Solid Mechanics and Its Applications

Volume 158

Founding Editor

G. M. L. Gladwell, University of Waterloo, Waterloo, ON, Canada

Series Editors

J. R. Barber, Department of Mechanical Engineering, University of Michigan,Ann Arbor, MI, USAAnders Klarbring, Mechanical Engineering, Linköping University, Linköping,Sweden

Aims and Scope of the Series

The fundamental questions arising in mechanics are: Why?, How?, and How much?The aim of this series is to provide lucid accounts written by authoritativeresearchers giving vision and insight in answering these questions on the subject ofmechanics as it relates to solids. The scope of the series covers the entire spectrumof solid mechanics. Thus it includes the foundation of mechanics; variationalformulations; computational mechanics; statics, kinematics and dynamics of rigidand elastic bodies; vibrations of solids and structures; dynamical systems andchaos; the theories of elasticity, plasticity and viscoelasticity; composite materials;rods, beams, shells and membranes; structural control and stability; soils, rocks andgeomechanics; fracture; tribology; experimental mechanics; biomechanics andmachine design. The median level of presentation is the first year graduate student.Some texts are monographs defining the current state of the field; others areaccessible to final year undergraduates; but essentially the emphasis is onreadability and clarity.

More information about this series at http://www.springer.com/series/6557

Richard B. Hetnarski • M. Reza Eslami

Thermal Stresses—AdvancedTheory and ApplicationsSecond Edition

123

Richard B. HetnarskiNaples, FL, USA

M. Reza EslamiDepartment of Mechanical EngineeringAmirkabir University of TechnologyTehran, Iran

ISSN 0925-0042 ISSN 2214-7764 (electronic)Solid Mechanics and Its ApplicationsISBN 978-3-030-10435-1 ISBN 978-3-030-10436-8 (eBook)https://doi.org/10.1007/978-3-030-10436-8

Library of Congress Control Number: 2018966845

1st edition: © Springer Science+Business Media, B.V. 20092nd edition: © Springer Nature Switzerland AG 2019This 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, expressed or implied, with respect to the material containedherein or for any errors or omissions that may have been made. The publisher remains neutral with regardto jurisdictional 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

The first paper ever published on thermal stresses and thermoelasticity: Duhamel, J.-M.-C., Secondmémoire sur les phénomènes thermo-mécaniques, J. de l’École Polytechnique, tome 15, cahier 25,1837, pp. 1–57

The authorsdedicate this book

to the memory of their Parents

Jan Hetnarski Mohammad Sadegh Eslami(1884–1966) (1900–1980)

and and

Izabela Hetnarska Zinat Shahrestani(1893–1980) (1925–2006)

Love of their sons was overwhelming.

Non Omnis Moriar.

Preface

Verba volant, scripta manent

The authors are pleased to present to readers the second edition of the book ThermalStresses—Advanced Theory and Applications. The book has been expanded andimproved. It is to serve a wide range of readers, in particular, graduate students,Ph.D. candidates, professors, scientists, researchers in various industrial and gov-ernment institutes, and engineers. Thus, the book should be considered not only as agraduate textbook, but also as a reference handbook to those working or interested inareas of applied mathematics, continuum mechanics, stress analysis, and mechanicaldesign. In addition, the book provides extensive coverage of great many theoreticalproblems and numerous references to the literature.

The field of thermal stresses lies at the crossroads of stress analysis, theory ofelasticity, thermodynamics, heat conduction theory, and advanced methods ofapplied mathematics. Each of these areas is covered to the extent it is necessary.Therefore, the book is self-contained, so that the reader should not need to consultother sources while studying the topic. The book starts from basic concepts andprinciples, and these are developed to more advanced levels as the text progresses.Nevertheless, some basic preparation on the part of the reader in classicalmechanics, stress analysis, and mathematics, including vector and Cartesian tensoranalysis, is expected.

While selecting material for the book, the authors made every effort to presentboth classical topics and methods, and modern, or more recent, developments in thefield. The second edition of the book comprises eleven chapters.

Chapter 1 treats, among other topics, the basic laws of thermoelasticity, withdescriptions and mathematical formulations of stresses, deformations, constitutivelaws, the equations of equilibrium and motion, the compatibility conditions, and anintroduction to two-dimensional thermoelasticity.

Chapter 2 is devoted to the necessary topics of thermodynamics. Detailedattention is given to the first and second laws of thermodynamics, Fourier’s law ofheat conduction, and more advanced topics, namely generalized thermoelasticityand second sound phenomenon, thermoelasticity without energy dissipation,

ix

variational formulation of mechanics, the reciprocity theorem, and the discussion ofinitial and boundary conditions. In the generalized theory of thermoelasticity, theLord–Shulman, the Green–Lindsay, and the Green–Naghdi models are treated, andthese models are then presented in a unified formulation for heterogeneous/anisotropic materials. The variational principle for the Green–Lindsay and Green–Naghdi models and uniqueness theorem for the Green–Naghdi model are added toSect. 2.14 in the second edition. These two subsections now present the Lord–Shulman, Green–Lindsay, and Green–Naghdi models.

Basic problems of Thermoelasticity are discussed in Chap. 3, where the analogyof thermal gradient and body forces is presented, and general solutions are derivedin rectangular Cartesian, cylindrical, and spherical coordinate systems. The equa-tions of motion and compatibility equations in spherical coordinates are added inthe second edition.

Chapter 4 is devoted to problems of heat conduction, again treated in variouscoordinate systems. Steady-state one-, two-, and three-dimensional problems arediscussed, and necessary mathematical methods, like the use of Fourier series andBessel functions, are introduced.

Engineering applications are treated in Chap. 5. Various kinds of beams,including rectangular, bimetallic, and curved beams, are discussed in detail, andmore advanced or modern aspects, such as functionally graded beams, are treated.

In Chap. 6, thermal stresses in disks, cylinders, and spheres are treated, includingfunctionally graded cylinders and spheres. The expressions for the radial dis-placement and stresses for thick spheres and disks are given for the specifiedtemperatures.

Chapter 7 presents an analysis of thermal expansion in piping systems, a uniqueintroduction to this frequently encountered engineering application, a topic ofimportance, treated by advanced design codes. The method of stiffness for pipingflexibility analysis is added to the second edition. The advantage of this method,compared to the elastic center method, is that the number of pipe branches can beselected larger than one and the pipe sections may have any orientation, not nec-essarily parallel to the coordinate axes.

