Advanced Structured Materials

Konstantin NaumenkoMarcus Aßmus Editors

Advanced Methods of Continuum Mechanics for Materials and Structures

Advanced Structured Materials

Volume 60

Series editors

Andreas Öchsner, Southport Queensland, AustraliaLucas F.M. da Silva, Porto, PortugalHolm Altenbach, Magdeburg, Germany

More information about this series at http://www.springer.com/series/8611

Konstantin Naumenko • Marcus AßmusEditors

Advanced Methodsof Continuum Mechanicsfor Materials and Structures

123

EditorsKonstantin NaumenkoInstitute of MechanicsOtto von Guericke UniversityMagdeburgGermany

Marcus AßmusInstitute of MechanicsOtto von Guericke UniversityMagdeburgGermany

ISSN 1869-8433 ISSN 1869-8441 (electronic)Advanced Structured MaterialsISBN 978-981-10-0958-7 ISBN 978-981-10-0959-4 (eBook)DOI 10.1007/978-981-10-0959-4

Library of Congress Control Number: 2016936648

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On the occasion of the 60th birthdayof Holm Altenbach

Preface

This volume of the Advanced Structured Materials Series is dedicated to Prof. HolmAltenbach, the leading researcher and teacher in the field of applied mechanics fromthe Faculty of Mechanical Engineering, Institute of Mechanics, Otto-von-Guericke-University Magdeburg (Germany) on the occasion of his 60th birthday.

Holm has made contributions in many fields of applied mechanics, includingtheory of shells, mechanics of composite materials, yield and failure criteria, con-stitutive models for inelastic behavior, continuum damage mechanics, micro-polarcontinua as well as mechanics of nano-sized structures. His achievements in theseareas have led to numerous national and international awards.

Holm holds honorary doctorates from the National Technical University,“Kharkiv Polytechnical Institute”, Kharkiv (Ukraine), the Ovidius University,Constanta (Romania) and I. Javakhishvili Tbilisi State University (Georgia).Amongst numerous awards he received, two notable medals are Gold-Medal of theMechanical Engineering Faculty, Politechnika Lubelska and Semko-Medal,National Technical University, “Kharkiv Polytechnical Institute”.

vii

He is Editor-in-Chief of Zeitschrift für Angewandte Mathematik und Mechanik(ZAMM, Journal of Applied Mathematics and Mechanics) and member of theeditorial boards of The Journal of Strain Analysis for Engineering Design,Mechanics of Composite Materials, Continuum Mechanics and Thermodynamicsand Technische Mechanik.

Among many international conferences and advanced courses Holm organizedor co-organized, some are as follows: the CISM-Courses in Udine, Italy Creep andDamage in Materials and Structures (1998), Modern Trends in CompositeLaminates Mechanics (2002), Cellular and Porous Materials: Modeling - Testing -Application (2009), Generalized Continua - From the Theory to EngineeringApplications (2011), Failure and Damage Analysis of Advanced Materials (2013),Shell-like Structures - Advanced Theories and Applications (2014), EUROMECHColloquia 444 Critical Review of the Theories of Plates and Shells and NewApplications (2002), 527 Shell-like Structures - Nonclassical Theories andApplications (2011) as well as international seminars on different aspects ofMechanics of Generalized Continua (2010) and (2012) in Lutherstadt Wittenbergand (2015) in Magdeburg.

Holm Altenbach was born in 1956 in Leipzig and grew up in Magdeburg, wherehe attended the Polytechnic Advanced High School. He passed the high schoolgraduation exam, “Reifeprüfung”, at the Martin-Luther-University Halle-Wittenbergin 1974. He went to Leningrad Polytechnical Institute (today St. Petersburg StatePolytechnical University) to study Energy Engineering (1974–1975) and Dynamicand Strength of Machines (1975–1980), where he graduated with distinction.

