absolute value of electric displacement after the thermaldisturbance has left the rod. For classical coupled thermoelasticity, an increase in kdiminishes the dynamic response of displacement, temperature and electricpotential. Nonetheless, variations in non-homogeneity have no effect on the con-stant temperature reached once the heat source exits the rod [3].

Fig. 4.21 Effect of thenon-homogeneity index onthe displacement history.[Reproduced from [3] withpermission from WorldScientific Publishing Co.,Inc.]

Fig. 4.22 Effect of thenon-homogeneity index onthe temperature history.[Reproduced from [3] withpermission from WorldScientific Publishing Co.,Inc.]


4.4.5 Introduction of Dual Phase Lag Models

As previously mentioned, the work of Green and Lindsay (G-L) introduced a newversion of generalized thermoelasticity which incorporated two relaxation times forthe relations of stress and entropy [28, 40]. More recently, Chandrasekharaiah andTzou (C-T) proposed another generalized theory that considers the dual phase lag ofheat flux and temperature gradient [41, 42]. Essentially, this theory establishes thehypothesis that the temperature gradient or the heat flux may precede one another[40]. Given that it has had close agreements with a variety of experiments on bothmicroscale and macroscale ranges, this new theory has gained prevalence [43, 44].

In the previous section, results were presented from the response of a ther-mopiezoelectric, one-dimensional, functionally graded rod on the basis of L-Sgeneralized theory. However, in certain applications such as pulsed laser heating

Fig. 4.23 Effect of thenon-homogeneity index onthe electric displacementhistory. [Reproduced from [3]with permission from WorldScientific Publishing Co.,Inc.]

Fig. 4.24 Effect of thenon-homogeneity index onthe distribution of stress.[Reproduced from [3] withpermission from WorldScientific Publishing Co.,Inc.]


and ultra-fast heating sources, there is a delay in the response of the object withrespect to the heat source. Furthermore, the delay of the heat flux may not beidentical to the delay of the temperature gradient [40]. It is therefore very beneficialto consider the dual-phase-lag model in thermoelastic analysis in order to fullyunderstand the behaviour of piezoelectric media. It is with this reasoning that thefollowing section is presented. The same problem from the beginning of Sect. 4.4 isconsidered, only this time based on C-T thermoelasticity theory with two phase lagspresent [40]. The equations, unknowns, and solution procedures are identical inboth cases, with a few exceptions that will be outlined.


The constitutive equations for the C-T generalized thermoelasticity for piezoelectricmaterials are defined below [26, 34, 35]:

rij ¼ cijklekl � bijh� eijkEk ð4:82aÞ

Di ¼ eijkejk þ pihþ 2ij Ej ð4:82bÞ

1þ s@

@tþ t22

@2

@t2

� �qi ¼ �Kij 1þ t1

@

@t

� �h;j ð4:82cÞ

qS ¼ bijeij þqCh0

hþ piEi ð4:82dÞ

The variables contained in the above equations have been previously describedin Sect. 4.3.2 and so their definitions are omitted here. The additions to theseequations include t1 and t2, which are defined according to the following approx-imations of the modification of Fourier’s law based on C-T theory [26, 45]:

t1 ¼ sh [ 0; s ¼ sq [ 0; t22 ¼ 0; sq [ sh [ 0 ð4:83aÞ

t1 ¼ sh [ 0; s ¼ sq [ 0; t22 ¼12s2q ð4:83bÞ

where sq is the heat flux phase-lag and sh is the temperature gradient phase-lag.Giving the phase lags in this structure allows us to investigate the generalizedthermoelasticity based on the Lord-Shulman (L-S) theory as well. From this pointonward, the solution procedure used to solve these equations is identical to thatfound in Sect. 4.4.3 for the coupled theories. For this reason, the process is omittedfor brevity and the results are directly shown below.


4.4.6 Results of Dual Phase Lag Model Analysis

The problem analyzed in this section is identical to that of Sect. 4.4.4. The onlymodification is the addition of the temperature gradient phase lag, which has anon-dimensional value of 0.04. Since the solution in this section was based on ageneral approximation of C-T thermoelasticity theory, it is possible to study theresponse of the rod based on L-S theory by setting t1 and t2 equal to zero. In thiscase, the problem truly is identical to the preceding one and so these results are usedin this section for comparison purposes [12]. The analysis based on C-T theory,however, is performed for both t2 ¼ 0 and t2 ¼ sqffiffi

2p in order to investigate the effects

of this parameter on the results.In Figs. 4.25, 4.26, 4.27 and 4.28, the distributions of displacement, temperature

change, stress, and electric potential are depicted based on two approximations of

Fig. 4.25 Comparison of thedisplacement distributionbased on C-T and L-Stheories. [Reproduced from[40] with permission fromSAGE Publications Ltd.]

Fig. 4.26 Comparison of thetemperature distribution basedon C-T and L-S theories.[Reproduced from [40] withpermission from SAGEPublications Ltd.]


C-T theory as well as L-S theory. The results are shown for non-dimensional timet ¼ 0:1333, thus the non-dimensional location of the heat source isx ¼ tt ¼ 0:0667. As seen previously, the maximum temperature in the rod occursat this point for L-S theory, but the same is not true for C-T. Nonetheless, it will beseen in Fig. 4.29 that as the temperature gradient phase lag decreases, the tem-perature maximum tends to occur at the location of the thermal disturbance.Figures 4.26 and 4.27 clearly show that the thermal wavefront based on C-T theorywith t2 6¼ 0 is located farther ahead than the wavefront based on the L-S theory, andfurthermore that when t2 ¼ 0, no wavefront is observed.

In the following two figures, the effect of the temperature gradient phase lagsh ¼ t1 on temperature is studied for the C-T theory with t2 6¼ 0 and t2 ¼ 0. The

Fig. 4.27 Comparison of thestress distribution based onC-T and L-S theories.[Reproduced from [40] withpermission from SAGEPublications Ltd.]

Fig. 4.28 Comparison of theelectric potential distributionbased on C-T and L-Stheories. [Reproduced from[40] with permission fromSAGE Publications Ltd.]


material is still considered homogeneous at this point and the results are shown atthe same non-dimensional time t ¼ 0:1333. It can be concluded that as the phaselag increases, the wave propagation speed increases, thus forcing the wave fronts tomove ahead farther. Moreover, the wavefronts weaken as sh decreases and theyeventually disappear at t1 ¼ 0:01, as seen in Fig. 4.29. In Fig. 4.30, no wavefrontsare observed at all due to the heat flux phase lag being equal to zero. We can alsoconclude that a decrease in t1 increases the absolute values of the extrema for thetemperature distribution in the C-T theory whether or not t2 ¼ 0.

We will now study this problem on the basis of functionally graded media.As seen below in Fig. 4.31 for the elastic constant, the non-homogeneity index kholds an exponential relationship with the material properties of the thermopiezo-electric rod.

Fig. 4.29 Effect of thephase-lag of temperaturegradient on the temperaturedistribution for C-T theorywith t2 6¼ 0. [Reproducedfrom [40] with permissionfrom SAGE PublicationsLtd.]

Fig. 4.30 Effect of thephase-lag of temperaturegradient on the temperaturedistribution for C-T theorywith t2 ¼ 0. [Reproducedfrom [40] with permissionfrom SAGE PublicationsLtd.]


In Figs. 4.32, 4.33, 4.34 and 4.35, the distributions of displacement, tempera-ture, electric potential, and stress are depicted with varying non-homogeneityindices for C-T theory with sh ¼ t1 ¼ 0:04 and t2 6¼ 0 at non-dimensional timet ¼ 0:1333. An increase in k lowers the absolute value of displacement, temperatureand electric potential and also reduces the heights of the wavefronts seen in thedistributions. The locations of the wavefronts, however, remains the same forvarying values of non-homogeneity. Finally, when k is increased, the absolute valueof stress decreases before it reaches its maximum and increases after its maximum.For C-T theory with t2 ¼ 0, as well as L-S theory, similar results can be observed.

Through the dual phase lag results presented in this section, it is possible tostudy C-T theory with different values for phase lags, as well as generalized L-S

Fig. 4.31 Effect ofnon-homogeneity index onthe distribution of elasticconstant. [Reproduced from[40] with permission fromSAGE Publications Ltd.]

Fig. 4.32 Effect ofnon-homogeneity index onthe displacement distribution.[Reproduced from [40] withpermission from SAGEPublications Ltd.]


theory. In the C-T theory, sq and sh can be interpreted as two relaxation times,whereas in L-S theory we only account for one relaxation time [40]. The resultsfound are reduced to those for coupled L-S theory when t1 ¼ t2 ¼ 0. However,using C-T theory with two phase lags sh and sq allows the consideration of the factthat the heat flux vector may precede the temperature gradient or vice versa. Inaddition, non-equilibrium thermodynamic transitions and microscope effects ofenergy exchange in high-rate heating applications are significant setbacks which areaddressed by the dual phase lag C-T theory [40]. This approach to thermopiezo-electric problems provides a multiphysical description of functionally gradedmaterials on microscopic and macroscopic scales while including other moregeneralized thermoelasticity theories.

Fig. 4.33 Effect ofnon-homogeneity index onthe temperature distribution.[Reproduced from [40] withpermission from SAGEPublications Ltd.]

Fig. 4.34 Effect ofnon-homogeneity index onthe electric potentialdistribution. [Reproducedfrom [40] with permissionfrom SAGE PublicationsLtd.]


4.5 Remarks

This chapter introduce the analysis of advanced smart materials under thermalstresses. As previously discussed, many potential applications of thermopiezo-electric structures include high-temperature conditions [2]. As such, it is importantto understand how these materials behave when exposed to thermal disturbances. Inthe first section, a homogeneous piezoelectric rod is subjected to a moving heatsource and solutions are obtained according to the Lord-Shulman theory of ther-moelasticity [5]. Section 4.3 analyze the behaviour of a functionally gradedpiezoelectric cylinder under thermal shock [17], while Sect. 4.4 revisite the rodproblem, only this time assuming non-homogeneity and three different thermoe-lasticity theories [3, 40]. Through the Laplace transform, successive elimination ofvariables, finite element method, and numerical inversion method, solutions fordifferent material variables are plotted and examined. It is found that thenon-homogeneity of the material has profound effects on the solutions, as did thegoverning theories chosen for analysis. The presence of thermal wavefronts intemperature and stress, as well as the overall behaviour of distributions dependheavily on whether the equations are coupled, uncoupled or generalized.Furthermore, the degree to which the material properties are varied change theamplitudes and shapes of the distributions drastically. These two factors representintrinsic details that determine the useful role of advanced and smart materials inthermal applications.

Fig. 4.35 Effect ofnon-homogeneity index onthe stress distribution.[Reproduced from [40] withpermission from SAGEPublications Ltd.]


References

1. Ray AK et al (2005) Bamboo—a functionally graded composite-correlation betweenmicrostructure and mechanical strength. J Mater Sci 40(19):5249–5253

2. Rubio WM et al (2010) Functionally graded piezoelectric material systems—a multiphysicsperspective, in advanced computational materials modeling. Wiley-VCH Verlag GmbH &Co. KGaA, pp 301–339

3. Akbarzadeh A, Babaei M, Chen Z (2011) Thermopiezoelectric analysis of a functionallygraded piezoelectric medium. Int J Appl Mech 3(01):47–68

4. Akbarzadeh A, Cui Y, Chen Z (2017) Thermal wave: from nonlocal continuum to moleculardynamics. RSC Adv 7(22):13623–13636

5. Babaei M, Chen Z (2008) Dynamic response of a thermopiezoelectric rod due to a movingheat source. Smart Mater Struct 18(2):025003

6. Parton V, Kudryavtsev B (1988) Electromagnetoelasticity, vol 2. Gordon and Breach SciencePublishers, New York, p 90059-0

7. Vernotte P (1958) Les paradoxes de la théorie continue de léquation de la chaleur. C R HebdSeances Acad Sci 246(22):3154–3155

8. Cattaneo C (1948) Sullaconduzione de calore. Atti del. Seminar 3:83–1019. Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27(3):

240–25310. Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech

Phys Solids 15(5):299–30911. Chandrasekharaiah D (1988) A generalized linear thermoelasticity theory for piezoelectric

media. Acta Mech 71(1):39–4912. He T, Cao L, Li S (2007) Dynamic response of a piezoelectric rod with thermal relaxation.

J Sound Vib 306(3):897–90713. Boyce WE, DiPrima RC, Haines CW (1969) Elementary differential equations and boundary

value problems, vol 9. Wiley, New York14. Durbin F (1974) Numerical inversion of Laplace transforms: an efficient improvement to

Dubner and Abate’s method. Comput J 17(4):371–37615. Wojnar R, Bytner S, Galka A (1999) Effective properties of elastic composites subject to

thermal Fields. Therm Stresses, 516. Landolt HH et al (1980) Numerical data and functional relationships in science and

technology: new series. Nuclear and particle physics. Elastic and charge exchange scatteringof elementary particles. Nucleon nucleon and kaon nucleon scattering, vol 9. Springer, Berlin

17. Babaei M, Chen Z (2009) The transient coupled thermo-piezoelectric response of afunctionally graded piezoelectric hollow cylinder to dynamic loadings. Proc R Soc London:Math Phys Eng Sci (The Royal Society)

18. Liu G et al (2003) Dispersion of waves and characteristic wave surfaces in functionally gradedpiezoelectric plates. J Sound Vib 268(1):131–147

19. Takagi K et al (2002) Design and fabrication of functionally graded PZT/Pt piezoelectricbimorph actuator. Sci Technol Adv Mater 3(2):217–224

20. Richard BH, Eslami MR (2008) Thermal stresses-advanced theory and applications. Springer,The Netherlands

21. Babaei MH, Akhras G (2011) Temperature-dependent response of radially polarizedpiezoceramic cylinders to harmonic loadings. J Intell Mater Syst Struct 22(7):645–654

22. Eslami M (2003) A first course in finite element analysis. Amirkabir University Press, Tehran23. Tsamasphyros G, Song Z (2006) The general solution for a finite thermopiezoelectric plate

containing a hole and a crack. Arch Appl Mech 76(1):1–1724. Liu W et al (2003) Noise and specific detectivity of pyroelectric detectors using lead titanate

zirconate (PZT) thin films. Microelectron Eng 66(1):785–79125. Ootao Y, Akai T, Tanigawa Y (2008) Transient piezothermoelastic analysis for a functionally

graded thermopiezoelectric hollow cylinder. J Therm Stresses 31(10):935–955

References 169

26. Hetnarski RB, Ignaczak J (1999) Generalized thermoelasticity. J Therm Stresses 22(4–5):451–476

27. Hetnarski R, Ignaczak J (2000) Nonclassical dynamical thermoelasticity. Int J Solids Struct37(1):215–224

28. Green A, Lindsay K (1972) Thermoelasticity. J Elast 2(1):1–729. Tzou DY (1993) An engineering assessment to the relaxation time in thermal wave

propagation. Int J Heat Mass Transf 36(7):1845–185130. Al-Huniti NS, Al-Nimr M, Naji M (2001) Dynamic response of a rod due to a moving heat

source under the hyperbolic heat conduction model. J Sound Vib 242(4):629–64031. He T, Tian X, Shen Y (2002) State space approach to one-dimensional thermal shock problem

for a semi-infinite piezoelectric rod. Int J Eng Sci 40(10):1081–109732. Aouadi M (2007) Generalized thermoelastic-piezoelectric problem by hybrid Laplace

transform-finite element method. Int J Comput Methods Eng Sci Mech 8(3):137–14733. Babaei M, Chen Z (2010) Transient thermopiezoelectric response of a one-dimensional

functionally graded piezoelectric medium to a moving heat source. Arch Appl Mech80(7):803–813

34. El-Karamany AS, Ezzat MA (2005) Propagation of discontinuities in thermopiezoelectric rod.J Therm Stresses 28(10):997–1030

35. Mindlin R (1961) On the equations of motion of piezoelectric crystals. Prob ContinuumMech, 282–290

36. Davis S (2009) A finite algorithm for the solution to an algebraic equation. Int J Algebra3(10):449–460

37. Mikhalkin EN (2009) Solution of fifth-degree equations. Russ Math (Iz VUZ) 53(6):15–2338. Tignol J (1987) Theory of algebraic equations. Wiley, New York39. Hetnarski RB, Eslami MR, Gladwell G (2009) Thermal stresses: advanced theory and

applications, vol 41. Springer, Berlin40. Akbarzadeh A, Babaei M, Chen Z (2011) Coupled thermopiezoelectric behaviour of a

one-dimensional functionally graded piezoelectric medium based on C-T theory. Proc InstMech Eng, Part C: J Mech Eng Sci 225(11):2537–2551

41. Chandrasekharaiah D (1998) Hyperbolic thermoelasticity: a review of recent literature. ApplMech Rev 51:705–730

42. Tzou DY (1995) The generalized lagging response in small-scale and high-rate heating. Int JHeat Mass Transf 38(17):3231–3240

43. Antaki PJ (1998) Solution for non-Fourier dual phase lag heat conduction in a semiinfiniteslab with surface heat flux. Int J Heat Mass Transf 41(14):2253–2258

44. Tzou DY (1995) Experimental support for the lagging behavior in heat propagation.J Thermophys Heat Transfer 9(4):686–693

45. Tzou D (1997) Macro- to micro-scale heat transfer: the lagging behavior. Taylor & Francis,Washington, DC


Chapter 5Thermal Fracture of AdvancedMaterials Based on Fourier HeatConduction

5.1 Introduction

In this chapter, we introduce a so-called extended displacement discontinuityapproach to deal with three-dimensional (3D) thermoelastic plane crack problems inadvanced materials. The method can be used to treat general 3D thermoelastic crackproblems in advanced materials, including 3D interface crack problems. As theinterface crack problem is more general and can be reduced to an embedded crackproblem in a single material, we will directly introduce the methodology in the firstsection for a 3D interface plane crack based on general thermoelasticity. Then, weshow that the methodology can be extended readily for advanced smart materials inthe subsequent sections, such as piezoelectric materials, electromagnetic materials,and quasi-crystals. To illustrate the application of the method, a boundary integralapproach based on the analytical results is introduced to deal with arbitrarilyshaped, 3D cracks in advanced materials. Some numerical results are presented toillustrate the interaction among different physical fields.

5.2 Extended Displacement Discontinuity Methodand Fundamental Solutions for Thermoelastic CrackProblems

Due to the increasing use of composite materials in thermomechanical environ-ments, the study on the fracture behavior of interfacial cracks is of great importancein engineering and has attracted much attention. As the failure of composite lami-nates is dominated by the development of interface cracks, understanding theinterface fracture behavior is of great importance. Sih [1] (1962) pointed out that inhomogeneous elastic media, the thermal stress near the crack tip has the classicalsingularity r−1/2 as mechanical stresses. Chen and Ting [2] pointed out that the


171




https://doi.org/10.1007/978-3-030-25201-4_5

temperature is proportional to r1/2, while temperature gradient and stresses to r−1/2

near the crack tip for an insulated crack. The discontinuities of the material prop-erties and geometries leads to a much complex stress field around the interface cracktip with an oscillatory singularity, r−1/2+e, with e being the bi-material constant evenunder uniaxial tensile loading about the crack. For two dimensional (2D) problems,Brown and Erdogan [3] initially studied an insulated, Griffith interfacial crack underuniform heat flow and obtained the stress fields. Herrmann et al. [4] compared theexperimental results with numerical results of thermal cracking in dissimilar mate-rials. Herrmann and Grebner [5] studied a curved, thermal crack problem in a brittle,two-phase, compound material and built a closed form solution for the stress field.Martin-Moran et al. [6] and Barber and Comninou [7] studied a penny-shapedinterface crack subjected to a heat flow with either perfect or imperfect contact andcompared the difference between the two contact conditions. Later, Takakuda et al.[8] used the complex function method to solve an external interface crack subjectedto a uniform temperature change or heat flow, and obtained the distributions ofdisplacements and stresses on the interface. Similar work on interface crack prob-lems can aslo be found in [9–16]. Contact crack faces of interface cracks in ther-momechanical analysis may reflect actual crack face boundary conditions. Thermalstress analysis of interface cracks based on contact zone models can be found in[17–20]. Ratnesh and Chandra [21] employed the weight function method to analyzea 2D interface and found that the general expression of the stress field for theinterface crack is in the same form as that of the homogeneous one. Pant et al. [22]extended the element free, Galerkin method and employed jump function to solveinterfacial crack problems in bi-materials. Khandelwal and Chandra [23] utilizedbody analogy method to analyze an interfacial crack subjected to thermal loads andobtained the analytical solution by computing the thermal weight function, withwhich the stress intensity factors are computed as well. Ma et al. [24] studied theZener-Stroh model of an interface crack subjected to a uniform temperature shift,and evaluated the interface defect tolerant size, which can be used to assess theinterface integrity and reliability under thermal loading.

For 3D cases, Bregman and Kassir [25] employed the Muskhelishvili’s method[26] to study a penny-shaped, interface crack subjected to a uniform heat flow andgot the stress intensity factors and energy release rate. Andrzej and Stanislaw [27]used the potential theory method to study a plane crack on an interface in amicroperiodic, two-layered composite under a uniform, vertical heat flow. Johnsonand Qu [28] extended the interaction integral method to analyze curvilinear cracksin a bimaterial interface under a non-uniform, temperature distribution and obtainedthe induced stress intensity factors. Nomura et al. [29] developed a numericalmethod using a path-independent, H-integral to analyze the singular stress field of a3D interface corner between anisotropic bimaterials subjected to thermal stresses.Guo et al. [30] investigated a plane crack problem of inhomogeneous materials withinterfaces subjected to thermal loading using a modified, interaction energy integralmethod, and obtained the thermal stress intensity factors. Li et al. [31] used theweight function method to study a 3D, interface crack in a bi-material undercombined, thermomechanical loading.

172 5 Thermal Fracture of Advanced Materials Based on Fourier Heat …

The displacement discontinuity boundary integral equation method proves to bevery efficient in solving crack problems as it grasps the intrinsic characteristics ofcrack problems that physical fields are discontinuous across crack faces [32]. Thismethod has also been extended to solve interface crack problems in elastic media[33–35], piezoelectric media [36] and magnetoelectroelastic media [37, 38].

Motivated by the current research on interface crack problems, we developed thedisplacement and temperature discontinuity, boundary hyper-singular,integral-differential equation method for interface cracks in dissimilar, isotropic,thermal elastic bi-materials [39].

The displacement discontinuity method was first proposed by Tang et al. [33] todeal with planar crack problems in a three dimensional (3D) solid. It provides anefficient way to approach the complicated, 3D crack problems via a simple,numerical algorithm.

In the absence of body forces, the governing equations for a 3D, homogeneousthermoelastic medium in a steady state are [1]

rij;j ¼ 0 ð5:1aÞ

hi;i ¼ 0 ð5:1bÞ

where rij and hi are the stress and heat flux, respectively, i; j ¼ 1; 2; 3 ori; j ¼ x; y; z, and the index i or j after the comma denotes differentiation with respectto the coordinate.

In the Cartesian coordinates x; y; zð Þ and cylindrical coordinates r;/; zð Þ, theconstitutive equations are expressed, respectively, in the form

rx ¼ 2l@u@x

þ k@u@x

þ @v@y

þ @w@z

� �� E1� 2m

ah;

ry ¼ 2l@v@y

þ k@u@x

þ @v@y

þ @w@z

� �� E1� 2m

ah;

rz ¼ 2l@w@z

þ k@u@x

þ @v@y

þ @w@z

� �� E1� 2m

ah;

sxy ¼ l@u@y

þ @v@x

� �; syz ¼ l

@w@y

þ @v@z

� �; szx ¼ l

@w@x

þ @u@z

� �;

hx ¼ �b@h@x

; hy ¼ �b@h@y

; hz ¼ �b@h@z

;

ð5:2aÞ

5.2 Extended Displacement Discontinuity Method and Fundamental … 173

rr ¼ kþ 2lð Þ @ur@r

þ k1r@u/@/

þ urr

� �þ k

@w@z

� E1� 2m

ah;

r/ ¼ k@ur@r

þ kþ 2lð Þ 1r@u/@/

þ urr

� �þ k

@w@z

� E1� 2m

ah;

rz ¼ k@ur@r

þ kþ 2lð Þ @w@z

þ k1r@u/@/

þ urr

� �� E1� 2m

ah;

rr/ ¼ l1r@ur@/

þ @u/@r

� u/r

� �; r/z ¼ l

1r@w@/

þ @u/@z

� �;

rzr ¼ l@w@r

þ @ur@z

� �;

ð5:2bÞ

where u, v, and w are displacements, rij is stress, and h is temperature change(hereafter it is referred to as “temperature” for simplicity) with h ¼ 0 correspondingto the free stress state; E; m; a, and b are the elastic modulus, Poisson’s ratio,coefficient of linear thermal expansion, and coefficient of thermal conductivity,respectively, and k and l are the Lame constants, which are expressed in thefollowing forms:

k ¼ Ev1þ vð Þ 1� 2vð Þ ; l ¼ E

2 1þ vð Þ ð5:3Þ

5.2.1 Fundamental Solutions for Unit Point Loadingon a Penny-Shaped Interface Crack

In order to build the solution for general loading and crack geometry, solutions fordisplacement and temperature discontinuities can be constructed first as funda-mental solutions, which can then be used as Green’s functions for general loadingand interface crack geometry.

Consider a penny-shaped crack with radius a lying at the interface of two bondeddissimilar materials as illustrated in Fig. 5.1. The crack is located in the plane xoy,and the two bonded solids are assumed to occupy the upper and lower half-space,respectively. Then the relation between the Cartesian and cylindrical coordinatescan be expressed as

x ¼ r cos/;y ¼ r sin/;R2 ¼ r2 þ z2 ¼ x2 þ y2 þ z2:

8<: ð5:4Þ

According to Zhao et al. [40] and Zhao and Liu [41], if the radius of the cracka approaches zero, one can obtain the fundamental solutions corresponding to a


unit, concentrated temperature discontinuity and displacement discontinuity satis-fying the governing equations of thermoelasticity and the following conditions

lima!0

ZS

uk k; vk k; wk k; hk kf gdS ¼ 1; 0; 0; 0f g; ð5:5aÞ

lima!0

ZS

uk k; vk k; wk k; hk kf gdS ¼ 0; 1; 0; 0f g; ð5:5bÞ

lima!0

ZS

uk k; vk k; wk k; hk kf gdS ¼ 0; 0; 1; 0f g; ð5:5cÞ

lima!0

ZS

uk k; vk k; wk k; hk kf gdS ¼ 0; 0; 0; 1f g: ð5:5dÞ

where uk k; vk k; wk k; and hk k are the elastic displacement and temperature discon-tinuities across the crack face. In the following sections, we present the fundamentalsolutions for each unit point loading case, which would be readily used as weightfunctions for general loading cases of arbitrary crack shapes. It is worth mentioningthat as the point solutions are independent of the original shape of the penny-shapedcrack, these solutions can be used to build the solutions for any interface crack ofarbitrary shape under various loading conditions. We will show a few exampleslater, where fundamental solutions have been extended and used to build analyticalsolutions for interface crack problems in piezoelectric, piezoelectromagnetic, andquasi-crystalline materials.

Interface

z

x

yoa

r

S+

S

S -

Fig. 5.1 A penny-shaped crack with radius of a lying in the interface plane [39]


5.2.1.1 Solution for Unit-Point Displacement Discontinuityin the Z-Direction of the Crack

In this case, the boundary condition in Eq. (5.5c) can be rewritten as:

uðn; gÞk k ¼ 0; vðn; gÞk k ¼ 0; wðn; gÞk k ¼ dðn; gÞ; hðn; gÞk k ¼ 0: ð5:6Þ

where d(n,η) is the Dirac delta function. This is a non-torsional axisymmetricproblem. Introducing the potential functions proposed by Hou et al. [42], and therelatively completed non-torsional axisymmetric general solution around the z-axisis expressed in the following form:

2lur ¼ @w1

@rþ z

@w2

@r; ð5:7aÞ

2lw ¼ @w1

@z� 3� 4vð Þw2 þ z

@w2

@zþ 4 1� vð Þw3; ð5:7bÞ

rz ¼ @2w1

@z2� 2 1� vð Þ @w2

@zþ z

@2w2

@z2þ 2 1� vð Þ @w3

@z; ð5:7cÞ

rzr ¼ @2w1

@r@z� 1� 2vð Þ @w2

@rþ z

@2w2

@r@zþ 2 1� vð Þ @w3

@r; ð5:7dÞ

2lC

h ¼ @w3

@z; ð5:7eÞ

hz ¼ � bC2l

@2w3

@z2; ð5:7fÞ

where C ¼ 2 1�vð Þa 1þ vð Þ.

Using the zero-order Hankel transformation technique, the potential functionscan be expressed as

wþ1 ¼

Z10

nA1e�nzJ0 nrð Þdn; ð5:8aÞ

w�1 ¼

Z10

nA2enzJ0 nrð Þdn; ð5:8bÞ


wþ2 ¼

Z10

nB1e�nzJ0 nrð Þdn; ð5:8cÞ

w�2 ¼

Z10

nB2enzJ0 nrð Þdn; ð5:8dÞ

wþ3 ¼

Z10

nC1e�nzJ0 nrð Þdn; ð5:8eÞ

w�3 ¼

Z10

nC2enzJ0 nrð Þdn; ð5:8fÞ

where the superscript “+” and “_” denote the upper and lower domain, respectively.When the unit displacement discontinuity in z-direction is applied on the interfacialcrack, the corresponding boundary conditions are given in the cylindrical coordi-nate system as

uþr r; 0ð Þ � u�r r; 0ð Þ ¼ 0;

wþ r; 0ð Þ � w� r; 0ð Þ ¼ d rð Þ;hþ r; 0ð Þ � h� r; 0ð Þ ¼ 0;

8<:

rþz r; 0ð Þ � r�z r; 0ð Þ ¼ 0;

rþzr r; 0ð Þ � r�zr r; 0ð Þ ¼ 0;

hþz r; 0ð Þ � h�z r; 0ð Þ ¼ 0:

8><>: ð5:9Þ

where 0� r� a belongs to the crack region. After inserting Eq. (5.8) into Eq. (5.9),one can obtain

A1l1¼ A2

l2;

12l1

�nA1 � 3� 4v1ð ÞB1 þ 4 1� v1ð ÞC1½ �� 1

2l2nA2 � 3� 4v2ð ÞB2 þ 4 1� v2ð ÞC2½ � ¼ 1

2p;

� C1l1C1 ¼ C2

l2C2;

8>>><>>>:

ð5:10aÞ

nA1 þ 2 1� v1ð ÞB1 � 2 1� v1ð ÞC1 ¼ nA2 � 2 1� v2ð ÞB2 þ 2 1� v2ð ÞC2;nA1 þ 1� 2v1ð ÞB1 � 2 1� v1ð ÞC1 ¼ �nA2 þ 1� 2v2ð ÞB2 � 2 1� v2ð ÞC2;C1b1l1

C1 ¼ C2b2l2

C2;

8<:

ð5:10bÞ

where the subscripts “1” and “2” denote the upper and lower domains, respectively.Solving these six equations one can get


C1 ¼ C2 ¼ 0 ð5:11aÞ

A1 ¼ � l1 2� 3v1 � 3v2 þ 4v1v2ð Þ3l2 þ l1 � 4l2v1ð Þ �l2 � 3l1 þ 4l1v2ð Þ

2p1n¼ A�

11n

ð5:11bÞ

A2 ¼ � l2 2� 3v1 � 3v2 þ 4v1v2ð Þ�3l2 � l1 þ 4l2v1ð Þ l2 þ 3l1 � 4l1v2ð Þ

2p1n¼ A�

21n

ð5:11cÞ

B1 ¼ l1l22 �l1 � 3l2 þ 4l2v1ð Þ

2p¼ � l1l2

p l1 þ k1l2ð Þ ð5:11dÞ

B2 ¼ l1l22 l2 þ 3l1 � 4l1v2ð Þ

2p¼ l1l2

p l2 þ l1k2ð Þ ð5:11eÞ

where ka ¼ 3� 4va, A�1, A

�2, B1 and B2 are all constants, and the potential functions

are obtained as:

wþ1 ¼ A�

11R

ð5:12aÞ

w�1 ¼ A�

21R

ð5:12bÞ

wþ2 ¼ B1

zR3 ð5:12cÞ

w�2 ¼ �B2

zR3 ð5:12dÞ

wþ3 ¼ 0 ð5:12eÞ

w�3 ¼ 0 ð5:12fÞ

where R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2

p. Then the corresponding displacements and stresses are

obtained, for example, the stresses of an arbitrary point in the upper domain areobtained as:

rþz ¼ l1l2

2p l1 þ k1l2ð Þ 1þ g2ð Þ 1R3 þ 3 5� g2ð Þ z

2

R5 � 30z4

R7

� �ð5:13aÞ

rþzr ¼ l1l2

2p l1 þ k1l2ð Þ 3 3� g2ð Þ rzR5 � 30

rz3

R7

� �ð5:13bÞ


and the forms of the stresses in the Cartesian coordinates are:

rþz ¼ l1l2

2p l1 þ k1l2ð Þ 1þ g2ð Þ 1R3 þ 3 5� g2ð Þ z

2

R5 � 30z4

R7

� �ð5:14aÞ

rþzx ¼ l1l2

2p l1 þ k1l2ð Þ 3 3� g2ð Þ xzR5 � 30

xz3

R7

� �ð5:14bÞ

rþyz ¼ l1l2

2p l1 þ k1l2ð Þ 3 3� g2ð Þ yzR5 � 30

yz3

R7

� �ð5:14cÞ

where g2 ¼ l1 þ l2k1l2 þ l1k2

.