In Chap. 8, the theories of coupled and generalized thermoelasticity are pre-sented. To the authors’ knowledge, such extensive treatment of these topics hasnever before been given in a textbook.

Finite and boundary element methods are the topic of Chap. 9. The Galerkinfinite element is introduced, and the methods of generalized thermoelasticity areapplied to disks and spheres. Also, problems of functionally graded beams andlayers are presented. Section 9.9 on thermally nonlinear generalized thermoelas-ticity is added in the second edition. This analysis is useful when the temperaturedifference compared to the reference temperature is significant.

Chapter 10 was not a part of the first edition of the book. It treats thermallyincluded vibrations in isotropic beams, in FGM beams, and in shallow arches.

x Preface

The last chapter, Chap. 11, is devoted to the analysis of creep. First, generaldefinitions and the theory are presented, and then, the problems related to thermaleffects are discussed. This chapter deals with useful and efficient numerical tech-niques to handle creep problems of structures subjected to thermal stresses.

At the end of Chaps. 1–8 and 11, there are given a number of problems forstudents to solve. In total, there are 63 problems. Also, at the end of each chapter,there is a list of literature.

Since the publication of the first edition, there have passed 10 years. The authorsfelt that a new, expanded, and improved edition would even better serve theinterested readers.

The authors express their thanks to Nathalie Jacobs, Publishing Editor,Mechanical Engineering, Springer, for her kind undertaking of the publishing of thesecond edition of the book.

Naples, Florida, USA Richard B. HetnarskiTehran, Iran M. Reza EslamiJanuary 2019

Preface xi

Contents

1 Basic Laws of Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Stresses and Tractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Coordinate Transformation. Principal Axes . . . . . . . . . . . . . . . 61.5 Principal Stresses and Stress Invariants . . . . . . . . . . . . . . . . . . 81.6 Displacement and Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . 101.7 Compatibility Equations. Simply Connected Region . . . . . . . . 151.8 Compatibility Conditions. Multiply Connected Regions . . . . . . 181.9 Constitutive Laws of Linear Thermoelasticity . . . . . . . . . . . . . 221.10 Displacement Formulation of Thermoelasticity . . . . . . . . . . . . 251.11 Stress Formulation of Thermoelasticity . . . . . . . . . . . . . . . . . . 261.12 Two-Dimensional Thermoelasticity . . . . . . . . . . . . . . . . . . . . 301.13 Michell Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2 Thermodynamics of Elastic Continuum . . . . . . . . . . . . . . . . . . . . . 452.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2 Thermodynamics Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 462.3 First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 472.4 Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 482.5 Variational Formulation of Thermodynamics . . . . . . . . . . . . . 502.6 Thermodynamics of Elastic Continuum . . . . . . . . . . . . . . . . . 512.7 General Theory of Thermoelasticity . . . . . . . . . . . . . . . . . . . . 562.8 Free Energy Function of Hookean Materials . . . . . . . . . . . . . . 602.9 Fourier’s Law and Heat Conduction Equation . . . . . . . . . . . . . 622.10 Generalized Thermoelasticity, Second Sound . . . . . . . . . . . . . 642.11 Thermoelasticity Without Energy Dissipation . . . . . . . . . . . . . 722.12 A Unified Generalized Thermoelasticity . . . . . . . . . . . . . . . . . 77

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2.13 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.14 Variational Principle of Thermoelasticity . . . . . . . . . . . . . . . . 922.15 Reciprocity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.16 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 1022.17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3 Basic Problems of Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . 1093.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.2 Temperature Distribution for Zero Thermal Stress . . . . . . . . . . 1103.3 Analogy of Thermal Gradient with Body Forces . . . . . . . . . . . 1133.4 General Solution of Thermoelastic Problems . . . . . . . . . . . . . . 1173.5 Solution of Two-Dimensional Navier Equations . . . . . . . . . . . 1213.6 General Solution in Cylindrical Coordinates . . . . . . . . . . . . . . 1243.7 Solution of Problems in Spherical Coordinates . . . . . . . . . . . . 1273.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4 Heat Conduction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.2 Problems in Rectangular Cartesian Coordinates . . . . . . . . . . . . 134

4.2.1 Steady-State One-Dimensional Problems . . . . . . . . . . 1344.2.2 Steady Two-Dimensional Problems–Separation

of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.2.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.2.4 Double Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 1434.2.5 Bessel Functions and Fourier–Bessel Series . . . . . . . . 1454.2.6 Nonhomogeneous Differential Equations

and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 1564.2.7 Lumped Formulation . . . . . . . . . . . . . . . . . . . . . . . . 1614.2.8 Steady-State Three-Dimensional Problems . . . . . . . . . 1664.2.9 Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 169

4.3 Problems in Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . 1794.3.1 Steady-State One-Dimensional Problems

(Radial Flow) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.3.2 Steady-State Two-Dimensional Problems . . . . . . . . . . 1834.3.3 Steady-State Three-Dimensional Problems . . . . . . . . . 1974.3.4 Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.4 Problems in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . 2054.4.1 Steady-State One-Dimensional Problems . . . . . . . . . . 2054.4.2 Steady-State Two- and Three-Dimensional

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.4.3 Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 217

xiv Contents

4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5 Thermal Stresses in Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255.2 Thermal Stresses in Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.3 Deflection Equation of Beams . . . . . . . . . . . . . . . . . . . . . . . . 2295.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2325.5 Shear Stress in a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2325.6 Beams of Rectangular Cross Section . . . . . . . . . . . . . . . . . . . 2335.7 Transient Stresses in Rectangular Beams . . . . . . . . . . . . . . . . 2385.8 Beam with Internal Heat Generation . . . . . . . . . . . . . . . . . . . . 2395.9 Bimetallic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.10 Functionally Graded Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 2425.11 Transient Stresses in FGM Beams . . . . . . . . . . . . . . . . . . . . . 2475.12 Thermal Stresses in Thin Curved Beams and Rings . . . . . . . . 2505.13 Deflection of Thin Curved Beams and Rings . . . . . . . . . . . . . 2515.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