During his graduation, Holm started his research in applied mechanics. His firstexperience was to find effective properties of a composite plate from the givenproperties of constituents. Holm continued to explore composite plates and shellsduring his Ph.D. study (1980–1983) under the supervision of Profs. P.A. Zhilin andV.A. Palmov from the Chair “Mechanics and Control Processes” at St. PetersburgState Polytechnical University. At that time, this was a prestigious school in thefield of solid mechanics, founded by A.I. Lurie, the famous Russian and Sovietscientist. First paper by Holm in co-authorship with Zhilin was on the stability ofshells (Altenbach and Shilin 1982) and then with Palmov was on the topicCosserat-type plate theory (Palmow and Altenbach 1982) appeared in the localjournal, Journal of TH “Otto von Guericke” Magdeburg.1 In this university, Holmheld research fellowship as assistant (1980–1987) and senior assistant (1987–1995).Holm returned to St. Petersburg in 1987 to defend his second doctoral thesis andreceived the degree “Doctor of Technical Sciences”.2 The objective of the thesiswas to develop a nonlinear theory of shells under consideration of visco-elasticmaterial properties. Holm presented the main results in Ingenieur-Archiv

1TH “Otto von Guericke” Magdeburg was founded on August 6, 1953 as Hochschule (highereducational institute) of heavy machinery, in 1961 it was renamed as TH (technical higher edu-cational institute) Magdeburg, in 1987 in Technical University Otto von Guericke. In 1993 theOtto-von-Guericke-University Magdeburg was founded upon the former TH.2This degree is equivalent the habilitation in Germany.

viii Preface

(Altenbach 1988)—it was his first publication in the western part of Germany.3 Onefeature of his approach is that the theory is developed from the basic principles ofrational mechanics, directly for the thin shells. This is in contrast to the diverse shelltheories which are derived from equations of three-dimensional continuummechanics by mathematical or numerical techniques, for example, asymptoticmethods or variational principles. The direct approach is robust and elegant as thebalance laws are applied for shells. On the other hand, it is rather complex, sinceconstitutive equations must be formulated for stress resultants (Altenbach andZhilin 1988). This requires to extend the classical concepts of material and physicalsymmetries and to develop new approaches as how to identify the material prop-erties of shells. At that time it was not quite clear, why such an expensive directapproach might have been advantageous. In the last two decades, engineers havebeen dealing with materials which cannot be modeled by the three-dimensionalCauchy continuum and interestingly some materials even do not exist in a“three-dimensional form”. Examples include coatings, (organic) light-emittingdiodes, silicon photovoltaic cells, and thin films.

In 1996, Holm was appointed Full Professor of Engineering Mechanics at theMartin-Luther-University Halle-Wittenberg. His chair was integrated with theDepartment of Materials Sciences and the Institute of Materials Science.4 Holmspecialized in the research of mechanics of thin-walled structures. One aspect wasto consider inelastic material behavior including creep and damage (Altenbach et al.1997; Altenbach 2002). Another point was the modeling of laminates and shellsmade from short-fiber-reinforced materials. For such modeling, the manufacturingprocess should be analyzed first to predict the orientation of short fibers during theflow of fiber suspensions. Here the micro-polar (Cosserat-type) continuum theory isrequired to capture independent rotations of short fibers (Altenbach et al. 2003b,2007). Knowing the orientation states of fibers, one should estimate anisotropicmaterial properties to investigate deformations and stress state in thin-walledcomponents (Altenbach et al. 2003a; Kröner et al. 2009).

3In those years it was not easy to publish research results outside the Eastern Bloc countries.4This combination: Department of Materials Sciences (Fachbereich Werkstoffwissenschaften) andInstitute of Materials Science (Institut für Werkstoffwissenschaft) was the result of severalrestructuring stages of the former Technische Hochschule für Chemie Leuna-Merseburg. In 1998the department was renamed as Department of Engineering Sciences. In 2003 the department andthe chair were moved to Halle. In 2011, Holm returned with his group toOtto-von-Guericke-University Magdeburg (www.ifme.ovgu.de/ltm).