5.2.1.2 Unit Point Temperature Discontinuity of the Crack

The boundary condition in Eq. (5.5d) can be rewritten as

uðn; gÞk k ¼ 0; vðn; gÞk k ¼ 0; wðn; gÞk k ¼ 0; hðn; gÞk k ¼ dðn; gÞ: ð5:15Þ

It is also a non-torsional axisymmetric problem. We can use the same form ofpotential functions in Eq. (5.7) and adopt the same zero-order Hankel transfor-mation in Eq. (5.8). When the unit temperature discontinuity is applied on theinterface crack, the corresponding boundary conditions are as follows

uþr r; 0ð Þ � u�r r; 0ð Þ ¼ 0

wþ r; 0ð Þ � w� r; 0ð Þ ¼ 0hþ r; 0ð Þ � h� r; 0ð Þ ¼ d rð Þ

8<:

rþz r; 0ð Þ � r�z r; 0ð Þ ¼ 0

rþzr r; 0ð Þ � r�zr r; 0ð Þ ¼ 0

hþz r; 0ð Þ � h�z r; 0ð Þ ¼ 0

8><>: ð5:16Þ

thus we can obtain

A1l1¼ A2

l21l1

�nA1 � 3� 4v1ð ÞB1 þ 4 1� v1ð ÞC1½ � ¼ 1l2

nA2 � 3� 4v2ð ÞB2 þ 4 1� v2ð ÞC2½ �� C1

2l1C1 � C2

2l2C2 ¼ 1

2p1n

8><>:

ð5:17aÞ

nA1 þ 2 1� v1ð ÞB1 � 2 1� v1ð ÞC1 ¼ nA2 � 2 1� v2ð ÞB2 þ 2 1� v2ð ÞC2

nA1 þ 1� 2v1ð ÞB1 � 2 1� v1ð ÞC1 ¼ �nA2 þ 1� 2v2ð ÞB2 � 2 1� v2ð ÞC2C1b1l1

C1 ¼ C2b2l2

C2

8<:

ð5:17bÞ


Solving these six equations, we can get

A1 ¼ v1 � 1ð ÞC2b2l1 l2 þ k2l1ð Þ � v2 � 1ð ÞC1b1l2 l1 þ k1l2ð ÞpC1C2 b1 þ b2ð Þ l1 þ k1l2ð Þ l2 þ k2l1ð Þ

2l1n2

¼ A�11

n2

ð5:18aÞ

A2 ¼ v1 � 1ð ÞC2b2l1 l2 þ k2l1ð Þ � v2 � 1ð ÞC1b1l2 l1 þ k1l2ð ÞpC1C2 b1 þ b2ð Þ l1 þ k1l2ð Þ l2 þ k2l1ð Þ

2l2n2

¼ A�21

n2

ð5:18bÞ

B1 ¼ 4l1l2 v1 � 1ð Þb2pC1 b1 þ b2ð Þ l1 þ k1l2ð Þ

1n¼ B�

11n

ð5:18cÞ

B2 ¼ 4l1l2 v2 � 1ð Þb1pC2 b1 þ b2ð Þ l2 þ k2l1ð Þ

1n¼ B�

21n

ð5:18dÞ

C1 ¼ � l1b2pC1 b1 þ b2ð Þ

1n¼ C�

11n

ð5:18eÞ

C2 ¼ � l2b1pC2 b1 þ b2ð Þ

1n¼ C�

21n

ð5:18fÞ

where A�1, A

�2, B

�1, B

�2, C

�1 and C�

2 are constants, and the potential functions aredetermined as:

wþ1 ¼ �A�

1 ln Rþ zð Þ ð5:19aÞ

w�1 ¼ �A�

2 ln R� zð Þ ð5:19bÞ

wþ2 ¼ B�

11R

ð5:19cÞ

w�2 ¼ B�

21R

ð5:19dÞ

wþ3 ¼ C�

11R

ð5:19eÞ

w�3 ¼ C�

21R

ð5:19fÞ

Furthermore, the corresponding displacements, stresses, temperature and heatflux caused by the unit temperature discontinuity can be obtained. For example, thestresses and heat flux of an arbitrary point in the upper domain are given as:


rþz ¼ l1l2

2p l1 þ k1l2ð Þ2 a2 1þ v2ð Þb1g2 þ a1 1þ v1ð Þb2½ �

b1 þ b2

zR3 �

12a1 1þ v1ð Þb2b1 þ b2

z3

R5

� �ð5:20aÞ

rþzr ¼ l1l2


b1 þ b2

rR3 �


rz2

R5

� �ð5:20bÞ

hþz ¼ � b1b2

2p b1 þ b2ð Þ1R3 �

3z2

R5

� �ð5:20cÞ

In terms of the Cartesian coordinates, they are:

rþz ¼ l1l2


b1 þ b2

zR3 �


z3

R5

� �ð5:21aÞ

rþzx ¼ l1l2


b1 þ b2

xR3 �


xz2

R5

� �ð5:21bÞ

rþyz ¼ l1l2


b1 þ b2

yR3 �


yz2

R5

� �ð5:21cÞ

hþz ¼ � b1b2

2p b1 þ b2ð Þ1R3 �

3z2

R5

� �ð5:21dÞ

5.2.1.3 Unit-Point Displacement Discontinuity in the y-Directionof the Crack

The boundary condition in Eq. (5.5b) can be rewritten as:

uðn; gÞk k ¼ 0; vðn; gÞk k ¼ dðn; gÞ; wðn; gÞk k ¼ 0; hðn; gÞk k ¼ 0: ð5:22Þ

Obviously this problem is no longer non-torsional axisymmetric, which requiresa complete, general solution. The relatively completed, general solution around thez-axis is expressed in the following form [42]:


2lur ¼ � @w0

r@/þ @w1

@rþ z

@w2

@r; ð5:23aÞ

2lu/ ¼ @w0

@rþ @w1

r@/þ z

@w2

r@/; ð5:23bÞ

2lw ¼ @w1

@z� 3� 4vð Þw2 þ z

@w2

@zþ 4 1� vð Þw3; ð5:23cÞ

rz ¼ @2w1

@z2� 2 1� vð Þ @w2

@zþ z

@2w2

@z2þ 2 1� vð Þ @w3

@z; ð5:23dÞ

rzr ¼ � 12r

@2w0

@/@zþ @2w1

@r@z� 1� 2vð Þ @w2

@rþ z

@2w2

@r@zþ 2 1� vð Þ @w3

@r; ð5:23eÞ

rz/ ¼ 12@2w0

@r@zþ 1

r@2w1

@/@z� 1� 2vð Þ 1

r@w2

@/þ z

r@2w2

@/@zþ 2 1� vð Þ 1

r@w3

@/; ð5:23fÞ

2lC

h ¼ @w3

@z; ð5:23gÞ

hz ¼ � bC2l

@2w3

@z2; ð5:23hÞ

When the unit displacement discontinuity in y-direction is applied on theinterfacial crack, the corresponding boundary conditions are:

uþr r; 0ð Þ � u�r r; 0ð Þ ¼ d rð Þ sin/;

uþ/ r; 0ð Þ � u�/ r; 0ð Þ ¼ d rð Þ cos/;

wþ r; 0ð Þ � w� r; 0ð Þ ¼ 0;hþ r; 0ð Þ � h� r; 0ð Þ ¼ 0;

8>><>>:

rþz r; 0ð Þ � r�z r; 0ð Þ ¼ 0;

rþzr r; 0ð Þ � r�zr r; 0ð Þ ¼ 0;

rþz/ r; 0ð Þ � r�z/ r; 0ð Þ ¼ 0;

hþz r; 0ð Þ � h�z r; 0ð Þ ¼ 0:

8>>>><>>>>:

ð5:24Þ

According to the boundary conditions, the first-order Hankel transformation isintroduced, and the corresponding potential functions are:

wþ1 ¼

Z10

nA1e�nzJ1 nrð Þdn � sin/; ð5:25aÞ

w�1 ¼

Z10

nA2enzJ1 nrð Þdn � sin/; ð5:25bÞ


wþ2 ¼

Z10

nB1e�nzJ1 nrð Þdn � sin/; ð5:25cÞ

w�2 ¼

Z10

nB2enzJ1 nrð Þdn � sin/; ð5:25dÞ

wþ3 ¼

Z10

nC1e�nzJ1 nrð Þdn � sin/; ð5:25eÞ

w�3 ¼

Z10

nC2enzJ1 nrð Þdn � sin/; ð5:25fÞ

wþ0 ¼

Z10

nD1e�nzJ1 nrð Þdn � cos/; ð5:25gÞ

w�0 ¼

Z10

nD2enzJ1 nrð Þdn � cos/; ð5:25hÞ

where superscripts “+” and “_” denote the upper and lower domains, respectively.Substituting Eq. (5.25) into the boundary conditions in Eq. (5.24) yields:

D1�A1l1

� D2�A2l2

¼ 0D1 þA12l1

� D2 þA22l2

¼ 1pn

1l1

�nA1 � 3� 4v1ð ÞB1 þ 4 1� v1ð ÞC1½ � ¼ 1l2

nA2 � 3� 4v2ð ÞB2 þ 4 1� v2ð ÞC2½ �� C1

l1C1 ¼ C2

l2C2

8>>>><>>>>:

ð5:26aÞ

nA1 þ 2 1� v1ð ÞB1 � 2 1� v1ð ÞC1 ¼ nA2 � 2 1� v2ð ÞB2 þ 2 1� v2ð ÞC2;

�n D1þ 2A1ð Þ � 2 1� 2v1ð ÞB1þ 4 1� v1ð ÞC1 ¼ n D2þ 2A2ð Þ � 2 1� 2v2ð ÞB2 þ 4 1� v2ð ÞC2;

�n D1þ 2A1ð Þþ 2 1� 2v1ð ÞB1 � 4 1� v1ð ÞC1 ¼ n D2 � 2A2ð Þþ 2 1� 2v2ð ÞB2 � 4 1� v2ð ÞC2;

C1b1l1

C1 ¼ C2b2l2

C2;

8>>>>><>>>>>:

ð5:26bÞ

where subscripts “1” and “2” denote the upper and lower domains, respectively.Solving Eq. (5.26), one can get


A1 ¼ � l2 �3þ 4v1ð Þþ l1 �5þ 6v1 þ 6v2 � 8v1v2ð Þl1 þ l2k1ð Þ l2 þ l1k2ð Þ

1pn

¼ A�11n; ð5:27aÞ

A2 ¼ � l1 3� 4v2ð Þþ l2 5� 6v2 � 6v2 þ 8v1v2ð Þl1 þ l2k1ð Þ l2 þ l1k2ð Þ

1pn

¼ A�21n; ð5:27bÞ

B1 ¼ l1l2�l2 � 3l1 þ 4l1v2ð Þ

1p¼ � l1l2

p l2 þ l1k2ð Þ ; ð5:27cÞ

B2 ¼ l1l2�l1 � 3l2 þ 4l2v1ð Þ

1p¼ � l1l2

p l1 þ l2k1ð Þ ; ð5:27dÞ

C1 ¼ C2 ¼ 0; ð5:27eÞ

D1 ¼ � l1l2l1 þ l2

� 1pn

¼ D�11n; ð5:27fÞ

D2 ¼ l1l2l1 þ l2

� 1pn

¼ D�21n; ð5:27gÞ

where ka ¼ 3� 4va, A�1, A

�2, D

�1, D

�2, B1 and B2 are all constants, and the potential

functions are obtained as:

wþ0 ¼ D�

11r

1� zR

cos/; ð5:28aÞ

w�0 ¼ D�

21r

1þ zR

cos/; ð5:28bÞ

wþ1 ¼ A�

11r

1� zR

sin/; ð5:28cÞ

w�1 ¼ A�

21r

1þ zR

sin/; ð5:28dÞ

wþ2 ¼ B1

rR3 sin/; ð5:28eÞ

w�2 ¼ �B2

rR3 sin/; ð5:28fÞ

wþ3 ¼ 0; ð5:28gÞ

w�3 ¼ 0; ð5:28hÞ


where R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2

p. Then, the corresponding displacements and stresses are all

obtained. For example, the stresses of an arbitrary point in the upper domain areobtained in the Cartesian coordinates:

rþz ¼ l1l2

2p l1 þ k1l2ð Þ3 1þ g2ð Þyz

R5 � 30yz3

R7

� �; ð5:29aÞ

rþzx ¼ l1l2

2p l1 þ k1l2ð Þ3 1� g1 þ g2ð Þxy

R5 � 30xyz2

R7

� �; ð5:29bÞ

rþyz ¼ l1l2

2p l1 þ k1l2ð Þ2g1 � g2 � 1

R3 þ 31� g1 þ g2ð Þy2 þ 2� g2ð Þz2

R5 � 30y2z2

R7

� �ð5:29cÞ

5.2.1.4 Unit Point Displacement Discontinuity in the x-Directionof the Crack

Similar to Sect. 5.2.1.3, as the problem is symmetric about x and y-axis, under unitpoint displacement discontinuity in the x-direction, the stresses of an arbitrary pointin the upper domain are obtained in the Cartesian coordinates

rþz ¼ l1l2

2p l1 þ k1l2ð Þ3 1þ g2ð Þxz

R5 � 30xz3

R7

� �; ð5:30aÞ

rþzx ¼ l1l2

2p l1 þ k1l2ð Þ2g1 � g2 � 1

R3 þ 31� g1 þ g2ð Þx2 þ 2� g2ð Þz2

R5 � 30x2z2

R7

� �;

ð5:30bÞ

rþyz ¼ l1l2

2p l1 þ k1l2ð Þ3 1� g1 þ g2ð Þxy

R5 � 30xyz2

R7

� �ð5:30cÞ

5.2.2 Boundary Integral-Differential Equationsfor Interfacial Cracks

Consider a three-dimensional, isotropic, thermoelastic bimaterial with the twomaterials denoted as 1 and 2, respectively. A Cartesian coordinate system is set upwith the xoy plane lying in the interface, and an arbitrarily shaped, planar crack S islocated on the interface, as shown in Fig. 5.2.


With the aid of the fundamental solutions obtained above, one can get thefundamental solutions for the arbitrarily shaped, interface crack S. The corre-sponding stresses and heat flux at an arbitrary internal point in domain 1 can beexpressed in terms of the displacement and temperature discontinuities across crackfaces as follows:

rþzx ¼ l1l2

2p l1þ k1l2ð ÞZSþ

2g1 � g2 � 1R3 þ 3 2� g2ð Þ z

2

R5 þ 31� g1 þ g2

R5 � 30z2

R7

� �n� xð Þ2

� �uk k

þ 31� g1 þ g2

R5� 30

z2

R7

� �n� xð Þ g� yð Þ vk kþ 30

z3

R7� 3

3� g1ð ÞzR5

� �n� xð Þ wk k

þ 12a1 1þ v1ð Þb2b1 þ b2

z2

R5� a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2

b1 þ b2ð Þ=21R3

� �n� xð Þ hk k

�dS

ð5:31aÞ

Fig. 5.2 A planar interfacecrack S of arbitrary shape in abi-material system underthermomechanical loading[39]


rþyz ¼ l1l2

2p l1 þ k1l2ð ÞZSþ

31� g1 þ g2

R5 � 30z2

R7

� �n� xð Þ g� yð Þ uk k

�

þ 2g1 � g2 � 1R3 þ 3 2� g2ð Þ z

2

R5 þ 31� g1 þ g2

R5 � 30z2

R7

� �g� yð Þ2

� �vk k

þ 30z3

R7 � 33� g1ð ÞzR5

� �g� yð Þ wk k

þ 12a1 1þ v1ð Þb2b1 þ b2

z2

R5 �a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2

b1 þ b2ð Þ=21R3

� �g� yð Þ hk k

�dS

ð5:31bÞ

rþz ¼ l1l2

2p l1 þ k1l2ð ÞZSþ

30z3

R7 � 31þ g2ð ÞzR5

� �n� xð Þ uk k

�

þ 30z3

R7 � 31þ g2ð ÞzR5

� �g� yð Þ vk k

þ 1þ g2R3 þ 3 5� g2ð Þz2

R5 � 30z4

R7

� �wk k

þ a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2b1 þ b2ð Þ=2

zR3 �


z2

R5

� �hk k�dS

ð5:31cÞ

hþz ¼ � b1b2

2p b1 þ b2ð ÞZSþ

1R3 �

3z2

R5

� �hk kdS ð5:31dÞ

where R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� xð Þ2 þ g� yð Þ2 þ z2

q.

5.2.2.1 Hypersingular Integral-Differential Equations

As the internal point approaches the crack face, the integrals in Eq. (5.34) willbecome hypersingular. These integrals must be evaluated using finite-part integrals,and the following formulas are applied [33–35]:

I1 ¼ limz!0

ZSþ

u n; gð Þk k n� xð ÞzR5 dndg ¼ 2p

3@ u x; yð Þk k

@xð5:32aÞ

I2 ¼ limz!0

ZSþ

w n; gð Þk k z2

R5 dndg ¼ 0 ð5:32bÞ


After some algebraic manipulations, one can obtain the boundaryintegral-differential equations for an arbitrarily shaped, interfacial crack:

ZSþ

2g1 � g2 � 1r3

þ 3 1� g1 þ g2ð Þ cos2 /r3

� ��u n; gð Þk k

þ 3 1� g1 þ g2ð Þ sin/ cos/r3

v n; gð Þk k

�a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2b1 þ b2ð Þ=2

cos/r2

h n; gð Þk k�dS

þ 2p g2 � 1ð Þ @ w x; yð Þk k@x

¼ � 2p l1 þ k1l2ð Þl1l2

px x; yð Þ

ð5:33aÞ

ZSþ

2g1 � g2 � 1r3

þ 3 1� g1 þ g2ð Þ sin2 /r3

� ��v n; gð Þk k


u n; gð Þk k

� a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2b1 þ b2ð Þ=2

sin/r2

h n; gð Þk k�dS

þ 2p g2 � 1ð Þ @ w x; yð Þk k@y

¼ � 2p l1 þ k1l2ð Þl1l2

py x; yð Þ

ð5:33bÞ

ZSþ

1þ g2r3

w n; gð Þk kdSþ 2p 1� g2ð Þ @ u x; yð Þk k@x

þ @ v x; yð Þk k@y

� �

¼ � 2p l1 þ k1l2ð Þl1l2

pz x; yð Þð5:33cÞ

� b1b22p b1 þ b2ð Þ

ZSþ

1r3

h n; gð Þk kdS ¼ �hz x; yð Þ ð5:33dÞ

where

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� xð Þ2 þ g� yð Þ2

q; cos/ ¼ n� xð Þ=r; sin/ ¼ g� yð Þ=r:

5.2.2.2 Singular Behavior Near the Interface Crack Front

The singular behavior of interfacial crack in a three-dimensional, two-phase elasticmedium were analyzed by Tang et al. [33]. Following the similar procedure, theinterfacial crack in a three-dimensional, two-phase thermoelastic medium will be


discussed. An arbitrary point P is chosen on the crack edge C of crack S. The edgeC is smooth at point P. Without loss of generality, the Cartesian coordinate systemoxyz is oriented so that the x-direction and y-direction are normal and tangent to C,respectively, and a local orthogonal, intrinsic coordinate system (n, s, m) =(x, y, z) at point P along the smooth periphery of the planar crack is set up. A givensmall circular area R contained in plane S with its radius defined as e is centered atpoint P, as shown in Fig. 5.3.

The hypersingular parts of Eq. (5.33) should be finite in R for finite prescribedmechanical and heat flux loadings:

Fx ¼ � l1l22p l1 þ k1l2ð Þ

ZR

2g1 � g2 � 1r3

þ 3 1� g1 þ g2ð Þ cos2 /r3

� ��uk k


vk k � a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2b1 þb2ð Þ=2

cos/r2

hk k�dS

� l1l2 g2 � 1ð Þl1 þ k1l2ð Þ

@ wk k@x

ð5:34aÞ

ε

Fig. 5.3 The local intrinsiccoordinate system at theinterface crack front [39]


Fy ¼ � l1l22p l1 þ k1l2ð Þ

ZR

2g1 � g2 � 1r3

þ 3 1� g1 þ g2ð Þ sin2 /r3

� ��vk k


uk k � a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2b1 þ b2ð Þ=2

sin/r2

hk k�dS

� l1l2 g2 � 1ð Þl1 þ k1l2ð Þ

@ wk k@y

ð5:34bÞ

Fz ¼ � l1l22p l1 þ k1l2ð Þ

ZR

1þ g2r3

wk kdSþ l1l2 g2 � 1ð Þl1 þ k1l2ð Þ

@ uk k@x

þ @ vk k@y

� �

ð5:34cÞ

Fh ¼ b1b22p b1 þ b2ð Þ

ZR

1r3

hk kdS ð5:34dÞ

where Fx, Fy, Fz and Fh are finite functions of x; yð Þ 2 R.In the neighborhood of point P, for a small SR, the displacement and temperature

discontinuities are related only to the coordinates x in the normal plane throughpoint P as well as the position of point P along the contour of the crack [33, 36]

uk k ¼ A1 Pð Þna1 ;vk k ¼ A2 Pð Þna2 ;wk k ¼ A3 Pð Þna3 ;hk k ¼ A4 Pð Þna4 ;

8>><>>: ð5:35Þ

where A1, A2, A3 and A4 are all complex constants [33], and 0\Re ai½ �\1, insertingEq. (5.35) into Eq. (5.34), and using the following identities [33]

lime!0

ZR

uk kr3

dndg ¼ �2A1 Pð Þpa1 cot pa1ð Þna1�1 ð5:36aÞ

lime!0

ZR

n� xð Þ2 uk kr5

dndg ¼ � 43A1 Pð Þpa1 cot pa1ð Þna1�1 ð5:36bÞ

lime!0

ZR

n� xð Þ g� yð Þ uk kr5

dndg ¼ 0 ð5:36cÞ

lime!0

ZR

g� yð Þ2 uk kr5

dndg ¼ � 23A1 Pð Þpa1 cot pa1ð Þna1�1 ð5:36dÞ


lime!0

ZR

g� yð Þ hk kr3

dndg ¼ 0 ð5:36eÞ

lime!0

ZR

n� xð Þ hk kr3

dndg ¼ �2A4 Pð Þ cot pa4ð Þna4 ð5:36fÞ

assuming n ! 0, yields

cot pbð ÞA1 Pð Þþ cA3 Pð Þ ¼ 0cot pa2ð ÞA2 Pð Þ ¼ 0cA1 Pð Þ � cot pbð ÞA3 Pð Þ ¼ 0cot pa4ð ÞA4 Pð Þ ¼ 0

8>><>>: ð5:37Þ

where b ¼ a1 ¼ a3, c ¼ 1� g2ð Þ= 1þ g2ð Þ. As the constants Ai Pð Þ are generallyassumed to be non-zero, from Eq. (5.37), the characteristic equations to determinethe indices of singular behavior, ai, are found as

cot pa2ð Þ ¼ 0cot pa4ð Þ ¼ 0cot pbð Þ ¼ �ci

8<: ð5:38Þ

where i ¼ ffiffiffiffiffiffiffi�1p

, and the acceptable roots are

a2 ¼ a4 ¼ 12; b1;2 ¼

12� ie; e ¼ 1

2pln g2 ð5:39Þ

The roots show that both the displacement discontinuity vk k and temperaturediscontinuity hk k have the classical singularity index 1/2, while uk k or wk k has thesame singularity index 1

2 þ ie as for pure elastic materials. Considering that thedisplacement and temperature discontinuities in the neighborhood of the interfacialcrack tip, SR, they should have the following forms

uk k ¼ Re A1 Pð Þna½ �;vk k ¼ A2 Pð Þn1=2;wk k ¼ Re A3 Pð Þna½ �;hk k ¼ A4 Pð Þn1=2;

8>><>>: ð5:40Þ

where a ¼ b1 ¼ 12 þ ie, A2 Pð Þ and A4 Pð Þ are arbitrary real constants, while A1 Pð Þ

and A3 Pð Þ are arbitrary complex constants which satisfy the relation ofA1 Pð Þ ¼ iA3 Pð Þ. So they can also be written as


wk kþ i uk k ¼ AR Pð Þþ iAI Pð Þ½ �na ¼ A Pð Þna ð5:41Þ

It should be pointed out that the above results are necessary for studying thesingular stress fields near the interface crack front exactly and guaranteeing aunique solution for the hypersingular, integral-differential equations expressedabove.

5.2.2.3 Singular Stress and Heat Flux Fields Ahead of the InterfacialCrack Front

The stress and heat flux fields of an arbitrary point in the upper domain induced bythe interface crack have been derived already in Eq. (5.34). In order to obtain theexact expressions for the singular stress and heat flux fields of an arbitrary pointN x; y; zð Þ ahead of the periphery of the interface crack, the local nature coordinatesystem P; n;mð Þ and the local polar coordinates q;xð Þ with PN as the negativenormal axis are introduced as shown in Fig. 5.3. We then have

x ¼ �q cosx; y ¼ 0; z ¼ q sinx; R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ q cosxð Þ2 þ g2 þ q sinxð Þ2

qð5:42Þ

For simplicity, we only study the singular stress and heat flux fileds of N0 in theinterface crack front area (x ¼ 0). Substituting Eq. (5.42) into Eq. (5.31) andcombining with the singular index obtained in Eq. (5.39) as well as the form of theexpressions for the displacement and temperature discontinuities in Eq. (5.40), onecan obtain the following dominant-part integrals [33]

limRe!0

ZSe

u n; gð Þk kR3 dS ¼ ip

2 cosh peð Þ A Pð Þ 1� 2ieð Þq�1=2�ie � A Pð Þ 1þ 2ieð Þq�1=2þ ieh i

ð5:43aÞ

limRe!0

ZSe

n� xð Þ2 u n; gð Þk kR5 dS ¼ ip

3 cosh peð Þ A Pð Þ 1� 2ieð Þq�1=2�ie � A Pð Þ 1þ 2ieð Þq�1=2þ ieh i

ð5:43bÞ

limRe!0

ZSe

n� xð Þ g� yð Þ u n; gð Þk kR5 dS ¼ 0 ð5:43cÞ


limRe!0

ZSe

v n; gð Þk kR3 dS ¼ pA2 Pð Þq�1=2 ð5:43dÞ

limRe!0

ZSe

g� yð Þ2 v n; gð Þk kR5 dS ¼ p

3A2 Pð Þq�1=2 ð5:43eÞ

limRe!0

ZSe

n� xð Þ g� yð Þ v n; gð Þk kR5 dS ¼ 0 ð5:43fÞ

limRe!0

ZSe

w n; gð Þk kR3 dS ¼ p

2 cosh peð Þ A Pð Þ 1� 2ieð Þq�1=2�ie þA Pð Þ 1þ 2ieð Þq�1=2þ ieh i

ð5:43gÞ

limRe!0

ZSe

h n; gð Þk kR3 dS ¼ pA4 Pð Þq�1=2 ð5:43hÞ

limRe!0

ZSe

n� xð Þ h n; gð Þk kR3 dS ¼ �2A4 Pð Þq1=2 ð5:43iÞ

limRe!0

ZSe

g� yð Þ h n; gð Þk kR3 dS ¼ 0 ð5:43jÞ

where A Pð Þ is the conjugate of A Pð Þ. Substituting the above integrals intoEq. (5.31), we can get the singular stress and heat flux fields in the neighborhood ofpoint P

rzx q; 0ð Þ ¼ il1l2 1þ g2ð Þ

4 l1 þ k1l2ð Þ cosh peð Þ A Pð Þ 1� 2ieð Þq�1=2�ie � A Pð Þ 1þ 2ieð Þq�1=2þ ieh i

þ l1l2p l1 þ k1l2ð Þ

a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2b1 þ b2ð Þ=2 A4 Pð Þq1=2

ð5:44aÞ

ryz q; 0ð Þ ¼ l1l22 l1 þ l2ð ÞA2 Pð Þq�1=2 ð5:44bÞ

rz q; 0ð Þ ¼ l1l2 1þ g2ð Þ4 l1 þ k1l2ð Þ cosh peð ÞA Pð Þ 1� 2ieð Þq�1=2�ie þA Pð Þ 1þ 2ieð Þq�1=2þ ieh i ð5:44cÞ


hz q; 0ð Þ ¼ � b1b22 b1 þ b2ð ÞA4 Pð Þq�1=2 ð5:44dÞ

Clearly both ryz q; 0ð Þ and hz q; 0ð Þ have the classical square root singularity,r�1=2, while the stresses rzx q; 0ð Þ and rz q; 0ð Þ are oscillatory with the singularity ofr�1=2þ ie. It is also noted that the temperature term induces the stress rzx q; 0ð Þ whilethe influence will vanish as the point approaching the crack front. In other words,the temperature term does not contribute to the singular stress, rzx q; 0ð Þ.

5.2.3 Stress Intensity Factor and Energy Release Rate

Based on the obtained stress and heat flux fields in the neighborhood of the crackfront, and using the definition of stress intensity factors for interfacial cracks byHutchinson et al. [43], we can get the stress and heat flux intensity factors at thecrack front

KI ¼ limq!0

ffiffiffiffiffiffiffiffi2pq

pq�ierz q; 0ð Þ ð5:45aÞ

KII ¼ limq!0


pq�ierzx q; 0ð Þ ð5:45bÞ

K ¼ KI þ iKII ¼ limq!0


pq�ie rz q; 0ð Þþ irzx q; 0ð Þ½ � ð5:45cÞ

KIII ¼ limq!0


pryz q; 0ð Þ ð5:45dÞ

Kh ¼ limq!0


phz q; 0ð Þ ð5:45eÞ

They are expressed in terms of the displacement and temperature discontinuitiesas

K ¼ffiffiffiffiffiffi2p

pl1l2 1þ g2ð Þ 1þ 2ieð Þ

2 l1 þ k1l2ð Þ cosh peð Þ limq!0

w q; 0ð Þk kþ i u q; 0ð Þk kq1=2þ ie

ð5:46aÞ

KIII ¼ffiffiffiffiffiffi2p

pl1l2

2 l1 þ l2ð Þ limq!0

v q; 0ð Þk kffiffiffiq

p ð5:46bÞ

Kh ¼ �ffiffiffiffiffiffi2p

pb1b2

2 b1 þ b2ð Þ limq!0

h q; 0ð Þk kffiffiffiq

p ð5:46cÞ


When the bimaterial is reduced to a homogeneous solid, the stress intensityfactors are the same as those for homogeneous materials [39].

It is worth noting that the oscillatory singularity appeared in the stress intensityfactors is a physically unrealistic phenomenon, thus the well-defined quantity, strainenergy release rate having a relevance to conventional fracture mechanics, is nec-essary to be evaluated. From the above expressions for intensity factors inEq. (5.46), we can get the displacement jumps across the crack surfaces at theinterface crack front edge in the form

DU q; 0ð Þ ¼ w q; 0ð Þk kþ i u q; 0ð Þk k ¼ffiffiffi2p

rl1 þ k1l2ð Þ cosh peð Þ

l1l2 1þ g2ð Þ 1þ 2ieð ÞKq1=2þ ie

ð5:47aÞ

v q; 0ð Þk k ¼ffiffiffi2p

r1l1

þ 1l2

� �KIII

ffiffiffiq

p ð5:47bÞ

Based on the virtual crack closure method and following the same procedureproposed in [33, 34], let the interfacial crack front advance locally a very small newcrack surface area, DS, at point P, as shown in Fig. 5.4, the local energy release ratefor a unit area of interface to debond can be readily evaluated in the integral formutilizing the obtained tractions and displacement fields ahead of the interface crackfront

Fig. 5.4 An interface crackwith virtual incremental areaDS of the crack surface [39]


G Pð Þ ¼ limDS!0

12DS

ZDS

rz x; yð Þþ irzx x; yð Þ½ � � DU x; yð Þþ ryz x; yð Þ v x; yð Þk k� �dS

ð5:48Þ

The infinitesimal dx normal to the planar interfacial crack front edge @S is usedto describe the virtual extension of the crack front. The total changes of the cracksurface are expressed as

DS ¼Z

@S yð Þ

dxdy ð5:49Þ

Substituting Eq. (5.49) into Eq. (5.48), one can get

G Pð Þ ¼ limDS!0

12R@s yð Þ dxdy

Z@s yð Þ

Zdx0

K Pð ÞK Pð Þ4p 1þ 2ieð Þ cosh pe

1þ k1l1

þ 1þ k2l2

� �dx� x

x

� �1=2þ ie"

þ l1 þ l2ð ÞK2III Pð Þ

pl1l2

ffiffiffiffiffiffiffiffiffiffiffiffiffidx� x

x

r #dxdy

ð5:50Þ

where the inner integral of the first term is recognized as the complex beta functionB 1=2þ ie; 3=2� ieð Þ and the inner integral of the second term as the real betafunction B 1=2; 3=2ð Þ. Upon evaluating of the beta function by the gamma functionC zð Þ, one can obtain [39]:

G Pð Þ ¼ l1 þ k1l2ð Þþ l2 þ k2l1ð Þ16l1l2 cosh

2 peð Þ K Pð ÞK Pð Þþ l1 þ l24l1l2

K2III Pð Þ ð5:51aÞ

or, alternatively

G Pð Þ ¼ l1 þ k1l2ð Þþ l2 þ k2l1ð Þ16l1l2 cosh

2 peð Þ K2I Pð ÞþK2

II Pð Þ �þ l1 þ l24l1l2

K2III Pð Þ ð5:51bÞ

The expression for the energy release rate shows that it does not include the heatflux intensity factor, which is the same as the elastic one [33].


5.3 Interface Crack Problems in ThermopiezoelectricMaterials

Due to their excellent piezoelectric, dielectric and pyroelectric properties,piezothermoelastic materials are being widely used in smart structural systems assensors, actuators, transducers and intelligent structures, etc. Because of the pyro-electric effect, the response characteristics may change considerably when piezo-electric structures work in environment where temperature varies notably. Sincetemperature variation in piezoelectric materials will seriously affect the overallsensing and controlling performance of a distributed control system [44], a wideand thorough understanding of the mechanical-electro-thermal coupling behavior inpiezoelectric materials is essential to better use of piezothermoelastic-based intel-ligent structures.

For homogeneous materials, Mindlin [45] gave the equations describing smallvibrations of piezoelectric plates to reveal the relationship between the thermal,elastic and electric fields. The analytical theory for piezothermoelastic materials hasalso been comprehensively studied [46–48], and different methods were applied toobtain the general solutions for linear or nonlinear problems of piezothermoelas-ticity [49–52]. As it is very important to understand the fracture behaviors of apiezothermoelastic solid with defects under thermal loadings, many efforts havebeen made in this regard. Yu and Qin [53] analyzed the singularities of the nearcrack-tip thermoelectroelastic fields. Chen [54] derived a general solution of 3Dpiezothermoelasticity for both static and dynamic cases. Gao and Wang [55] dealtwith 2D N-collinear permeable cracks in a thermopiezoelectric medium. Chen et al.[56] gave the explicit solution for a penny-shaped crack subjected to an arbitrarilypoint-temperature loading by utilizing the elementary functions. Making use ofStroh’s formalism and conformal mapping technique, Qin [57] presented the Greenfunction for a thermopiezoelectric material containing an elliptic hole by usingStroh’s formalism and conformal mapping technique. Based on the given Greenfunction, the boundary element method (BEM) was used to analyze various 2Dcrack problems [58–60]. In addition, many researchers have investigated 2D and3D crack problems in piezothermoelastic materials under different thermal condi-tions of stationary or dynamic cases [61–74].

In order to get a stronger electro-mechanical-thermal coupling effect, laminatedpiezoelectric structures are often used. Therefore, interface cracking problems arefrequently encountered in piezoelectric structures. Shen et al. [75] combined theextended version of Eshelby-Stroh’s formulation and the method of analyticalcontinuation to study interface cracks and obtained a general, explicit, closed formsolution. Later, Shen and Kuang [76] analyzed an electrically impermeable, inter-face crack in an infinite, piezothermoelastic bi-material under a remote heat flux.Qin and Mai [77] gave the Green functions for thermoelectroelastic bi-materialssubjected to a temperature discontinuity in terms of the Stroh formalism in a 2Dpiezoelectric plate. Qin and Mai [78] employed the Lekhnitskii-Eshelby-Strohformalism to study an interface crack in a thermopiezoelectric bi-material with the

5.3 Interface Crack Problems in Thermopiezoelectric Materials 197

assumption of a contact zone model. Later, Gao and Wang [79] used the sameclassical interface crack model proposed by Shen and Kuang [76] to investigate anelectrically permeable, interface crack. In the frame of this crack model, the solutionprocessed the oscillatory singularities at the crack tips as in elastic materials foundin [80]. Herrmann and Loboda [81, 82] adopted the contact-zone model and studiedinterface cracks under applied thermoelectromechanical loadings in piezother-moelastic bi-materials using electrically permeable and impermeable conditions,respectively. Ueda [83] adopted the Fourier transform technique to study a crack ina piezoelectric laminate under uniform electric and temperature fields. Herrmannand Loboda [84] extended their proposed contact-zone crack model to studymoving interface crack problems. Hou and Leung [85] obtained 3D Green’sfunctions for two-phase piezothermoelastic bi-materials expressed in terms ofharmonic functions.