6 Disks, Cylinders, and Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2616.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2616.2 Cylinders with Radial Temperature Variation . . . . . . . . . . . . . 2626.3 Thermal Stresses in Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . 2666.4 Thick Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2696.5 Thermal Stresses in a Rotating Disk . . . . . . . . . . . . . . . . . . . . 2736.6 Non-axisymmetrically Heated Cylinders . . . . . . . . . . . . . . . . . 2766.7 Method of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . 2826.8 Functionally Graded Thick Cylinders . . . . . . . . . . . . . . . . . . . 2896.9 Axisymmetric Stresses in FGM Cylinders . . . . . . . . . . . . . . . . 3006.10 Transient Thermal Stresses in Thick Spheres . . . . . . . . . . . . . 3076.11 Functionally Graded Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 3156.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

7 Thermal Expansion in Piping Systems . . . . . . . . . . . . . . . . . . . . . . 3297.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3297.2 Definition of the Elastic Center . . . . . . . . . . . . . . . . . . . . . . . 3307.3 Piping Systems in Two Dimensions . . . . . . . . . . . . . . . . . . . . 3347.4 Piping Systems in Three Dimensions . . . . . . . . . . . . . . . . . . . 3407.5 Pipelines with Large Radius Elbows . . . . . . . . . . . . . . . . . . . 3477.6 Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3587.7 Rotation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3637.8 Transformation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

Contents xv

7.9 Flexibility Matrix of a Single Member . . . . . . . . . . . . . . . . . . 3657.10 Flexibility Matrix of a Branch . . . . . . . . . . . . . . . . . . . . . . . . 3667.11 Flexibility Matrix of a Straight Member . . . . . . . . . . . . . . . . . 3687.12 Flexibility Matrix of a Bend Member . . . . . . . . . . . . . . . . . . . 3717.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

8 Coupled and Generalized Thermoelasticity . . . . . . . . . . . . . . . . . . . 3778.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3778.2 Governing Equations of Coupled Thermoelasticity . . . . . . . . . 3788.3 Coupled Thermoelasticity for Infinite Space . . . . . . . . . . . . . . 3818.4 Variable Heat Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3838.5 One-Dimensional Coupled Problem . . . . . . . . . . . . . . . . . . . . 3878.6 Propagation of Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . 3938.7 Half-Space Subjected to a Harmonic Temperature . . . . . . . . . . 4048.8 Coupled Thermoelasticity of Thick Cylinders . . . . . . . . . . . . . 4088.9 Green–Naghdi Model of a Layer . . . . . . . . . . . . . . . . . . . . . . 4138.10 Generalized Thermoelasticity of Layers . . . . . . . . . . . . . . . . . 4228.11 Generalized Thermoelasticity in Spheres and Cylinders . . . . . . 4308.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

9 Finite and Boundary Element Methods . . . . . . . . . . . . . . . . . . . . . . 4399.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4399.2 Galerkin Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4409.3 Functionally Graded Layers . . . . . . . . . . . . . . . . . . . . . . . . . . 4499.4 Coupled Thermoelasticity of Thick Spheres . . . . . . . . . . . . . . 4569.5 Generalized Thermoelasticity of FG Spheres . . . . . . . . . . . . . . 4669.6 Generalized Thermoelasticity of FG Disk . . . . . . . . . . . . . . . . 4789.7 Higher-Order Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4909.8 Functionally Graded Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 4949.9 Thermally Nonlinear Generalized Thermoelasticity . . . . . . . . . 5059.10 Boundary Element Formulation . . . . . . . . . . . . . . . . . . . . . . . 515References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

10 Thermally Induced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53310.2 Thermally Induced Vibrations of Isotropic Beams . . . . . . . . . . 53710.3 Thermally Induced Vibration of FGM Beams . . . . . . . . . . . . . 54610.4 Thermally Induced Vibration of Shallow Arches . . . . . . . . . . . 562Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

xvi Contents

11 Creep Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57911.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57911.2 Creep of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58211.3 Constitutive Equation of Uniaxial Creep . . . . . . . . . . . . . . . . . 58911.4 Creep Relaxation, Linear Rheological Models . . . . . . . . . . . . . 59111.5 Three-Dimensional Governing Equations . . . . . . . . . . . . . . . . 59311.6 Creep Potential, General Theory of Creep . . . . . . . . . . . . . . . . 59711.7 Stress Function for Creep Problems . . . . . . . . . . . . . . . . . . . . 60111.8 Creep Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60411.9 Creep Relaxation of Axisymmetric Stresses . . . . . . . . . . . . . . 61011.10 Creep Relaxation of Non-axisymmetric Stresses . . . . . . . . . . . 61511.11 Thermoelastic Creep Relaxation in Beams . . . . . . . . . . . . . . . 61911.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

Contents xvii

Notation—A Short List

The list contains basic notation. In particular chapters, the additional local notationis used. The system of notation is similar to that used in other engineering text-books. In particular, scalars appear as light face letters, and vectors and tensors asboldface straight letters.

A AreaB Boussinesq’s functionbi Components of body force vector per unit masscy Specific heat on path ycr Specific heat at constant stressce Specific heat at constant strainD Domain; Biot’s free energy functionDij Rate of deformation tensor componentseijk Permutation symboleij Green strain tensor componentsE Young’s modulusE Strain tensorF Free energy function per unit volumeF Forcegij Euclidean metric tensor componentsG Shear modulus = l; gravitational energy;

Gibbs thermodynamic potentialh Heat transfer coefficient by convectionH(.) Heaviside functionI Intrinsic energy per unit massI Moment of inertia of an area; intrinsic energy per unit volumeI1, I2, I3 Invariants of stress tensorJ Polar moment of inertiaJ1, J2, J3 Invariants of deviatoric stress tensork Thermal conductivityK Kinetic energy; bulk modulus

xix

L Velocity gradient tensorM Bending moment; Michell’s functionn Unit outer normal vectorp Pressurepi Entropy flux vector componentsq Heat flux vectorq Specified heat fluxQ Heatr Heat produced per unit time per unit massR Heat produced per unit time per unit volumes Entropy per unit mass; Laplace transform parameterS Entropy per unit volumesij Second Piola–Kirchhoff stress tensor componentst Timet0, t1, t2 Relaxation timestn Traction vectorT Absolute temperatureT0 Initial temperatureu, U Displacement vectorU Internal energy; generalized strain energy functionx, y, z Cartesian coordinatesr, u, z Cylindrical coordinatesr, u, h Spherical coordinatesW WorkX Body force vector per unit volumea Coefficient of linear thermal expansiond(.) Dirac delta functiondij Kronecker symboleij Strain tensor componentsj Diffusivityk, l Lamé constants (l = G)m Poisson’s ratioq Densityrij Stress tensor componentsh Temperature changeU Airy stress functionw Displacement potentialxij Rotation tensor components