Preface ix

One of the favorite discussion topics of Holm is the question how to compute theeffective transverse shear stiffness of a plate. This is a tricky task even for platesmade of linear-elastic materials. Indeed, to find the transverse shear stiffness withinthe linear theory of elasticity, a nonlinear equation should be solved (Altenbach2000; Altenbach et al. 2015). As an example, consider a three-layer laminate platewith skin layers made from the same material and a core layer. Let h be thethickness of the plate and hc be the thickness of the core layer. Assume that theshear moduli are Gc and Gs for the core and skin layers, respectively. Thenaccording to Altenbach (2000), the effective transverse shear stiffness C of the platecan be computed as follows

C ¼ 13Gshk

2 1� a3ð1� lÞ� �; l ¼ Gc

Gs; a ¼ hc

h; ð1Þ

where k is the least positive root of the following equation

sin ka sin kð1� aÞ ¼ l cos ka cos kð1� aÞ: ð2Þ

For homogeneous plates with l ¼ 1, Eqs. (1) and (2) yield

C ¼ jGh; G ¼ Gs ¼ Gc;

where the factor j takes the value p2=12 as originally derived by Mindlin (1951).Sandwich plates applied in lightweight structures are usually composed of relativelythick soft core layer and thin skin layers with a relatively high stiffness. As shownin Altenbach (2000) for sandwich plates, the Reissner’s formula (Reissner 1947)follows from Eqs. (1) and (2)

C ¼ Gch:

For laminated glass plates having a thin core layer with the lower shear modulus,the approximate solution of Eq. (2) reads

k2 ¼ lað1� aÞ : ð3Þ

With Eq. (1), the approximate value of the transverse shear stiffness is

C ¼ 13Gch

1� a3ð1� lÞað1� aÞ : ð4Þ

For thin-walled structures undergoing creep and damage, special iterative pro-cedures are required in order to compute transverse shear deformation (Altenbachand Naumenko 2002).

Beyond the activities on shear correction factors, let us take the opportunity tomention some recent researches by Holm, his students, and collaborators. These

x Preface

include analysis of shells from functionally graded materials (Altenbach andEremeyev 2008), photovoltaic modules (Schulze et al. 2012; Weps et al. 2013;Eisentrager et al. 2015a, b), nanoscale shell structures, where surface effects have tobe taken into account (Altenbach et al. 2009; Altenbach and Eremeyev 2011;Altenbach et al. 2012), inelastic behavior of advanced heat-resistant materials(Altenbach et al. 2008; Langler et al. 2014), micro-mechanics of grain boundarycavitation under creep conditions (Ozhoga-Maslovskaja et al. 2015) and inelasticmicro-polar materials (Altenbach and Eremeyev 2014).

It is not straightforward to identify the exact number of books published and/oredited by Holm. We guess that this number would be in the range between 40 and50. Let us mention some of his textbooks which are used by many professors forteaching applied mechanics and are popular among students as well. These includeEngineering Mechanics (Altenbach 2014), Continuum Mechanics (Altenbach2015), Theory of Plates (Altenbach et al. 1998), and Mechanics of CompositeStructural Elements (Altenbach et al. 2004).

Holm has directly supervised over 30 doctoral students, and countless, doctoraland postdoctoral fellows from different countries. For many years, he has been theboard member of the interdisciplinary Graduate School 1554 on Micro-Macro-Interactions in Structured Media and Particle Systems5 funded by the GermanResearch Foundation (Deutsche Forschungsgemeinschaft, abbreviated DFG). ManyPh.D. holders from this school and from the former one6 have managed academicpositions in different universities.

Professor Altenbach with assistants and students. Merseburg, 2002

5see homepage: www.grk1554.ovgu.de.6Ph.D. schools (“Graduiertenkollegs”, abbreviated GRK) are established by German universities topromote young scientists. Their key emphasis is on the qualification of doctoral researchers withinthe framework of a focused research program and a structured training strategy. The first Ph.D.school at the TU Magdeburg entitled Modellierung, Berechnung und Identification mechanischerSysteme was founded in 1992.

Preface xi

This volume contains a collection of contributions on advanced approaches ofcontinuum mechanics written by leading scientists and collaborators, former Ph.D.students, and friends of Holm around the globe. These articles cover not onlymodern high-impact research areas but also historical essays and fundamentals. Wethank all these distinguished authors for sharing this celebration.