Similar as discussed in Sect. 5.1, the displacement discontinuity method areextended to include the electric potential, and temperature discontinuities to provide thefundamental solutions of interface crack problem in piezoelectric biomaterials [86].

5.3.1 Basic Equations

For a stationary process, in the absence of body forces, free electric charges, electriccurrent and body heat source, when the xoy-plane is parallel to the plane of isotropyin the Cartesian coordinates (x, y, z), the corresponding constitutive relations oflinear, transversely isotropic, piezothermoelastic materials are given by Mindlin[44] in the form

rx ¼ c11@u@x

þ c12@v@y

þ c13@w@z

þ e31@u@z

� k11h;

ry ¼ c12@u@x

þ c11@v@y

þ c13@w@z

þ e31@u@z

� k11h;

rz ¼ c13@u@x

þ c13@v@y

þ c33@w@z

þ e33@u@z

� k33h;

syz ¼ c44@v@z

þ @w@y

� �þ e15

@u@y

;

szx ¼ c44@u@z

þ @w@x

� �þ e15

@u@x

;

sxy ¼ c66@u@y

þ @v@x

� �;

ð5:52aÞ


Dx ¼ e15@u@z

þ @w@x

� �� e11

@u@x

;

Dy ¼ e15@v@z

þ @w@y

� �� e11

@u@y

;

Dz ¼ e31@u@x

þ @v@y

� �þ e33

@w@z

� e33@u@z

þ p3h;

ð5:52bÞ

hx ¼ �b11@h@x

; hy ¼ �b11@h@y

; hz ¼ �b33@h@z

; ð5:52cÞ

where u, v, w, / and h are elastic displacements, electric potential and temperaturechange (with respect to the reference temperature), respectively, and can be referredto as extended displacements; cij, eij, eij, kii and p3 are elastic, piezoelectric,dielectric, thermal modules and pyroelectric constants, respectively, and bij are thecoefficients of heat conduction. The relation c66 ¼ c11 � c12ð Þ=2 holds for piezo-electric materials with transverse isotropy. Compared with (5.2), although theanisotropy of piezothermoelectricity makes the problem more complicated, a sim-ilar solution method can be employed here.

The equilibrium equations are given by [50]

@rx@x

þ @sxy@y

þ @szx@z

¼ 0;

@sxy@x

þ @ry@y

þ @syz@z

¼ 0;

@szx@x

þ @syz@y

þ @rz@z

¼ 0;

ð5:53aÞ

@Dx

@xþ @Dy

@yþ @Dz

@z¼ 0; ð5:53bÞ

@hx@x

þ @hy@y

þ @hz@z

¼ 0; ð5:53cÞ

where rij, Di, and hi, with i, j = x, y, z, are the components of stress, electricdisplacement and heat flux, respectively, which are all referred to as extendedstresses.


5.3.2 Fundamental Solutions for Unit-Point ExtendedDisplacement Discontinuities

Consider a three-dimensional, transversely isotropic, piezothermoelastic bi-materialwith the two bonded materials occupying the upper (denoted as Material 1) andlower domains (denoted as Material 2), respectively. The poling directions are allperpendicular to the interface. A Cartesian coordinate system is set up with the xoyplane lying in the interface. Again, a penny-shaped crack of radius a is centered atthe origin on the interface, as illustrated in Fig. 5.1. The upper and lower surfaces ofthe crack are denoted by S+ and S−, respectively, and

x ¼ r cos/; y ¼ r sin/ ð5:54Þ

The extended displacement discontinuities across the faces of the crack can beexpressed as

uk k ¼ u x; y; 0þð Þ � u x; y; 0�ð Þ; vk k ¼ v x; y; 0þð Þ � v x; y; 0�ð Þ;wk k ¼ w x; y; 0þð Þ � w x; y; 0�ð Þ; uk k ¼ u x; y; 0þð Þ � u x; y; 0�ð Þ;hk k ¼ h x; y; 0þð Þ � h x; y; 0�ð Þ; x; yð Þ 2 S:

ð5:55Þ

According to Zhao et al. [40] (1988) and Zhao and Liu [41], if we let the radiusof the crack a approach zero, the fundamental solutions corresponding tounit-concentrated extended displacement discontinuities can be obtained, as dis-cussed in the preceding section.

Using the operator theory and the generalized Almansi’s theorem, Ding et al.[50] derived the general solutions for piezothermoelasticity satisfying the equilib-rium and constitutive equations in Eqs. (5.52) and (5.53). According to the generalsolutions given by Ding et al. [50], we have

U ¼ �K iW0 þX4j¼1

Wj

!; wm ¼

X4j¼1

amj@Wj

@zj; h ¼ a34

@2W4

@z24;

r1 ¼X4j¼1

c3j@2Wj

@z2j; r2 ¼ �2c66K

2 iW0 þX4j¼1

Wj

!;

rzm ¼X4j¼1

cmj@2Wj

@z2j; hx ¼ �b33a34

@3W4

@x@z34;

hy ¼ �b33a34@3W4

@y@z24; hz ¼ �b33s4a34

@3W4

@z34;

sm ¼ KX4j¼1

sjcmj@Wj

@zj�is0qm

@W0

@z0

!; m ¼ 1; 2ð Þ;

ð5:56Þ


The corresponding notations for the components in the Cartesian coordinates (x,y, z) and cylindrical coordinates (r, /, z) are both introduced as [50]

zj ¼ sjz; K ¼ @

@xþ i

@

@y;

U ¼ uþ iv ¼ ei/ ur þ iu/� �

; w1 ¼ w; w2 ¼ u;

r1 ¼ rx þ ry ¼ rr þ r/;

r2 ¼ rx � ry þ 2isxy ¼ ei/ rr � r/ þ 2isr/� �

;

rz1 ¼ rz; rz2 ¼ Dz;

sm ¼ sxm þ isym ¼ ei/ srm þ is/m� �

sx1 ¼ szx; sy1 ¼ syz; sr1 ¼ szr; s/1 ¼ s/z;

sx2 ¼ Dx; sy2 ¼ Dy; sr2 ¼ Dr; s/2 ¼ D/;

ð5:57Þ

where zj= sjz (j = 0,1,2,3,4), with all material-related constants concerned givenin [86].

The harmonic functions Wj j ¼ 0; 1; 2; 3; 4ð Þ satisfy

Dþ @2

@z2j

!Wj ¼ 0; ð5:58Þ

where

D ¼ @2

@x2þ @2

@y2in the Cartesian coordinatesðx; y; zÞ; ð5:59aÞ

and

D ¼ @2

@r2þ @

r@rþ @2

r2@/2 in cylindrical coordinatesðr;/; zÞ: ð5:59bÞ

5.3.2.1 Fundamental Solution for a Unit-Point TemperatureDiscontinuity

The boundary condition in this case can be rewritten as:

uðn; gÞk k ¼ 0; vðn; gÞk k ¼ 0; wðn; gÞk k ¼ 0; uðn; gÞk k ¼ 0;hðn; gÞk k ¼ dðn; gÞ; ð5:60aÞ

where d(n, η) is the Dirac delta function. It can be seen that this is non-torsionalaxisymmetric problem, thus all the quantities depend only on (r, z) while inde-pendent of angle /. Hence, the general solutions can be simplified as:


ur ¼ �X4j¼1

@Wj

@r; wm ¼

X4j¼1

amj@Wj

@zj;

hz ¼ �s4b33a34@3W4

@z34; h ¼ a34

@2W4

@z24;

rzm ¼X4j¼1

cmj@2Wj

@z2j; rzr ¼

X4j¼1

sjc1j@2Wj

@r@zj;

ð5:61aÞ

and@2

@r2þ @

r@rþ @2

@z2j

!Wj ¼ 0: ð5:61bÞ

Employing the zero-order Hankel transform, the potential functions can beset as:

Wþj r; zð Þ ¼

Z10

nAþj nð Þe�nzj J0 nrð Þdn; for z[ 0 ð5:62aÞ

W�j r; zð Þ ¼

Z10

nA�j nð Þenzj J0 nrð Þdn; for z\0 ð5:62bÞ

where the superscript “+” and “−” denote the upper and lower domains,respectively.

When the unit temperature discontinuity is applied on the interface crack, thecorresponding boundary conditions are as follows:

uþr � u�r ¼ 0;

wþ � w� ¼ 0;

uþ � u� ¼ 0;

hþ � h� ¼ d;

8>>><>>>:

rþzr � r�zr ¼ 0;

rþz � r�z ¼ 0;

Dþz � D�

z ¼ 0;

hþz � h�z ¼ 0;

8>>><>>>:

ð5:63Þ

where 0� r� a belongs to the crack region. Inserting Eq. (5.61a) into Eq. (5.62),and the resultant expressions into Eq. (5.63), one can get:


P4j¼1

Aþj nð Þ ¼P4

j¼1A�j nð Þ;

�P4j¼1

aþ1j A

þj nð Þ ¼P4

j¼1a�1jA

�j nð Þ;

�P4j¼1

aþ2j A

þj nð Þ ¼P4

j¼1a�2jA

�j nð Þ;

n2 aþ34A

þ4 nð Þ � a�34A

�4 nð Þ � ¼ 1

2p ;

8>>>>>>>>>><>>>>>>>>>>:

ð5:64aÞ

P4j¼1

sþj cþ1j Aþj nð Þ ¼ �P4

j¼1s�j c

�1jA

�j nð Þ;

P4j¼1

cþ1j Aþj nð Þ ¼P4

j¼1c�1jA

�j nð Þ;

P4j¼1

cþ2j Aþj nð Þ ¼P4

j¼1c�2jA

�j nð Þ;

bþ33 s

þ4 aþ

34Aþ4 nð Þ ¼ �b�33s

�4 a

�34A

�4 nð Þ:

8>>>>>>>>>><>>>>>>>>>>:

ð5:64bÞ

Solving Eq. (5.64), the eight coefficients are determined:

Aþj ¼ Aþ�

j1

n2; A�

j ¼ A��j

1

n2; j ¼ 1; 2; 3; 4; ð5:65Þ

where Aþ�j and A��

j are constants, and then the potential functions are obtained:

wþj ðr; zÞ ¼ �Aþ�

j ln Rj þ zj� �

; ð5:66aÞ

w�j ðr; zÞ ¼ �A��

j ln Rj � zj� �

; ð5:66bÞ

where Rj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2j

q: With the obtained potential functions, the corresponding

extended displacements and stresses in the whole space can all be determined. Forinstance, the extended stresses of an arbitrary field point in the upper domain areobtained as

rþz ¼

X4j¼1

Aþ�j cþ1j

zjR3j; ð5:67aÞ

Dþz ¼

X4j¼1

Aþ�j cþ2j

zjR3j; ð5:67bÞ

rþzr ¼

X4j¼1

Aþ�j sþj cþ1j

rR3j; ð5:67cÞ


rþzx ¼

X4j¼1

Aþ�j sþj cþ1j

xR3j; ð5:67dÞ

rþyz ¼

X4j¼1

Aþ�j sþj cþ1j

yR3j; ð5:67eÞ

hþz ¼ Aþ�

4 sþ4 bþ33a

þ34

3z24R54� 1R34

� �: ð5:67fÞ

Through the same approach, one can obtain the fundamental solutions satisfyingthe boundary conditions of other unit point, extended displacement discontinuities,respectively. For simplicity, the detailed derivation is omitted which can be foundin [86].

5.3.3 Boundary Integral-Differential Equationsfor an Interfacial Crack in PiezothermoelasticMaterials

Consider an arbitrarily shaped, planar crack, S, lying on the interface of two dif-ferent piezoelectric materials. With the aid of the fundamental solutions obtained inSect. 5.2.1.2, one can get the fundamental solutions for the interface crack S. Byvirtue of the obtained fundamental solutions and superposition, the correspondingextended stresses caused by the interface crack, S at an arbitrary, internal field point(x, y, z) in the upper domain can be expressed in terms of the extended displacementdiscontinuities across the interface crack as

rþzx ¼

ZSþ

X3j¼1

pþ�1j sþj cþ1j 3

1R3j� z2jR5j

!n� xð Þ2r2

� 1R3j

" #� sþ0 qþ

1 pþ�2 3

1R30� z20R50

� �g� yð Þ2r2

� 1R30

" # !(

uk kþ 3n� xð Þ g� yð Þ

r2X3j¼1

pþ�1j sþj cþ1j

1R3j� z2jR5j

!þ sþ0 qþ

1 pþ�2

1R30� z20R50

� �" #vk k

�X3j¼1

Bþ�j sþj cþ1j

3 n� xð ÞzjR5j

wk k �X3j¼1

Cþ�j sþj cþ1j

3 n� xð ÞzjR5j

uk k

�X4j¼1

Aþ�j sþj cþ1j

n� xð ÞR3j

hk k)dS n; gð Þ;

ð5:68aÞ


rþyz ¼

ZSþ

3n� xð Þ g� yð Þ

r2

� X3j¼1

pþ�1j sþj cþ1j

1R3j� z2jR5j

!þ sþ0 qþ

1 pþ�2

1R30� z20R50

� �" #uk k

þX3j¼1

pþ�1j sþj cþ1j 3

1R3j� z2jR5j

!g� yð Þ2r2

� 1R3j

" #

� sþ0 qþ1 pþ�

2 31R30� z20R50

� �n� xð Þ2r2

� 1R30

" #!vk k

�X3j¼1

Bþ�j sþj cþ1j

3 g� yð ÞzjR5j

wk k�X3j¼1

Cþ�j sþj cþ1j

3 g� yð ÞzjR5j

uk k

�X4j¼1

Aþ�j sþj cþ1j

g� yð ÞR3j

hk k)dS n; gð Þ;

ð5:68bÞ

rþz ¼

ZSþ

�X3j¼1

pþ�1j cþ1j

3 n� xð ÞzjR5j

uk k(

�X3j¼1

pþ�1j cþ1j

3 g� yð ÞzjR5j

vk k

þX3j¼1

Bþ�j cþ1j

3z2jR5j

� 1R3j

!wk kþ

X3j¼1

Cþ�j cþ1j

3z2jR5j

� 1R3j

!uk k

þX4j¼1

Aþ�j cþ1j

zjR3j

hk k)dS n; gð Þ;

ð5:68cÞ

Dþz ¼

ZSþ

�X3j¼1

pþ�1j cþ2j

3 n� xð ÞzjR5j

uk k(

�X3j¼1

pþ�1j cþ2j

3 g� yð ÞzjR5j

vk k

þX3j¼1

Bþ�j cþ2j

3z2jR5j

� 1R3j

!wk kþ

X3j¼1

Cþ�j cþ2j

3z2jR5j

� 1R3j

!uk k

þX4j¼1

Aþ�j cþ2j

zjR3j

hk k)dS n; gð Þ;

ð5:68dÞ

hþz ¼ �Aþ�

4 sþ4 bþ33a

þ34

ZSþ

1R34� 3z24

R54

� �hk kdS n; gð Þ; ð5:68eÞ

where Rj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� xð Þ2 þ g� yð Þ2 þ z2j

q, r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� xð Þ2 þ g� yð Þ2

q, zj ¼ sjz.


5.3.4 Hyper-Singular Integral-Differential Equations

As the field point approaches the crack face, the integrals in Eq. (5.68) will becomehyper-singular. These integrals must be evaluated using finite-part integrals, and thefollowing integral formulas are applied [33]:

I1 ¼ limz!0

ZSþ

u n; gð Þk k n� xð ÞzR5 dndg ¼ 2p

3@ u x; yð Þk k

@x; ð5:69aÞ

I2 ¼ limz!0

ZSþ

w n; gð Þk k z2

R5 dndg ¼ 0; ð5:69bÞ

After some algebraic manipulations, the boundary integral-differential equationsfor an arbitrarily shaped, interface crack are obtained:

px x; yð Þ ¼ �ZSþ

L11 3 cos2 /� 1� �� L12 3 sin2 /� 1

� � �� 1r3

u n; gð Þk k

þ 3 L11 þ L12ð Þ sin/ cos/r3

v n; gð Þk k�L13cos/r2

h n; gð Þk k�dS n; gð Þ

þ 2p L14@ w x; yð Þk k

@xþ L15

@ u x; yð Þk k@x

� �;

ð5:70aÞ

py x; yð Þ ¼ �ZSþ

L11 3 sin2 /� 1� �� L12 3 cos2 /� 1

� � �� 1r3

v n; gð Þk k


u n; gð Þk k�L13sin/r2

h n; gð Þk k�dS n; gð Þ

þ 2p L14@ w x; yð Þk k

@yþ L15

@ u x; yð Þk k@y

� �;

ð5:70bÞ

pz x; yð Þ ¼ZSþ

L31 w n; gð Þk kþ L32 u n; gð Þk kð Þ 1r3dS n; gð Þ

þ 2pL1@ u x; yð Þk k

@xþ @ v x; yð Þk k

@y

� �;

ð5:70cÞ


x ¼ �ZSþ

L41 w n; gð Þk kþ L42 u n; gð Þk kð Þ 1r3dS n; gð Þ

� 2pL2@ u x; yð Þk k

@xþ @ v x; yð Þk k

@y

� �;

ð5:70dÞ

hz x; yð Þ ¼ �L5

ZSþ

1r3

h n; gð Þk kdS n; gð Þ; ð5:70eÞ

where px, py, and pz are the mechanical tractions; x and hz are the applied electricdisplacement and heat flux, respectively, and all of them are called the extendedtractions, and the related coefficients are:

L11 ¼X3j¼1

pþ�1j sþj cþ1j ; L12 ¼ sþ0 qþ

1 pþ�2 ; L13 ¼

X4j¼1

Aþ�j sþj cþ1j ;

L14 ¼X3j¼1

Bþ�j sþj cþ1j ; L15 ¼

X3j¼1

Cþ�j sþj cþ1j ; L5 ¼ Aþ�

4 sþ4 bþ33a

þ34 ;

L1 ¼X3j¼1

pþ�1j cþ1j ; L31 ¼

X3j¼1

Bþ�j cþ1j ; L32 ¼

X3j¼1

Cþ�j cþ1j ;

L2 ¼X3j¼1

pþ�1j cþ2j ; L41 ¼

X3j¼1

Bþ�j cþ2j ; L42 ¼

X3j¼1

Cþ�j cþ2j ;

ð5:71Þ

where r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� xð Þ2 þ g� yð Þ2

q, cos/ ¼ n� xð Þ=r, sin/ ¼ g� yð Þ=r, zj ¼ sjz.

It should be pointed out that Eq. (5.70) are the expressions of the electric andthermal impermeable boundary conditions in the complete form. The electricpotential discontinuity or the temperature discontinuity will be zero for electricallyor thermally permeable boundary condition. In addition, there are also electricallysemi-permeable boundary condition [87] and thermally semi-permeable boundarycondition [74, 88]. In these kinds of boundary conditions, the electric displacementin the crack cavity can be determined by the electric potential discontinuity andcrack opening displacement as Dc

z ¼ �jc uk k= wk k, while the heat flux in the crackcavity can be determined by the temperature discontinuity and crack opening dis-placement as hcz ¼ �bc hk k= wk k, where ec and bc represent the dielectric and heatconduction coefficients in the crack interior, respectively. The difference inboundary conditions is obvious, for brevity, the detailed discussion about theinfluence of different electric and thermal boundary conditions was omitted here.Details can be found in [86]. Here we only present the results for electrically and


thermally impermeable boundary conditions. When the bi-material becomeshomogeneous, the differential terms will disappear. Therefore, the boundaryintegral-differential equations are reduced to hyper-singular boundary integralequations.

5.3.4.1 Solution Method for the Extended Displacement Discontinuity||W|| + G||u||

Combining Eqs. (5.70c) and (5.70d), one obtains

L1L2

L41 � L31

� � ZSþ

wk kþ g uk kf g 1r3dS ¼ �pz þ L1

L2x; ð5:72Þ

where

g ¼ L1L42 � L2L32L1L41 � L2L31

; ð5:73Þ

Equation (5.72) is the hyper-singular, boundary integral equation for theextended displacement discontinuity wj jj j þ gjjujj. For the same crack in an iso-tropic, thermoelastic bi-material, the boundary integral equation of the displacementdiscontinuity ||W|| in the z-direction takes the same form [39]

E8p 1� v2ð Þ

ZSþ

Wk kr3

dS ¼ �tz x; yð Þ; ð5:74Þ

where v and E are, respectively, the Poisson’s ratio and Young’s modulus, and tz(x,y) is the prescribed traction along the z-direction on the crack surface. Using

E8p 1� v2ð Þ ¼

L1L2

L41 � L31; ð5:75aÞ

�tz x; yð Þ ¼ �pz þ L1L2

x ð5:75bÞ

We can find that Eqs. (5.72) and (5.74) are in an identical form, and therefore,they have the same solution

wk kþ g uk k ¼ Wk k ð5:76Þ


Equation (5.76) indicates that the solution of the extended displacement dis-continuity wj jj j þ gjjujj can be directly obtained from the corresponding thermoe-lastic solution.

5.3.4.2 Extended Stress rZ − L1Dz/L2 and Extended Intensity FactorKI1

The stress for isotropic thermoelastic bi-materials in the crack plane is givenby [39]:

rz ¼ E8p 1� v2ð Þ

ZSþ

Wk kR3 dS; ð5:77Þ

where R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� xð Þ2 þ g� yð Þ2

q[ 0.

From Eqs. (5.72) and (5.77), it can be seen that the extended stress rz − L1Dz/L2has the classical singularity r−1/2 near the interface crack front in the piezother-moelastic bi-material.

In a purely elastic problem, the Mode I stress intensity factor is defined as:

KMI ¼ lim

r!0

ffiffiffiffiffiffiffiffi2pr

prz; ð5:78aÞ

and can be expressed in terms of the displacement discontinuity

KMI ¼ E

8p 1� v2ð Þ limq!0

Wk kffiffiffiq

p : ð5:78bÞ

In the sam fashion, the first Mode I extended intensity factor in piezothermoe-lastic medium is defined as [86]:

KI1 ¼ limr!0

ffiffiffiffiffiffiffiffi2pr

prz � L1Dz=L2ð Þ; ð5:79aÞ

Substituting Eqs. (5.72) into (5.79a), one obtains:

KI1 ¼ L1L2

L41 � L31

� �limq!0

wk kþ g uk kffiffiffiq

p ; ð5:79bÞ


Finally, utilizing Eqs. (5.76) and (5.77) and comparing Eqs. (5.78b) and (5.79b)leads to:

KI1 ¼ KMI : ð5:80Þ

The Mode I1 intensity factor KI1 is one of the parameters characterizing theextended stresses near the interface crack front.

5.3.5 Solution Method of the Integral-DifferentialEquations

Equations (5.70a, b and f) can be rewritten as:ZSþ

L11 3 cos2 /� 1� �� L12 3 sin2 /� 1

� � �� 1r3

uk k


vk k�L13cos/r2

hk k�dS n; gð Þ

� 2pL14@

@xwk kþ L15

L14uk k

� �¼ �px;

ð5:81aÞ

ZSþ

L11 3 sin2 /� 1� �� L12 3 cos2 /� 1

� � �� 1r3

vk k


uk k�L13sin/r2

hk k�dS n; gð Þ

� 2pL14@

@ywk kþ L15

L14uk k

� �¼ �py;

ð5:81bÞ

�L5

ZSþ

1r3

h n; gð Þk kdS ¼ �hz x; yð Þ: ð5:81cÞ

Combining Eqs. (5.75a) and (5.75b) yields

L31 þ fL41ð ÞZSþ

wk kþ L32 þ fL42L31 þ fL41

uk k� �

1r3dS

þ 2p L1 þ fL2ð Þ @ uk k@x

þ @ vk k@y

� �¼ pz þ fx;

ð5:82Þ


In Eqs. (5.81) and (5.82), assuming

L32 þ fL42L31 þ fL41

¼ L15L14

; ð5:83Þ

the constant f is then solved as

f ¼ L15L31 � L32L14L42L14 � L41L15

: ð5:84Þ

Therefore, Eqs. (5.81) and (5.82) can be rewritten as:

1pxL14

¼ZSþ

1L11L14

1� 3 cos2 /� �� 1L12

L141� 3 sin2 /� ��

1r3

uk k

� 31 L11 þ L12ð Þ

L14

sin/ cos/r3

vk kþ 1L13L14

cos/r2

hk k�dS n; gð Þ

þ 2p@ w�k k@x

;

ð5:85aÞ

1pyL14

¼ZSþ

1L11L14

1� 3 sin2 /� �� 1L12

L141� 3 cos2 /� ��

1r3

vk k

� 31 L11 þ L12ð Þ

L14

sin/ cos/r3

uk kþ 1L13L14

sin/r2

hk k�dS n; gð Þ

þ 2p@ w�k k@y

;

ð5:85bÞ

pz þ fxð ÞL1 þ fL2ð Þ ¼

L31 þ fL411 L1 þ fL2ð Þ

ZSþ

w�k kr3

dSþ 2p@ uk k@x

þ @ vk k@y

� �; ð5:85cÞ

hz x; yð Þ ¼ �L5

ZSþ

1r3

h n; gð Þk kdS; ð5:85dÞ

where 1 is a constant to be determined, and w�k k is the extended displacementdiscontinuity defined as

w�k k ¼ 1 wk kþ L15L14

uk k� �

: ð5:86Þ

On the other hand, the boundary integral-differential equations of the same crackin a 3D, two-phase, isotropic, thermoelastic bi-material are given by Zhao et al.[39], as


� 2p l1 þ k1l2ð Þl1l2 g2 � 1ð Þ p1 ¼

ZSþ

2g1 � g2 � 1g2 � 1

1r3

þ 3 1� g1 þ g2ð Þg2 � 1

cos2 /r3

� ��~uk k

þ 3 1� g1 þ g2ð Þg2 � 1

sin/ cos/r3

~vk k


cos/r2

h n; gð Þk k��

dS

þ 2p@ ~wk k@x

;

ð5:87aÞ

� 2p l1 þ k1l2ð Þl1l2 g2 � 1ð Þ p2 ¼

ZSþ

2g1 � g2 � 1g2 � 1

1r3

þ 3 1� g1 þ g2ð Þg2 � 1

sin2 /r3

� ��~vk k

þ 3 1� g1 þ g2ð Þg2 � 1

sin/ cos/r3

~uk k


sin/r2

h n; gð Þk k��

dS

þ 2p@ ~wk k@y

;

ð5:87bÞ

� 2p l1 þ k1l2ð Þl1l2 1� g2ð Þ p3 ¼

ZSþ

1þ g21� g2

~wk kr3

dSþ 2p@ ~uk k@x

þ @ ~vk k@y

� �; ð5:87cÞ

hz x; yð Þ ¼ � b1b22p b1 þ b2ð Þ

ZSþ

1r3

~h�� dS; ð5:87dÞ

where

g1 ¼l1 þ k1l2l1 þ l2

; g2 ¼l1 þ k1l2l2 þ k2l1

; k1 ¼ 3� 4v1; k2 ¼ 3� 4v2; ð5:88Þ

and p1, p2 and p3 are the tractions along the x-, y- and z-directions on the crack facesin the thermal-elastic medium, hz is the heat flux along the z-direction. l1, v1, l2 andv2 are, respectively, the shear modulus and Poisson’s ratio of Material 1 in the upperdomain and Material 2 in the lower domain; a1, b1, a2 and b2 are, respectively,coefficient of linear thermal expansion, and coefficient of thermal conductivity ofthe two bonded half spaces. In addition, ~uk k, ~vk k, ~wk k and ~h

�� are the displace-ment discontinuities in the x-, y- and z-directions and temperature discontinuity,respectively.


Comparing the coefficients in Eqs. (5.85) and (5.87), and letting:

1 L11 þ 2L12ð ÞL14

¼ 2g1 � g2 � 1g2 � 1

; ð5:89aÞ

� 1 L11 þ L12ð ÞL14

¼ 1� g1 þ g2g2 � 1

; ð5:89bÞ

L31 þ fL41ð Þ1 L1 þ fL2ð Þ ¼ 1þ g2

1� g2; ð5:89cÞ

1L13L14

¼ � a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2b1 þ b2ð Þ=2 ; ð5:89dÞ

L5 ¼ b1b22p b1 þ b2ð Þ ; ð5:89eÞ

andpx ¼ }p1; py ¼ }p2; ‘ pz þ fxð Þ ¼ }p3; ð5:89fÞ

the corresponding material constants are obtained:

1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL14L11

L31 þ fL41L1 þ fL2

s; ð5:90aÞ

g1 ¼ � 21L121L11 þ L14

; ð5:90bÞ

g2 ¼1L11 � L141L11 þ L14

; ð5:90cÞ

� a1 1þ v1ð Þb2 þ a2 1þ v2ð Þb1g2b1 þ b2ð Þ=2 ¼ 1L13

L14; ð5:90dÞ

b1b22p b1 þ b2ð Þ ¼ L5: ð5:90eÞ

From Eqs. (5.87) and (5.89f), one can obtain the coefficients:

} ¼ � 2pL14 l1 þ k1l2ð Þl1l2 g2 � 1ð Þ1 ; ð5:91aÞ

‘ ¼ � L141 L1 þ fL2ð Þ : ð5:91bÞ


Comparing Eqs. (5.85) and (5.87), it can be seen that the boundaryintegral-differential equations for interface cracks between the piezothermoelasticmedia and thermoelastic media have the identical forms. Therefore, the corre-sponding solutions are the same, namely

uk k ¼ ~uk k; vk k ¼ ~vk k; w�k k ¼ ~wk k; hk k ¼ ~h�� : ð5:92Þ

Equation (5.92) shows that the solution for the extended displacement discon-tinuity of the interface cracks in piezothermoelastic, bi-materials can be obtaineddirectly from the corresponding solution of the isotropic, thermoelastic bi-materials.

5.3.5.1 Singular Behavior Near the Interface Crack Front

The singular behavior of interface crack in a 3D, two-phase, isotropic, thermoelasticbi-material was analyzed by Zhao et al. [39]. Following the same procedure, thesingularity of the interface crack in a 3D transversely isotropic, piezothermoelasticbi-material is presented here. An arbitrary point P is chosen on the crack edge C ofcrack S. The edge C is smooth at point P. Without loss of generality, the Cartesiancoordinate system oxyz is oriented so that the x-direction and y-direction are normaland tangent to C, respectively. A small circular area R in S with a radius e iscentered at point P, as shown in Fig. 5.3.

The hyper-singular parts of Eq. (5.85) shall be finite in R for finite, prescribed,extended loadings, namely:

Fx ¼ZR

L11 þ 2L12r3

þ 3L12 � L11ð Þ cos2 /

r3

� ��uk k

� 3 L11 þ L12ð Þ sin/ cos/r3

vk kþ L13 cos/r2

hk k�dS n; gð Þ

þ 2pL141

@ w�k k@x

;

ð5:93aÞ

Fy ¼ZR

L11 þ 2L12r3

þ 3L12 � L11ð Þ sin2 /

r3

� ��vk k

� 3 L11 þ L12ð Þ sin/ cos/r3

uk kþ L13 sin/r2

hk k�dS n; gð Þ

þ 2pL141

@ w�k k@y

;

ð5:93bÞ


FMD ¼ L31 þ fL411

ZR

w�k kr3

dSþ 2p L1 þ fL2ð Þ @ uk k@x

þ @ vk k@y

� �; ð5:93cÞ

Fh ¼ L5

ZR

1r3

hk kdS; ð5:93dÞ

where Fx, Fy, FMD and Fh are functions of finite values of x; yð Þ 2 R.In the neighborhood of point P, for a small SR, the extended displacement

discontinuities are related only to the coordinates x in the normal plane throughpoint P as well as the position of point P along the contour of the crack [33, 40]:

uk k ¼ A1 Pð Þna1 ;vk k ¼ A2 Pð Þna2 ;w�k k ¼ A3 Pð Þna3 ;hk k ¼ A4 Pð Þna4 ;

8>><>>: ð5:94Þ

where A1, A2, A3 and A4 are all complex constants [33], and 0\Re ai½ �\1. InsertingEqs. (5.94) into (5.93), and making use of the following integral formulas:

lime!0

ZR

uk kr3

dndg ¼ �2A1 Pð Þpa1 cot pa1ð Þna1�1; ð5:95aÞ

lime!0

ZR

n� xð Þ2 uk kr5

dndg ¼ � 43A1 Pð Þpa1 cot pa1ð Þna1�1; ð5:95bÞ

lime!0

ZR

n� xð Þ g� yð Þ uk kr5

dndg ¼ 0; ð5:95cÞ

lime!0

ZR

g� yð Þ2 uk kr5

dndg ¼ � 23A1 Pð Þpa1 cot pa1ð Þna1�1; ð5:95dÞ

lime!0

ZR

g� yð Þ hk kr3

dndg ¼ 0; ð5:95eÞ

lime!0

ZR

n� xð Þ hk kr3

dndg ¼ �2A4 Pð Þ cot pa4ð Þna4 : ð5:95fÞ


Letting n ! 0, yields

cot pbð ÞA1 Pð Þþ cA3 Pð Þ ¼ 0;cot pa2ð ÞA2 Pð Þ ¼ 0;cA1 Pð Þ � cot pbð ÞA3 Pð Þ ¼ 0;cot pa4ð ÞA4 Pð Þ ¼ 0;

8>><>>: ð5:96Þ

where b ¼ a1 ¼ a3, c ¼ 1� g2ð Þ= 1þ g2ð Þ.As the constants Ai Pð Þ are generally assumed to be non-zero, the characteristics

to determine singularity behavior indices ai are found from Eqs. (5.96):

cot pa2ð Þ ¼ 0;cot pa4ð Þ ¼ 0;cot pbð Þ ¼ �ci,

8<: ð5:97Þ

where i ¼ ffiffiffiffiffiffiffi�1p

, and the acceptable roots are:

a2 ¼ a4 ¼ 12; b1;2 ¼

12� ij; j ¼ 1

2pln g2: ð5:98Þ

The roots show that the displacement discontinuity vk k and temperature dis-continuity hk k both have the classical singularity index 1/2, while uk k or w�k k hasthe singularity index 1=2þ ij as the case in purely elastic materials [33]. Takinginto consideration that the extended displacement discontinuities are in the neigh-borhood of the interface crack tip SR, they should have the following forms

uk k ¼ Re A1 Pð Þna½ �;vk k ¼ A2 Pð Þn1=2;w�k k ¼ Re A3 Pð Þna½ �;hk k ¼ A4 Pð Þn1=2;

8>><>>: ð5:99Þ

where a ¼ b1 ¼ 1=2þ ie, A2 Pð Þ and A4 Pð Þ are arbitrary real constants, whilstA1 Pð Þ and A3 Pð Þ are arbitrary complex constants which satisfy the relation ofA1 Pð Þ ¼ iA3 Pð Þ. So they can also be written as

w�k kþ i uk k ¼ AR Pð Þþ iAI Pð Þ½ �na ¼ A Pð Þna: ð5:100Þ

It should be pointed out that the above results are necessary for studying thesingular extended stress fields near the interface crack front exactly and for a uniquesolution for the hyper-singular, integral-differential Eq. (5.70).