xx Notation—A Short List

Historical Note. Beginnings of Thermal StressesAnalysis

Compared to the history of the theory of elasticity, which is traced to Robert Hookeand Edmé Mariotte in the seventeenth century or, even earlier, to Galileo Galilei inthe sixteenth–seventeenth century, the history of thermoelasticity and thermalstresses is much younger. The first paper on thermoelasticity, by Duhamel, was readbefore the French Academy of Sciences in Paris on February 23, 1835, and pub-lished in the Journal de l’Ècole Polytechnique in 1837 [1]. Duhamel’s paper con-tained the formulation of boundary value problems and also the derivation ofequations for the coupling of the temperature field and the body’s deformation. Tokeep this in perspective, the appearance of Duhamel’s paper occurred not very longafter Navier’s seminal paper on the foundations of the theory of elasticity [2] wasread to the Academy on May 14, 1821, and published in 1827, and Fourier’streatise on the theory of heat [3] was published in 1822. At present, we are in thethird century of development of this field of mechanics.

The formulation of thermoelasticity equations is due to Neumann [4] in 1885, toAlmansi [5] in 1897, to Tedone [6] in 1906, and to Voigt [7] in 1910. Most of theearly works were devoted to static problems. Integration of thermoelasticityequations was reduced to problems of action of body forces with the potential ofwhich density is the temperature of the body. Thermoelasticity problems werereduced to elasticity problems, where a body is acted upon by volume and surfaceforces. The method was developed by creators of the theory of elasticity, B. deSaint-Venant, G. Lamé, and P. S. Laplace. Namely, problems of thermoelasticitywere solved by finding solutions of Lamé displacement equations when a body isacted upon by arbitrary mass forces. Thus, many basic thermoelasticity problemswere considered within classic theory of elasticity.

Besides the development of the theory, a number of specific problems weresolved. We should mention the work of Borchardt [8] of 1873 on a solution inintegral representation for a sphere acted upon by an arbitrarily distributed tem-perature, and also papers by Hopkinson [9] of 1874 on thermal stresses in a sphere,by Leon [10] of 1905 on a hollow cylinder, and by Timoshenko [11] of 1925 onbimetallic strips.

xxi

Further progress was made by Biot [12] who, in 1935, analyzed properties oftwo-dimensional distributions of thermal stresses, and by Goodier [13] who, in1937, introduced the notion of the thermoelastic potential and considered the effectof non-continuous temperature fields. A paper by Signorini [14] of 1930 on finitethermoelastic deformations should also be mentioned. Of the papers publishedbefore the Second World War, papers by Muschelishvili [15] of 1923 and byPapkovich [16] of 1937 should be cited. We should also list the following threepapers that appeared after the War: by Lighthill and Bradshaw [17] of 1949 onthermal stresses in turbine blades, by Manson [18] of 1947 on gas turbine disks, andby Aleck [19] of 1949 on thermal stresses in rectangular plates.

During and after the Second World War, the requirements associated with newtechnologies contributed to a wave of research on thermal stresses. Gainingknowledge of the distribution of temperature in specific situations, finding thermalstresses in parts of complex mechanical systems, assessment of allowed stresses invarious materials and in various loading conditions, matters of stability, problemsof viscoelasticity, of fatigue, and thermal shock, became topics of active researchboth theoretical and experimental. Of the theoretical nature, a paper by Maysel [20]of 1941 on the generalization of the Betti-Maxwell theorem to thermal stresses, aswell as the work by Myklestadt [21] of 1942 on thermal stresses in a body withellipsoidal inclusion should be mentioned.

Steady-state problems of thermal stresses in a half-space were considered byMindlin and Cheng [22] in 1950, and by Sen [23] who, in 1951, introduced thenotion of nucleus of thermoelastic displacements, a fundamental development in thetheory of steady-state problems. A dynamic counterpart to the static nucleus wasproposed by Ignaczak [24].

An important development was the publication in 1953 of the first book onthermal stresses by Melan and Parkus [25]. The book provided a consistentdescription and analysis of most of the results received in steady-state thermoe-lasticity up to that time, and its publication provided a source from which teachingof the subject could be conducted. The book was subsequently complemented bythe publication, in 1959, of the book by Parkus [26] on non-stationary thermalstresses. Within the next few years, a number of other extensive texts appeared.Among them:

1. A very informative and carefully written monograph by Boley and Weiner [27],published in 1960. Besides its clear treatment of linear thermoelasticity, thebook contained also some nonlinear effects.

2. Somewhat earlier, in 1957, a practical description of thermal stressesApplications for airplanes etc. by Gatewood [28] was published. The bookcontains also many theoretical analyses.

3. The first book on thermoelasticity by Nowacki [29], in Polish, appeared in 1960.4. The publication of the book by W. Nowacki in 1960 was followed two years

later by the arrival of Nowacki’s monograph [30] in English, Thermoelasticity,an extensive and complete treatise on the subject. The second edition of thebook, revised and enlarged, appeared in 1986.

xxii Historical Note. Beginnings of Thermal Stresses Analysis

5. Also, Nowacki [31] published a more specialized monograph on DynamicalProblems of Thermoelasticity, first in Polish in 1966, and then its translationappeared in English in 1975.

6. A large monograph on thermoelasticity was published by Nowiński [32] in 1978.7. It was only many years later that a book specifically written as a textbook on

thermal stresses was prepared by Noda, Hetnarski, and Tanigawa [33].

Returning to research papers related to a half-space, detailed analysis ofsteady-state problem for half-space was provided by Sternberg and McDowell [34]in 1957. Some results contained in that paper were independently received, also in1957, by Nowacki [35] who considered a problem with discontinuous boundaryconditions for temperature on the surface of the half-space. Three years later, in1960, appeared a work by Sneddon and Lockett [36] in which steady-state prob-lems for a half-space and a layer were considered, and the authors receivedclose-form solutions for some types of surface heating.

While considering papers related to a half-space, we should mention the pub-lications on quasi-static problems. When changing the temperature is a slow pro-cess, one can disregard inertia terms in the equations of motion, provided that thegeometric parameters accept such assumption. To a group of first publications onsuch problems belong papers by Sadovsky [37] of 1955, Bailey [38] of 1958, andthe analysis of this type of problems by Nowacki [29] in his first book on ther-moelasticity in Polish in 1960. Boley [39] in 1956 showed that for a thin beamunder applied thermal load, thermally induced vibrations may occur if the beam isthin. He suggested a nondimensional inertia parameter B, of which numerical valueindicates whether this behavior may occur.