We would like to acknowledge the series editor Prof. Andreas Öchsner forgiving us the opportunity to publish this volume. We thank Helal Chowdhury,Johanna Eisenträger, Oliver Junge, and Barbara Renner from the Institute ofMechanics, Otto-von-Guericke-University Magdeburg, for their careful readings ofparts of the volume. We would like to acknowledge Dr. Christoph Baumann fromSpringer Publisher for the assistance and support during the preparation of thebook.

We wish Holm a wonderful 60th birthday, continued success, many new sci-entific papers and books, happiness, as well as excellent health for many years tocome.

Magdeburg Konstantin NaumenkoMarch 2016 Marcus Aßmus

References

Altenbach, H.: Eine direkt formulierte lineare Theorie für viskoelastische Platten und Schalen.Ingenieur-Archiv 58(3), 215–228 (1988). doi:10.1007/BF00534332

Altenbach, H.: An alternative determination of transverse shear stiffnesses for sandwich andlaminated plates. Int. J. Solids Struct. 37(25), 3503–3520 (2000). doi:10.1016/S0020-7683(99)00057-8

Altenbach, H.: Creep analysis of thin-walled structures. ZAMM—Journal of Applied Mathematicsand Mechanics (Zeitschrift für Angewandte Mathematik und Mechanik) 82(8), 507–533(2002). doi:10.1002/1521-4001(200208)82:8<507::AID-ZAMM507>3.0.CO;2-Y

Altenbach, H.: Holzmann/Meyer/Schumpich Technische Mechanik Festigkeitslehre. SpringerVieweg, Wiesbaden (2014). doi:10.1007/978-3-658-06041-1

Altenbach, H.: Kontinuumsmechanik: Einführung in die materialunabhängigen undmaterialabhängigen Gleichungen, 3rd edn. Springer, Berlin Heidelberg (2015). doi:10.1007/978-3-662-47070-1, (originally appeared at Teubner publisher, Leipzig, 1994; publishedtogether with his father Johannes Altenbach)

Altenbach, H., Eremeyev, V.: Strain rate tensors and constitutive equations of inelastic micropolarmaterials. Int. J. Plast 63, 3–17 (2014). doi:10.1016/j.ijplas.2014.05.009

Altenbach, H., Eremeyev, V.A.: Analysis of the viscoelastic behavior of plates made offunctionally graded materials. ZAMM—Journal of Applied Mathematics and Mechanics(Zeitschrift für Angewandte Mathematik und Mechanik) 88(5), 332–341 (2008). doi:10.1002/zamm.200800001

Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int.J. Eng. Sci. 49(12), 1294–1301 (2011). doi:10.1016/j.ijengsci.2011.03.011

Altenbach, H., Naumenko, K.: Shear correction factors in creep-damage analysis of beams, platesand shells. JSME International Journal Series A, Solid Mech. Mater. Eng. 45, 77–83 (2002).doi:10.1299/jsmea.45.77

xii Preface

Altenbach, H., Shilin, P.: Eine nichtlineare Theorie dünner Dreischichtschalen und ihreAnwendung auf die Stabilitätsuntersuchung eines dreischichtigen Streifens. TechnischeMechanik 3(2), 23–30 (1982)

Altenbach, H., Zhilin, P.A.: A general theory of elastic simple shells (in Russian). Adv. in Mech.11(4), 107–148 (1988)

Altenbach, H., Morachkovsky, O., Naumenko, K., Sychov, A.: Geometrically nonlinear bendingof thin-walled shells and plates under creep-damage conditions. Arch. Appl. Mech. 67, 339–352 (1997). doi:10.1007/s004190050122

Altenbach, H., Altenbach, J., Naumenko, K.: Ebene Flächentragwerke. Springer, BerlinHeidelberg (1998). doi:10.1007/978-3-642-58721-4

Altenbach, H., Naumenko, K., L’vov, G.I., Pylypenko, S.: Numerical estimation of the elasticproperties of thin-walled structures manufactured from short-fiber reinforced thermoplastics.Mech. Compos. Mater. 39, 221–234 (2003a). doi:10.1023/A:1024566026411