5.3.5.2 Singular Fields Around Interfacial Cracksin Piezoethermoelastic Materials

Based on the expressions of the extended stress fields at an arbitrary field point inthe upper domain induced by the interface crack given in Eq. (5.68), the sametreatment proposed in [39] is adopted to study the singular extended stress fieldsahead of the crack front. In order to get the exact expressions for the singularextended stress fields at an arbitrary point N x; y; zð Þ near point P of the periphery Гof the interface crack, the local nature coordinate system Pnm, and the local polarcoordinates, Pqx, with PN as the negative normal axis are introduced as shown inFig. 5.2. The transformational relation between the two coordinate systems are

x ¼ �q cosx; y ¼ 0; z ¼ q sinx; R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ q cosxð Þ2 þ g2 þ q sinxð Þ2

q:

ð5:101Þ

For brevity, we only study the singular extended stresses at N0 in the interfacecrack front area (x ¼ 0). Substituting Eq. (5.101) into Eq. (5.68) and combiningwith the singular index in Eq. (5.98) as well as the form of the expressions for theextended displacement discontinuities in Eq. (5.99), one can have the followingdominant-part integrals [33, 39]:

limRe!0

ZSe

u n; gð Þk kR3 dS

¼ ip2 cosh pjð Þ A Pð Þ 1� 2ijð Þq�1=2�ij � A Pð Þ 1þ 2ijð Þq�1=2þ ij

h i;

ð5:102aÞ

limRe!0

ZSe

n� xð Þ2 u n; gð Þk kR5 dS

¼ ip3 cosh pjð Þ A Pð Þ 1� 2ijð Þq�1=2�ij � A Pð Þ 1þ 2ijð Þq�1=2þ ij

h i;

ð5:102bÞ

limRe!0

ZSe

n� xð Þ g� yð Þ u n; gð Þk kR5 dS ¼ 0; ð5:102cÞ

limRe!0

ZSe

v n; gð Þk kR3 dS ¼ pA2 Pð Þq�1=2; ð5:102dÞ

limRe!0

ZSe

g� yð Þ2 v n; gð Þk kR5 dS ¼ p

3A2 Pð Þq�1=2; ð5:102eÞ


limRe!0

ZSe

n� xð Þ g� yð Þ v n; gð Þk kR5 dS ¼ 0; ð5:102fÞ

limRe!0

ZSe

w� n; gð Þk kR3 dS

¼ p2 cosh pjð Þ A Pð Þ 1� 2ijð Þq�1=2�ij þA Pð Þ 1þ 2ijð Þq�1=2þ ij

h i;

ð5:102gÞ

limRe!0

ZSe

h n; gð Þk kR3 dS ¼ pA4 Pð Þq�1=2; ð5:102hÞ

limRe!0

ZSe

n� xð Þ h n; gð Þk kR3 dS ¼ �2A4 Pð Þq1=2; ð5:102iÞ

limRe!0

ZSe

g� yð Þ h n; gð Þk kR3 dS ¼ 0; ð5:102jÞ

where A Pð Þ is the conjugate of A Pð Þ, Re denotes the fan-shaped integral area withP as the origin and e as the radius. Substituting Eqs. (5.102) into (5.68), theextended stress fields of point P(q, 0) ahead of the crack front are obtained as

rzx q; 0ð Þ ¼ iL14epj

2 1� g2ð Þ1 A Pð Þ 1� 2ijð Þq�1=2�ij � A Pð Þ 1þ 2ijð Þq�1=2þ ijh i

þ 2L131� g2ð ÞA4 Pð Þq1=2;

ð5:103aÞ

ryz q; 0ð Þ ¼ pL14g11� g2ð Þ1A2 Pð Þq�1=2; ð5:103bÞ

‘ rz q; 0ð Þþ fDz q; 0ð Þ½ �¼ 2pL14epj

1� g1ð Þ1 A Pð Þ 1� 2ijð Þq�1=2�ij þA Pð Þ 1þ 2ijð Þq�1=2þ ijh i

;ð5:103cÞ

hz q; 0ð Þ ¼ �pL5A4 Pð Þq�1=2; ð5:103dÞ

which are analogous to the expressions in the isotropic, thermoelastic bi-materials[39].


5.3.6 Extended Stress Intensity Factors

In isotropic thermoelastic bi-materials, the stress and heat flux intensity factors aredefined as [39]:

KI ¼ limq!0


pq�ierz q; 0ð Þ; ð5:104aÞ

KII ¼ limq!0


pq�ierzx q; 0ð Þ; ð5:104bÞ

KIII ¼ limq!0


pryz q; 0ð Þ; ð5:104cÞ

Kh ¼ limq!0


phz q; 0ð Þ; ð5:104dÞ

and can be expressed in terms of the displacement and temperature discontinuities:

KI þ iKII ¼ffiffiffiffiffiffi2p

p l1l2 1þ 2ieð Þepel1 þ k1l2ð Þ lim

q!0

~w q; 0ð Þk kþ i ~u q; 0ð Þk kq1=2þ ie

; ð5:105aÞ


pl1l2

2 l1 þ l2ð Þ limq!0

~v q; 0ð Þk kffiffiffiq

p ; ð5:105bÞ


pb1b2

2 b1 þ b2ð Þ limq!0

~h q; 0ð Þ�� ffiffiffiq

p : ð5:105cÞ

Similarly, the extended stress intensity factors in piezothermoelastic bi-materialscan be defined as [86]:

KI2 ¼ limq!0


pq�ie‘ rz q; 0ð Þþ fx q; 0ð Þ½ �; ð5:106aÞ

KII ¼ limq!0


pq�ierzx q; 0ð Þ; ð5:106bÞ

KIII ¼ limq!0


pryz q; 0ð Þ; ð5:106cÞ

Kh ¼ limq!0


phz q; 0ð Þ: ð5:106dÞ


After inserting Eqs. (5.103) into (5.106), one can obtain the intensity factorsexpressed in terms of the extended displacement discontinuities:

KI2 þ iKII ¼ 2ffiffiffiffiffiffi2p

ppL14 1þ 2ijð Þepj

1� g2ð Þ1 limq!0

wk kþ L15 uk k=L14 þ i uk kq1=2þ ij

; ð5:107aÞ


pp

L14g11� g2ð Þ1 limq!0

v q; 0ð Þk kffiffiffiq

p ; ð5:107bÞ


ppL5 lim

q!0

h q; 0ð Þk kffiffiffiq

p ; ð5:107cÞ

The Mode I2 intensity factor KI2 is another new fracture parameter near the cracktip in piezothermoelastic bi-materials. And the new fracture parameters can be usedin the fracture criterion for piezothermoelastic interface crack problem. It can beobserved that the extended stress intensity factors are in the equivalent form withthe piezoelectric ones [37].

5.4 Fundamental Solutionsfor Magnetoelectrothermoelastic Bi-Materials

Magnetoelectrothermoelastic materials has an extra magnetic field coupled withelectric and thermoelastic field in comparison with piezothermoelastic materials.Electromagnetic coupling can be found in most electric conductors when magneticfield is applied. Integrity of magnetoelectrothermoelastic materials under coupledmultifield environment is essential to the application of these materials in advancedcontrol and actuation of modern microelectromechanical systems (MEMSs). Forhomogeneous materials, a lot of researches have been conducted [89–98].However, laminated structures are often used to get stronger, multiphysical cou-pling effect. Therefore, interface problems occur frequently, and the interface crackproblems of dissimilar mangetoelectroelastic bi-materials were investigated bymany researchers [99–106]. Here, employing the similar approaches as discussed inthe preceding sections, we present the fundamental solution framework for mane-toelectrothermoelastic materials.

When the xy-plane is parallel to the plane of isotropy in the Cartesian coordinates(x,y,z), the corresponding constitutive relations of transversely isotropic,thermo-magneto-electroelastic materials are [107, 108]


rx ¼ c11@u@x

þ c12@v@y

þ c13@w@z

þ e31@u@z

þ f31@w@z

� k11h

ry ¼ c12@u@x

þ c11@v@y

þ c13@w@z

þ e31@u@z

þ f31@w@z

� k11h

rz ¼ c13@u@x

þ c13@v@y

þ c33@w@z

þ e33@u@z

þ f33@w@z

� k33h

syz ¼ c44@v@z

þ @w@y

� �þ e15

@u@y

þ f15@w@y

szx ¼ c44@u@z

þ @w@x

� �þ e15

@u@x

þ f15@w@x

sxy ¼ c66@u@y

þ @v@x

� �

ð5:108aÞ

Dx ¼ e15@u@z

þ @w@x

� �� e11

@u@x

� g11@w@x

Dy ¼ e15@v@z

þ @w@y

� �� e11

@u@y

� g11@w@y

Dz ¼ e31@u@y

þ @v@x

� �þ e33

@w@z

� e33@u@z

� g33@w@z

þ p3h

ð5:108bÞ

Bx ¼ f15@u@z

þ @w@x

� �� g11

@u@x

� l11@w@x

By ¼ f15@v@z

þ @w@y

� �� g11

@u@y

� l11@w@y

Bz ¼ f31@u@y

þ @v@x

� �þ f33

@w@z

� g33@u@z

� l33@w@z

þm3h

ð5:108cÞ

hx ¼ �b11@h@x

; hy ¼ �b11@h@y

; hz ¼ �b33@h@z

ð5:108dÞ

where u, v, w, u, w and h are the mechanical elastic displacements, electricpotential, magnetic potential and temperature change (with respect to the referencetemperature), respectively, and referred to as extended displacements; cij, eij, fij, eij,gij, lij, kii, p3 and m3 are elastic, piezoelectric, piezomagnetic, dielectric, electro-magnetic, magnetic, thermal modulus, pyroelectric and pyromagnetic constants,respectively, and bij are the coefficients of heat conduction. The relation c66 ¼c11 � c12ð Þ=2 holds for materials with transverse isotropy.In the absence of body force, electric charge, electric current and body heat

source, the governing equations of TMEE materials in the Cartesian coordinatesystem o-xyz are given by Shechtman et al. [110]

5.4 Fundamental Solutions for Magnetoelectrothermoelastic … 221

@rx@x

þ @sxy@y

þ @szx@z

¼ 0

@sxy@x

þ @ry@y

þ @syz@z

¼ 0

@szx@x

þ @syz@y

þ @rz@z

¼ 0

ð5:109aÞ

@Dx

@xþ @Dy

@yþ @Dz

@z¼ 0 ð5:109bÞ

@Bx

@xþ @By

@yþ @Bz

@z¼ 0 ð5:109cÞ

@hx@x

þ @hy@y

þ @hz@z

¼ 0 ð5:109dÞ

where rij, Di, Bi, and hi, with i, j = x, y, z, are the components of stress, electricdisplacement, magnetic induction and heat flux, respectively, which are referred toas extended stresses here.

Assuming a penny-shaped crack with radius a centered at the origin of thecoordinate system is oriented at the interface of two bonded dissimilarmagneto-electro-thermo-elastic materials perpendicular to the poling direction, as isshown in Fig. 5.5. The crack lies in the plane xoy, and two bonded solids areassumed to be perfectly combined in the interface and occupy the upper and lowerhalf-space respectively. The upper and lower surfaces of the crack are denoted byS+ and S−, respectively, and

x ¼ r cos/; y ¼ r sin/ ð5:110Þ

Fig. 5.5 A penny-shaped interface crack of radius a perpendicular to the poling direction [86]


The EDDs across the surfaces of the crack can be expressed as:

uk k ¼ u x; y; 0þð Þ � u x; y; 0�ð Þ; vk k ¼ v x; y; 0þð Þ � v x; y; 0�ð Þwk k ¼ w x; y; 0þð Þ � w x; y; 0�ð Þ; uk k ¼ u x; y; 0þð Þ � u x; y; 0�ð Þwk k ¼ w x; y; 0þð Þ � w x; y; 0�ð Þ; hk k ¼ h x; y; 0þð Þ � h x; y; 0�ð Þ; x; yð Þ 2 S

ð5:111Þ

In analogy with Zhao et al. [40] and Zhao and Liu [41], let the radius of the cracka approach zero, the fundamental solutions corresponding to a unit concentratedEDD can all be obtained. It is required that the fundamental solution should satisfythe governing equations of thermo-magneto-electro-elasticity [37] and the follow-ing conditions [109], respectively:

lima!0

ZS

uk k; vk k; wk k; uk k; wk k; hk kf gdS ¼ 1; 0; 0; 0; 0; 0f g ð5:112aÞ

lima!0

ZS

uk k; vk k; wk k; uk k; wk k; hk kf gdS ¼ 0; 1; 0; 0; 0; 0f g ð5:112bÞ

lima!0

ZS

uk k; vk k; wk k; uk k; wk k; hk kf gdS ¼ 0; 0; 1; 0; 0; 0f g ð5:112cÞ

lima!0

ZS

uk k; vk k; wk k; uk k; wk k; hk kf gdS ¼ 0; 0; 0; 1; 0; 0f g ð5:112dÞ

lima!0

ZS

uk k; vk k; wk k; uk k; wk k; hk kf gdS ¼ 0; 0; 0; 0; 1; 0f g ð5:112eÞ

lima!0

ZS

uk k; vk k; wk k; uk k; wk k; hk kf gdS ¼ 0; 0; 0; 0; 0; 1f g ð5:112fÞ

Using the operator theory and the generalized Almansi’s theorem, Chen et al.[95] derived the general solution satisfying Eqs. (5.108) and (5.112). According tothis general solution, we have


U ¼ K iW0 þX5j¼1

Wj

!; wm ¼

X5j¼1

sjkmj@Wj

@zj; h ¼ k45

@2W5

@z25

r1 ¼ 2X5j¼1

c66 � x1js2j

@2Wj

@z2j; r2 ¼ 2c66K

2 iW0 þX5j¼1

Wj

!

rzm ¼X5j¼1

xmj@2Wj

@z2j; hx ¼ �b33k45

@3W5

@x@z35

hy ¼ �b33k45@3W5

@y@z25; hz ¼ �b33s5k45

@3W5

@z35

sm ¼ K is0qm@W0

@z0þX5j¼1

sjxmj@Wj

@zj

!; m ¼ 1; 2; 3

ð5:113Þ

The corresponding notations for all components in the Cartesian coordinates (x,y, z) and cylindrical coordinates (r, /, z) are defined as:

zj ¼ sjz; K ¼ @

@xþ i

@

@y

U ¼ uþ iv ¼ ei/ ur þ iuu� �

w1 ¼ w; w2 ¼ u; w3 ¼ w

r1 ¼ rx þ ry ¼ rr þ r/

r2 ¼ rx � ry þ 2isxy ¼ ei/ rr � r/ þ 2isr/� �

rz1 ¼ rz; rz2 ¼ Dz; rz3 ¼ Bz

sm ¼ sxm þ isym ¼ ei/ srm þ is/m� �

sx1 ¼ szx; sy1 ¼ syz; sr1 ¼ szr; s/1 ¼ s/zsx2 ¼ Dx; sy2 ¼ Dy; sr2 ¼ Dr; s/2 ¼ D/

sx3 ¼ Bx; sy3 ¼ By; sr3 ¼ Br; s/3 ¼ B/

ð5:114Þ

where all the concerning material related constants can be found in [109].The harmonic functions Wj (j = 0, 1, 2, 3, 4, 5) satisfy

Dþ @2

@z2i

� �Wj ¼ 0 ð5:115Þ

where

D ¼ @2

@x2þ @2

@y2in the Cartesian coordinates(x; y; zÞ ð5:116aÞ


and

D ¼ @2

@r2þ @

r@rþ @2

r2@/2 in cylindrical coordinatesðr;/; zÞ: ð5:116bÞ

The fundamental solution for the interface crack problem in thermoelectro-magnetic bi-materials can be obtained using similar procedures in the precedingsection, and details can be found in [109].

5.5 Fundamental Solutions for Interface Crack Problemsin Quasi-Crystalline Materials

The first quasicrystal (QC) was found by Shechtman et al. [110]. Since then, theatomic structure and physical properties of the QC material have been attractingincreasing attentions. As a promising solid structure, QCs possess a series of idealproperties such as low adhesion, low coefficient of friction, low porosity, lowthermal conductivity, high abrasion resistance, and high resistivity [111, 112], andhave been adopted progressively in high-tech industries, such as the automotive,aerospace and energy industries. Due to its quasi-periodic symmetry, concepts ofhigh-dimensional space have been introduced instead of the classical crystallo-graphic theory to describe the physical properties of QC materials. The phonon fieldrepresents the lattice vibrations while the phason field depicts the quasi-periodicrearrangement of atoms, and both fields are used to describe the elasticity of QCs.As a typical QC material, 1D, hexagonal QCs exhibit just one quasi-periodic axis,while the perpendicular plane of the axis exhibits the classical crystalline properties.The properties of QCs are very sensitive to defects, which are inevitable in QCmaterials. Extensive research has been performed on quasi-crystals with variousforms of defects [113–135]. Green’s functions for modelling the interface defects inquasi-crystal structures has recently been proposed as laminated structures isincreasingly introduced to elevate the coupling effect of multiphysical fields [136,137]. Employing the similar approaches in the preceding sections, we present thefundamental solutions for interface crack problems in quasi-crystals in this section.

In the absence of body forces, the constitutive relations for 1D hexagonal QCswith thermo-electro effect, referred to the Cartesian coordinate (x, y, z) with xoycoincident with the periodic plane and the z-axis identical to the quasi-periodicdirection, can be expressed as [138, 139]


rxx ¼ c11@ux@x

þ c12@uy@y

þ c13@uz@z

þR1@wz

@zþ e31

@u@z

� b1h;

ryy ¼ c12@ux@x

þ c11@uy@y

þ c13@uz@z

þR1@wz

@zþ e31

@u@z

� b1h;

rzz ¼ c13@ux@x

þ c13@uy@y

þ c33@uz@z

þR2@wz

@zþ e33

@u@z

� b3h;

ryz ¼ c44@uz@y

þ @uy@z

� �þR3

@wz

@yþ e15

@u@y

;

rzx ¼ c44@uz@x

þ @ux@z

� �þR3

@wz

@xþ e15

@u@x

;

rxy ¼ c66@ux@y

þ @uy@x

� �;

ð5:117aÞ

Hzy ¼ R3@uz@y

þ @uy@z

� �þK2

@wz

@yþ e015

@u@y

;

Hzx ¼ R3@uz@x

þ @ux@z

� �þK2

@wz

@xþ e015

@u@x

;

Hzz ¼ R1@ux@x

þ @uy@y

� �þR2

@uz@z

þK1@wz

@zþ e033

@u@z

;

ð5:117bÞ

Dx ¼ e15@uz@x

þ @ux@z

� �þ e015

@wz

@x� f11

@u@x

;

Dy ¼ e15@uz@y

þ @uy@z

� �þ e015

@wz

@y� f11

@u@y

;

ð5:117cÞ

Dz ¼ e31@ux@x

þ @uy@y

� �þ e33

@uz@z

þ e033@wz

@z� f33

@u@z

þ p3h;

qx ¼ �K11@h@x

; qy ¼ �K11@h@y

; qz ¼ �K33@h@z

;

ð5:117dÞ

where ui, wi, and u are, respectively, phonon displacements, phason displacements,and electric potential; rij, Hij, and cij (Ki) are, respectively, phonon stresses, phasonstresses, and elastic stiffness constants; Ri are phonon-phason relevant elasticconstants; bi are thermal constants; p3 denotes the pyroelectric constant; h repre-sents the temperature variation and h = 0 corresponds to a reference state; Di, Ei,and fii are electric displacements, electric fields, and dielectric coefficients,respectively; eij and e′ij are piezoelectric coefficients; and K11 and K33 are coeffi-cients of thermal conductivity.

Without body forces and free charges, the equilibrium equations for 1Dhexagonal piezoelectric QCs can be expressed as


@rxx@x

þ @rxy@y

þ @rxz@z

¼ 0;@ryx@x

þ @ryy@y

þ @ryz@z

¼ 0; ð5:118aÞ

@rzx@x

þ @rzy@y

þ @rzz@z

¼ 0;@Hzx

@xþ @Hzy

@yþ @Hzz

@z¼ 0; ð5:118bÞ

@Dx

@xþ @Dy

@yþ @Dz

@z¼ 0;

@qx@x

þ @qy@y

þ @qz@z

¼ 0: ð5:118cÞ

Comparing the basic equations for 1D QC materials to those for 3D transverselyisotropic magnetoelectrothermoelastic (METE) materials [109], we can find thespecial equivalent relations as listed in Table 5.1. That is to say, if the variables andthe coefficients in the governing equations for METEs are replaced respectively bythose for 1D QCs based on the equivalent relations in Table 5.1, we can obtain thegoverning equations for 1D QCs.

In column 3 of Table 5.1, u, v, w, u, w and h are the mechanical elastic dis-placements, electric potential, magnetic potential and temperature change, respec-tively, for METE material. cij, eij, fij, eij, gij, lij, kii, p3 and m3 are elastic,piezoelectric, piezomagnetic, dielectric, electromagnetic, magnetic, thermal mod-ulus, pyroelectric and pyromagnetic constants, respectively, and bij are the coeffi-cients of heat conduction. With the above analogy relation, one can get thesolutions for 1D QC materials directly from those for METEs [109].

Table 5.1 The analogy relation between 1D quasicrystal and magnetoelectrothermoelasticmaterials [145]

Material 1D QC 3D METE

Extended displacements ux; uy; uz u; v;w

wz;u; h w;u; h

Extended stresses rij; i; j ¼ x; y; z rij; i; j ¼ x; y; z

Dx;Dy;Dz Dx;Dy;Dz

Hzx;Hzy;Hzz Bx;By;Bz

qx; qy; qz hx; hy; hzCoefficients cij, R1;R2;R3 cij, f31; f33; f15

e15; e31; e33; e015; e033 e15; e31; e33;�g11;�g33

K11;K33; 0 b11;b33;m3

b1;b3; p3 k11; k33; p3K1;K2; f11; f33 �l33;�l11, e11; e33

5.5 Fundamental Solutions for Interface Crack Problems … 227

5.5.1 Fundamental Solutions for Unit-Point EDDs

Consider a penny-shaped crack with radius a centered at the origin of the coordinatesystem lies at the interface of two bonded dissimilar QC materials perpendicular tothe quasi-periodic direction, as illustrated in Fig. 5.4. The crack is located in theperiodic plane xoy. The two solids are assumed to be perfectly bonded along theinterface except the cracked segment and occupy the upper and lower half-space,respectively. The upper and lower surfaces of the crack are denoted by S+ and S−,respectively, and the relations between the Cartesian coordinates and cylindricalcoordinates are

x ¼ r cos/;y ¼ r sin/;R2 ¼ r2 þ z2 ¼ x2 þ y2 þ z2:

8<: ð5:119Þ

The EDDs across the interface crack faces can be expressed as:

uxk k ¼ ux x; y; 0þð Þ � ux x; y; 0�ð Þ; uy�� ¼ uy x; y; 0þð Þ � uy x; y; 0�ð Þ;

uzk k ¼ uz x; y; 0þð Þ � uz x; y; 0�ð Þ; wzk k ¼ wz x; y; 0þð Þ � wz x; y; 0�ð Þ;uk k ¼ u x; y; 0þð Þ � u x; y; 0�ð Þ; hk k ¼ h x; y; 0þð Þ � h x; y; 0�ð Þ; x; yð Þ 2 S:

ð5:120Þ

For unit-point phonon displacement discontinuity in z-direction, the fundamentalsolutions are given by Zhao et al. [140]

rþz ¼

X4j¼1

Aþ�j cþ1j

3z2jR5j

� 1R3j

!; ð5:121aÞ

H þzz ¼

X4j¼1

Aþ�j cþ2j

3z2jR5j

� 1R3j

!; ð5:121bÞ

Dþz ¼

X4j¼1

Aþ�j cþ3j

3z2jR5j

� 1R3j

!; ð5:121cÞ

rþzr ¼ 3

X4j¼1

Aþ�j bþ

1jrzjR5j

; ð5:121dÞ

rþzx ¼ 3

X4j¼1

Aþ�j bþ

1jxzjR5j

; ð5:121eÞ


rþyz ¼ 3

X4j¼1

Aþ�j bþ

1jyzjR5j

; ð5:121fÞ

qþz ¼ 0; ð5:121gÞ

where Rj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2j

q; and the coefficients Aþ�

j can be obtained by solving the

following equations [140]:

P5j¼1

Aþ�j ¼P5

j¼1A��j ;

�P5j¼1

aþ1j A

þ�j þ a�1jA

��j

h i¼ 1

2n ;

�P5j¼1

aþ2j A

þ�j ¼P5

j¼1a�2jA

��j ;

�P5j¼1

aþ3j A

þ�j ¼P5

j¼1a�3jA

��j ;

aþ45A

þ5 ¼ a�45A

�5 ;

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

P5j¼1

bþ1j A

þ�j ¼ �P5

j¼1b�1jA

��j ;

P5j¼1

cþ1j Aþ�j ¼P5

j¼1c�1jA

��j ;

P5j¼1

cþ2j Aþ�j ¼P5

j¼1c�2jA

��j ;

P5j¼1

cþ3j Aþ�j ¼P5

j¼1c�3jA

��j ;

K þ33 s

þ5 Aþ�

5 aþ45 ¼ �K�

33s�5 A

��5 a�45:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð5:122Þ

For unit-point phason displacement discontinuity in z-direction, the fundamentalsolutions are given by Zhao et al. [140]:

rþz ¼

X4j¼1

Bþ�j cþ1j

3z2jR5j

� 1R3j

!; ð5:123aÞ

H þzz ¼

X4j¼1

Bþ�j cþ2j

3z2jR5j

� 1R3j

!; ð5:123bÞ

Dþz ¼

X4j¼1

Bþ�j cþ3j

3z2jR5j

� 1R3j

!; ð5:123cÞ

rþzr ¼ 3

X4j¼1

Bþ�j bþ

1jrzjR5j

; ð5:123dÞ

rþzx ¼ 3

X4j¼1

Bþ�j bþ

1jxzjR5j

; ð5:123eÞ


rþyz ¼ 3

X4j¼1

Bþ�j bþ

1jyzjR5j

; ð5:123fÞ

qþz ¼ 0; ð5:123gÞ

where the coefficients Bþ�j can be obtained by solving the following equations:

P5j¼1

Bþ�j ¼P5

j¼1B��j ;

�P5j¼1

aþ1j B

þ�j ¼P5

j¼1a�1jB

��j ;

�P5j¼1

aþ2j B

þ�j þ a�2jB

��j

h i¼ 1

2p ;

�P5j¼1

aþ3j B

þ�j ¼P5

j¼1a�3jB

��j ;

aþ45B

þ�j ¼ a�45B

��j ;

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

P5j¼1

bþ1j B

þ�j ¼ �P5

j¼1b�1jB

��j ;

P5j¼1

cþ1j Bþ�j ¼P5

j¼1c�1jB

��j ;

P5j¼1

cþ2j Bþ�j ¼P5

j¼1c�2jB

��j ;

P5j¼1

cþ3j Bþ�j ¼P5

j¼1c�3jB

��j ;

K þ33 s

þ5 Bþ�

5 aþ45 ¼ �K�

33s�5 B

��5 a�45:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð5:124Þ

For unit-point electric potential discontinuity, the fundamental solutions aregiven by [140]:

rþz ¼

X4j¼1

Cþ�j cþ1j

3z2jR5j

� 1R3j

!; ð5:125aÞ

H þzz ¼

X4j¼1

Cþ�j cþ2j

3z2jR5j

� 1R3j

!; ð5:125bÞ

Dþz ¼

X4j¼1

Cþ�j cþ3j

3z2jR5j

� 1R3j

!; ð5:125cÞ

rþzr ¼ 3

X4j¼1

Cþ�j bþ

1jrzjR5j

; ð5:125dÞ

rþzx ¼ 3

X4j¼1

Cþ�j bþ

1jxzjR5j

; ð5:125eÞ


rþyz ¼ 3

X4j¼1

Cþ�j bþ

1jyzjR5j

; ð5:125fÞ

qþz ¼ 0; ð5:125gÞ

where coefficients Cþ�j can be obtained by solving the following equations:

P5j¼1

Cþ�j ¼P5

j¼1C��j ;

�P5j¼1

aþ1j C

þ�j ¼P5

j¼1a�1jC

��j ;

�P5j¼1

aþ2j C

þ�j ¼P5

j¼1a�2jC

��j ;

�P5j¼1

aþ3j C

þ�j þ a�3jC

��j

h i¼ 1

2p ;

aþ45C

þ�5 ¼ a�45C

��5 ;

P5j¼1

bþ1j C

þ�j ¼ �P5

j¼1b�1jC

��j ;

P5j¼1

cþ1j Cþ�j ¼P5

j¼1c�1jC

��j ;

P5j¼1

cþ2j Cþ�j ¼P5

j¼1c�2jC

��j ;

P5j¼1

cþ3j Cþ�j ¼P5

j¼1c�3jC

��j ;

K þ33 s

þ5 Cþ�

5 aþ45 ¼ �K�

33s�5 C

��5 a�45:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð5:126Þ

For unit-point temperature discontinuity, the fundamental solutions are given byZhao et al. [140]:

rþz ¼

X5j¼1

Dþ�j cþ1j

zjR3j; ð5:127aÞ

H þzz ¼

X5j¼1

Dþ�j cþ2j

zjR3j; ð5:127bÞ

Dþz ¼

X5j¼1

Dþ�j cþ3j

zjR3j; ð5:127cÞ

rþzr ¼

X5j¼1

Dþ�j bþ

1jrR3j; ð5:127dÞ

rþzx ¼

X5j¼1

Dþ�j bþ

1jxR3j; ð5:127eÞ


rþyz ¼

X5j¼1

Dþ�j bþ

1jyR3j; ð5:127fÞ

qþz ¼ K þ

33 sþ5 aþ

45Dþ�5

3z25R55

� 1R35

� �; ð5:127gÞ

where the coefficients Dþ�j can be obtained by solving the following equations:

P5j¼1

Dþ�j ¼P5

j¼1D��

j ;

�P5j¼1

aþ1j D

þ�j ¼P5

j¼1a�1jD

��j ;

�P5j¼1

aþ2j D

þ�j ¼P5

j¼1a�2jD

��j ;

�P5j¼1

aþ3j D

þ�j ¼P5

j¼1a�3jD

��j ;

aþ45D

þ�5 � a�45D

��5 ¼ 1

2p ;

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

P5j¼1

bþ1j D

þ�j ¼ �P5

j¼1b�1jD

��j ;

P5j¼1

cþ1j Dþ�j ¼P5

j¼1c�1jD

��j ;

P5j¼1

cþ2j Dþ�j ¼P5

j¼1c�2jD

��j ;

P5j¼1

cþ3j Dþ�j ¼P5

j¼1c�3jD

��j ;

K þ33 s

þ5 Dþ�

5 aþ45 ¼ �K�

33s�5 D

��5 a�45:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð5:128Þ

Meanwhile, the extended stresses for unit-point, in-plane displacement discon-tinuities are given as [140]:

rþz ¼ sin/ �

X4j¼1

pþ�1j cþ1j

3rzjR5j

; ð5:129aÞ

H þzz ¼ sin/ �

X4j¼1

pþ�1j cþ2j

3rzjR5j

; ð5:129bÞ

Dþz ¼ sin/ �

X4j¼1

pþ�1j cþ3j

3rzjR5j

; ð5:129cÞ

rþzr ¼ sin/ �

X4j¼1

bþ1j p

þ�1j

2r2 � z2jR5j

þ sþ0 qþ1 pþ�

21R30

!; ð5:129dÞ

rþzu ¼ � cos/ � sþ0 qþ

1 pþ�2

2r2 � z20R50

þX4j¼1

bþ1j p

þ�1j

1R3j

!; ð5:129eÞ


rþzx ¼ 3 sin/ cos/ �

X4j¼1

pþ�1j bþ

1j1R3j� z2jR5j

!þ sþ0 qþ

1 pþ�2

1R30� z20R50

� �" #;

ð5:129fÞ

rþyz ¼

X4j¼1

pþ�1j bþ

1j � 31R3j� z2jR5j

!sin2 /� 1

R3j

" #


2 31R30� z20R50

� �cos2 /� 1

R30

� �; ð5:129gÞ

qþz ¼ 0; ð5:129hÞ

for the unit-point phonon displacement discontinuity in y-direction, and

rþz ¼ cos/ �

X4j¼1

pþ�1j cþ1j

3rzjR5j

; ð5:130aÞ

H þzz ¼ cos/ �

X4j¼1

pþ�1j cþ2j

3rzjR5j

; ð5:130bÞ

Dþz ¼ cos/ �

X4j¼1

pþ�1j cþ3j

3rzjR5j

; ð5:130cÞ

rþzr ¼ cos/ �

X4j¼1

bþ1j p

þ�1j

2r2 � z2jR5j

þ sþ0 qþ1 pþ�

21R30

!; ð5:130dÞ

rþzu ¼ sin/ � sþ0 qþ

1 pþ�2

2r2 � z20R50

þX4j¼1

bþ1j p

þ�1j

1R3j

!; ð5:130eÞ

rþyz ¼ 3 sin/ cos/ �

X4j¼1

pþ�1j bþ

1j1R3j� z2jR5j

!þ sþ0 qþ

1 pþ�2

1R30� z20R50

� �" #;

ð5:130fÞ

rþzx ¼

X4j¼1

pþ�1j bþ

1j � 31R3j� z2jR5j

!cos2 /� 1

R3j

" #


2 31R30� z20R50

� �sin2 /� 1

R30

� � ð5:130gÞ

qþz ¼ 0 ð5:130hÞ


for the unit-point phonon displacement discontinuity in x-direction. The coefficientscan be obtained by solving the following equations [140]:

p��2 � pþ�

2 þ P5j¼1

p��1j � pþ�

1j

h i¼ 1

p ;

p��2 � pþ�

2 �P5j¼1

p��1j � pþ�

1j

h i¼ 0;

�P5j¼1

aþ1j p

þ�1j ¼P5

j¼1a�1jp

��1j ;

�P5j¼1

aþ2j p

þ�1j ¼P5

j¼1a�2jp

��1j ;

�P5j¼1

aþ3j p

þ�1j ¼P5

j¼1a�3jp

��1j ;

aþ45p

þ�15 ¼ a�45p

��15 ;

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

sþ0 qþ1 pþ�

2 þ s�0 q�1 p

��2 ¼ 0;P5

j¼1bþ1j p

þ�1j þ b�1jp

��1j

h i¼ 0;

P5j¼1

cþ1j pþ�1j ¼P5

j¼1c�1jp

��1j ;

P5j¼1

cþ2j pþ�1j ¼P5

j¼1c�2jp

��1j ;

P5j¼1

cþ3j pþ�1j ¼P5

j¼1c�3jp

��1j ;

�K þ33 s

þ5 aþ

45pþ�15 ¼ K�

33s�5 a

�45p

��15 :

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

ð5:131Þ

5.6 Application of General Solution in the Problemof an Interface Crack of Arbitrary Shape

The fundamental, general solutions obtained in the previous sections can be readilyused to obtain the analytical solutions for a regularly-shaped, interface crack in asmart medium under general, multiphysical loading, as long as the integral of thepoint force solution over the crack face has a closed form result. In other words, theobtained, point loading results works as the Green functions, and the exact solutioncan be obtained directly through integration over the crack area R provided. Forirregularly shaped interface crack, numerical method can be developed based onboundary integral equations of a general triangular element in the crack plane [141].Detailed derivations, discussions and related results for irregularly shaped, interfacecrack problems in thermoelastic bimaterials can be found in [141].