A pioneering work on dynamic thermoelasticity was a paper in Russian byDanilovskaya [40] of 1950, where she solved a one-dimensional problem ofstresses in a half-space due to a thermal shock applied to the bounding plane.Similar problems were subsequently considered by Mura [41] in 1952, and later bySternberg and Chakravorty [42] who in 1959 analyzed the behavior of a stress waveas a result of a change of boundary condition for temperature. In the same categoryare papers by Ignaczak [43] of 1957, and by Boley and Barber [44] of 1957. A largeamount of research on dynamic problems of thermoelasticity followed.

A broad subject to which much attention was given in the 1950s was plates anddisks. We should mention papers by Horvay [45] of 1952 on perforated plates, and[46] of 1954, on rectangular strips. Also, in 1957 appeared a paper by Parkus [47]on a heated disk. Of other problems with practical applications are problemsof thermal stresses in shells, like those treated by Parkus [48–50] in 1950–1951, andby Nowacki [51] in 1956. Other problems of that time were complex problemsof thermal stresses in axisymmetric bodies, such as circular cylinders with stressescaused by a non-steady and discontinuous temperature applied to the cylinder’ssurface. To this group belong papers by Trostel [52] and by Mura [53], both of1956, and by Ignaczak [54] and by Sokołowski [55], both of 1958.

Another area of research that was initiated in the 1950s was problems of ani-sotropic bodies. A method of operator determinants allowed reduction of the

Historical Note. Beginnings of Thermal Stresses Analysis xxiii

problem of solving of complex differential equations to quasi-biharmonic equationsand to the application of displacement functions similar to Galerkin’s displacementfunctions. We will mention works by Pell [56] of 1946, and a series of works byPolish authors: Nowiński [57] of 1955, Nowiński, Olszak, and Urbanowski [58] of1956, Mossakowski [59] of 1957, and also Nowacki [60] of 1958, and from otherworks, by Sharma [61] of 1958.

A number of researchers turned their attention to the investigation of inelasticeffects, such as visco-thermoelasticity effects. A book by Alfrey [62] on polymersof 1948 should be mentioned. We list papers by Tsien [63] of 1950, Read [64] of1950, Freudenthal [65] and [66] both of 1954, and Prager [67] of 1956, and [68] of1958.

Although J. M. C. Duhamel presented equations of thermoelasticity with cou-pling of field of deformation with field of temperature already in 1837, only paperspublished 120 years later by Biot [69] of 1956 and Lessen [70] of 1957 gave a newimpulse to do research in this area. In classic thermoelasticity, a problem of tem-perature was solved first, and then stresses were received from Duhamel–Neumannequations. But both theoretical considerations and simple experiments show that achange of displacements in a body accompanies a change of temperature, and achange of temperature is accompanied by a change of displacements. Thus, treatinga dynamic problem of thermoelasticity in stresses requires simultaneous solutionof the stress equation of motion and the heat conduction equation in which appearsthe time derivative of first stress invariant. A number of important papers oncoupled thermoelasticity were published in the next few years after Biot’s paperappeared in 1956. For example, Weiner [71] published in 1957 a proof onuniqueness of solutions of coupled equations of thermoelasticity. Analysis of wavepropagation, including Rayleigh waves, in thermoelastic bodies was the subject ofpapers that appeared in 1958: by Chadwick and Sneddon [72], Deresiewicz [73],and Lockett [74] and published in 1959 [75]. Of other papers published on thesubject in 1959, we should mention papers by Sneddon [76], Eason and Sneddon[77], Nowacki [78], and Paria [79].

The theory of coupled thermoelasticity, as well as developed later theory ofgeneralized thermoelasticity which was initiated by the paper by Lord and Shulman[80] of 1967, is extensively treated in this book in Chaps. 2, 8, and 9, and numerousreferences are provided. In this Note, we do not attempt to describe the develop-ment of the theory of thermal stresses for the time after 1960. If we did, the Notewould take substantially more space. However, we direct interested readers to thefollowing sources of information:

(A) A review of papers on thermal stresses edited by one of the authors(RBH) contained in two special issues of Applied Mechanics Reviews:

Vol. 44, issue No. 8–9, August–September 1991, with four review articles:

T. R. Tauchert, Thermally induced flexure, buckling, and vibration of plates.D. H. Allen, Thermomechanical coupling in inelastic solids.

xxiv Historical Note. Beginnings of Thermal Stresses Analysis

J. Ignaczak, Domain of influence results in generalized thermoelasticity—asurvey.N. Noda, Thermal stresses in materials with temperature-dependent properties.

Vol. 50, issue No. 9, September 1997, with three review articles:

E. A. Thornton, Aerospace thermal-structural testing technology.S. A. Dunn, Using nonlinearities for improved stress analysis by thermoelastictechniques.K. K. Tamma and R. R. Namburu, Computational approaches with applica-tions to non-classical and classical thermomechanical problems.

(B) In the years 1986–1999, five volumes of Thermal Stresses handbook, alledited by the first author (RBH), were published. First four volumes werepublished by North-Holland, Elsevier, Amsterdam, and the fifth volume waspublished by Lastran in Rochester, N.Y. Total number of pages in all fivevolumes is 2129. Each volume contains chapters with state-of-the-art coverageof specific areas of research and extensive bibliography. The lists of contentsare as follows:

Thermal Stresses I, North-Holland, Elsevier, Amsterdam, 1986, 547 pages.

1. R. B. Hetnarski, Basic Equations of the Theory of Thermal Stresses, pp. 1–21.2. T. R. Tauchert, Thermal Stresses in Plates—Statical Problems, pp. 24–141.3. J. Padovan, Anisotropic Thermal Stress Analysis, pp. 143–262.4. D. P. H. Hasselman and J. P. Singh, Criteria for the Thermal Stress Failure

of Brittle Structural Ceramics, pp. 263–298.5. L. Karlsson, Thermal Stresses in Welding, pp. 299–389.6. N. Noda, Thermal Stresses in Materials with Temperature-Dependent

Properties, pp. 391–483.

Thermal Stresses II, North-Holland, Elsevier, Amsterdam, 1987, 441 pages.