Altenbach, H., Naumenko, K., Zhilin, P.: A micro-polar theory for binary media with applicationto phase-transitional flow of fiber suspensions. Continuum Mech. Thermodyn. 15, 539–570(2003 b). doi:10.1007/s00161-003-0133-5

Altenbach, H., Altenbach, J.W., Kissing, W.: Mechanics of composite structural elements.Springer, Berlin Heidelberg (2004). doi:10.1007/978-3-662-08589-9

Altenbach, H., Naumenko, K., Pylypenko, S., Renner, B.: Influence of rotary inertia on the fiberdynamics in homogeneous creeping flows. ZAMM—Journal of Applied Mathematics andMechanics (Zeitschrift für Angewandte Mathematik und Mechanik) 87(2), 81–93 (2007).doi:10.1002/zamm.200610303

Altenbach, H., Naumenko, K., Gorash, Y.: Creep analysis for a wide stress range based on stressrelaxation experiments. Int. J. Mod. Phys. B 22(31–32), 5413–5418 (2008). doi:10.1142/S0217979208050589

Altenbach, H., Eremeyev, V., Morozov, N.: Linear theory of shells taking into account surfacestresses. Dokl. Phys. 54(12), 531–535 (2009). doi:10.1134/S1028335809120039

Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Surface viscoelasticity and effective properties ofthin-walled structures at the nanoscale. Int. J. Eng. Sci. 59, 83–89 (2012). doi:10.1016/j.ijengsci.2012.03.004

Altenbach, H., Eremeyev, V.A., Naumenko, K.: On the use of the first order shear deformationplate theory for the analysis of three-layer plates with thin soft core layer. ZAMM—Journal ofApplied Mathematics and Mechanics (Zeitschrift für Angewandte Mathematik und Mechanik)95(10), 1004–1011 (2015). doi:10.1002/zamm.201500069

Eisenträger, J., Naumenko, K., Altenbach, H., Köppe, H.: Application of the first-order sheardeformation theory to the analysis of laminated glasses and photovoltaic panels. Int. J. Mech.Sci. 96, 163–171 (2015a). doi:10.1016/j.ijmecsci.2015.03.012

Eisenträger, J., Naumenko, K., Altenbach, H., Meenen, J.: A user-defined finite element forlaminated glass panels and photovoltaic modules based on a layer-wise theory. Compos. Struct.133, 265–277 (2015b). doi:10.1016/j.compstruct.2015.07.049

Kröner, C., Altenbach, H., Naumenko, K.: Coupling of a structural analysis and flow simulationfor short-fiber-reinforced polymers: property prediction and transfer of results. Mech. Compos.Mater. 45(3), 249–256 (2009). doi:10.1007/s11029-009-9086-5

Längler, F., Naumenko, K., Altenbach, H., Ievdokymov, M.: A constitutive model for inelasticbehavior of casting materials under thermo-mechanical loading. J. Strain Anal. Eng. Des. 49(6), 421–428 (2014). doi:10.1177/0309324714522034

Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elasticplates. J. Appl. Mech. 18(2), 31–38 (1951)

Ozhoga-Maslovskaja, O., Naumenko, K., Altenbach, H., Prygorniev, O.: Micromechanicalsimulation of grain boundary cavitation in copper considering non-proportional loading.Comput. Mater. Sci. 96, 178–184 (2015). doi:10.1016/j.commatsci.2014.08.050

Palmow, W.A., Altenbach, H.: Über eine Cosseratsche Theorie für elastische Platten. TechnischeMechanik 3(3), 5–9 (1982)

Preface xiii

Reissner, E.: On the bending of elastic plates. Quarterly of Applied Mathematics 5, 55–68 (1947)Schulze, S.H., Pander, M., Naumenko, K., Altenbach, H.: Analysis of laminated glass beams for

photovoltaic applications. Int. J. Solids Struct. 49(15), 2027–2036 (2012). doi:10.1016/j.ijsolstr.2012.03.028

Weps, M., Naumenko, K., Altenbach, H.: Unsymmetric three-layer laminate with soft core forphotovoltaic modules. Compos. Struct. 105, 332–339 (2013). doi:10.1016/j.compstruct.2013.05.029

xiv Preface

Contents

Part I Fundamentals and Elasticity

Time Derivatives in Material and Spatial Description—WhatAre the Differences and Why Do They Concern Us? . . . . . . . . . . . . . . 3Elena A. Ivanova, Elena N. Vilchevskaya and Wolfgang H. Müller