Similar approaches can be applied to interface crack problems in piezother-moelastic materials and magnetoelectrothermoelastic materials, as illustrated in[142, 143]. Figures 5.6 and 5.7 show some of the results from the numericalcalculations for an elliptical, interface crack under fixed multiphysical loading inpiezoelectric materials and magnetoelectrothermoelastic materials. The results wereobtained using the triangular elements and the numerical algorithms developed in[142, 143]. Clearly, the energy release rate at the minor axis tip of the ellipticalcrack shows a much higher value than that at the major axis tip, indicating that thecrack will prefer to grow in the direction of the minor axis and tends to become apenny-shaped one in the end. This finding coincides with what repeatedly seen inclassical texts of linear elastic fracture mechanics [144]. It is worth noting that when


the ratio of major to minor axis, a/b, equals 1, the results will become the solutionfor penny-shaped crack, which can be directly obtained from the analytical, fun-damental solutions through direct integration [141, 142]. Similar results forquasi-crystals can be found in [145].

5.7 Summary

The displacement and temperature discontinuity boundary integral-differentialequation method is developed to analyze an interface crack in an isotropic ther-moelastic bi-material, and the fundamental solutions for a unit point displacementand temperature discontinuities on the interface are proposed and the correspondinghyper-singular integral-differential equations for an arbitrarily shaped interfacial

Fig. 5.7 Energy release rateat the tips of major and minoraxes of an interface, ellipticalcrack under combinedloadings versus the ellipticityratio a/b formagnetoelectrothermoelasticmaterials. When a/b = 1, theresults represent the solutionfor a penny-shaped crack[143]

Fig. 5.6 Energy release rateat the tips of major and minoraxes under fixed loadingsversus the ellipticity ratio a/b under electric and thermalsemi-permeable boundaryconditions for an ellipticalinterface crack in athermopiezoelectricbi-material [142]

5.6 Application of General Solution in the Problem of an Interface … 235

crack are obtained. By analyzing the singular behavior, the stress and heat fluxintensity factors as well as the energy release rate are obtained. It can be observedthat the oscillatory singularity of the stress intensity factors are the same as theelastic one, independent of the thermal properties. It is also worth noting that theenergy release rate is in the same form as the elastic one, independent of the heatflux intensity factor.

References

1. Sih GC (1962) On the singular character of thermal stress near a crack tip. J Appl Mech29:587–589

2. Chen WH, Ting K (1985) Finite element analysis of mixed-mode thermoelastic fractureproblems. Nucl Eng Des 90:55–65

3. Brown EJ, Erdogan F (1968) Thermal stresses in bonded materials containing cuts on theinterface. Int J Eng Sci 6:517–529

4. Herrmann K, Braun H, Kemeny P (1979) Comparison of experimental and numericalinvestigations concerning thermal cracking of dissimilar materials. Int J Fract 15:187–190

5. Herrmann K, Grebner H (1982) Curved thermal crack growth in a bounded brittle two-phasesolid. Int J Fract 19:69–74

6. Martin-Moran CJ, Barber JR, Comninou M (1983) The penny-shaped interface crack withheat flow Part 1: perfect contact. ASME J Appl Mech 50:29–36

7. Barber JR, Comninou M (1983) The penny-shaped interface crack with heat flow part 2:imperfect contact. J Appl Mech 50:770–776

8. Takakuda K, Aoyagi K, Shibuya T, Koizumi T (1984) Steady state thermal stresses in abonded dissimilar medium with an external crack on the interface. JSME 27:2643–2650

9. Yuuki R, Cho SB (1989) Efficient boundary element analysis of stress intensity factors forinterface cracks in dissimilar materials. Eng Fract Mech 34:179–188

10. Lee KY, Shul CW (1991) Determination of thermal stress intensity factors for an interfacecrack under vertical uniform heat flow. Eng Fract Mech 40(6):1067–1074

11. Munz D, Yang YY (1992) Stress singularities at the interface in bonded dissimilar materialsunder mechanical and thermal loading. J Appl Mech 59:857–861

12. Chao CK, Chang RC (1994) Analytical solutions and numerical examples for thermoelasticinterface crack problems in dissimilar anisotropic media. J Thermal Stresses 17:285–299

13. Lee KY, Park SJ (1995) Thermal-stress intensity factors for partially insulated interfacecrack under uniform heat-flow. Eng Fract Mech 50:475–482

14. Wilson RI, Meguid SA (1995) On the determination of mixed mode stress intensity factorsof an angled cracks in a disc using FEM. Finite Ele Anal Des 18:433–438

15. Martynyak RM (1999) Thermal opening of an initially closed interface crack underconditions of imperfect thermal contact between its lips. Mater Sci 35:612–622

16. Zhang SC (2000) Thermal stress intensities at an interface crack between two elastic layers.Int J Fract 106:277–290

17. Herrmann KP, Loboda VV (2001) Contact zone models for an interface crack in athermomechanically loaded anisotropic bimaterial. J Thermal Stresses 24:479–506

18. Kharun IV, Loboda VV (2004) A thermoelastic problem for interface cracks with contactzones. Int J Solids Struct 41:159–175

19. Kharun IV, Loboda VV (2004) A problem of thermoelasticity for a set of interface crackswith contact zones between dissimilar anisotropic materials. Mech Mater 36:585–600

20. Herrmann KP, Loboda VV, Kharun IV (2004) Interface crack with a contact zone in anisotropic bimaterial under thermomechanical loading. Theor Appl Fract Mech 42:335–348


21. Ratnesh K, Chandra JM (2008) Thermal weight functions for bi-material interface cracksystem using energy principles. Int J Solids Struct 45:6157–6176

22. Pant M, Singh IV, Mishra BK (2010) Numerical simulation of thermo-elastic fractureproblems using element free Galerkin method. Int J Mech Sci 52:1745–1755

23. Khandelwal R, Chandra JM (2011) Thermal weight functions and stress intensity factors forbonded dissimilar media using body analogy. J Appl Mech 78:1–9

24. Ma LF, He R, Zhang JS, Shaw B (2013) A simple model for the study of the tolerance ofinterfacial crack under thermal load. Acta Mech 224:1571–1577

25. Bregman AM, Kassir MK (1974) Thermal fracture of bonded dissimilar media containing apenny-shaped crack. Int J Fract 10:87–98

26. Muskhelishvili NI (1958) Singular integral equations. Springer, Dordrecht27. Andrzej K, Stanislaw JM (2003) On the three-dimensional problem of an interface crack

under uniform heat flow in a bimaterial periodically-layered space. Int J Fract 123:127–13828. Johnson J, Qu J (2007) An interaction integral method for computing mixed mode stress

intensity factors for curved bimaterial interface cracks in non-uniform temperature fields.Eng Fract Mech 74:2282–2291

29. Nomura Y, Ikeda T, Miyazaki N (2010) Stress intensity factor analysis of a three-dimensional interfacial corner between anisotropic bimaterials under thermal stress. Int JSolids Struc 47:1775–1784

30. Guo JH, Yu J, Xing YM (2013) Anti-plane analysis on a finite crack in a one-dimensionalhexagonal quasicrystal strip. Mech Res Commun 52:40–45

31. Li L, Wu HP, Bao YM, Lu YL, Chai GZ (2013) Analysis of weight function method forthree dimensional interface crack under mechanical and thermal loading. Acta Mech Sinica34(4):401–409 (In Chinese)

32. Crouch SL (1976) Solution of plane elasticity problems by the displacement discontinuitymethod. I. Infinite body solution. Int J Numer Methods Eng 10:301–343

33. Tang RJ, Chen MC, Yue JC (1998) Theoretical analysis of three-dimensional interfacecrack. Sci Chin 41(4):443–448

34. Chen MC, Noda NA, Tang RJ (1999) Application of finite-part integrals to planar interfacialfracture problems in three-dimensional bimaterials. J Appl Mech 66:885–890

35. Chen MC, Gao C, Tang RJ (1999) A study on stress intensity factors and singular stressfields of 3D interface crack. Acta Mech Sinica 20(1):8–15 (In Chinese)

36. Zhao MH, Fang PZ, Shen YP (2004) Boundary integral-differential equations and boundaryelement method for interfacial cracks in three-dimensional piezoelectric media. Eng AnalBound Elem 28:753–762

37. Zhao MH, Li N, Fan CY, Xu GT (2008) Analysis method of planar interface cracks ofarbitrary shape in three-dimensional transversely isotropic magnetoelectroelastic bimaterials.Int J Solids Struct 45:1804–1824

38. Zhao YF, Zhao MH, Pan EN, Fan CY (2015) Green’s functions and extended displacementdiscontinuity method for interfacial cracks in three-dimensional transversely isotropicmagneto-electro-elastic bi-materials. Int J Solids Struct 52:56–71

39. Zhao M, Dang H, Fan C, Chen Z (2016) Analysis of an arbitrarily shaped interface cracks ina three-dimensional isotropic thermoelastic bi-material. Part 1: theoretical solution. Int JSolids Struct 97–98:168–181

40. Zhao MH, Shen YP, Liu YJ, Liu GN (1988) The method of analysis of cracks inthree-dimensional transversely isotropic media: boundary integral equation approach. EngAnal Bound Elem 21:169–178

41. Zhao MH, Liu YJ (1994) Boundary integral equations and the boundary element method forthree-dimensional fracture mechanics. Eng Anal Bound Elem 13:333–338

42. Hou PF, Jiang HY, Li QH (2013) Three-dimensional steady-state general solution forisotropic thermoelastic materials with applications 1: general solutions. J Therm Stresses36:727–747

43. Hutchinson JW, Mear ME, Rice JR (1987) Crack paralleling an interface between dissimilarmaterials. ASME J Appl Mech 54:828–832

References 237

44. Rao SS, Sunar M (1993) Analysis of distributed thermopiezoelectric sensors and actuatorsinadvanced intelligent structures. AIAA J 31:1280–1086

45. Mindlin RD (1974) Equations of high frequency vibrations of thermopiezoelectric crystalplates. Int J Solids Struct 10:625–637

46. Nowacki W (1978) Some general theorems of thermopiezoelectricity. J Therm Stresses1:171–182

47. Chandrasekharaiah DS (1988) A generalized linear thermoelasticity theroy for piezoelectricmedia. Acta Mech 71:39–49

48. Altay GA, Dokmeci MC (2003) Some comments on the higher order theories ofpiezoelectric, piezothermoelastic and thermopiezoelectric rods and shells. Int J Solids Struct40:4699–4706

49. Ashida F, Tauchert TR, Noda N (1994) A general solution technique for piezothermoe-lasticity of hexagonal solids of class 6 mm in cartesian coordinates. Math Mech 74:87–95

50. Ding HJ, Guo FL, Hou PF (2000) A general solution for piezothermoelasticity oftransversely isotropic piezoelectric materials and its applications. Int J Eng Sci38:1415–1440

51. Tarn JQ (2002) A state space formalism for piezothermoelasticity. Int J Solids Struct39:5173–5184

52. Hou PF, Leung AYT, Chen CP (2009) Three-dimensional fundamental solution fortransversely isotropic piezothermoelastic material. Int J Num Methods Eng 78:84–100

53. Yu SW, Qin QH (1996) Damage analysis of thermopiezoelectric properties: part I-crack tipsingularities. Theor Appl Fract Mech 25:263–277

54. Chen WQ (2000) On the general solution for piezothermoelasticity for transverse isotropywith application. ASME J Appl Mech 67:705–711

55. Gao CF, Wang MZ (2001) Collinear permeable cracks in thermopiezoelectric materials.Mech Mater 33:1–9

56. Chen WQ, Lim CW, Ding HJ (2005) Point temperature solution for a penny-shaped crack inan infinite transversely isotropic thermo-piezo-elastic medium. Eng Anal Bound Elem29:524–532

57. Qin QH (1998) Thermoelectroelastic Green’s function for a piezoelectric plate containing anelliptic hole. Mech Mater 30:21–29

58. Qin QH (1999) Thermoelectroelastic analysis of cracks in piezoelectric half-plane by BEM.Com Mech 23:353–360

59. Qin QH, Lu M (2000) BEM for crack-inclusion problems in thermopiezoelectric materials.Eng Fract Mech 69:577–588

60. Qin QH, Mai YW (2002) BEM for crack-hole problems in thermopiezoelectric materials.Eng Fract Mech 69:577–588

61. Shang FL, Wang ZK, Li ZH (1996) Thermal stresses anaysis of a three-dimensional crack ina thermopiezoelectric solid. Eng Fract Mech 55:737–750

62. Shang FL, Kuna M, Scherzer M (2002) Analytical solution for two penny-shaped crackproblems in thermo-piezoelectric materials and their finite element comparisons. Int J Fract117:113–128

63. Shang FL, Kuna M, Scherzer M (2003) Development of finite element techniques forthree-dimensional analyses of thermo-piezoelectric materials. ASME J Eng Mater Tech125:18–21

64. Shang FL, Kuna M (2003) Thermal stress around a penny-shaped crack in a thermopiezo-electric solid. Com Mater Sci 26:197–201

65. Niraula OP, Noda N (2002) Thermal stress analysis in thermopiezoelectric strip with an edgecrack. J Therm Stresses 25:389–405

66. Gao CF, Zhao YT, Wang MZ (2002) An exact and explicit treatment of an elliptic holeproblem in thermopiezoelectric media. Int J Solids Struct 39:2665–2685

67. Tsamasphyros G, Song ZF (2005) Analysis of a crack in a finite thermopiezoelectric plateunder heat flux. Int J Solids Struct 136:143–166


68. Ueda S (2006) Thermal stress intensity factors for a normal crack in a piezoeelctricstrip. J Therm Stresses 29:1107–1125

69. Ueda S, Hatano H (2012) T-shaped crack in a piezoelectric material underthermo-electro-mechanical loadings. J Therm Stresses 35:12–29

70. Ueda S, Tani Y (2008) Thermal stress intensity factors for two coplanar cracks in apiezoelectric strip. J Therm Stresses 31:403–415

71. Ueda S, Ikeda Y, Ishii A (2012) Transient thermoelectromecahnical response of apiezoelectric strip with two parallel cracks of different lengths. J Therm Stresses 35:534–549

72. Wang BL, Noda N (2010) Exact thermoelectroelasticity solution for a penny-shaped crack inpiezoelectric materials. J Therm Stresses 27:241–251

73. Wang BL, Sun YG, Zhu Y (2011) Fracture of a finite piezoelectric layer with apenny-shaped crack. Int J Fract 172:19–39

74. Zhong XC, Zhang KS (2013) An opening crack model for thermopiezoelectric solids. Eur JMech A-Solids 41:101–110

75. Shen S, Wang X, Kuang ZB (1995) Fracture mechanics for piezothermoelastic materials.Acta Mech Sinica 16(4):283–293 (In Chinese)

76. Shen S, Kuang ZB (1998) Interface crack in bi-piezothermoelastic media and the interactionwith a point heat source. Int J Solids Struct 35:3899–3915

77. Qin QH, Mai YW (1998) Thermoelectroelastic Green’s function and its application forbimaterial of piezoelectric materials. Arch Appl Mech 68:433–444

78. Qin QH, Mai YW (1999) A closed crack tip model for interface cracks in thermopiezo-electric materials. Int J Solids Struct 36:2463–2479

79. Gao CF, Wang MZ (2001) A permeable interface crack between dissimilar thermopiezo-electric media. Acta Mech 149:85–95

80. Williams ML (1959) The stresses around a fault or cracks in dissimilar media. Bull SeismolSoc Am 49:199–204

81. Herrmann KP, Loboda VV (2003) Fracture mechanical assessment of interface cracks withcontact zones in piezoelectric bimaterials under thermoelectromechanical loadings I.Electrically permeable interface cracks. Int J Solids Struct 40:4191–4217

82. Herrmann KP, Loboda VV (2003) Fracture mechanical assessment of interface cracks withcontact zones in piezoelectric bimaterials under thermoelectromechanical loadings II.Electrically impermeable interface cracks. Int J Solids Struct 40:4291–4237

83. Ueda S (2003) Thermally induced fracture of a piezoelectric laminate with a crack normal tointerfaces. J Therm Stresses 26:311–331

84. Herrmann KP, Loboda VV (2006) Contact zone approach for a moving interface crack in apiezoelectric bimaterial under thermoelectromechanical loading. Arch Appl Mech75:665–677

85. Hou PF, Leung AYT (2009) Three-dimensional Green’s functions for two-phasetransversely isotropic piezothermoelastic media. J Intell Mat Syst Struct 20:11–21

86. Zhao MH, Dang HY, Fan CY, Chen ZT (2017) Extended displacement discontinuity methodfor an interface crack in a three-dimensional transversely isotropic piezothermoelasticbi-material. Part 1: Theoretical solution. Int J Solids Struct 117:14–25

87. Fan CY, Dang HY, Zhao MH (2014) Nonlinear solution of the PS model for asemi-permeable crack in a 3D piezoelectric medium. Eng Anal Bound Elements 46:23–29

88. Li XF, Lee KY (2015) Effect of heat conduction of penny-shaped crack interior on thermalstress intensity factors. Int J Heat Mass Transf 91:127–134

89. Wang BL, Niraula OP (2007) Transient thermal fracture analysis of transversely isotropicmagneto-electro-elastic materials. J Therm Stresses 30:297–317

90. Feng WJ, Pan E, Wang X (2008) Stress analysis of a penny-shaped crack in a magneto-electro-thermo-elastic layer under uniform heat flow and shear loads. J Therm Stresses31:497–514

91. Aboudi J (2001) Micromechanical analysis of fully coupled electro-magneto-thermo-elasticmultiphase composites. Smart Mater Struct 10:867–877

References 239

92. Gao CF, Kessler H, Balke H (2003) Fracture analysis of electromagnetic thermoelasticsolids. Eur J Mech A-Solid 22:433–442

93. Niraula OP, Wang BL (2006) Thermal stress analysis in magneto-electro-thermo-elasticitywith a penny-shaped crack under uniform heat flow. J Therm Stresses 29:423–437

94. Zhong XC, Huang QA (2014) Thermal stress intensity factor for an opening crack inthermomagnetoelectroelastic solids. J Therm Stresses 37:928–946

95. Chen WQ, Lee KY, Ding HJ (2004) General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. Int J Eng Sci 42:1361–1379

96. Hou PF, Yi T, Wang L (2008) 2D general solution and fundamental solution for orthotropicelectro-magneto-thermo-elastic materials. J Therm Stresses 31:807–822

97. Hou PF, Chen HR, He S (2009) Three-dimensional fundamental solution for transverselyisotropic electro-magneto-thermo-elastic materials. J Therm Stresses 32:887–904

98. Li Y, Dang HY, Xu GT, Fan CY, Zhao MH (2016) Extended displacement discontinuityboundary integral equation and boundary element method for cracks in thermo-magneto-electro-elastic media. Smart Mater Struct 25:085048

99. Gao CF, Tong P, Zhang TY (2003) Interfacial crack problems in magneto-electric solids.Int J Eng Sci 41:2105–2121

100. Chue CH, Liu TJ (2005) Magneto-electro-elastic antiplane analysis of a bimaterial BaTiO3-CoFe2O4 composite wedge with an interface crack. Theor Appl Fract Mech 44:275–296

101. Li R, Kardomateas GA (2006) The mode III interface crack in piezo-electro-magneto-elasticdissimilar bimaterials. J Appl Mech Trans ASME 73:220–227

102. Zhong XC, Li XF (2006) A finite length crack propagating along the interface of twodissimilar magnetoelectroelastic materials. Int J Eng Sci 44:1394–1407

103. Ma P, Feng WJ, Su RKL (2011) Fracture assessment of an interface crack between twodissimilar magnetoelectroelastic materials under heat flow and magnetoelectromechanicalloadings. Acta Mechanica Solids Sinica 24:0894–9166

104. Herrmann KP, Loboda VV, Khodanen TV (2010) An interface crack with contact zones in apiezoelectric/piezomagnetic bimaterial. Arhive Appl Mech 80:651–670

105. Feng WJ, Ma P, Su RKL (2012) An electrically impermeable and magnetically permeableinterface crack with a contact zone in magneto-electroelastic bimaterials under a thermal fluxand magnetoelectromechanical loads. Int J Solids Struct 49:3472–3483

106. Ma P, Jing J (2013) Further analysis for fracture behaviors of an interfacial crack betweenpiezoelectric and piezomagnetic layers. Engs Mech 30:327–333

107. Li JY, Dunn ML (1998) Anisotropic coupled-field inclusion and inhomogeneity problems.J Intell Mater Syst Struct 7:404

108. Huang JH, Kuo WS (1997) The analysis of piezoelectric/piezomagnetic composite materialscontaining ellipsoidal inclusions. J Appl Phys 81:1378

109. Zhao MH, Dang HY, Fan CY, Chen ZT (2017) Analysis of an interface crack of arbitraryshape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bi-material.Part 1: theoretical solution. J Therm Stresses 40(8):929–952

110. Shechtman D, Blech I, Gratias D, Cahn JW (1984) Metallic phase with long-rangeorientational order and no translational symmetry. Phys Rev Lett 53:1951–1953

111. Dubois JM, Kang SS, Stebut JV (1991) Quasicrystalline low-friction coatings. J Mater SciLett 10:537–541

112. Wu JS, Brien V, Brunet P, Dong C, Dubois JM (2000) Scratch-induced surfacemicrostructures on the deformed surface of Al-Cu-Fe icosahedral quasicrystals. Mater SciEng 4:294–296

113. Fan TY, Tang ZY, Chen WQ (2012) Theory of linear, nonlinear and dynamic fracture ofquasicrystals. Eng Fract Mech 82:185–194

114. Li PD, Li XY, Kang GZ (2015) Crack tip plasticity of half-infinite Dugdale crack embeddedin an infinite space of one-dimensional hexagonal quasicrystal. Mech Res Commun70:72–78


115. Wang YW, Wu TH, Li XY, Kang GZ (2015) Fundamental elastic field in an infinite mediumof two-dimensional hexagonal quasicrystal with a planar crack: 3D exact analysis. Int JSolids Struct 66:171–183

116. Tupholme GE (2015) Row of shear cracks moving in one-dimensional hexagonalquasicrystalline materials. Eng Fract Mech 134:451–458

117. Sladek J, Sladek V, Atluri SN (2015) Path-independent integral in fracture mechanics ofquasicrystals. Eng Fract Mech 140:61–71

118. Fan CY, Li Y, Xu GT, Zhao MH (2016) Fundamental solutions and analysis of three-dimensional cracks in one-dimensional hexagonal piezoelectric quasicrystals. Mech ResCommun 74:39–44

119. Wang X (2004) Eshelby’s problem of an inclusion of arbitrary shape in a decagonalquasicrystalline plane or half-plane. Int J Eng Sci 42:1911–1930

120. Gao Y, Ricoeur A (2012) Three-dimensional analysis of a spheroidal inclusion in atwo-dimensional quasicrystal body. Phil Mag 92:4334–4353

121. Guo JH, Zhang ZY, Xing YM (2016) Antiplane analysis for an elliptical inclusion in 1Dhexagonal piezoelectric quasicrystal composites. Phil Mag 96:349–369

122. Guo JH, Pan E (2016) Three-phase cylinder model of one-dimensional hexagonalpiezoelectric quasicrystal composites. J Appl Mech 83:0810071

123. Fan TY, Trebin HR, Messerschmidt U, Mai YW (2004) Plastic flow coupled with a crack insome one- and two-dimensional quasicrystals. J Phys: Condens Matter 16:5229–5240

124. Li HL, Liu GT (2012) Stroh formulism for icosahdral quasicrystal and its applications. PhysLett A 376:987–990

125. Lazar M, Agiasofitou E (2014) Fundamentals in generalized elasticity and dislocation theoryof quasicrystals: green tensor, dislocation key-formulas and dislocation loops. Phil Mag94:4080–4101

126. Chen WQ, Ma YL, Ding HJ (2004) On three-dimensional elastic problems of one-dimensional hexagonal quasicrystals bodies. Mech Res Commun 31:633–641

127. Wang X (2006) The general solution of one-dimensional hexagonal quasicrystal. Mech ResComm 33:576–580

128. Li XY, Fan TY (2008) Exact solutions of two semi-infinite collinear cracks in a strip of onedimensional hexagonal quasicrystal. Appl Math Comput 196:1–5

129. Guo JH, Lu ZX (2011) Exact solution of four cracks originating from an elliptical hole inone-dimensional hexagonal quasicrystals. Appl Math Comput 217:9397–9403

130. Li XY, Li PD (2012) Three-dimensional thermo-elastic general solutions of one-dimensionalhexagonal quasi-crystal and fundamental solutions. Phys Lett A 376:2004–2009

131. Li XY, Deng H (2013) On 2D Green’s functions for 1D hexagonal quasi-crystals. Phys B430:45–51

132. Li XY, Wang T, Zheng RF, Kang GZ (2015) Fundamental thermo-electro-elastic solutionsfor 1D hexagonal QC. Z Angew Math Mech 95:457–468

133. Yang J, Li X (2014) The anti-plane shear problem of two symmetric cracks originating froman elliptical hole in 1D hexagonal piezoelectric QCs. Adv Mater Res 936:127–135

134. Yang J, Li X (2015) Analytic solutions of problem about a circular hole with a straight crackin one-dimensional hexagonal quascrystals with piezoelectric effects. Theor Appl FractMech 82:17–24

135. Yu J, Guo JH, Pan E, Xing YM (2015) General solutions of plane problem inone-dimensional quasicrystal piezoelectric materials and its application on fracturemechanics. Appl Math Mech 36:793–814

136. Li PD, Li XY, Zheng RF (2013) Thermo-elastic Green’s functions for an infinite bi-materialof one-dimensional hexagonal quasi-crystals. Phys Lett A 377:637–642

137. Zhang LL, Wu D, Xu WS, Yang LZ, Ricoeur A, Wang ZB, Gao Y (2016) Green’s functionsof one-dimensional quasicrystal bi-material with piezoelectric effect. Phys Lett A380:3222–3228

138. Ding HJ, Yang WG, Hu CZ, Wang RH (1993) Generalized elasticity theory of quasicrystals.Phys Rev B 48:7003–7010

References 241

139. Wang X, Pan E (2008) Analytical solutions for some defect problems in 1D hexagonal and2D octagonal quasicrystals. Pramana J Phys 70:911–933

140. Zhao MH, Dang HY, Fan CY, Chen ZT (2017) Analysis of a three-dimensional arbitrarilyshaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystalbi-material. Part 1: theoretical solution. Eng Fract Mech 179:59–78

141. Dang HY, Zhao MH, Fan CY, Chen ZT (2016) Analysis of an arbitrarily shaped interfacecracks in a three-dimensional isotropic thermoelastic bi-material. Part 2: numerical method.Int J Solids Struct 99:48–56

142. Dang HY, Zhao MH, Fan CY, Chen ZT (2017) Extended displacement discontinuity methodfor an interface crack in a three-dimensional transversely isotropic piezothermoelasticbi-material. Part 2: numerical method. Int J Solids Struct 109:199–209

143. Dang HY, Zhao MH, Fan CY, Chen ZT (2017) Analysis of an interface crack of arbitraryshape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bi-material.Part 2: numerical method. J Therm Stresses 40(8):953–972

144. Anderson TL (1991) Fracture mechanics, fundamentals and applications. CRC Press145. Dang HY, Zhao MH, Fan CY, Chen ZT (2017) Analysis of a three-dimensional arbitrarily

shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystalbi-material. Part 2: numerical method. Eng Fract Mech 180:268–281


Chapter 6Advanced Thermal Fracture AnalysisBased on Non-Fourier Heat ConductionModels

6.1 Introduction

In this chapter, the non-Fourier heat conduction models such as the hyperbolic heatconduction, dual phase lag heat conduction, and the memory-dependent fractionalheat conduction models are used to deal with crack problems in advanced com-posite materials. A few typical examples, such as cracks in a half-plane with a thinfilm coating, partially-insulated crack with thermal insulation interior, circumfer-ential crack in a hollow cylinder and viscoelastic materials are be presented toillustrate the use of the models and the unique features of the heat conductionmodels revealed in these prolems.

6.2 Hyperbolic Heat Conduction in a Cracked Half-Planewith a Coating

High-rate heat transfer has become a major concern in modern industries especiallyin material processing, such as the application of pulsed laser heating in additivemanufacturing. Recently, very strong substrate/coating interfaces have beenobtained via pulsed laser coating of bioceramic/metal nanomaterials on metalsubstrates [1]. Investigation of the temperature field is essential to calculatingthermal stresses within materials fabricated by advanced manufacturing, which isnecessary to understand the problem of thermal damage, and accurate heat con-duction analysis is of great importance for the structural integrity.

As discussed intensively in the literature, the Fourier, parabolic heat conductionmodel, although provides sufficient accuracy for many engineering applications,implies infinite thermal wave propagation speed and is ineffective at the very smalllength and time scales associated with small-scale systems [2, 3]. For many tech-nological applications that involving high thermal energy with extremely short


243




https://doi.org/10.1007/978-3-030-25201-4_6

time, the results predicted using the parabolic heat conduction model differ sig-nificantly from the experimental results [4, 5]. Examples include the transienttemperature field caused by pulsed laser heating of thin structures, and the measuredsurface temperature of a slab immediately after an intense thermal shock, which was300 °C higher than that obtained from the parabolic heat conduction model [6, 7].

As discussed before, when relaxation, or the time lag of heat flux is considered ina thermal process, one has a hyperbolic heat conduction equation, which implies afinite speed for heat transport [8]. Consideration of the hyperbolic heat conductionmodel becomes important if irreversible physical processes, such as crack or voidinitiation in a solid, are involved in the process of heat transport. In applicationsinvolving high rate heating where extremely small time scales are concerned, it isappropriate to use the hyperbolic heat conduction model [9, 10].

Inherent defects in materials such as dislocations and cracks may disturb thetemperature distribution when thermal loading is applied to the material, and sin-gular stress and thermal fields may be developed in the neighborhood of discon-tinuities. Some studies have been devoted to studying the singular behavior oftemperature gradient around crack tip based on the classical Fourier heat conductionmodel [11–13]. Many researchers have paid attention to the effect of cracks, holesand other defects under thermal loading in advanced materials using the Fourierheat conduction model [14–23].

A few investigations on crack problems in thermo-elastic materials have beenmade using the hyperbolic heat conduction model. Among them are Manson andRosakis [24], who derived a solution of the hyperbolic heat conduction equation fora travelling point heat source around a propagating crack tip, and measured thetemperature distribution at the tip of a dynamically propagating crack. Tzou [25]investigated the near-tip, thermal field around a moving crack, and evaluated theeffect of crack velocity on the thermal shock waves. The transient thermal stressesaround a crack in a strip and a half-plane were recently investigated in [26, 27]under thermal impact loading.

During laser manufacturing processes, high-energy pulsed laser beams arerapidly moving along the workpiece to generate various surface topologies asdesigned, leading to extremely high, local temperature gradients. Occasionally,structures are suffered from damage due to the applied thermal loading. The localthermal stresses near the crack can be elevated by the intensified temperaturegradient, which may initiate crack propagation or breakdown of the structure evenunder normal thermal conditions. An accurate analysis of the intensification oftemperature gradient near the crack is essential to predicting the failure behaviorunder thermal loading [28].

In this section, we present the theoretical framework to investigate the transienttemperature field around a crack in a substrate bonded by a coating under a thermalimpact using the hyperbolic heat conduction model. Considering that the substrateis much thicker than coating, the crack problem of a half-plane bonded to a coatingstrip is investigated, approximately equivalent to the actual situation. Fourier andLaplace transforms are employed to establish the singular integral equation aboutthe temperature field. The singular integral equation is solved numerically and the

244 6 Advanced Thermal Fracture Analysis Based on Non-Fourier …

asymptotic fields around the crack tips are obtained. Laplace inversion is thenapplied to get the temperature field in the time domain. The effect of the parametersof the hyperbolic heat conduction model and the geometric size of the composite onthe temperature disturbance is demonstrated. The results of the current problem leadto the existing solution based on parabolic heat conduction model when therelaxation time vanishes. The results based on hyperbolic heat conduction modelshow much higher dynamic temperature disturbance in the very early stages ofimpact comparing to the parabolic model. The theoretical framework can be easilyextended to investigate the thermomechanical behavior of cracked structures ofdifferent geometries or thermal loading conditions.

6.2.1 Basic Equations

Consider a thermoelastic half-plane containing a crack of length 2c parallel to theinterface between the half-plane and the coating, as shown in Fig. 6.1. Thehalf-plane is initially at the uniform temperature zero, and the free surface of thecoating at y ¼ �ðaþ bÞ is suddenly heated to a temperature T0. The crack surfacesare assumed to be thermally insulated, which indicates no heat flux can go throughthe crack surfaces. This kind of thermal conditions can be observed in saturated,porous materials where the interior of cracks is filled by thermally insulated fluids.In this study, the effects of inertia and thermal-elastic coupling are neglected whichleads to an uncoupled, quasi-static problem.

In the heat-transfer process involving high temperature gradients, large heatfluxes or short, transient durations, the heat propagation speed is finite. Fourier’slaw can be modified with a time lag of heat flux in response to a temperaturedisturbance in the following form [29]

cc

a

x

y

b

T0 · H(t)

Fig. 6.1 A thin layer(coating) on top of a crackedhalf-plane substrate under asudden thermal shock [34]

6.2 Hyperbolic Heat Conduction in a Cracked Half-Plane … 245

qþ s@q@t

¼ �k � rT ð6:1Þ

where q is the heat flux (thermal heat current density), T is the temperature, k is thethermal conductivity of the material, r is the spatial gradient operator, and s is theso-called relaxation time (a non-negative constant), or build-up period for thecommencement of heat flow after a temperature gradient has been imposed on themedium.

The local energy balance equation with vanishing heat source can be expressedas [30]

�rq ¼ qC � @T@t

ð6:2Þ

where q and C are the mass density and the specific heat capacity, respectively.Incorporating Eq. (6.1) with Eq. (6.2) leads to the hyperbolic heat conductionequation for the substrate and the coating,

di � DT ðiÞ ¼ @TðiÞ

@tþ si

@2TðiÞ

@t2; ði ¼ 1; 2Þ ð6:3Þ

where D is Laplace’s differential operator, si are the relaxation times for the sub-strate and the coating, respectively; ki are the thermal conductivity of the materialsði ¼ 1; 2Þ, and di ¼ ki

qiCiði ¼ 1; 2Þ are the thermal diffusivities for the substrate and

the coating, respectively.It should be noted that the relaxation time for most engineering materials are of

the order of 10�14 to 10�6 s, but experiments have shown that some nonhomoge-neous materials have relaxation time up to 10 s which are very important materialsoften used as thermal insulators [31–33].

Introducing the following dimensionless variables

�x;�y; �a; �bð Þ ¼ x; y; a; bð Þ=cT ¼ T=T0; �t ¼ td0=c

2ð6:4Þ

where T0 is the reference temperature and d0 is the reference thermal diffusivity (wecan choose either d1 or d2), the governing Eq. (6.3) have the following dimen-sionless forms:

r2TðiÞ ¼ @TðiÞ

@td0di

þ sid20c2di

@2T ðiÞ

@t2; ði ¼ 1; 2Þ ð6:5Þ

It is noted that in the Eq. (6.5) and hereafter, the hat “ ” of the dimensionlessvariables is omitted for simplicity.