1. T. R. Tauchert, Thermal Stresses in Plates—Dynamical Problems, pp. 1–56.2. H. Sekine, Thermal Stress Singularities, pp. 57–117.3. F. Ziegler and H. Irschik, Thermal Stress Analysis Based on Maysel’s

Formula, pp. 119–188.4. R. A. Heller and S. Thangjitham, Probabilistic Methods in Thermal Stress

Analysis, pp. 189–268.5. R. S. Dhaliwal, Micropolar Thermoelasticity, pp. 269–328.6. G. R. Halford, Low-Cycle Thermal Fatigue, pp. 329–428.

Thermal Stresses III, North-Holland, Elsevier, Amsterdam, 1989, 573 pages.

1. J. R. Barber andM. Comninou, Thermoelastic Contact Problems, pp. 1–106.2. F. Ziegler and F. G. Rammerstorfer, Thermoelastic Stability, pp. 107–189.

Historical Note. Beginnings of Thermal Stresses Analysis xxv

3. T. Inoue, Inelastic Constitutive Relationships and Applications to SomeThermomechanical Processes Involving Phase Transformations,pp. 191–278.

4. J. Ignaczak,GeneralizedThermoelasticity and itsApplications, pp. 279–354.5. S. A. Łukasiewicz, Thermal Stresses in Shells, pp. 355–553.

Thermal Stresses IV, North-Holland, Elsevier, Amsterdam, 1996, 546 pages.

1. E. A. Thornton, Experimental Methods for High-Temperature AerospaceStructures, pp. 1–89.

2. S. A. Dunn, Non-Linear Effects in Stress Measurement by ThermoelasticTechniques, pp. 91–154.

3. G. L. England and Chiu M. Tsang, Thermally Induced Problems in CivilEngineering Problems, pp. 155–275.

4. K. K. Tamma, An Overview of Non-Classical/Classical Thermal structuralModels and Computational Methods for Analysis of EngineeringStructures, pp. 277–378.

5. L. Librescu and Weiqing Lin, Thermomechanical Postbuckling of Platesand Shells Incorporating Non-Classical Effects, pp. 379–452.

6. L. G. Hector, Jr., and R. B. Hetnarski, Thermal Stresses in Materials Due toLaser Heating, pp. 453–531.

Thermal Stresses V, Lastran, Rochester, NY, 1999, 542 pages.

1. C. T. Herakovich and J. Aboudi, Thermal Effects in Composites, pp. 1–142.2. K. K. Tamma and A. F. Avila, An Integrated Micro/Macro Modeling and

Computational Methodology for High Temperature Composites,pp. 143–256.

3. R. Wojnar, S. Bytnar, and A. Gałka, Effective Properties of ElasticComposites Subject to Thermal Fields, pp. 257–466.

4. N. Rajic, Material Characterization Using the Thermoplastic Effect,pp. 467–534.

In the year 2014, the 11-volume Encyclopedia of Thermal Stresses, edited byRichard B. Hetnarski, was published by Springer. It was a result of a 3-year effort,in which 708 entries were prepared by 614 authors. In the second edition of thisbook, there are numerous references to the entries that appeared in theEncyclopedia.

We should include in the list two review papers that were also used in thepreparation of the Note:

1. J. Ignaczak, Development of Thermoelasticity in years 1945–1960 (in Polish),Rozprawy Inżynierskie (Engineering Disertations), vol. 8, 3, 1960, pp. 581–600.

2. B. A. Boley, Thermal Stresses: A Survey, Thermal Stresses in SevereEnvironments, Proc. of Int. Conference, Virginia Polytechnic Institute and StateUniversity, Blacksburg, Virginia, published by Plenum Press, 1980, pp. 1–11.

xxvi Historical Note. Beginnings of Thermal Stresses Analysis

Also, two special issues of the Journal of Thermal Stresses contain bibliogra-phies on thermal stresses:

1. Bibliography on Thermal Stresses; Compiled by T. R. Tauchert andR. B. Hetnarski, Journal of Thermal Stresses, Vol. 9, Supplement, 1986,pp. (i)–(v), 1–128. The issue covers publications in JTS in Vol. 1–7 and partiallyVol. 8 (years 1978–1985).

2. Bibliography on Thermal Stresses in Shells; Compiled by F. W. Keene andR. B. Hetnarski, Journal of Thermal Stresses, Vol. 13, No. 4, 1990,pp. 341–545. The issue contains alphabetical listing with abstracts of somearticles, listing of particular types of shells (thin, thick, composite, etc.), theauthors index, list of journals, etc.

Finishing this Note, it is worth mentioning that two developments in recentdecades provided additional push to the growth of the field of thermal stresses:

• In 1978, an international scientific journal, the Journal of Thermal Stresses, waslaunched by R. B. Hetnarski, who was its Editor for the following 40 years, thatis, until 2018. The Journal, initially a quarterly publication, is now a monthly.

• In 1995, the First International Congress on Thermal Stresses (then called aSymposium), was held at Shizuoka University, Hamamatsu, Japan. Since then,the Congresses have been in principle held every two years, consecutively onthree continents, that is, in Asia, America, and Europe. International Congresseson Thermal Stresses (ICTS) are affiliated with the International Union ofTheoretical and Applied Mechanics (IUTAM).

For each of the Congresses extensive proceeding volumes have been publishedwhich present actual developments in the area of thermal stresses. Moreover,invited lectures presented at each Congress have been published in special issuesof the Journal of Thermal Stresses.

The list of Congresses is as follows:

International Congresses on Thermal Stresses (ICTS) (first two were calledSymposia)

First International Symposium on Thermal Stresses and Related TopicsThermal Stresses ’95—now called the First International Congress on ThermalStressesShizuoka University, Hamamatsu, JapanChair and Principal Local Organizer: Naotake NodaJune 5–7, 1995

Second International Symposium on Thermal Stresses and Related Topics—nowcalled the Second International Congress on Thermal StressesThermal Stresses ’97Rochester Institute of Technology, Rochester, New YorkChair and Principal Local Organizer: Richard B. HetnarskiJune 8–11, 1997

Historical Note. Beginnings of Thermal Stresses Analysis xxvii

Third International Congress on Thermal StressesThermal Stresses ’99Cracow University of Technology, Cracow, PolandChair and Principal Local Organizer: Jacek SkrzypekJune 13–17, 1999

Fourth International Congress on Thermal StressesThermal Stresses 2001Osaka Prefecture University in cooperation withOsaka Institute of TechnologyChair and Principal Local Organizer: Yoshinobu TanigawaJune 8–11, 2001