The Cosserats’ Memoir of 1896 on Elasticity . . . . . . . . . . . . . . . . . . . . 29Gérard A. Maugin

On Equivalent Inhomogeneities for Particleswith Multiple-Component Interphases . . . . . . . . . . . . . . . . . . . . . . . . . 49Lidiia Nazarenko and Henryk Stolarski

The Influence of Distributed Dislocations on Large Deformationsof an Elastic Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Evgeniya V. Zhbanova and Leonid M. Zubov

Part II Visco-Elastic Media and Coupled Phenomena

A Closed-Form Solution for a Linear ViscoelasticSelf-gravitating Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Wolfgang H. Müller and Elena N. Vilchevskaya

Constitutive Modelling of the Glass Transition and RelatedPhenomena: Relaxation of Shear Stress Under Pressure . . . . . . . . . . . . 103Alexander Lion, Michael Johlitz and Christoph Mittermeier

A Material Model for Electroactive Polymers . . . . . . . . . . . . . . . . . . . . 119Kerstin Weinberg and Anna Pandolfi

Part III Thermo-Elasto-Plasticity

Finite Elastoplasticity with Thermal Effects—Some UnexpectedPhenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Otto T. Bruhns

xv

Crystal-Plasticity Simulation of Micromachining of Single-CrystalMetal: Methodology and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Qiang Liu, Srihari Dodla, Anish Roy and Vadim V. Silberschmidt

Warpage Variation Analysis of Si/Solder/Cu Layered PlatesSubjected to Cyclic Thermal Loading . . . . . . . . . . . . . . . . . . . . . . . . . 185Nobutada Ohno, Satoshi Mizushima, Dai Okumura and Hisashi Tanie

A Note on Plasticity with Additional Internal Variables . . . . . . . . . . . . 205Michael Wolff and Michael Böhm

Finite Inelastic Deformations of Compressible Soft Solidswith the Mullins Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Heng Xiao, Hao Li, Zhao-Ling Wang and Zheng-Nan Yin

Effect of Geometric Dimension on the Dissipative Propertyof the Structures Consisting of NiTi Shape Memory Alloy Wires . . . . . 243Chao Yu, Guozheng Kang and Qianhua Kan

Part IV Continuum Damage and Phase Fields

A Discussion on Gradient Damage and Phase-Field Modelsfor Brittle Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263René de Borst and Clemens V. Verhoosel

Experiments on Damage and Failure Mechanisms in DuctileMetals at Different Loading Conditions . . . . . . . . . . . . . . . . . . . . . . . . 279Michael Brünig, Steffen Gerke and Marco Schmidt

Modeling of Damage Deactivation in Concrete Under CyclicCompression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Artur Ganczarski and Marcin Cegielski

Phase-Field Modelling of Damage and Fracture—Convergenceand Local Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307Markus Kästner, Paul Hennig, Thomas Linse and Volker Ulbricht

Part V Fracture Mechanics and Failure Criteria

Prestressed Orthotropic Material Containing an Elliptical Hole . . . . . . 327Eduard-Marius Craciun

Generalized Limit Surfaces—With an Example of Hard Foams . . . . . . 337Nina-Carolin Fahlbusch, Vladimir A. Kolupaev and Wilfried Becker

On the Problem of Cracking in 2-Phase Ceramic MatrixComposite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367Tomasz Sadowski, Liviu Marsavina and Eduard-Marius Craciun

xvi Contents

On Failure Theories for Composite Materials. . . . . . . . . . . . . . . . . . . . 379Ramesh Talreja

Part VI Rods, Plates and Shells

On the Dislocation Density Tensor in the CosseratTheory of Elastic Shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391Mircea Bîrsan and Patrizio Neff

Direct Approach Versus Consistent Approximation . . . . . . . . . . . . . . . 415Reinhold Kienzler and Patrick Schneider