The hyperbolic heat Eq. (6.5) is subjected to the following boundary and initialconditions in dimensionless forms

Tðx;�ðaþ bÞÞ ¼ T0; ð xj j\1; t[ 0Þ ð6:6Þ

Tðx;�aþ Þ ¼ Tðx;�a�Þ; ð xj j\1; t[ 0Þ ð6:7Þ

k1@Tðx;�aþ Þ

@y¼ k2

@Tðx;�a�Þ@y

; ð xj j\1; t[ 0Þ ð6:8Þ

@Tðx; 0Þ@y

¼ 0; ð xj j\1Þ ð6:9Þ

Tðx; 0þ Þ ¼ Tðx; 0�Þ; ð xj j � 1Þ ð6:10Þ

@Tðx; 0þ Þ@y

¼ @Tðx; 0�Þ@y

; ð xj j � 1Þ ð6:11Þ

T ¼ 0; ðt ¼ 0Þ ð6:12Þ@T@t

¼ 0; ðt ¼ 0Þ ð6:13Þ

6.2.2 Temperature Field

Apply Laplace transform to Eq. (6.5):

TðiÞ�ðx; y; pÞ ¼ L T ðiÞðx; y; tÞ� �

¼Z10

T ðiÞðx; y; tÞ expð�ptÞdt

TðiÞðx; y; tÞ ¼ L�1 T ðiÞ�ðx; y; pÞ� �

¼ 12pi

ZBr

T ðiÞ�ðx; y; pÞ expðptÞdpð6:14Þ

where Br stands for the Bromwich path of integration. Considering the initialconditions (6.12) and (6.13), we have:

r2T ðiÞ� ¼ AipTðiÞ� þBip

2T ðiÞ�; ði ¼ 1; 2Þ ð6:15Þ


with Ai and Bi defined as

Ai ¼ d0di; Bi ¼ sid20

c2diði ¼ 1; 2Þ ð6:16Þ

The boundary conditions in the Laplace transform plane (p-plane) are:

T�ðx;�ðaþ bÞÞ ¼ T0=p; ð xj j\1Þ ð6:17Þ

T�ðx;�aþ Þ ¼ T�ðx;�a�Þ; ð xj j\1Þ ð6:18Þ

k1@T�ðx;�aþ Þ

@y¼ k2

@T�ðx;�a�Þ@y

; ð xj j\1Þ ð6:19Þ

@T�ðx; 0Þ@y

¼ 0; ð xj j\1Þ ð6:20Þ

T�ðx; 0þ Þ ¼ T�ðx; 0�Þ; ð xj j � 1Þ ð6:21Þ

@T ðx; 0þ Þ@y

¼ @T�ðx; 0�Þ@y

; ð xj j � 1Þ ð6:22Þ

The appropriate temperature field in the Laplace domain satisfying the boundarycondition and regularity condition can be expressed as

Tð1Þ�ðx; y; pÞ ¼Z1�1

E1ðnÞ expð�ryÞ expð�ixnÞdn

þWðy; pÞ; for y� 0

ð6:23Þ

Tð1Þ�ðx; y; pÞ ¼Z1�1

E2ðnÞ expðryÞþE3ðnÞ expð�ryÞ½ � expð�ixnÞdn

þWðy; pÞ; or � a� y� 0

ð6:24Þ

and

T ð2Þ�ðx; y; pÞ ¼Z1�1

D1ðnÞ expðnyÞþD2ðnÞ expð�nyÞ½ � expð�ixnÞdn

þVðy; pÞ; for � ðaþ bÞ� y� � a

ð6:25Þ


where EiðnÞ ði ¼ 1; 2; 3Þ and DjðnÞ ðj ¼ 1; 2Þ are unknowns to be determined;functionsWðy; pÞ, Vðy; pÞ, r and n can be found in [34]. Application of Eqs. (6.15)–(6.16) leads to the expressions of D2ðnÞ and EiðnÞ (i = 1, 2) as the functions ofD1 nð Þ [34].

We introduce the temperature density function as:

/ðxÞ ¼ @Tð1Þ�ðx; 0þ Þ@x

� @T ð1Þ�ðx; 0�Þ@x

; ð6:26Þ

It is clear from the boundary conditions (6.21) and (6.22) that

Z1�1

/ðtÞdt ¼ 0 ð6:27Þ

and

/ðxÞ ¼ 0; ð xj j � 1Þ ð6:28Þ

Substituting Eqs. (6.23) and (6.24) into Eq. (6.26) considering Eqs. (6A3a–d) andusing Fourier inverse transform, we have:

D1ðnÞ ¼ 1i4pk1n

Z1�1

/ðsÞ expðisnÞds ð6:39Þ

Substituting Eq. (6.29) into Eq. (6.20) and applying the relation (6A3a–d), we getthe singular integral equation for /ðxÞ as follows

Z1�1

/ tð Þ 1t � x

þH x; tð Þ� �

dt ¼ 2pqf ; ð xj j\1Þ ð6:30Þ

where the kernel function Hðx; tÞ is given as

Hðx; tÞ ¼Z10

1� rðk1 � k2Þk1n

� �sin nðx� tÞ½ �dn ð6:31Þ

and k1; k2 are defined in [34].


The integral Eq. (6.30) under singled-value condition (6.27) has the followingform of solution [34]:

/ðxÞ ¼ UðxÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p ; xj j\1 ð6:32Þ

where UðxÞ is bounded and continuous on the interval [−1, 1]. From the propertiesof symmetry or from the condition (6.27), it is seen that UðxÞ is an odd function ofx, i.e.,

Uð�xÞ ¼ �UðxÞ ð6:33Þ

Following the numerical techniques of Erdogan [34], Eqs. (6.30) and (6.27) can besolved at discrete points as

XNk¼1

1NU tkð Þ 1

tk � xrþH xr; tkð Þ

� �¼ 2fq; r ¼ 1; 2; . . .;N � 1 ð6:34Þ

XNk¼1

pNU tkð Þ ¼ 0 ð6:35Þ

tk ¼ cos ð2k � 1Þp=2N½ �; k ¼ 1; 2; . . .;N ð6:36Þ

xr ¼ cosðrp=NÞ; r ¼ 1; 2; . . .;N � 1 ð6:37Þ

Once function UðtÞ is obtained, function D1ðnÞ can be calculated by using theChebyshev quadrature for integration as

D1ðnÞ ffi 14pnk1

XNi¼1

wiU xið Þ sin nxið Þ ð6:38Þ

xi ¼ cos2i� 12N

p

; i ¼ 1; 2; . . .;N ð6:39Þ

wi ¼ p=N ð6:40Þ

Substituting Eq. (6.38) into Eqs. (6.23)–(6.25), we can get the temperature field inthe p-plane. The temperature in the time domain can be obtained by applying theLaplace inverse transform.


6.2.3 Temperature Gradients

From Eqs. (6.38) and (6.29), we can get the expressions for D2ðnÞ and EiðnÞ. Byconsidering the asymptotic nature of the integrands in Eqs. (6.29) for large valuesof the integration variable n and using the asymptotic formula [35]:

Z1�1

FjðtÞffiffiffiffiffiffiffiffiffiffiffiffi1� t2

p eintdt¼ffiffiffiffiffiffiffiffip2 nj j

rFjð1Þ exp i n� pn

4 nj j � ��

þFjð�1Þ exp �i n� pn4 nj j

� �þO

1nj j

�;

ð6:41Þ

Z10

xl�1 exp �sxð Þ sin

cos

� �bxð Þdx

¼ CðlÞs2 þ b2� �l=2 sin

cos

� �l tan�1 b

s

; s[ 0; l[ 0;

ð6:42Þ

the singular temperature gradients near the crack tip in Laplace domain can beobtained as

T�;y r; h; pð Þ ¼ � Uð1Þ

2ffiffiffiffiffi2r

p cosh2

; ð6:43Þ

T�;x r; h; pð Þ ¼ Uð1Þ

2ffiffiffiffiffi2r

p sinh2

; ð6:44Þ

T�;r r; h; pð Þ ¼ � Uð1Þ

2ffiffiffiffiffi2r

p sinh2

ð6:45Þ

where ðr; hÞ are the polar coordinates measured from the crack tip defined by

r2 ¼ ðx� 1Þ2 þ y2; tanðhÞ ¼ y=ðx� 1Þ ð6:46Þ

Right at the crack tip, r ! 0, and h ¼ �p, the temperature gradient reaches amaximum value and the intensity factor of the temperature gradient (IFTG) at thecrack tip can be defined as [12]

K�TðpÞ ¼ lim

r!02ffiffir

pT�;r r; h; pð Þ

h¼�p

¼ Uð1Þffiffiffi2

p ð6:47Þ


By applying the inverse Laplace transform to Eqs. (6.43)–(6.45), the crack-tiptemperature gradients in the time domain can be obtained as

T;y r; h; tð Þ ¼ �KTðtÞ2ffiffir

p cosh2

; ð6:48Þ

T;x r; h; tð Þ ¼ KTðtÞ2ffiffir

p sinh2

; ð6:49Þ

T;r r; h; tð Þ ¼ �KTðtÞ2ffiffir

p sinh2

; ð6:50Þ

where the IFTGs in the time domain, KTðtÞ, is given by

KTðtÞ ¼ L�1 K�TðpÞ

� � ð6:51Þ

It can be seen from Eq. (6.34) that the dynamic temperature gradients present anr�1=2 singularity at the crack tip, which is in agreement with the correspondingstatic thermal crack problem [11, 12, 28]. The dynamic effect is merely introducedby the IFTGs, which are time-dependent as shown in Eq. (6.51).

6.2.4 Numerical Results

The temperature field in the time domain can be obtained from Eqs. (6.23)–(6.25)by using the numerical inversion of Laplace transform, as detailed in Miller andGuy [36], with the following parameters: N ¼ 8 10; b ¼ 0; 0:2� d� 0:3. Bychoosing the geometric size of the composite to be a=c ¼ 2b=c ¼ 1, the materialparameters to be d2 ¼ 2d1 ¼ 2:0; k1 ¼ 1:0; k2 ¼ 0:5; s1 ¼ 1:0; s2 ¼ 0:4 and theboundary condition T0 ¼ 1:0, the temperature field can be obtained by solving thesingular integral Eq. (6.27) and substituting Eq. 6.28 into Eq. 6.19.

The steady temperature distribution in the cracked half-plane bonded to a coatingis shown in Fig. 6.2 as t ! 1. The disturbance of the crack on the temperaturefield can be observed from the iso-temperature lines, and there is a temperaturejump across the crack faces.

The dynamic IFTGs can be obtained from Eqs. (6.47) and (6.51) once thealgebraic Eq. (6.34) are solved and the numerical inverse of Laplace transform isperformed.

The variation of dynamic IFTGs versus time is shown in Fig. 6.3 for hyperbolicheat conduction model and parabolic model. For hyperbolic heat conduction model,


-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5 0.6 0.6 0.6

0.65

0.65 0.65

0.65

0.7

0.7 0.7

0.7

0.75 0.750.75

0.8 0.8 0.8

0.85

0.85 0.85

0.850.9 0.9

0.9

0.9

0.9

0.9

0.95 0.95 0.951 1 1 1

1 1 1 1

x/c

y/c

Fig. 6.2 Steady-state temperature distribution in the cracked half-plane with a coating underthermal shock [34]

Fig. 6.3 Intensity factors of temperature gradients vary with time [34]


the IFTGs fluctuate and increase with time and reach peak values then oscillate forsome time to get the steady value. The IFTGs for parabolic model increasesmoothly with time until get the peak value then decrease gradually to the steadyvalue. The magnitudes of the IFTGs for hyperbolic model are bigger than that forparabolic model, which shows the effect of the relaxation time on the temperaturefield.

The effect of the geometric size b on the dynamic IFTGs is shown in Fig. 6.4when other parameters are kept unchanged. As the values of b increase the dynamicIFTGs decrease, which means the temperature perturbation in the crackedhalf-plane can be reduced by increasing the thickness of the coating. Figure 6.7shows the dynamic IFTGs for different a values with the thickness of the coating beb=c ¼ 1. It can be observed that the magnitudes of the IFTGs decrease asa increases.

By setting the relaxation time of the substrate (haf-plane) to be naught, the effectof the relaxation time of the coating s1 on the dynamic IFTGs is shown in Fig. 6.5.The magnitude of the dynamic IFTGs decreases as the relaxation time decreasesand the limiting case of s1 ¼ 0 correspond to the parabolic heat conduction model.It is clearly seen that there is much difference in the very early stages of the thermalloading impact. In other words, the big difference is obvious in the very small timescales, but as the time increases, the values of IFTGs for different relaxation timesconverge.

Fig. 6.4 Dynamic IFTGs for different b values when a=c ¼ 1:0 [34]


6.3 Thermoelastic Analysis of a Partially Insulated Crackin a Strip

In the previous section, we investigate the thermal response of a cracked,semi-infinite half-plane with a coating under thermal impact using the hyperbolicheat conduction model. The results exhibit a strong overshooting phenomenon incomparison to the results based on the classical, Fourier’s heat conduction model.When the temperature field is obtained, thermoelastic analysis can be readily per-formed based on the classical thermoelasticity. As the final failure of crackedstructure is usually governed by the mechanical stress or strain, thermoelasticanalysis of cracked structure based on non-Fourier heat conduction model willprovide a more conservative prediction to the reliability of materials and structuresunder transient thermal disturbances. Here we present a piece of work on thethermoelastic response of a cracked strip of a finite width based on the hyperbolicheat conduction.

Many structural components are often subjected to severe thermal loading,leading to intense thermal stresses in the components, especially around cracks andother defects. Materials become brittle when thermal stresses appear quickly as theresult of a high temperature gradient in an unsteady temperature field. Thermalstresses combining with mechanical loadings can give rise to cracking and catas-trophic failure of materials and structures [37].

The distribution of thermal stress in the vicinity of a crack in an elastic body hasbeen extensively studied since 1950s using the classical Fourier heat conduction[38–40]. If the effects of both the inertial term and the thermo-elastic coupling term

Fig. 6.5 Dynamic IFTGs for different s1 values when s2 ¼ 0:0 [34]

6.3 Thermoelastic Analysis of a Partially Insulated Crack … 255

are neglected, superposition can be applied to solve the thermoelastic fractureproblem [41, 42]. More general, coupled, thermoelastic theories have been adoptedto study the fracture problems in thermoelasticity [43, 44]. In most of the usualengineering applications it is appropriate to use the uncoupled, thermoelastic theorywithout significant error [45–47].

Enriched research studies have been accumulated on crack problems underthermal loading in advanced materials based on the classical, Fourier heat con-duction [14–17, 21]. The Fourier heat conduction model, although with sufficientaccuracy for many engineering applications, implies infinite, thermal wave speedand is ineffective at the very small spatial and time scales associated withsmall-scale systems [2]. For many technological applications that involving highthermal energy with extremely short time, the results obtained from the Fourier heatconduction model differ significantly from the experimental results [4, 5]. Thehyperbolic heat conduction model becomes more applicable than the Fourier heatconduction when irreversible physical processes, such as crack or void initiation ina solid, are involved in heat transport [9, 10]. A thermoelastic analysis of a crackedhalf-plane under a thermal shock impact was recently given by Chen and Hu [27]based on the hyperbolic heat conduction theory.

In this section, we present the transient temperature and thermal stresses arounda crack in a thermoelastic strip under a temperature impact loading using thehyperbolic heat conduction. The theoretical framework established in Sect. 6.1 isused to solve the temperature field first; and the resultant thermal field is thenapplied to solve the transient, thermoelastic crack problem in the strip.

6.3.1 Definition of the Problem

Consider a thermo-elastic strip containing a crack of length 2c parallel to the freesurface, as shown in Fig. 6.6. The strip is initially at the uniform temperature zero,and its free surface, y ¼ �ha and y ¼ hb are suddenly heated to temperature Ta and

Fig. 6.6 Geometry of thecracked strip and coordinates[26]


Tb, respectively. The crack surfaces are assumed to be partially insulate, whichindicates a temperature drop across the crack surfaces is contributed by the thermalresistance Rc of the crack region [21]. The effects of inertia and thermal-elasticcoupling are neglected which leads to an uncoupled and quasi-static problem. Thehyperbolic heat conduction equations are adopted in the thermal stress analysis.

6.3.1.1 Thermal-Elastic Field Equations

The basic equations of plane thermal stress problems for thermal elastic body arethe equilibrium equations:

@rx@x

þ @rxy@y

¼ 0;@rxy@x

þ @ry@y

¼ 0 ð6:52Þ

the strain-displacement relations:

ex ¼ @u@x

; ey ¼ @v@y

; exy ¼ 12

@u@y

þ @v@x

ð6:53Þ

the compatibility equation:

@2ex@y2

þ @2ey@x2

¼ 2@2exy@x@y

; ð6:54Þ

and the constitutive law:

ex ¼ 1E

rx � mry� �þ aT ;

ey ¼ 1E

ry � mrx� �þ aT ;

exy ¼ 1þ mE

rxy;

ð6:55Þ

where E, m and a are the Young’s modulus, the Poisson’s ratio and the coefficient oflinear thermal expansion, respectively.

Let Uðx; yÞ be the Airy stress function, then the stresses can be expresses interms of U as

rx ¼ @2U@y2

; ry ¼ @2U@x2

; rxy ¼ @2U@x@y

; ð6:56Þ


Substituting Eq. (6.56) into the compatibility condition (6.65) and considering theconstitutive law (6.56) leads to

r2r2UþE � r2 aTð Þ ¼ 0; ð6:57Þ

By introducing the dimensionless variables

U ¼ U= Eac2T0� �

; �rij ¼ rij= EaT0ð Þ; �eij ¼ eij= aT0ð Þ�u;�vð Þ ¼ u; vð Þ= acT0ð Þ; �x;�y; �hð Þ ¼ x; y; hð Þ=cT ¼ T=T0; �t ¼ a=tc2

ð6:58Þ

where T0 is the reference temperature.The governing Eq. (6.57) has the following dimensionless form:

r2r2Uþr2ðTÞ ¼ 0; ð6:59Þ

where T can be solved by employing the same approach as detailed in Sect. 6.1 viathe thermal boundary conditions. It is noted that in Eq. (6.59) and hereafter, the hat“ ” of the dimensionless variables is omitted for simplicity.

Considering the following boundary and initial conditions for the thermal field inthe dimensionless form

Tðx;�haÞ ¼ Ta; ð xj j\1; t[ 0Þ ð6:60Þ

Tðx; hbÞ ¼ Tb; ð xj j\1; t[ 0Þ ð6:61Þ

@Tðx; 0þ Þ@y

¼ @Tðx; 0�Þ@y

¼ H Tðx; 0þ Þ � Tðx; 0�Þ½ �; ð xj j\1Þ ð6:62Þ

Tðx; 0þ Þ ¼ Tðx; 0�Þ; ð xj j � 1Þ ð6:63aÞ

@Tðx; 0þ Þ@y

¼ @Tðx; 0�Þ@y

; ð xj j � 1Þ ð6:63bÞ

T ¼ 0; ðt ¼ 0Þ ð6:64aÞ@T@t

¼ 0; ðt ¼ 0Þ ð6:64bÞ

where the quantity, H, is the dimensionless thermal conductivity of the cracksurface defined as H ¼ c= Rckð Þ [30, 48, 49]. The limiting value of H ¼ 0 corre-sponds to the completely insulated crack surface condition and H ! 1 corre-sponds to the conducting crack surface.


The mechanical conditions can be expressed as

rxyðx;�haÞ ¼ ryðx;�haÞ ¼ 0; ð xj j\1Þ ð6:65aÞ

rxyðx; hbÞ ¼ ryðx; hbÞ ¼ 0; ð xj j\1Þ ð6:65bÞ

rxyðx; 0Þ ¼ ryðx; 0Þ ¼ 0; ð xj j\1Þ ð6:66Þ

rxyðx; 0þ Þ ¼ rxyðx; 0�Þ; ð xj j � 1Þ ð6:67aÞ

ryðx; 0þ Þ ¼ ryðx; 0�Þ; ð xj j � 1Þ ð6:67bÞ

uðx; 0þ Þ ¼ uðx; 0�Þ; ð xj j � 1Þ ð6:67cÞ

vðx; 0þ Þ ¼ vðx; 0�Þ; ð xj j � 1Þ ð6:67dÞ

Similar to the derivations for the temperature field as discussed in Sect. 6.1, thetemperature field of the cracked strip under thermal impact can be readily obtained.Then we can investigate the thermoelastic field. Details of the derivations can befound in [38].

6.3.2 Thermal Stresses

Substituting the temperature expressions [26], the governing equation for the Airyfunction U� (6.26) becomes:

r2r2U� ¼Z1�1

ðn2 � m2ÞDðnÞ expðmyÞ � exp mð2hb � yÞ½ �f g

expð�ixnÞdn� q2Wðy; pÞ ðy� 0Þð6:68aÞ

r2r2U� ¼Z1�1

rabðn2 � m2ÞDðnÞ exp m 2ha þ yð Þ½ � � expð�myÞ½ �

expð�ixnÞdn� q2Wðy; pÞ ðy� 0Þð6:68bÞ

The general solution of Eq. (6.68) satisfying the regularity condition at infinity canbe expressed as


U� ¼Z1�1

ðA1 þA2yÞ expð nj jyÞþ ðA3 þA4yÞ expð� nj jyÞ½ � expð�ixnÞdn

þZ1�1

C11ðnÞ expðmyÞþC12ðnÞ expð�myÞ½ � expð�ixnÞdn

�Wðy; pÞ=q2 ðy� 0Þ

ð6:69aÞ

U� ¼Z1�1

ðB1 þB2yÞ expð nj jyÞþ ðB3 þB4yÞ expð� nj jyÞ½ � expð�ixnÞdn

þZ1�1

C21ðnÞ expðmyÞþC22ðnÞ expð�myÞ½ � expð�ixnÞdn

�Wðy; pÞ=q2 ðy� 0Þ

ð6:69bÞ

where Ai and Bi ði ¼ 1; 2; 3; 4Þ are unknowns to be determined, and

C11ðnÞ ¼ �DðnÞ=ðm2 � n2Þ ð6:70aÞ

C12ðnÞ ¼ exp 2mhbð ÞDðnÞ=ðm2 � n2Þ ð6:70bÞ

C21ðnÞ ¼ rab exp 2mhað ÞD nð Þ= n2 � m2� � ð6:70cÞ

C22 nð Þ ¼ � exp �2mhað ÞC21 nð Þ ¼ rabD nð Þ= m2 � n2� � ð6:70dÞ

The stresses in Laplace domain can be obtained by substituting Eq. (6.69) intoEq. (6.56) as

r�y ¼@U@x2

¼Z1�1

�n2 A1 þA2yð Þ exp nj jyð Þþ A3 þA4yð Þ exp � nj jyð Þ½ � exp �inxð Þdn

�Z1�1

n2 C11 exp myð ÞþC12 exp �myð Þ½ � exp �inxð Þdn;

ðy>0Þ

ð6:71aÞ


r�y ¼ @U@x2

¼Z1�1

�n2 B1 þB2yð Þ exp nj jyð Þþ B3 þB4yð Þ exp � nj jyð Þ½ � exp �inxð Þdn

� R1�1

n2 C21 exp myð ÞþC22 exp �myð Þ½ � exp �inxð Þdn; ðy60Þ

ðy60Þ

ð6:71bÞ

r�x ¼Z1�1

2A2 nj j þ n2 A1 þA2yð Þ� �exp nj jyð Þþ n2 A3 þA4yð Þ � 2A4 nj j� �

exp � nj jyð Þ� �

exp �inxð ÞdnþZ1�1

m2 C11 exp myð ÞþC12 exp �myð Þ½ � exp �inxð Þdn�Wðy; pÞ

ðy>0Þð6:71cÞ

r�x ¼Z1�1

2B2 nj j þ n2 B1 þB2yð Þ� �exp nj jyð Þþ n2 B3 þB4yð Þ � 2B4 nj j� �

exp � nj jyð Þ� �


m2 C21 exp myð ÞþC22 exp �myð Þ½ � exp �inxð Þdn�Wðy; pÞ

ðy60Þð6:71dÞ

r�xy ¼Z1�1

in A2 þ nj j A1 þA2yð Þ½ � exp nj jyð Þþ A4 � nj j A3 þA4yð Þ½ � exp � nj jyð Þ½ �


inm C11 exp myð Þ � C12 exp �myð Þ½ � exp �inxð Þdn

ðy>0Þð6:71eÞ

r�xy ¼Z1�1

in B2 þ nj j B1 þB2yð Þ½ � exp nj jyð Þþ B4 � nj j B3 þB4yð Þ½ � exp � nj jyð Þ½ �


inm C21 exp myð Þ � C22 exp �myð Þ½ � exp �inxð Þdn

ðy60Þð6:71fÞ


Denote the jumps of displacements across the line y ¼ 0 by uh i and vh i,

uh i ¼ uðx; 0þ Þ � uðx; 0�Þvh i ¼ vðx; 0þ Þ � vðx; 0�Þ ð6:72Þ

Following the procedure in [14] and introducing two dislocation density functionsfjðxÞ ðj ¼ 1; 2Þ as

f1ðxÞ ¼ @ uh i@x

ð6:73aÞ

f2ðxÞ ¼ @ vh i@x

ð6:73bÞ

By applying the boundary conditions (6.65)–(6.67), it can be shown that fiðxÞði ¼ 1; 2Þ satisfy the following singular integral equations:

Z1�1

f1ðtÞ 1t � x

þM11ðx; tÞ� �

dtþZ1�1

f2ðtÞM12ðx; tÞdt ¼ pL1ðxÞ ð6:74aÞ

Z1�1

f1ðtÞM21ðx; tÞdtþZ1�1

f2ðtÞ 1t � x

þM22ðx; tÞ� �

dt ¼ pL2ðxÞ ð6:74bÞ

where

L1ðxÞ ¼ �8Z10

n � l1ðnÞ sinðxnÞdn ð6:75aÞ

L2ðxÞ ¼ 8Z1�1

n2 � l2ðnÞ cosðxnÞdn ð6:75bÞ

M11ðx; tÞ ¼Z10

1þ 4<11ðnÞ½ � sin nðx� tÞ½ �dn ð6:75cÞ

M12ðx; tÞ ¼ �4Z10

<12ðnÞ cos nðt � xÞ½ �dn ð6:75dÞ


M21ðx; tÞ ¼ �4Z10

<21ðnÞ cos nðt � xÞ½ �dn ð6:75eÞ

M22ðx; tÞ ¼Z10

1� 4<22ðnÞ½ � sin nðx� tÞ½ �dn ð6:75fÞ

and the functions l1ðnÞ, l2ðnÞ and <ijðnÞ ði; j ¼ 1; 2Þ are defined as given in [37].The functions fjðxÞ ðj ¼ 1; 2Þ also satisfy the singled-value equations

Z1�1

fjðxÞdx ¼ 0; ðj ¼ 1; 2Þ ð6:76aÞ

fjðxÞ ¼ 0; ðj ¼ 1; 2Þ xj j � 1 ð6:76bÞ

The solution of the integral Eq. (6.76) of fjðxÞ ðj ¼ 1; 2Þ can be expressed asfollows

fjðtÞ ¼ FjðtÞffiffiffiffiffiffiffiffiffiffiffiffi1� t2

p ; ðj ¼ 1; 2Þ ð6:77Þ

Using the Lobatto-Chebyshev method [50, 51], we can reduce the integral equa-tions to the following algebraic equations:

Xni¼1

wi1

ðti � xkÞ þM11ðxk; tiÞ� �

F1ðtiÞ

þXni¼1

wiM12ðxk; tiÞF2ðtiÞ ¼ L1ðxkÞð6:78aÞ

Xni¼1

wiF1ðtiÞ ¼ 0 ð6:78bÞ

Xni¼1

wiM21ðxk; tiÞF1ðtiÞ

þXni¼1

wi1

ðti � xkÞ þM22ðxk; tiÞ� �

F2ðtiÞ ¼ L2ðxkÞð6:78cÞ

Xni¼1

wiF2ðtiÞ ¼ 0 ð6:78dÞ


where,

ti ¼ cosi� 1ð Þpn� 1

; i ¼ 1; 2; . . .; n; xk ¼ cos2k � 1ð Þp2 n� 1ð Þ ; k ¼ 1; 2; . . .; n� 1: ð6:79aÞ

wi ¼ p2 n� 1ð Þ ; i ¼ 1; n; wi ¼ p

n� 1; i ¼ 2; 3; . . .; n� 1: ð6:79bÞ

6.3.3 Asymptotic Stress Field Near Crack Tip

By considering the asymptotic nature of the integrands in Eq. (6.71) for largevalues of the integration variable, n, following the procedure given in references[14, 50] and using the asymptotic formula [35]:

Z1�1

FjðtÞffiffiffiffiffiffiffiffiffiffiffiffi1� t2

p eintdt¼ffiffiffiffiffiffiffiffip2 nj j

rFjð1Þ exp i n� pn

4 nj j � ��

þFjð�1Þ exp �i n� pn4 nj j

� �þO

1nj j

�;

ð6:80aÞ

Z10

xl�1 exp �sxð Þ sincos

� �bxð Þdx

¼ CðlÞs2 þ b2ð Þl=2

sincos

� �l tan�1 b

s

� �� ;

s[ 0; l[ 0; ð6:80bÞ

the singular stresses near the crack tip in Laplace domain can be obtained as

r�y r; h; pð Þ ¼ 1ffiffiffiffiffiffiffiffi2pr

p K�1ðpÞ cos

h2

1þ sin

h2

sin

3h2

� ��

þK�2ðpÞ sin

h2

cos

h2

cos

3h2

�

r�xy r; h; pð Þ ¼ 1ffiffiffiffiffiffiffiffi2pr

p K�1ðpÞ sin

h2

cos

h2

cos

3h2

�

þK�2ðpÞ cos

h2

1� sin

h2

sin

3h2

� ��

r�x r; h; pð Þ ¼ 1ffiffiffiffiffiffiffiffi2pr

p K�1ðpÞ cos

h2

1� sin

h2

sin

3h2

� ��

�K�2ðpÞ sin

h2

2þ cos

h2

cos

3h2

� ��

ð6:81Þ


where ðr; hÞ are the polar coordinates measured from the crack tip defined by

r2 ¼ ðx� 1Þ2 þ y2; tanðhÞ ¼ y=ðx� 1Þ ð6:82Þ

and the dimensionless stress intensity factors (SIFs) K�1ðpÞ and K�

2ðpÞ are

K�1 ðpÞ ¼ lim

r!0

ffiffiffiffiffi2r

pryðr; 0; pÞ ¼ �

ffiffiffip

pF2ð1; pÞ4

K�2 ðpÞ ¼ lim

r!0

ffiffiffiffiffi2r

prxyðr; 0; pÞ ¼ �

ffiffiffip

pF1ð1; pÞ4

ð6:83Þ

By applying the inverse Laplace transform to (6.83), the crack-tip stress fields in thetime domain can be obtained as

ry r; h; tð Þ ¼ 1ffiffiffiffiffiffiffiffi2pr

p K1ðtÞ cos h2

1þ sin

h2

sin

3h2

� ��

þK2ðtÞ sin h2

cos

h2

cos

3h2

�

rxy r; h; tð Þ ¼ 1ffiffiffiffiffiffiffiffi2pr

p K1ðtÞ sin h2

cos

h2

cos

3h2

�

þK2ðtÞ cos h2

1� sin

h2

sin

3h2

� ��

rx r; h; tð Þ ¼ 1ffiffiffiffiffiffiffiffi2pr

p K1ðtÞ cos h2

1� sin

h2

sin

3h2

� ��

�K2ðtÞ sin h2

2þ cos

h2

cos

3h2

� ��ð6:84Þ

where the SIFs in the time domain, K1ðtÞ and K2ðtÞ are given by

K1ðtÞ ¼ L�1 K�1ðpÞ

� �K2ðtÞ ¼ L�1 K�

2ðpÞ� � ð6:85Þ

From Eq. (6.81) we can see that the dynamic stress field exhibits the same form asthe static stress field around the crack tip. The dynamic effect is merely introducedby the SIFs, which are time-dependent as shown in Eq. (6.85).

The principal stresses near the crack tip can be expressed as

r1r2

¼ rx þ ry2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirx � ry

2

� �2þ r2xy

rð6:86Þ


The normalized principal stresses can be defined as

P1 ¼ r1ffiffiffiffiffiffiffiffi2pr

p

¼ K1ðtÞ cos h2

� K2ðtÞ sin h

2

þ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK21ðtÞ sin2ðhÞþK2

2ðtÞ 4� 3 sin2ðhÞ� �þ 2K1ðtÞK2ðtÞ sinð2hÞq ð6:87aÞ

P2 ¼ r2ffiffiffiffiffiffiffiffi2pr

p

¼ K1ðtÞ cos h2

� K2ðtÞ sin h

2

� 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK21ðtÞ sin2ðhÞþK2

2ðtÞ 4� 3 sin2ðhÞ� �þ 2K1ðtÞK2ðtÞ sinð2hÞq ð6:87bÞ

6.3.4 Numerical Results and Discussions

The temperature field in the time domain can be obtained through the same approachas discussed in Sect. 6.1, which exhibits the similar thermal wave behavior. Forsimplicity, details are omitted here. By considering the thermal loading boundaryconditions to be Ta ¼ 2; Tb ¼ 1 and the geometric size of the strip to be2ha=c ¼ hb=c ¼ 2, the stress field around the crack can be calculated. The dynamicSIFs can be obtained from (6.83) and (6.85) once the algebraic Eq. (6.78) are solved,and the numerical inversion of Laplace transform is performed.

The effect of the thermal conductivity, H, on the dynamic SIFs is shown inFig. 6.7 for the symmetric case ha=c ¼ hb=c ¼ 1 and R ¼ 0. It can be seen that K1

vanishes due to the geometric symmetry of the cracked strip. The magnitudes of K2

reduce as the thermal conductivity H increases, which means the crack disturbanceon the stress field decreases as the crack faces become more conducting. Forperfectly conducting crack case, there is no any disturbance around the crack and noany stress concentration.

The variationof dynamic SIFs versus time is shown inFig. 6.8 for different geometricsizewhenH ¼ 0 andR ¼ 0:5. The SIFs increase with time and reach their peak valuesthen oscillate for some time to get their steady value. The magnitudes of the peakvalues increase as the size ratio hb=ha increases. The limiting case hb ! 1 corre-sponds to the cracked half-plane problem and the results are in agreement with [27].

The variation of peak-values of dynamic SIFs versus hb for differentRwhen ha ¼ cand H ¼ 0 is shown in Fig. 6.9. The magnitudes of the peak-values of dynamic SIFsincrease as the size hb increases from 1 to about 3.5 and then get steady values.We cansee that for bigger relaxation time R corresponds to bigger magnitude of the SIFs.