Fifth International Congress on Thermal StressesThermal Stresses 2003Virginia Polytechnic Institute and State UniversityBlacksburg, VirginiaChair and Principal Local Organizer: Liviu LibrescuJune 8–11, 2003

Sixth International Congress on Thermal StressesThermal Stresses 2005TechnischeUniversität, Vienna, AustriaChair and Principal Local Organizers: Rudolf Heuer and Franz ZieglerMay 26–29, 2005

Seventh International Congress on Thermal StressesThermal Stresses 2007National Taiwan University of Science and Technology (NTUST)Taipei, TaiwanChair and Principal Local Organizer: Ching-Kong ChaoJune 4–7, 2007

Eighth International Congress on Thermal StressesThermal Stresses 2009University of Illinois Urbana-ChampaignUrbana-Champaign, IllinoisChair and Principal Local Organizer: Martin Ostoja-StarzewskiJune 1–4, 2009

Ninth International Congress on Thermal StressesThermal Stresses 2011Budapest University of Technology and Economics, and Hungarian Academy ofSciences, Budapest, HungaryChair and Principal Local Organizer: Andras SzekeresJune 6–9, 2011

xxviii Historical Note. Beginnings of Thermal Stresses Analysis

Tenth International Congress on Thermal StressesThermal Stresses 2013Nanjing University of Aeronautics and Astronautics, Nanjing, ChinaChair and Principal Local Organizer: Cun-Fa GaoMay 31–June 3, 2013

Eleventh International Congress on Thermal StressesThermal Stresses 2016University of Salerno, ItalyChair and Principal Local Organizer: Michele CiarlettaJune 5–9, 2016

Twelfth International Congress on Thermal StressesThermal Stresses 2019Zhejiang University, Hangzhou, ChinaChair and Principal Local Organizer: Weiqiu ChenJune 1–5, 2019

Bibliography

1. Duhamel J-M-C (1837) Second mémoire sur les phénoménes thermo-mécaniques. J del’École Polytechnique, tome 15, cahier 25, pp 1–57

2. Navier CLMH (1827) Mémoire sur les lois de l’équilibre et du movement des corps solidesélastiques. Mém Acad Sci, Paris, tome VII, pp 375–393

3. Fourier JBJ (1822) Théorie analytique de la chaleur. Firmin Didot, Paris4. Neumann F (1855) Vorlesung über die Theorie des Elasticität der festen Körper und des

Lichtäthers. Teubner, Leipzig5. Almansi E (1897) Use of the stress function in thermoelasticity. Mem Reale Accad Sci

Torino, Series 2, vol 476. Tedone O (1906) Allgemeine Theoreme der matematischen Elastizitätslehre (Integra-

tionstheorie). Encyklopädie der matematischen Wissenschaften 4:55–124; 125–214, Part D(second article written with A. Timpe)

7. Voigt W (1910) Lehrbuch der Kristallphysik. Teubner, Berlin8. Borchardt CW (1873) Untersuchungen über die Elasticität fester isotropen Körper unter

Bercksichtigung der Wärme. Mber Akad. der Wissenschaften, 9, Berlin9. Hopkinson J (1879) Thermal stresses in a sphere, whose temperature is a function of r only.

Mess Math 8:16810. Leon A (1904) On thermal stresses. Der Bautechniker 26:96811. Timoshenko S (1925) Bending and buckling of bi-metallic strips. J Opt Soc Am 11:23312. Biot MA (1935) A general property of two-dimensional thermal stress distribution. Philos

Mag VII13. Goodier JN (1937) On the integration of the thermo-elastic equations. Philos Mag VII,

23:101714. Signorini A (1930) Sulle deformazioni termoelastiche finite. In: Proceedings of 3rd interna-

tional congress on applied mechanics, vol 2, Stockholm, pp 80–89

Historical Note. Beginnings of Thermal Stresses Analysis xxix

15. Muschelishvili N (1923) Sur l’équilibre des corps élastiques soumis l’action de la chaleur.Bull de l’Université Tiflis 3

16. Papkovich PF (1937) On the general solution of thermal stresses (in Russian). Prikl Mat MechNew Series 1(2)

17. Lighthill J, Bradshaw J (1949) Thermal stresses in turbine blades. Philos Mag 40:77018. Manson SS (1947) Determination of elastic stresses in gas turbine discs. NACA-Rep 87119. Aleck J (1949) Thermal stresses in a rectangular plate clamped along an edge. J Appl Mech

16(118)20. Maysel VM (1941) A generalization of Betti-Maxwell theorem to the case of the state

of thermal stresses and some applications (in Russian). Dokl Acad Nauk SSSR 30:115–11821. Myklestadt NO (1942) Two problems of thermal stress in the infinite solid. J Appl Mech 9

(136)22. Mindlin RD, Cheng DH (1950) Thermoelastic stress in the semi-infinite solid. J Appl Phys

21:93123. Sen B (1951) Note on the stresses produced by nuclei of thermoelastic strain in a semi-infinite

solid. Q Appl Math 8:36524. Ignaczak J (1959) A dynamic nucleus of thermoelastic strains in an elastic infinite space and

semi-space. Bull Acad Pol Sci Serie Tech 5:30525. Melan E, Parkus H (1953) Wärmespannungen infolge stationärer Temperaturfelder. Springer,

Wien26. Parkus H (1959) Instationäre Wärmespannungen. Springer, Wien27. Boley BA, Weiner JH (1960) Theory of thermal stresses. Wiley, New York28. Gatewood BE (1957) Thermal stresses. McGraw-Hill, New York29. Nowacki W (1960) Problems of thermoelasticity (in Polish). PWN-Polish Scientific

Publishers, Warszawa30. Nowacki W (1962) Thermoelasticity. Pergamon Press, Oxford, and PWN Polish Scientific

Publishers, Warszawa (2nd edition, PWN-Polish Scientific Publishers, Warszawa, andPergamon Press, Oxford, 1986)

31. Nowacki W (1966) Dynamic problems of thermoelasticity (in Polish). PWN Polish ScientificPublishers, Warszawa (The English translation: Noordhoff, Leyden, 1975)