Modelling of a Rotating Active Thin-Walled CompositeBeam System Subjected to High Electric Fields . . . . . . . . . . . . . . . . . . 435Jarosław Latalski

On a Description of Deformable Junction in the ResultantNonlinear Shell Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457Wojciech Pietraszkiewicz

Analytical and Numerical Modelling of a Sub- and SupersonicMoving Load Front Along a Rod’s Skin . . . . . . . . . . . . . . . . . . . . . . . 469Wolfgang E. Weber, Yannick F. Fangye, Daniel Balzaniand Bernd W. Zastrau

Part VII Generalized Continua, Multi-Scales and Meta-Materials

Application of Multi-scale Approaches to the Investigationof Sealing Surface Deformation for the Improvement of LeakTightness in Pressure Relief Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Ali A. Anwar, Yevgen Gorash and William Dempster

Against the Fragmentation of Knowledge: The Powerof Multidisciplinary Research for the Design of Metamaterials . . . . . . . 523Francesco dell’Isola, Sara Bucci and Antonio Battista

On Equilibrium of a Second-Gradient Fluid Near Edgesand Corner Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547Victor A. Eremeyev

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

Contents xvii

Part IFundamentals and Elasticity

Part IIVisco-Elastic Media and Coupled

Phenomena

Part IIIThermo-Elasto-Plasticity

Part IVContinuum Damage and Phase Fields

Part VFracture Mechanics and Failure Criteria

Part VIRods, Plates and Shells

Part VIIGeneralized Continua, Multi-Scales

and Meta-Materials

Author Index

AAßmus, Marcus, xiiAnwar, Ali A., 493

BBalzani, Daniel, 469Battista, Antonio, 523Becker, Wilfried, 337Bîrsan, Mircea, 391Böhm, Michael, 205Bruhns, Otto T., 135Brünig, Michael, 279Bucci, Sara, 523

CCegielski, Marcin, 295Craciun, Eduard-Marius, 327, 367

DDe Borst, René, 263Dell’Isola, Francesco, 523Dempster, William, 493Dodla, Srihari, 165

EEremeyev, Victor A., 547

FFahlbusch, Nina-Carolin, 337Fangye, Yannick F., 469

GGanczarski, Artur, 295Gerke, Steffen, 279Gorash, Yevgen, 493

HHennig, Paul, 307

IIvanova, Elena A., 3

JJohlitz, Michael, 103

KKang, Guozheng, 243Kan, Qianhua, 243Kästner, Markus, 307Kienzler, Reinhold, 415Kolupaev, Vladimir A., 337

LLatalski, Jarosław, 435Li, Hao, 223Linse, Thomas, 307Lion, Alexander, 103Liu, Qiang, 165

© Springer Science+Business Media Singapore 2016K. Naumenko and M. Aßmus (eds.), Advanced Methods of Continuum Mechanicsfor Materials and Structures, Advanced Structured Materials 60,DOI 10.1007/978-981-10-0959-4

557

558 Author Index

MMarsavina, Liviu, 367Maugin, Gérard A., 29Mittermeier, Christoph, 103Mizushima, Satoshi, 185Müller, Wolfgang H., 3, 79

NNaumenko, Konstantin, x, xiiNazarenko, Lidiia, 49Neff, Patrizio, 391

OOhno, Nobutada, 185Okumura, Dai, 185

PPandolfi, Anna, 119Pietraszkiewicz, Wojciech, 457

RRoy, Anish, 165

SSadowski, Tomasz, 367Schmidt, Marco, 279Schneider, Patrick, 415Silberschmidt, Vadim V., 165Stolarski, Henryk, 49

TTalreja, Ramesh, 379Tanie, Hisashi, 185

UUlbricht, Volker, 307

VVerhoosel, Clemens V., 263Vilchevskaya, Elena N., 3, 79

WWang, Zhao-Ling, 223Weber, Wolfgang E., 469Weinberg, Kerstin, 119Wolff, Michael, 205

XXiao, Heng, 223

YYin, Zheng-Nan, 223Yu, Chao, 243

ZZastrau, Bernd W., 469Zhbanova, Evgeniya V., 61Zubov, Leonid M., 61