The dynamic effect on the stress field around the crack tip can be well expressedby studying the variation of the principal stress versus time. From Eq. (6.84), it canbe seen that the stress intensity factors introduce the dynamic effect to the stresses


Fig. 6.7 Dynamic SIFs versus time for different thermal conductivity H when ha=c ¼ hb=c ¼ 1and R ¼ 0 [26]

Fig. 6.8 Dynamic SIFs versus time for different geometric size ha; hb when H ¼ 0 and R ¼ 0:5[26]


near the crack tip. The angular variations of the maximum principal stresses P1 atdifferent time when 2ha=c ¼ hb=c ¼ 2, R ¼ 0:5 and H ¼ 0 are displayed inFig. 6.10. It is noted that the angle at which the maximum principal stress appears

Fig. 6.9 The variation of peak-values of dynamic SIFs versus hb for different R when ha ¼ c andH ¼ 0 [26]

Fig. 6.10 The variation of principal stresses P1 with angle at different time when2ha=c ¼ hb=c ¼ 2, R ¼ 0:5 and H ¼ 0 [26]


varies with time, which implies that the crack propagation direction may changeunder dynamic thermal loading. The magnitudes of both principal stresses are largerin the early loading period than the steady state values which indicate that the crackis more likely to propagate under dynamic thermal loading than the static thermalloading case.

6.4 Thermal Stresses in a Circumferentially CrackedHollow Cylinder Based on Memory-Dependent HeatConduction

The transient, thermoelastic fracture behavior of circumferentially cracked cylindershas been investigated by many researchers. Nied and Erdogan [41] solved thetransient, thermal stress problem of an internally cracked hollow cylinder under asudden cooling load, where the temperature field, parallel to the crack surfaces, wasassumed to be independent of the presence of the crack. Nabavi and Ghajar [52]obtained thermal stress intensity factors (SIFs) of a circumferentially cracked,hollow cylinder under steady thermo-mechanical loading using the weight functionmethod. The range of maximum values of SIFs of a circumferential crack in afinite-length, thick-walled cylinder with rotation-restrained edges under thermalstriping was obtained in [53]. It is worth mentioning and these studies are all basedon the Fourier’s law.

As discussed in Chap. 1, Fourier’s law leads to inaccurate results when dealingwith extremely low temperature, very short time duration or high heat flux [54]. Tobetter predict the heat transfer in solids, a number of non-Fourier heat conductionmodels, such as the Cattaneo and Vernotte (CV) model [55, 56], the inertial theory[57], the dual-phase lag (DPL) model [58], or the Green and Naghdi (GN) model[59, 60], etc. Employing these models, the mode I fracture problem of circumfer-ential cracks under various crack surface loadings in a solid cylinder has beenextensively studied [61–66].

The existence of the anomalous diffusion of a material leads to the abnormal heatconduction [67, 68], whereas the diffusion exhibits path dependence,memory-related behaviour. The standard, integer-order, time derivative is definedby the local limit, which cannot describe the memory-dependent process. Thefractional time derivative is essentially a differential-integral convolution operator,and the integral term in its definition reflects the history-dependence of the system.It is very suitable for describing the anomalous heat conduction under extremeconditions [69–73]. The memory-dependent, fractional heat conduction model canbe regarded the extension of the CV model with memory-dependent effect.Recently, Wang and Li [74] proposed a memory-dependent derivative (MDD),which is defined in an integral form of a common derivative with a kernel functionon a slipping interval. This model performs better than the fractional one inreflecting the memory effect. Subsequently, Yu et al. [75] first established a new,


memory-dependent, generalized thermoelasticity based upon MDD. More appli-cations on the MDD-based thermoelasticity model can be found in [76–78]. Here,the transient, thermoelastic response of a hollow cylinder with a circumferentialcrack under thermal shock is presented using the MDD based, memory-dependent,heat conduction model [79]. The results are compared with the results based on theCV model to illustrate the unique feature of the memory-dependent model inthermoelastic analysis.

6.4.1 Problem Formulation

Consider an infinitely long, hollow cylinder with an initial temperature, T0. Itcontains a circumferential crack in the z = 0 plane with z being the axis of thecylinder, as shown in Fig. 6.11. The crack occupies the region c\r\d, while theinner and outer radii of the cylinder are ri and ro, respectively. Suppose the innersurface of the cylinder suffers a sudden, thermal shock, Ti, and the outer surface isinsulated. In the present analysis, we only consider that the temperature changealters stress distribution, whereas the elastic deformation does not affect the tem-perature distribution. As the crack faces are parallel to the direction of heat

Fig. 6.11 The geometry of ahollow cylinder containing acircumferential crack [79]


conduction, the effect of crack of zero height is negligible. Therefore, the tem-perature change is independent of z. The procedure to solve the thermoelastic, crackproblem usually includes three steps, as illustrated in Sect. 5.1. First, the transientdistribution of temperature in a crack-free, hollow cylinder is obtained by solvingthe heat conduction equation. Then this temperature distribution is employed todetermine the thermostress field. Finally, the opposite of the transient, thermal stressobtained in the preceding step is applied on the crack surfaces to solve an elastic,crack problem.

6.4.2 Thermal Axial Stress in an Un-cracked HollowCylinder

According to Wang and Li [75] the mth order MDD of function f has the followingform

Dmxf tð Þ ¼ 1

x

Z t

t�x

K t � nð Þf mð Þ nð Þdn ð6:88Þ

in which, the kernel function, K t � nð Þ, and the time delay, x, can be chosen toreflect the real behavior of the material. Here, the following kernel function isadopted,

K t � nð Þ ¼ 1� 2bx

t � nð Þþ a2 t � nð Þ2x2 ¼

1 a ¼ b ¼ 01� t�nð Þ

x a ¼ 0 b ¼ 12

1� t�nx

� �2a ¼ b ¼ 1

8<: ð6:89Þ

where a and b are constants. Employing the first order MDD into the rate of heatflux, a new, memory-dependent CV model can be obtained as [75]

qþ sDwq ¼ �krT ð6:90Þ

For an axially symmetric problem of an isotropic elastic material in the cylindricalcoordinates r; h; zð Þ, substituting Eq. (6.90) into (6.2), the generalized heat con-duction equation with MDD:

kr2T ¼ 1þ sDxð ÞqEC _T ð6:91Þ

where r2 ¼ 1r@@r r @

@r

� �þ @2

@z2 is the Laplace operator, and qE is the density of thematerials. It is worthing noting that for the infinitely long hollow cylinder,@h=@z ¼ 0.

6.4 Thermal Stresses in a Circumferentially Cracked … 271

The initial and boundary conditions have the forms:

T r; tð Þ ¼ 0 _T r; tð Þ ¼ 0 t ¼ 0 ð6:92aÞ

T r; tð Þ ¼ H tð Þ Ti � T0ð Þ r ¼ ri ð6:92bÞ

@T r; tð Þ=@r ¼ 0 r ¼ ro ð6:92cÞ

where HðtÞ is the Heaviside step function.For convenience, introduce the following non-dimensional variables:

r0 ¼ 1r0r; t0; s0;x0ð Þ ¼ k

qCr2ot; s;xð Þ; T 0 ¼ T

Ti � Toð6:93Þ

the governing Eq. (6.91), boundary conditions (6.92b) and (6.92c) can be rewrittenas:

@2T 0

@r02þ 1

r0@T 0

@r0¼ 1þ s0Dx0ð Þ _T 0 ð6:94Þ

T 0 r0; t0ð Þ ¼ H t0ð Þ r0 ¼ rin ð6:95aÞ

@T 0 r0; t0ð Þ@r0

¼ 0 r0 ¼ 1 ð6:95bÞ

where rin ¼ ri=ro.Applying Laplace transform to both sides of Eqs. (6.94) and (6.95), we have

@2T 0 r0; sð Þ@r02

þ 1r0@T 0 r0; sð Þ

@r0¼ 1þGð ÞsT 0 r0; sð Þ ð6:96Þ

T 0 rin; sð Þ ¼ 1s

ð6:97aÞ

@T 0 1; sð Þ@r0

¼ 0 ð6:97bÞ

where

G ¼ s0

x0 1� exp �sx0ð Þð Þ 1� 2bx0s

þ 2a2

x02s2

� a2 � 2bþ 2a2

x0s

exp �sx0ð Þ

� �:

Considering initial conditions (6.92a), the solution of Eq. (6.96) can beexpressed as:


T 0 r0; sð Þ ¼ A1I0ffiffiffik

pr0

� �þA2K0

ffiffiffik

pr0

� �ð6:98Þ

where k ¼ 1þGð Þs, In and Kn represent the nth-order, modified Bessel functions ofthe first and second kinds, respectively.

The unknown coefficients A1 and A2 can be determined by the boundary con-ditions (6.97a) and (6.97b):

A1 ¼K1

ffiffiffik

p� �s I0

ffiffiffik

prin

� �K1

ffiffiffik

p� �þ I1

ffiffiffik

p� �K0

ffiffiffik

prin

� �h i

A2 ¼I1

ffiffiffik

p� �s I0

ffiffiffik

prin

� �K1

ffiffiffik

p� �þ I1

ffiffiffik

p� �K0

ffiffiffik

prin

� �h i

As there is no axial force over any cross section of the cylinder, we have:

Zrori

rrzzdr ¼ 0 ð6:99Þ

where the axial stress component rzz in an un-cracked hollow cylinder is governedby the following equation:

rzz ¼ 2Eat1� tð Þ r2o � r2i

� � Zro

ri

rdTdr � Eat1� tð ÞdT ð6:100Þ

where E, t, and at are Young’s modulus, Poisson’s ratio, and the coefficient oflinear thermal expansion, respectively. By introducing the followingnon-dimensional axial stress:

r0zz ¼rzz 1� tð Þ

Eat Ti � T0ð Þ ð6:101Þ

and then the non-dimensional, thermal axial stress in the Laplace domain can berewritten as:

�r0zz ¼2

1� r2in

Z1rin

r0T 0dr0 � T 0 ð6:102Þ


Substituting Eq. (6.98) into Eq. (6.102) leads to:

�r0zz ¼2ffiffiffi

kp

1� r2inð Þ A1 I1ffiffiffik

p� �� rinI1

ffiffiffik

prin

� �h iþA2 rinK1

ffiffiffik

prin

� �� K1

ffiffiffik

p� �h in o

� A1I0ffiffiffik

pr0

� �� A2K0

ffiffiffik

pr0

� �, �p r0; sð Þ

ð6:103Þ

Thus far, the temperature and thermal axial stress in the Laplace domain have beensolved.

6.4.3 Thermal Stress in the Axial Direction

With the thermal stress in an un-cracked hollow cylinder in hand, in this section wewill determine the transient thermal stress in a circumferentially cracked hollowcylinder. For simplicity, the following non-dimensional quantities are introduced:

u0r; u0z

� � ¼ 1� tat 1þ tð Þ Ti � T0ð Þro ur; uzð Þ r0rr;r

0rz

� � ¼ 1� tatE Ti � T0ð Þ rrr; rrzð Þ

z0; c0; d0ð Þ ¼ 1ro

z; c; dð Þ

where ur and uz are the displacements in the radial and axial directions, respec-tively. rrr and rrz denote the normal and shear stress components.

By superposition, the perturbation problem for the cracked cylinder can beformulated with the crack surface tractions equal to the negative of the thermal axialstress �p r0; sð Þ given in Eq. (6.103). As the problem is symmetric about plane z = 0,the non-dimensional, boundary conditions in the Laplace domain for the mode I,crack problem can be written as:

�r0rr 1; z0; sð Þ ¼ 0 �r0rz 1; z0; sð Þ ¼ 0 0� z0 �1 ð6:104aÞ

�r0rr rin; z0; sð Þ ¼ 0 �r0rz rin; z

0; sð Þ ¼ 0 0� z0 �1 ð6:104bÞ

�r0rz r0; 0; sð Þ ¼ 0 rin � r0 � 1 ð6:104cÞ

�u0z r0; 0; sð Þ ¼ 0 rin � r0\c0; d0\r0 � 1 ð6:104dÞ

�r0zz r0; 0; sð Þ ¼ ��p r0; sð Þ c0\r0\d0 ð6:104eÞ

The axisymmetric problem for an isotropic cylinder can be solved by introducingthe Love potential function U r; z; tð Þ. The non-dimensional governing equation,


displacement components, and stress components are expressed in terms ofU r; z; tð Þ as:

r2r2U0 ¼ 0 ð6:105Þ

�u0r ¼ � @2U0

@r0@z0ð6:106aÞ

�u0z ¼ 2 1� tð Þr2U0 � @2U

0

@z02ð6:106bÞ

�r0zz ¼@

@z02� tð Þr2U

0 � @2U0

@z02

!ð6:106cÞ

�r0rr ¼@

@z0tr2U

0 � @2U0

@r02

!ð6:106dÞ

�r0rz ¼@

@r01� tð Þr2U

0 � @2U0

@z02

!ð6:106eÞ

in which U0is a non-dimensional, Love potential function in the Laplace domain

U0r0; z0; sð Þ ¼ U r; z; sð Þ 1� tð Þ

atE Ti � T0ð Þr3oð6:107Þ

As the stress vanishes at z ! 1, the bi-harmonic Eq. (6.105) can be solved byusing the Fourier transform and the Hankel transform as:

U0 ¼ 2

p

Z10

C1I0 fr0ð Þ þC2K0 fr0ð Þ þ r0C3I1 fr0ð Þ þ r0C4K1 fr0ð Þ½ � sin fz0ð Þdf

þZ10

g C5 þC6z0ð ÞJ0 gr0ð Þ exp �gz0ð Þdg

ð6:108Þ

where Ci i ¼ 1; 2; . . .; 6ð Þ are unknown functions of s to be determined from theboundary conditions (6.105). Substituting Eq. (6.108) into Eq. (6.106), the dis-placement and stress components can be expressed as:


�u0r r0; z0; sð Þ ¼ � 2p

Z10

f2 C3r0I0 fr0ð Þ � C4r

0K0 fr0ð Þ þC1I1 fr0ð Þ � C2K1 fr0ð Þ½ �

� cos fz0ð ÞdfþZ10

g2 C6 � g C5 þC6z0ð Þ½ �J1 gr0ð Þ exp �gz0ð Þdg

ð6:109aÞ

�u0z r0; z0; sð Þ ¼ 2

p

Z10

f fC1 þ 4 1� tð ÞC3½ �I0 fr0ð Þ þ fC2 � 4 1� tð ÞC4½ �f

� K0 fr0ð Þ þ fC3r0I1 fr0ð Þ þ fC4r

0K1 fr0ð Þg sin fz0ð Þdf

þZ10

g2 2 2t� 1ð ÞC6 � g C5 þC6z0ð Þ½ �J0 gr0ð Þ exp �gz0ð Þdg

ð6:109bÞ

�r0rr r0; z0; sð Þ ¼ 2p

Z10

f2 �fC1 þ 2t� 1ð ÞC3½ �f I0 fr0ð Þ þ �fC2 � 2t� 1ð ÞC4½ �

� K0 fr0ð Þ þ C1

r0� fC3r

0

I1 fr0ð Þ � C2

r0þ fC4r

0

K1 fr0ð Þ�

� cos fz0ð Þdf�Z10

g2

r0C6 � g C5 þC6z

0ð Þ½ �J1 gr0ð Þ exp �gz0ð Þdg

þZ10

g3 1þ 2tð ÞC6 � g C5 þC6z0ð Þ½ �J0 gr0ð Þ exp �gz0ð Þdg

ð6:109cÞ

�r0zz r0; z0; sð Þ ¼ 2

p

Z10

f2 fC1 þ 4� 2tð ÞC3½ �I0 fr0ð Þ þ fC2 � 4� 2tð ÞC4½ �f

� K0 fr0ð Þ þ fC3r0I1 fr0ð Þ þ fC4r

0K1 fr0ð Þg cos fz0ð Þdf

þZ10

g3 1� 2tð ÞC6 þ g C5 þC6z0ð Þ½ �J0 gr0ð Þ exp �gz0ð Þdg

ð6:109dÞ


�r0rz r0; z0; sð Þ ¼ 2

p

Z10

f2 fC3r0I0 fr0ð Þ � fC4r

0K0 fr0ð Þ þ fC1 þ 2 1� tð ÞC3½ �f

� I1 fr0ð Þ � fC2 � 2 1� tð ÞC4½ �K1 fr0ð Þg sin fz0ð Þdf

þZ10

g3 �2tC6 þ g C5 þC6z0ð Þ½ �J1 gr0ð Þ exp �gz0ð Þdg

ð6:109eÞ

In order to determine the unknown coefficients, a dislocation density function canbe defined as:

ddr0

�u0z r0; 0; sð Þ ¼ / r0; sð Þ ð6:110Þ

In order to determine the unknown function / r0; sð Þ, the following singular integralequation can be obtained by substituting Eq. (6.109d) into (6.106c):

Zd0c0

/ x0; sð Þ 1x0 � r0

þ L r0; x0ð Þ þ 2x0Z10

M r0; x0; fð Þdf24

35dx0

¼ �2p 1� tð Þ�p r0; sð Þ c0\r0\d0

ð6:111Þ

where

M r0; x0; fð Þ ¼ f2

DfI0 fr0ð ÞD1 þ fK0 fr0ð ÞD2 þ fr0I1 fr0ð Þ � 2t� 4ð Þ½f

� I0 fr0ð Þ�D3 þ fr0K1 fr0ð Þ þ 2t� 4ð ÞK0 fr0ð Þ½ �D4gð6:112Þ

L r0; x0ð Þ ¼ m r0; x0ð Þ � 1x0 � r0

þ m r0; x0ð Þx0 þ r0

ð6:113Þ

and

m r0; x0ð Þ¼ E r0x0� �

r0\x0r0x0 E

x0r0� �þ x02�r02

x0r0 K x0r0� �

r0 [ x0

(ð6:114Þ

In the above equation, KðÞ and EðÞ are the complete elliptic integrals of the first andsecond kinds, respectively. Di i ¼ 1; 2; 3; 4ð Þ can be found in [79].


From the displacement boundary condition (6.104d) and the definition (6.110), itis clear that the integral equation must be solved under the following single-valuecondition:

Zd0c0

/ r0; sð Þdr0 ¼ 0 ð6:115Þ

Introduce the following normalized parameters:

r0 ¼ d0 � c0

2qþ d0 þ c0

2; x0 ¼ d0 � c0

2q0 þ

d0 þ c0

2

where �1\q; q0\1, the singular integral Eq. (6.111) and the single-value con-dition (6.115) could be rewritten as:

Z1�1

u q0; sð Þ 1q0 � q

þ L0 q; q0ð Þþ 2x0Z10

M0 q; q0; fð Þdf24

35dq0

¼ �2p 1� tð Þ�p0 q; sð Þð6:116Þ

Z1�1

u q0; sð Þdq0 ¼ 0 ð6:117Þ

in which

u q0; sð Þ ¼ / x0; sð Þ ð6:118aÞ

L0 q; q0ð Þ¼ d0 � c0

2L r0; x0ð Þ ð6:118bÞ

M0 q; q0; fð Þ ¼ d0 � c0

2M r0; x0; fð Þ ð6:118cÞ

�p0 q; sð Þ ¼ �p r0; sð Þ ð6:118dÞ

Using the numerical quadrature formulas mentioned in [80], the fundamentalsolution of Eq. (6.116) may be expressed as:

u q0; sð Þ ¼ f q0; sð Þ 1þ q0ð Þb1 1� q0ð Þb2 ð6:119Þ

where f q0; sð Þ is a bounded function in the interval �1\q0\1.


6.4.4 Stress Intensity Factors

In this section, the SIFs for an embedded crack, an outer edge crack, and an inneredge crack are defined, respectively for further numerical calculation.

6.4.4.1 Embedded Crack ðrin\c0\d0\1Þ

For an embedded crack, the Cauchy kernel 1= q0 � qð Þ is the dominant kernel andthe power-law exponents are taken to be b1 ¼ b2 ¼ �0:5 due to the thermalstresses at the crack tips r0 ¼ c0; d0 exhibiting singular behavior. In this case, theunknown function f q0; sð Þ in Eq. (6.119) is determined using the numerical tech-nique described in [52] and the approximate expression reads:

Z1�1

f q0; sð Þq0 � qð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� q20

p dq0 ¼Xnj¼0

pncjf q0j; s� �q0j � ql

ð6:120Þ

Thus, Eqs. (6.116) and (6.117) can be approximated by the following system ofn + 1 linear algebraic equations at n + 1 unknown discrete points of f q0j; s

� �Xnj¼0

cjf q0j; s� �n

1q0j � ql

þ L0 ql; q0j� �þ 2x0

Z10

M0 ql; q0j; f� �

df

24

35

¼ �2 1� tð Þ�p0 ql; sð Þ l ¼ 1; 2; . . .; n

ð6:121Þ

Xnj¼0

cjf q0j; s� �n

¼ 0 ð6:122Þ

where

q0j ¼ cos jn p� �

j ¼ 0; 1; . . .; nql ¼ cos 2l�1

2n p� �

l ¼ 1; 2; . . .; nc0 ¼ cn ¼ 0:5; c1 ¼ c2 ¼ � � � ¼ cn�1 ¼ 1:

After solving the integral equation, the SIFs in the Laplace domain are defined as:

�kcðsÞ ¼ limr!c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 c� rð Þ

p�rzz r; 0; sð Þ; inner crack tip ð6:123aÞ

�kdðsÞ ¼ limr!d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 r � dð Þ

p�rzz r; 0; sð Þ; outer crack tip ð6:123bÞ


By introducing the normalized SIFs:

k0ðsÞ ¼ 1� tð ÞkðsÞatE Ti � T0ð Þ ffiffiffiffi

rop ð6:124Þ

Equations (6.123a) and (6.123b) can be rewritten as:

�k0c0 ðsÞ ¼f �1; sð Þ2 1� tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffid0 � c0

2

rð6:125aÞ

�k0d0 ðsÞ ¼ � f 1; sð Þ2 1� tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffid0 � c0

2

rð6:125bÞ

where f �1; sð Þ and f 1; sð Þ can be obtained from f q0j; s� �

using the methoddescribed in [52].

6.4.4.2 Outer Edge Crack ðrin\c0\d0 ¼ 1Þ

For an outer edge crack, the single-value condition (6.117) is not valid anymoresince the surface crack has only one crack tip r0 ¼ c0, i.e. the crack tip r0 ¼ d0

disappears. Special attention should be paid to the integral Eq. (6.116). Theasymptotic analysis of the integrand in (6.112) for large values of f indicates thatthe kernel M r0; x0; fð Þ may be expressed as the sum of two parts as follows:

M r0; x0; fð Þ ¼ M1 r0; x0; fð ÞþDM r0; x0; fð Þ ð6:126Þ

where DM r0; x0; fð Þ is nonsingular in the corresponding interval and M1 r0; x0; fð Þbecomes singular as r0 and x0 approach the end point d0 ¼ 1. After some manipu-lations the asymptotic expressions for the integrand and the singular part of thekernel M r0; x0; fð Þ are found to be:

M1 r0; x0; fð Þ ¼ � 1

2ffiffiffiffiffiffiffir0x0

p 2f2 1� r0ð Þ�1� x0ð Þ � 3f 1� x0ð Þ

� f 1� r0ð Þ þ 2� exp�f 2�r0�x0ð Þð6:127Þ

Thus, one can get:

Z10

M1 r0; x0; fð Þdf ¼ 1


p 12� r0 � x0

� 6 1� r0ð Þ2� r0 � x0ð Þ2 þ 4 1� r0ð Þ2

2� r0 � x0ð Þ3" #

,W r0; x0ð Þð6:128Þ


As f approaches infinity, DM r0; x0; fð Þ convergence rapidly. Therefore, Eq. (6.121)can be rewritten as:

Z1�1

u q0; sð Þ 1q0 � q

þ L0 q; q0ð Þþ 2x0W 0 q; q0ð Þ�

þ 2x0Z10

M0 q; q0; fð Þ½

�M01 q; q0; fð Þ�df�dq0 ¼ �2p 1� tð Þ�p0 q; sð Þ

ð6:129Þ

in which

M01 q; q0; fð Þ ¼ d0 � c0

2M1 r0; x0; fð Þ

W 0 q; q0ð Þ ¼ d0 � c0

2W r0; x0ð Þ

For an outer surface crack, the singularity indices must satisfy b1 ¼ �b2 ¼ �0:5.Applying the Gauss-Jacobi quadrature formulas [81]:

Z1�1

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� q01þ q0

sf q0; sð Þq0 � q

dq0 ¼Xnj¼1

2p 1� q0j� �2nþ 1

f q0j; s� �q0j � ql

ð6:130Þ

One derives a system of linear algebraic equations at n collocation points to solveEq. (6.130) as:

Xnj¼1

1� q0j2nþ 1

f q0j; s� � 1

q0j � qlþ L0 ql; q0j

� �þ 2x0jW0 ql; q0j� �þ 2x0j

"

�Z10

M0 ql; q0j; f� �� M0

1 ql; q0j; f� ��

df� ¼ � 1� tð Þ�p0 ql; sð Þ

ð6:131Þ

where

q0j ¼ cos 2j2nþ 1 p� �

j ¼ 1; 2; . . .; n

ql ¼ cos 2l�12nþ 1 p� �

l ¼ 1; 2; . . .; n

Similar to the embedded crack, the non-dimensional SIF at the crack tip of the outeredge crack in the Laplace domain can be expressed as:


�k0c0 ðsÞ ¼ f �1; sð Þ2 1� tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1� c0ð Þ

pð6:132Þ

where f �1; sð Þ can be obtained from f q0j; s� �

using the interpolation formulas.

6.4.4.3 Inner Edge Crack ðrin ¼ c0\d0\1Þ

For an inner edge crack, the similar method for the outer edge crack is adopted here.In this case, the singularity indices must satisfy �b1 ¼ b2 ¼ �0:5. As the integralof M r0; x0; fð Þ is singular when r0 and x0 approach c0 ¼ rin simultaneously, theasymptotic analysis of M r0; x0; fð Þ leads to:

M1 r0; x0; fð Þ ¼ 1


p 2f2 r0 � rinð Þ�x0 � rinð Þ � 3f x0 � rinð Þ

� f r0 � rinð Þþ 2� exp�f r0 þ x0�2rinð Þð6:133Þ

and

Z10

M1 r0; x0; fð Þdf ¼ 1


p �1r0 þ x0 � 2rin

þ 6 r0 � rinð Þr0 þ x0 � 2rinð Þ2

"

� 4 r0 � rinð Þ2r0 þ x0 � 2rinð Þ3

#,W r0; x0ð Þ

ð6:134Þ

Considering the quadrature formulas [31]:

Z1�1

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ q01� q0

sf q0; sð Þq0 � q

dq0 ¼Xnj¼1

2p 1þ q0j� �2nþ 1

f q0j; s� �q0j � ql

ð6:135Þ

One can have the following system of algebraic equations:

Xnj¼1

1þ q0j2nþ 1

f q0j; s� � 1

q0j � qlþ L0 ql; q0j

� �þ 2x0jW0 ql; q0j� �þ 2x0j

"

�Z10

M0 ql; q0j; f� �� M0

1 ql; q0j; f� ��

df� ¼ � 1� tð Þ�p0 ql; sð Þ

ð6:136Þ


where

q0j ¼ cos 2j�12nþ 1 p� �

j ¼ 1; 2; . . .; n

ql ¼ cos 2l2nþ 1 p� �

l ¼ 1; 2; . . .; n

The non-dimensional SIF at the crack tip of the inner edge crack in the Laplacedomain can be expressed as:

�k0d0 ðsÞ ¼ � f 1; sð Þ2 1� tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 d0 � c0ð Þ

pð6:137Þ

where f 1; sð Þ can be obtained from f q0j; s� �

through interpolation.Thus far, all the results in the Laplace domain have been obtained. To obtain the

solutions in the time domain, numerical inversion of Laplace transform is needed.


In this section, our focuses are placed on the effects of time delay, kernel functionand crack geometry on the transient temperature field, thermal axial stress, and theSIFs for three different kinds of cracks in a hollow cylinder. Here, it is worth notingthat the apostrophes of dimensionless quantities have been left out for brevity. Inthe calculation process, the non-dimensional inner and outer radius of the hollowcylinder is chosen to be ri ¼ 0:5 and ro ¼ 1, non-dimensional relaxation times ¼ 0:1 and Poisson’s ratio t ¼ 0:32, unless otherwise specified. A numericalalgorithm for inverse Laplace transform proposed by Brancik [82], with a furtherimprovement using a quotient-difference algorithm in [83], is employed in thecalculation.

Figure 6.12 shows the variation of transient temperature over the thickness ofthe cylinder at time t = 0.05 for various values of time delay. Clearly, the larger thetime delay, the smoother the distribution of the temperature through the thickness ofthe cylinder, similar to the findings in [75].

Figure 6.13 illustrates the effects of time delay on the history of SIFs at the innerand outer crack tips of a hollow cylinder with the embedded crack lying in theregion 0:7\r\0:9 when Kðt � nÞ ¼ 1. As expected, the maximum SIF at theinner crack tip is larger than that at the outer crack tip, indicating the crack growthmay start at the inner tip. Furthermore, a larger value of x will result in a lowermaximum SIF and a shorter duration for the SIF to reach the steady value.

Figure 6.14 displays the effects of time delay and kernel function on the transientSIF history near the crack tip for an outer edge crack in a hollow cylinder, in whichthe crack depth is set to be l=h ¼ 0:4. Taking the time delay into account leads to alower SIF, and an increase in x decreases the magnitude of SIF.


In an analogous manner, the transient SIF history near the crack tip for an inneredge crack in a hollow cylinder when cooled inside is depicted in Fig. 6.15. Similareffects of time delay and kernel function on the SIFs as reported for the embeddedcrack shown in Figs. 6.13 can be observed for the inner edge crack. Obviously,when the kernel function is determined, the maximum SIF with a particular crackdepth decreases with the increase of x. Moreover, for given values of x andK t � nð Þ, the maximum SIF will increase first for shallow edge cracks and thendecrease as the crack depth increases. For the same time delay, the maximum SIF isdifferent under different kernel functions.

6.5 Transient Thermal Stress Analysis of a CrackedHalf-Plane of Functionally Graded Materials

Extensive research has been accumulated on functionally graded materials (FGMs)due to their increasing application in heat engineering, such as high temperaturechambers, heat exchanger tubes, thermoelectric generators, gas turbines etc. [84–90]. Compared to homogenous composite materials, FGMs possess gradual chan-ges in composition and microstructure with spatially continuous variations inphysical and mechanical properties. Invented as a thermal shield to sustain veryhigh temperature gradients in thin-wall structures [84], one primary advantage ofFGMs is their excellent performance in improving bonding strength and reducingresidual and thermal stresses [37]. Fracture of a cracked FGM may occur when the

0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Τ

r

CV modelω=0.01ω=0.02ω=0.03

Fig. 6.12 Effects of time delay on the temperature distribution in a hollow cylinder without crackwhen heated inside at t = 0.05 for Kðt � nÞ ¼ 1 [79]


crack propagation is induced by an external thermal shock. To investigate thethermal stress concentration around cracks under high temperature, numerousstudies have been reported on crack problems in FGMs under thermal loading[26, 91–93].

However, almost all the analyses of heat conduction in FGMs are based on theclassical Fourier’s Law. Although Fourier’s Law is practical in many engineeringapplications, it is incapable of dealing with heat conduction in micro or nano scales,

0.0 0.2 0.4 0.6 0.8 1.0-0.10

-0.05

0.00

0.05

0.10

0.15

k c

t

CV modelω =0.01ω =0.02ω =0.03

(a)

0.0 0.2 0.4 0.6 0.8 1.0-0.20

-0.15

-0.10

-0.05

0.00

0.05

k d

t

CV modelω =0.01ω =0.02ω =0.03

(b)

Fig. 6.13 Effects of time delay on the SIF history at a the inner crack tip rc ¼ 0:7, b the outercrack tip rd ¼ 0:9 for an embedded crack in a hollow cylinder when cooled inside for Kðt � nÞ ¼1 [79]

6.5 Transient Thermal Stress Analysis … 285

or the extremely low or high temperature conditions [94]. In these situations, themeasured results showed significant discrepancy with the temperature predicted byFourier’s Law [6, 7], attributed to the finite thermal wave speed, as discussed

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

k d

t

CV modelω =0.01ω =0.02ω =0.03

Fig. 6.15 Effects of time delay on the SIF history for an inner edge crack in a hollow cylinderwhen cooled inside for Kðt � nÞ ¼ 1 [79]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-0.1

0.0

0.1

0.2

0.3

k c

t

CV modelω=0.01ω=0.02ω=0.03

Fig. 6.14 Effects of time delay on the SIF history for an outer edge crack in a hollow cylinderwhen heated inside for Kðt � nÞ ¼ 1 [79]


before. To compensate the effect of this non-Fourier heat conduction effect, aso-called hyperbolic heat conduction equation is proposed [55, 56] by simplyintroducing the time lag of thermal wave propagation into the heat conductionequation. Due to its simplicity, the hyperbolic heat conduction model has beenwidely used in the theoretical analysis of thermomechanical problems. For crackproblems, application of the hyperbolic heat conduction model on homogeneousmaterial has increasingly been found in the literature [27, 93–96]. Thermal stressanalysis of cracked homogeneous materials based on non-Fourier heat conductioncan be found in [97, 98], among others.

The transient crack problem in FGMs under thermal loading conduction modelhas only been investigated by Eshraghi et al. [99] assuming the circumferentialcrack does not disturb the temperature field. In this section, we build athermo-elastic, analytical model for a FGM half-plane containing a crack under athermal shock impact using the hyperbolic heat conduction theory. The crack isparallel to the free surface and assumed to be thermally insulated, so its disturbanceto the temperature field could not be neglected. Employing the theoretical frame-work developed earlier in this chapter, the problem is solved to illustrate the effectof nn-Fourier heat conduction on the transient thermoelastic response of thecracked, FGM half-plane.

6.5.1 Formulation of the Problem and Basic Equations

As shown in Fig. 6.16, assume a crack of length 2c parallel to the free surface islocated in a nonhomogeneous, functionally graded half-plane subjected to a thermalshock impact T0HðtÞ on the free surface at time t ¼ 0, where HðtÞ is the Heavisidefunction. The crack is assumed to be fully thermally insulated so the redistributionof temperature field must to be taken into considerations. At time t ¼ 0, thetemperature of the entire half-plane is initially a constant, without loss of gener-ality, which is set as zero. For simplicity, inertia effects and body forces areneglected.

Fig. 6.16 Crack geometryand coordinates [100]


The heterogenous properties of the FGM are assumed to vary exponentially withy-coordinate, expressed as:

E ¼ E0 expðbyÞ;t ¼ t0ð1þ eyÞ expðbyÞ;a ¼ a0 expðcyÞ;k ¼ k0 expðdyÞ;j ¼ j0;

ð6:138Þ

where b is graded material constant for Young’s modulus and Poisson’s ratio, e isgraded material constant for Poisson’s ratio, c is graded material constant forthermal expansion coefficient and d is graded material constant for thermalconductivity; E; t; a; k and j are the Young’s modulus, Poisson’s ratio, ther-mal expansion coefficient, heat conductivity and thermal diffusivity, respectively.