32. Nowiński JL (1978) Theory of thermoelasticity with applications. Sijthoff and Noordhoff,Alpen aan den Rijn

33. Noda N, Hetnarski RB, Tanigawa Y (2003) Thermal stresses, 2nd edn. Taylor and Francis,New York

34. Sternberg E, McDowell EL (1957) On the steady-state thermoelastic problem for thehalf-space. Q Appl Math 14:381

35. Nowacki W (1957) A three-dimensional thermoelastic problem with discontinuous boundaryconditions. Arch Mech Stosow 9:319

36. Sneddon IN, Lockett FJ (1960) On the steady-state thermoelastic problem for the half-spaceand the thick plate. Q J Appl Math 18

37. Sadowsky MA (1955) Thermal shock on a circular surface of exposure of an elastichalf-space. J Appl Mech 22:177

38. Bailey JL (1958) A thermoelastic problem in the half-space. Dissertation, Michigan StateUniversity

39. Boley BA (1956) Thermally induced vibrations of beams. J Aeronaut Sci 23(2):17940. Danilovskaya VI (1950) Thermal stresses in an elastic half-space arising after a sudden

heating of its boundary (in Russian). Prikl Mat Mech 14(3)41. Mura T (1952) Thermal strains and stresses in transient state. In: Proceedings of the 2nd Japan

national congress for applied mechanics, vol 19, p 9

xxx Historical Note. Beginnings of Thermal Stresses Analysis

42. Sternberg E, Chakravorty JG (1959) On inertia effects in a transient thermoelastic problem.J Appl Mech 26(4)

43. Ignaczak J (1957) Thermal displacements in an elastic semi-space due to sudden heatingof the boundary plane. Arch Mech Stosow 9:395

44. Boley BA, Barber AD (1957) Dynamic response of beams and plates to rapid heating. J ApplMech 24:413

45. Horvay G (1952) Thermal stresses in perforated plates. In: Proceedings of the 1st U.S.national congress of applied mechanics, p 247

46. Horvay G (1954) Thermal stresses in rectangular strips, I, II. In: Proceedings of the 2nd U.S.national congress of applied mechanics, p 313

47. Parkus H (1954) Stress in a centrally heated disc. In: Proceedings of the 2nd U.S. nationalcongress of applied mechanics, p 307

48. Parkus H (1951) Die Grundgleichungen der allgemeinen Zylinderschale. Ősterr Ing Arch 6:3049. Parkus H (1950) Die Grundgleichungen der Schalentheorie in allgemeinen Koordinaten.

Ősterr Ing Arch 4:16050. Parkus H (1951) Wärmespannungen in Rotationschalen bei drehsymmetrischer

Temperaturverteilung, Sitsungsber. Ősterr Akad Wiss, Abt IIa, p 16051. Nowacki W (1956) Thermal stresses in cylindrical disks (in Polish). Arch Mech Stosow 8(1)52. Trostel R (1956) Instationäre Wärmespannungen in Hohlzylindern mit Kreisringquerschnitt.

Ing Arch 24(1)53. Mura T (1956) Dynamical thermal stresses due to thermal shocks. Res Rep, Faculty of

Engineering, Meiji University, p 854. Ignaczak J (1958) Thermal stresses in a long cylinder heated in a discontinuous manner over

the lateral surface. Arch Mech Stosow, p 2555. Sokołowski M (1958) Axially-symmetrical problems of thermo-elasticity for a cylinder of

unlimited length. Bull Acad Pol Sci Serie Tech 6:20756. Pell WH (1946) Thermal deflection of anisotropic thin plates. Quart Appl Math 4:2757. Nowiński J (1955) Thermal stresses in a thick-walled spherical vessel of transversally iso-

tropic material (in Polish). Arch Mech Stosow 7:363–37458. Nowiński J, Olszak W, Urbanowski W (1955) On the thermoelastic problem in the case of

bodies of any type of curvilinear orthotropy. Arch Mech Stosow 7(2):247–26559. Mossakowski J (1957) The state of stress and displacement in a thin anisotropic plate due to a

concentrated source of heat. Arch Mech Stosow 9(5):563–57760. Nowacki W (1954) Thermal stresses in anisotropic bodies (in Polish). Arch Mech Stosow

6:48161. SharmaB (1958) Thermal stresses in transversally isotropic semi-infinite solids. J ApplMech 2562. Alfrey T (1948) Mechanical behavior of high polymers. Interscience Publishers, New York63. Tsien HS (1950) A generalization of Alfrey’s theorem for viscoelastic media. Q Appl Math

8:104–10664. Read WT (1950) Stress analysis for compressible viscoelastic materials. J Appl Phys

21:671–67465. Freudenthal AM (1954) On inelastic thermal stresses in flight structures. J Aero Sci

21:772–77866. Freudenthal AM (1954) Effect of rheological behavior on thermal stresses. J Appl Phys 2567. Prager W (1956) Thermal stresses in viscoelastic structures. ZAMP 768. Prager W (1958) Non-isothermal plastic deformation. Proc Kong Nederl Akad van

Wetenschappen Ser B 61(3):176–18269. Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27

(3):240–253

Historical Note. Beginnings of Thermal Stresses Analysis xxxi

70. Lessen M (1956) Thermoelasticty and thermal shock. J Mech Phys Solids 5:5771. Weiner JH (1957) A uniqueness problem for coupled thermoelastic problems. Q Appl Math

15(1):102–10572. Chadwick P, Sneddon IN (1958) Plane waves in an elastic solid conducting heat. J Mech Phys

Solids 6:22373. Deresiewicz H (1958) Solutions of the equations of thermoelasticity. In: Proceedings of the

3rd U.S. national congress of applied mechanics74. Lockett FJ (1958) Effect of thermal properties of a solid on the velocity of Rayleigh waves.

J Mech Phys Solids 7:7175. Lockett FJ (1959) Longitudinal elastic waves in cylinders and tubes—including thermoelastic

effects. Proc Edinb Math Soc 11(3)76. Sneddon IN (1959) The propagation of thermal stresses in thin metallic rods. Proc R Soc

Edinb Sect A 6577. Eason G, Sneddon IN (1959) The dynamic stresses produced in elastic bodies by uneven

heating. Proc R Soc Edinb Sect A 6578. Nowacki W (1959) Some dynamic problems of thermoelasticity. Arch Mech Stosow 11(2)79. Paria G (1958) Coupling of elastic and thermal deformations. Appl Sci Res (A) 7:680. Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech

Phys Solids 15(5):299–309

xxxii Historical Note. Beginnings of Thermal Stresses Analysis