6.5.1.1 Heat Conduction Equations

For the hyperbolic the heat conduction model, when the inner heat source isnegligible, the governing equation of the temperature field in FGMs can beobtained as:

r2T þ d@T@y

¼ 1j@T@t

þ sj@2T@t2

: ð6:139Þ

By introducing the following dimensionless variables,

T ¼ T=T0;�t ¼ t=ðc2=jÞ; ð�x;�y; �hÞ ¼ ðx; y; hÞ=c; �d ¼ d � c;

Eq. (6.139) is converted into the following dimensionless form:

r2T þ d@T@y

¼ @T@t

þ jsc2

@2T@t2

: ð6:140Þ

Here and after, the hats of the dimensionless variables have been omitted forsimplicity. The dimensionless initial and boundary conditions for temperaturefield are:


T ¼ 0;@T@t

¼ 0; ðt ¼ 0Þ;Tðx;�hÞ ¼ 1; ðt[ 0; xj j\1Þ;T ¼ 0; ðy ! 1Þ;@T@y

¼ 0; ðy ¼ 0; xj j � 1Þ;Tðx; 0þ Þ ¼ Tðx; 0�Þ; ð xj j[ 1Þ;@Tðx; 0þ Þ

@y¼ @Tðx; 0�Þ

@y; ð xj j[ 1Þ:

ð6:141Þ

6.5.1.2 Thermal Stress Field Equations

In the following, we assume the functionally graded half plane is under plane stresscondition; i.e. rzz ¼ rzx ¼ rzy ¼ 0. Without considering the body force and inertiaeffect, the equilibrium equations are:

@rx@x

þ @rxy@y

¼ 0;@rxy@x

þ @ry@y

¼ 0; ð6:142Þ

and the strain-displacement relations are:

ex ¼ @u@x

; ey ¼ @v@y

; exy ¼ 12ð@u@y

þ @v@x

Þ; ð6:143Þ

The compatibility equation is:

@2ex@y2

þ @2ey@x2

¼ 2@2exy@x@y

; ð6:144Þ

while the thermoelastic constitutive equations are:

ex ¼ 1Eðrx � tryÞþ aT ;

ey ¼ 1Eðry � trxÞþ aT ;

exy ¼ 1þ tE

rxy:

ð6:145Þ


Again, employing Airy’s function U, the governing equation of elastic stress field inFGMs can be obtained as:

r2r2U � 2b@

@yðr2UÞþ b2

@2U@y2

þE0a0 expððbþ cÞyÞðr2T þ 2c@T@y

þ c2TÞ ¼ 0:ð6:146Þ

By introducing the other following dimensionless variables as follows,

rij ¼ rij=ðE0a0T0Þ;U ¼ U=ðE0a0T0c2Þ;

ðu; vÞ ¼ ðu; vÞ=ðca0T0Þ; eij ¼ eij=ða0T0Þ;ðb; e; cÞ ¼ ðb; e; cÞ � c;

the governing equations can be reduced to dimensionless forms:

r2r2U � 2b@

@yðr2UÞþ b2

@2U@y2

þ expððbþ cÞyÞðr2T þ 2c@T@y

þ c2TÞ ¼ 0:ð6:147Þ

Similarly, the hat of the dimensionless variables is omitted for simplicity. And theboundary conditions for mechanical conditions are:

rxyðx;�hÞ ¼ ryðx;�hÞ ¼ 0; ð xj j\1Þ;rxyðx; 0Þ ¼ ryðx; 0Þ ¼ 0; ð xj j � 1Þ;

rxyðx; 0þ Þ ¼ rxyðx; 0�Þ; ð xj j[ 1Þ;ryðx; 0þ Þ ¼ ryðx; 0�Þ; ð xj j[ 1Þ;uðx; 0þ Þ ¼ uðx; 0�Þ; ð xj j[ 1Þ;vðx; 0þ Þ ¼ vðx; 0�Þ; ð xj j[ 1Þ:

ð6:148Þ

6.5.2 Solution of the Temperature Field

The Laplace transform is employed against time variable, thus the governingEq. (6.140) and the corresponding boundary conditions can be transformed to:


r2T� þ d@T�

@y¼ pT� þ js

c2p2T�; ð6:149Þ

T�ðx;�hÞ ¼ 1=p; ð xj j\1Þ;T� ¼ 0; ðy ! 1Þ;@T�

@y¼ 0; ðy ¼ 0; xj j � 1Þ;

T�ðx; 0þ Þ ¼ T�ðx; 0�Þ; ð xj j[ 1Þ;@T�ðx; 0þ Þ

@y¼ @T�ðx; 0�Þ

@y; ð xj j[ 1Þ:

ð6:150Þ

Here and after, the superscript * denotes the variables in the Laplace domain, andp is the Laplace transform variable.

Applying Fourier transform to (6.149), the solution of temperature field sub-jected to the boundary conditions (6.151) in the Laplace domain can be obtained as:

T�ðx; y; pÞ ¼Z1�1

Dðn; pÞ expð�m2y� ixnÞdnþ 1pexpð�qðyþ hÞÞ; y[ 0;

T�ðx; y; pÞ ¼Z1�1

m2Dðn; pÞm1 � m2 expð�2mhÞ 1� expð�2mðhþ yÞÞf g expð�m1y� ixnÞdn

þ 1pexpð�qðyþ hÞÞ; y\0;

ð6:151Þ

where m1 ¼ d2 � m;m2 ¼ d

2 þm;m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipþ n2 þ d2

4 þBp2q

; q ¼ d2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipþ d2

4 þBp2q

;

B ¼ jsc2 ; and Dðn; pÞ is unknown and will be determined by the following density

function:

/�ðx; pÞ ¼ @T�ðx; 0þ ; pÞ@x

� @T�ðx; 0�; pÞ@x

: ð6:152Þ

Incorporating Eqs. (6.152) and (6.151), and employing Fourier inverse transform,we have

Dðn; pÞ ¼ � i½m1 � m2 expð�2mhÞ�4pnm

Z1�1

/�ðs; pÞ expðinsÞds: ð6:153Þ


Then from the continuity condition of (6.151), it is clear that

Z1�1

/�ðx; pÞdx ¼ 0; ð6:154Þ

/�ðx; pÞ ¼ 0; ð xj j[ 1Þ: ð6:155Þ

Substituting Eq. (6.15) into the temperature distribution (6.151), by using theboundary condition on the crack faces in (6.150), the following singular integralequation is obtained:

Z1�1

/�ðs; pÞ½ 1s� x

þ k�ðx; s; pÞ�ds ¼ 2pqp

expð�qhÞ; xj j � 1; ð6:156Þ

and the kernel function is given as:

k�ðx; s; pÞ ¼Z10

1þ m2½m1 � m2 expð�2mhÞ�mn

� �sin½ðx� sÞn�dn: ð6:157Þ

The numerical technique in [101] is employed to solve the integral Eqs. (6.156) and(6.154), and the following algebraic equation is obtained:

Xnk¼1

1nF�ðsk; pÞ 1

sk � xrþ k�ðxr; sk; pÞ

� �¼ 2pq

pexpð�qhÞ; xj j � 1; ð6:158aÞ

Xnk¼1

pnF�ðsk; pÞ ¼ 0: ð6:158bÞ

where sk ¼ cos ð2k�1Þp2n ; k ¼ 1; 2; . . .; n; xr ¼ cos rpn ; r ¼ 1; 2; . . .; n� 1 and

F�ðx; pÞ ¼ /�ðx; pÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p ; xj j � 1: ð6:159Þ

Once the integral equations are solved, the temperature field in the Laplace domaincan be obtained. The numerical technique in [36] is again used for the Laplaceinverse transform, thus the temperature field in the time domain is obtained.


6.5.3 Solution of Thermal Stress Field

Once the temperature field in the Laplace domain is obtained, the general solutionof Eq. (6.147) satisfying the regular condition at infinity can be obtained as:

U�ðx; y; pÞ ¼Z1�1

ðB1 þB2yÞ expð�s2y� ixnÞdn

�Z1�1

C1 exp½ðbþ c� m2Þy� ixn�dn; y[ 0;

U�ðx; y; pÞ ¼Z1�1

fðA1 þA2yÞþ ðA3 þA4yÞ expð�2syÞg expð�s1y� ixnÞdn

�Z1�1

fC21 þC22 expð�2lyÞ exp½ðbþ c� m1Þy� ixn�dn; y\0;

ð6:160Þ

where A1;A2;A3;A4;B1;B2 can be derived from the boundary conditions (6.150),and

s1 ¼ � b2� s; s1 ¼ � b

2þ s; s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ b2

4

s;

and

C1ðn; pÞ ¼ ½ðbþ c� m2Þðc� m2Þ � n2��2½c2 þ p� ð2c� dÞm2�Dðn; pÞ;

C21ðn; pÞ ¼ ½ðbþ c� m1Þðc� m1Þ � n2��2½c2 þ p� ð2c� dÞm1� m2Dðn; pÞm1 � m2 expð�2mhÞ ;

C22ðn; pÞ ¼ ½ðbþ c� m2Þðc� m2Þ � n2��2½ð2c� dÞm2 � c2 � p�m2Dðn; pÞ expð�2mhÞm1 � m2 expð�2mhÞ :

ð6:161Þ

Then the plane stresses in the Laplace domain can be obtained directly fromEq. (6.160) by taking some derivatives according to the definition of Airy’sfunction. Similarly, to solve the displacement field, two dislocation density func-tions are introduced here:

w�1 x; pð Þ ¼ @½u�ðx; pÞ�

@x; w�

2 x; pð Þ ¼ @½v�ðx; pÞ�@x

; ð6:162Þ


where ½u�ðx; pÞ�, and ½v�ðx; pÞ� are the displacement jumps across the crack faces.Considering the mechanical boundary condition on crack faces in Eq. (6.148), thefollowing singular integral equations can be obtained as:

Z1�1

X2j¼1

½ dijs� x

þ kijðx; sÞ�w�j ðs; pÞds ¼ 4pW�

i ðx; pÞ; i ¼ 1; 2; �1� x� 1;

ð6:163Þ

with

Z1�1

w�i ðx; pÞdx ¼ 0; i ¼ 1; 2: ð6:164Þ

The Fredholm-type kernels are given by:

k11ðx; sÞ ¼Z10

½1� 4nf11ðnÞ� sin½ðx� sÞn�dn;

k22ðx; sÞ ¼Z10

½1� 4n2f22ðnÞ� sin½ðx� sÞn�dn;

k12ðx; sÞ ¼Z10

�4nf12ðnÞ cos½ðx� sÞn�dn;

k21ðx; sÞ ¼Z10

�4n2f21ðnÞ cos½ðx� sÞn�dn;

ð6:165Þ

and

W�1 ðx; pÞ ¼ 2

Z10

nw�1ðn; pÞ sinðxnÞdn;

W�2 ðx; pÞ ¼ �2

Z10

n2w�2ðn; pÞ cosðxnÞdn;

w�1ðn; pÞ ¼ � h11ðbg1 þ 2g2Þþ 2sh12ðs2g1 � g2Þ

8s3� g3;

w�2ðn; pÞ ¼ � h21ðbg1 þ 2g2Þþ 2sh22ðs2g1 � g2Þ

8s3� g4;

ð6:166Þ


where the expressions of fijðnÞ; hijðnÞ ði; j ¼ 1; 2Þ and giðnÞ ði ¼ 1; 2; 3; 4Þ can befound in [100]. The solutions of the above integral equations can be expressed as:

w�i ðx; pÞ ¼

G�i ðx; pÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p ; ði ¼ 1; 2Þ; xj j � 1: ð6:167Þ

Using the Lobatto-Chebyshev method [102], the above singular integral equationscan be transformed to algebraic equations:

Xni¼1

Ai½ 1si � xk

þ k11ðxk; siÞ�G�1ðsi; pÞþ

Xni¼1

Aik12ðxk; siÞG�2ðsi; pÞ ¼ 4pW�

1 ðxk; pÞ;

Xni¼1

AiG�1ðsi; pÞ ¼ 0;

Xni¼1

Aik21ðxk; siÞG�1ðsi; pÞþ

Xni¼1

Ai½ 1si � xk

þ k22ðxk; siÞ�G�2ðsi; pÞ ¼ 4pW�

2 ðxk; pÞ;

Xni¼1

AiG�2ðsi; pÞ ¼ 0;

ð6:168Þ

where

si ¼ cosði� 1Þpn� 1

; i ¼ 1; 2; . . .n;

xk ¼ cosð2k � 1Þp2ðn� 1Þ ; k ¼ 1; 2; . . .n� 1;

Ai ¼ p2ðn� 1Þ ; i ¼ 1; n;Ai ¼ p

n� 1; i ¼ 2; 3; . . .n� 1:

From reference [88], the stress intensity factors (SIFs) in the Laplace domain can beobtained as:

K�I ðpÞ ¼ �

ffiffiffip

p4

G�2ð1; pÞ;K�

IIðpÞ ¼ �ffiffiffip

p4

G�1ð1; pÞ: ð6:169Þ

The dynamic stress intensity factors in the time domain can be obtained by theLaplace inverse transform via Eq. (6.39),

KIðtÞ ¼ 12pi

ZBr

�ffiffiffip

p4G�

2ð1; pÞ expðptÞdp;

KIIðtÞ ¼ 12pi

ZBr

�ffiffiffip

p4G�

1ð1; pÞ expðptÞdp;ð6:170Þ


where “Br” stands for the Bromwich path. In the following section, the numericalalgorithm of Laplace inverse transform proposed by Miller and Guy [36] will beused to obtain the SIFs in the time domain.

6.5.4 Numerical Results and Discussion

The temperature field distribution in the time domain can be obtained after takingthe inverse Laplace transform of Eq. (6.151). Since the crack is assumed to bethermally insulated, the existence of crack parallel to the free surface will disturbthe temperature field. At the beginning, the temperature variations of the mid-pointsof crack faces versus dimensionless time are investigated under the influence ofB ¼ js

c2 , which plays a vital role in the hyperbolic heat conduction theory. From Ref.[33], the thermal relaxation time for nonhomogeneous FGMs could be up to theorder of 10 s. If we take the typical crack size as 1 mm, the parameter B can be upto 10 according the experiment results in [33], which is much larger than that inhomogenous materials, such as metals.

To give a better illustration of the transient dynamic stress field around the cracktips, the variation of cleavage stresses defined by the following equation:

rh ¼ 1ffiffiffiffiffiffiffiffi2pr

p KI ½34 cosðh2Þþ 1

4cosð3h

2Þ� þKII ½� 3

4sinðh

2Þ � 3

4sinð3h

2Þ�

� �ð6:171Þ

are plotted against angle h 2 ð�180�; 180�Þ, as shown in Fig. 6.17. Two differenttime instants, t ¼ 3; t ¼ 15 are considered when B ¼ 1;B ¼ 10. Clearly thecleavage stresses reach their maximum at the same angle, which means the direc-tion of the possible crack propagation will always be the same at different timeinstants, independent of the thermal relaxation time (Fig. 6.17).

In FGMs, the nonhomogeneous material constants play a vital role as they affectthe SIFs significantly according to the literature. As a result, the parametricinvestigations are conducted under the framework of hyperbolic heat conductiontheory when B ¼ 0:5. From the singular integral equation in thermal stress field, thePoisson’s ratio would have no influence on the stress intensity factors, only thematerial constants b; c; d will affect the SIFs. As d; b play a rather more dominantrole than c in the thermoelastic response of the cracked structure [100], we presentonly the stress intensity factors history at various values of b in Fig. 6.18 and theirpeak values versus the gradient parameters d 2 ð�2; 2Þ and b 2 ð�2; 2Þ inFig. 6.19, respectively. It is noted the negative values of KI indicate crack faceswould be under compression (Figs. 6.18 and 6.19).


Fig. 6.17 Variation of the cleavage stress versus angle h 2 ð�180�; 180�Þ at time t ¼ 3; t ¼ 15when B ¼ 1;B ¼ 10 [100]

Fig. 6.18 The effect of the gradient parameter b on the SIFs when d ¼ 1; c ¼ 0:1;B ¼ 0:5 [100]


6.6 Summary

In this chapter, a systematic framework has been introduced to deal with crackproblems under transient thermomechanical loading based on the non-Fourier heatconduction models. Thermal field is assumed to be independent of the elastic fieldallowing to adopt an uncoupled thermoelasticity treatment of the two differentphysical fields. Integral transform and singular integral equation methods have beenemployed to construct the analysis. The transient SIFs and the thermal stressesunder a thermal shock in a cracked half-plane with a coating, a hollow cylinder, anda functionally graded, half plane have been calculated to illustrate the application ofthe developed methodology. Future works will see further extension of the methodto deal with the thermoelastic crack problems of other advanced functional mate-rials, such as piezoelectric materials, nano-composites, and magnetoelectroelasticmaterials.

References

1. Zhang MY, Cheng GJ (2011) Pulsed laser coating of bioceramic/metallic nanoparticles onmetal implants: multiphysics simulation and experiments. IEEE Trans Nanobiosci 99:1

2. Babaei MH, Chen ZT (2010) Transient hyperbolic heat conduction in a functionally gradienthollow cylinder. J Thermophys Heat Transf 24(2):325

Fig. 6.19 The variation of peak values of stress intensity factors with: (1) the gradient parameterb 2 ð�2; 2Þ when d ¼ 1; c ¼ 0:1. (2) the gradient parameter d 2 ð�2; 2Þ when b ¼ 1; c ¼ 0:1[100]


3. Roy S, Vasudeva Murthy AS, Kudenatti RB (2009) A numerical method for thehyperbolic-heat conduction equation based on multiple scale technique. Appl NumerMath 59:1419

4. Al-Nimr MA (1997) Heat transfer mechanisms during short-duration laser heating of thinmetal films. Int J Thermophys 18:1257

5. Naji M, Al-Nimr M, Darabseh T (2007) Thermal stress investigation in unidirectionalcomposites under the hyperbolic energy model. Int J Solids Struct 44:5111

6. Maurer MJ, Thompson HA (1973) Non-Fourier effects at high heat flux. ASME J HeatTransf 95:284

7. Babaei MH, Chen ZT (2008) Hyperbolic heat conduction in a functionally graded hollowsphere. Int J Thermophys 29:1457

8. Ozisik M, Tzou DY (1994) On the wave theory in heat conduction. ASME J Heat Transf116:526

9. Tzou DY (1989) The effects of thermal shock waves on the crack initiation around a movingheat source. Eng Fract Mech 34:1109

10. Al-Khairy RT, Al-Ofey ZM (2009) Analytical solution of the hyperbolic heat conductionequation for moving semi-infinite medium under the effect of time-dependent laser heatsource. J Appl Math 2009:1

11. Sih GC (1965) Heat conduction in the infinite medium with lines of discontinuities. ASME JHeat Transf 87:283

12. Tzou DY (1990) The singular behavior of the temperature gradient in the vicinity of amacrocrack tip. Int J Heat Mass Transf 33(12):2625

13. Tzou DY (1992) Characteristics of thermal and flow behavior in the vicinity ofdiscontinuities. Int J Heat Mass Transf 35(2):481

14. Jin Z-H, Noda N (1993) An internal crack parallel to the boundary of a nonhomogeneoushalf plane under thermal loading. Int J Eng Sci 31:793

15. Noda N, Jin Z-H (1993) Thermal stress intensity factors for a crack in a strip of afunctionally gradient material. Int J Solids Struct 30:1039

16. Erdogan F, Wu BH (1996) Crack problems in FGM layers under thermal stresses. J ThermStresses 19:237–265

17. Itou S (2004) Thermal stresses around a crack in the nonhomogeneous interfacial layerbetween two dissimilar elastic half-planes. Int J Solids Struct 41:923

18. El-Borgi S, Erdogan F, Hidri L (2004) A partially insulated embedded crack in an infinitefunctionally graded medium under thermo-mechanical loading. Int J Eng Sci 42:371–393

19. Wang BL, Mai Y-W (2005) A periodic array of cracks in functional graded materialssubjected to thermo-mechanical loading. Int J Eng Sci 43:432

20. Ueda S (2008) Transient thermoelectroelastic response of a functionally graded piezoelectricstrip with a penny-shaped crack. Eng Fract Mech 75:1204

21. Zhou YT, Li X, Yu DH (2010) A partially insulated interface crack between a gradedorthotropic coating and a homogeneous orthotropic substrate under heat flux supply. Int JSolids Struct 47:768

22. Qin QH (2000) General solutions for thermopiezoelectrics with various holes under thermalloading. Int J Solids Struct 37:5561

23. Gao CF, Noda N (2004) Thermal-induced interfacial cracking of magnetoelectroelasticmaterials. Int J Eng Sci 42:1347

24. Manson JJ, Rosakis AJ (1993) The effects of hyperbolic heat conduction around adynamically propagating crack tip. Mech Mater 15:263–278

25. Tzou DY (1990) Thermal shock waves induced by a moving crack. ASME J Heat Transf112:21

26. Hu KQ, Chen ZT (2012) Thermoelastic analysis of a partially insulated crack in a strip underthermal impact loading using the hyperbolic heat conduction theory. Int J Eng Sci51:144–160

27. Chen ZT, Hu KQ (2012) Thermo-elastic analysis of a cracked half-plane under a thermalshock impact using the hyperbolic heat conduction theory. J Therm Stresses 35:342–362

References 299

28. Chen W-H, Huang C-C (1992) On the singularity of temperature gradient near an inclinedcrack terminating at bimaterial interface. Int J Fract 58:319–324

29. Achenbach JD (1973) Wave propagation in elastic solids. North-Holland, Amsterdam30. Carslaw HS, Jaeger JC (1990) Conduction of heat in solids. Clarendon Press, Oxford31. Lewandowska M, Malinowski L (1998) Hyperbolic heat conduction in the semi-infinite

body with the heat source which capacity linearly depends on temperature. Heat Mass Transf33:389

32. Ali YM, Zhang LC (2005) Relativistic heat conduction. Int J Heat Mass Transf 48:239733. Kaminski W (1990) Hyperbolic heat conduction equation for materials with a nonhomo-

geneous inner structure. ASME J Heat Transf 112(3):55534. Chen ZT, Hu KQ (2012) Hyperbolic heat conduction in a cracked thermoelastic half-plane

bonded to a coating. Int J Thermophys 33:895–91235. Gradshteyn IS, Ryzhic IM (1965) Tables of integrals, series and products. Academic Press,

New York36. Miller MK, Guy WT (1966) Numerical inversion of the Laplace transform by use of Jacobi

polynomials. SIAM J Numer Anal 3:62437. Kovalenko AD (1969) Thermoelasticity: basic theory and applications. Noordhoff,

Groningen38. Williams ML (1959) The stress around a fault or crack in dissimilar media. Bull Seismol Soc

Am 49:199–20439. Sih GC (1962) On singular character of thermal stress near a crack tip. ASME J Appl Mech

51:587–58940. Tsai YM (1984) Orthotropic thermalelastic problem of uniform heat flow disturbed by a

central crack. J Compos Mater 18:122–13141. Nied HF, Erdogan F (1983) Transient thermal stress problem for a circumferentially cracked

hollow cylinder. J Therm Stress 6(1):1–1442. Noda N, Matsunaga Y, Nyuko H (1986) Stress intensity factor for transient thermal stresses

in an infinite elastic body with external crack. J Therm Stresses 9:119–13143. Chen TC, Weng CI (1991) Coupled transient thermo-elastic response in an edge-cracked

plate. Eng Fract Mech 39:915–92544. Atkinson C, Craster RV (1992) Fracture in fully coupled dynamic thermoelasticity. J Mech

Phys Solids 40:1415–143245. Sternberg E, Chakravorty JG (1959) On inertia effects in a transient thermoelastic problem.

ASME J Appl Mech 26:503–50846. Boley BA, Weiner JH (1985) Theory of thermal stresses. Wiley, New York47. Noda N, Matsunaga Y, Nyuko H (1990) Coupled thermoelastic problem of an infinite solid

containing a penny-shaped crack. Int J Eng Sci 28:347–35348. Arpaci VS (1966) Conduction heat transfer. Addison-Wesley Pub. Co., Reading49. Choi HJ, Thangjitham S (1993) Thermal-induced interlaminar crack-tip singularities in

laminated anisotropic composites. Int J Fract 60:327–34750. Delale F, Erdogan F (1979) Effect of transverse shear and material orthotropy in a cracked

spherical cap. Int J Solids Struct 15:907–92651. Theocaris P, Ioakimidis N (1977) Numerical integration methods for the solution of singular

integral equations. Quart Appl Math 35:173–18352. Nabavi SM, Ghajar R (2010) Analysis of thermal stress intensity factors for cracked

cylinders using weight function method. Int J Eng Sci 48(12):1811–182353. Meshii T, Watanabe K (2004) Stress intensity factor of a circumferential crack in a

thick-walled cylinder under thermal striping. J Press Vessel Technol Trans ASME126(2):157–162

54. Wang L (1994) Generalized Fourier law. Int J Heat Mass Transf 37:2627–263455. Cattaneo C (1958) A form of heat conduction equation which eliminates the paradox of

instantaneous propagation. Comp Rend 247(4):431–43356. Vernotte P (1958) Paradoxes in the continuous theory of the heat conduction. Comp Rend

246:3154–3155


57. Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61:41–7358. Tzou DY (1995) A unified field approach for heat conduction from macro to micro scales.

J Heat Transf Trans ASME 117:8–1659. Mallik SH, Kanoria M (2008) A two dimensional problem for a transversely isotropic

generalized thermoelastic thick plate with spatially varying heat source. Eur J Mech A/Solids27:607–621

60. Taheri H, Fariborz SJ, Eslami MR (2005) Thermoelastic analysis of an annulus using theGreen-Naghdi model. J Therm Stress 28:911–927

61. Keer LM, Freedmann JM, Watts HA (1977) Infinite tensile cylinder with circumferentialedge crack. Lett Appl Eng Sci 5:129–139

62. Erdol R, Erdogan F (1975) A thick-walled cylinder with an axisymmetric internal or edgecrack. J Appl Mech Trans ASME 45:281–286

63. Aydin L, Secil Altundag Artem H (2008) Axisymmetric crack problem of thick-walledcylinder with loadings on crack surfaces. Eng Fract Mech 75(6):1294–1309

64. Fu JW, Chen ZT, Qian LF, Xu YD (2014) Non-Fourier thermoelastic behavior of a hollowcylinder with an embedded or edge circumferential crack. Eng Fract Mech 128:103–120

65. Chen LM, Fu JW, Qian LF (2015) On the non-Fourier thermal fracture of an edge-crackedcylindrical bar. Theor Appl Fract Mech 80:218–225

66. Guo SL, Wang BL, Zhang C (2016) Thermal shock fracture mechanics of a cracked solidbased on the dual-phase-lag heat conduction theory considering inertia effect. Theor ApplFract Mech 86:309–316

67. Scher H, Montroll EW (1975) Anomalous transit-time dispersion in amorphous solid. PhysRev B 12:2455–2477

68. Koch DL, Brady JF (1988) Anomalous diffusion in heterogeneous porous media. Phys RevFluids 31:965–973

69. Li B, Wang J (2003) Anomalous heat conduction and anomalous diffusion inone-dimensional systems. Phys Rev Lett 91:044301–1–044301–4

70. Povstenko YZ (2004) Fractional heat conduction equation and associated thermal stress.J Therm Stress 28(1):83–102

71. Sherief HH, El-Sayed AMA, Abd El-Latief AM (2010) Fractional order theory ofthermoelasticity. Int J Solids Struct 47(2):269–275

72. Youssef HM (2010) Theory of fractional order generalized thermoelasticity. J Heat TransfTrans ASME 132(6):061301–1–7

73. Ezzat MA, El-Karamany AS, Ezzat SM (2012) Two-temperature theory inmagneto-thermoelasticity with fractional order dual-phase-lag heat transfer. Nucl Eng Des252:267–277

74. Wang JL, Li HF (2011) Surpassing the fractional derivative: concept of the memory-dependent derivative. Comput Math Appl 62(3):1562–1567

75. Yu YJ, Tian XG, Lu TJ (2013) Fractional order generalized electro-magneto-thermo-elasticity. Eur J Mech A/Solids 42:188–202

76. Yu YJ, Hu W, Tian XG (2014) A novel generalized thermoelasticity model based onmemory-dependent derivative. Int J Eng Sci 81:123–134

77. Ezzat MA, El-Karamany AS, El-Bary AA (2014) Generalized thermo-viscoelasticity withmemory-dependent derivatives. Int J Mech Sci 89:470–475

78. Ezzat MA, El-Karamany AS, El-Bary AA (2016) Electro-thermoelasticity theory withmemory-dependent derivative heat transfer. Int J Eng Sci 99:22–38

79. Xue Z, Chen Z, Tian X (2018) Transient thermal stress analysis for a circumferentiallycracked hollow cylinder based on a memory-dependent heat conduction model. TheoretAppl Fract Mech 96:123–133

80. El-Karamany AS, Ezzat MA (2016) Thermoelastic diffusion with memory-dependentderivative. J Therm Stress 39(9):1035–1050

81. Erdogan F, Gupta GD, Cook TS (1973) Numerical solution of singular integral equations.In: Sih GC (ed) Mechanics of fracture, methods of analysis and solutions of crack problems,vol 1. Noordhoff, Leyden, Netherlands, pp 368–425

References 301

82. Brancik L (1999) Programs for fast numerical inversion of Laplace transforms in MATLABlanguage environment. In: Proceedings of the 7th conference MATLAB’99, CzechRepublic, Prague, pp 27–39

83. Brancık L (2001) Utilization of quotient-difference algorithm in FFT-based numerical ILTmethod. In: Proceedings of the 11th international Czech-Slovak SCIENTIfiCCONFERENCERADIOELEKTRONIKA, Czech Republic, Brno, pp 352–355

84. Erdogan F (1995) Fracture mechanics of functionally graded materials. Compos Eng5:753–770

85. Singh AK, Siddhartha (2018) A novel technique for manufacturing polypropylene basedfunctionally graded materials. Int Polym Process 33:197–205

86. Singh AK, Vashishtha S (2018) Mechanical and tribological peculiarity of nano-TiO2-augmented, polyester-based homogeneous nanocomposites and their functionally gradedmaterials. Adv Polym Technol 37:679–696

87. Parameswaran V, Shukla A (1998) Dynamic fracture of a functionally gradient materialhaving discrete property variation. J Mater Sci 33:3303–3311

88. Jin ZH, Noda N (1994) Transient thermal stress intensity factors for a crack in a semi-infiniteplate of a functionally gradient material. Int J Solids Struct 31:203–218

89. Erdogan F, Wu BH (1996) Crack problems in FGM layers under thermal stresses. J ThermStress 19:237–265

90. Zhou YT, Li X, Qin JQ (2007) Transient thermal stress analysis of orthotropic functionallygraded materials with a crack. J Therm Stress 30:1211–1231

91. Rao BN, Kuna M (2010) Interaction integrals for thermal fracture of functionally gradedpiezoelectric materials. Eng Fract Mech 77:37–50

92. Chen ZT, Hu KQ (2014) Thermoelastic analysis of a cracked substrate bonded to a coatingusing the hyperbolic heat conduction theory. J Therm Stress 37:270–291

93. Chang DM, Wang BL (2012) Transient thermal fracture and crack growth behavior in brittlemedia based on non-Fourier heat conduction. Eng Fract Mech 94:29–36

94. Wang BL, Li JE (2013) Thermal shock resistance of solids associated with hyperbolic heatconduction theory. Proc R Soc A 469:20120754

95. Fu J, Chen ZT, Qian L, Hu K (2014) Transient thermoelastic analysis of a solid cylindercontaining a circumferential crack using the C-V heat conduction model. J Therm Stress37:1324–1345

96. Wang BL (2013) Transient thermal cracking associated with non-classical heat conductionin cylindrical coordinate system. Acta Mech Sin 29:211–218

97. Zhang XY, Li XF (2017) Transient thermal stress intensity factors for a circumferentialcrack in a hollow cylinder based on generalized fractional heat conduction. Int J Therm Sci121:336–347

98. Keles I, Conker C (2011) Transient hyperbolic heat conduction in thick-walled FGMcylinders and spheres with exponentially-varying properties. Eur J Mech A Solids30(3):449–455

99. Eshraghi I, Soltani N, Dag S (2018) Hyperbolic heat conduction based weight functionmethod for thermal fracture of functionally graded hollow cylinders. Int J Pres Ves Pip165:249–262

100. Yang W, Chen Z (2019) Investigation of the thermoelastic problem in cracked semi-infiniteFGM under thermal shock using hyperbolic heat conduction theory. J Therm Stresses42:993–1010. https://doi.org/10.1080/01495739.2019.1590170

101. Chen EP, Sih GC (1977) Mechanics of fracture, elastodynamic crack problems, vol 4.Noordhoff International Publishers

102. Erdogan F (1975) Complex function technique. In: Eringen AC (ed) Continuum physics,vol 2. Academic Press, New York, pp 523–603


http://dx.doi.org/10.1080/01495739.2019.1590170

Chapter 7Future Perspectives

7.1 Heat Conduction Theories

Non-Fourier heat conduction theories have seen wide range of applications in boththeoretical and applied science in recent years. However, a better understanding ofthe mechanism of non-Fourier heat conduction and its impact on multiphysicalresponses of advanced materials is still needed, in particular with respect to theindustrial applications. A few points deserve further exploration before these theoriescan be applied in a comparable fashion as the classical Fourier heat conduction:

• Selection of appropriate non-Fourier heat conduction theories for particularapplications. Non-Fourier heat conduction theories of various forms show theiradvantages in dealing with transient, time-dependent, heat processes as well asgenerated thermal wave. For instant, for transient processes on metals with shortphase lags in the order of less than 100 ps, a single phase lag model such as C-Vcan be the best choice to address the thermal wave effect. On the other hand, forbiological materials such as mammal skins where thermal wave travels at arelatively low speed, and the phase lag is relatively large (in an order frommilliseconds up to seconds), a memory-based fractional differential model canbe ideal to capture the history of heat conduction and its effect on thethermo-mechanical behavior of advanced materials.

• Application to nanoscale materials. Advances in ultrafast, laser-assisted manu-facturing have enabled the fabrication of miniaturized, nano/microscale devicesfor applications in electronics, optics, medicine, and energy applications, wherea non-equilibrium heat transfer model incorporating the size-dependent multi-physical properties is required to evaluate temperature rise during laser-assistedmanufacturing. In these applications a non-local, non-Fourier heat conduction,e.g. NL C-V or NL TPL, should be considered for thermo-mechanical analysis.To also avoid the field singularity around wavefronts, a fractional version ofnon-local, non-Fourier heat conduction, e.g. NL FTPL, can be considered forultrafast heat transfer in nanomaterials.


303




https://doi.org/10.1007/978-3-030-25201-4_7

• Determination of appropriate phase lags for different non-Fourier theories. Theexisting non-Fourier heat conduction theories have their specific advantages inparticular applications from advanced manufacturing to medicine. However,determination of the thermal phase lags for various types of advanced materialsis still a major challenge and requires dedicated experimentation to first capturethe thermal wave, and then correlate it to phase-lags and microstructural featuresof advanced materials through hybrid, experimental-multiscale modellingapproaches.

7.2 Application in Advanced Manufacturing Technologies

Application of non-Fourier theories in advanced manufacturing technologies, suchas laser-based additive manufacturing and 3D printing of high-melting temperaturematerials such as metals and ceramics, will be more frequently seen in the academicand industrial community due to the recent paradigm shift in manufacturing anddesign caused by advances in additive manufacturing (3D printing). Non-Fourierheat conduction provides a more viable tool to capture the transient, thermome-chanical behavior of advanced materials and offers a more conservative, butaccurate prediction of the integrity of structures under thermal shocks and/orhigh-strain rate mechanical deformation. In particular, with the development of softmachines and robotics in biomedical engineering and advanced manufacturing,transient heat processes with non-negligible thermal phase lags will be encounteredmore often where non-Fourier heat conduction will turn out to be the best choice forthermomechanical analysis in the design and analysis of advanced material andstructures.

304 7 Future Perspectives

Documents

Zengtao Chen Abdolhamid Akbarzadeh