Internal Variables in Thermoelasticity PDF
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Solid Mechanics and Its Applications Arkadi Berezovski Peter Ván Internal Variables in Thermoelasticity Solid Mechanics and Its Applications Volume 243 Series editors J.R. Barber, Ann Arbor, USA Anders Klarbring, Linköping, Sweden Founding editor G.M.L. Gladwell, Waterloo, ON, Canada Aims and Scope of the Series Thefundamentalquestionsarisinginmechanicsare:Why?,How?,andHowmuch? The aim of this series is to provide lucid accounts written by authoritative researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. Themedianlevelofpresentationistothefirstyeargraduatestudent.Sometexts aremonographs defining thecurrentstateofthe field; othersareaccessibletofinal year undergraduates; but essentially the emphasis is on readability and clarity. More information about this series at http://www.springer.com/series/6557 á Arkadi Berezovski Peter V n (cid:129) Internal Variables in Thermoelasticity 123 Arkadi Berezovski PeterVán Department ofCybernetics, Schoolof Institute of Particle andNuclear Physics Science MTAWIGNERResearchCentreforPhysics Tallinn University of Technology Budapest Tallinn Hungary Estonia and Department ofEnergy Engineering, MontavidThermodynamic Research Group BudapestUniversity of Technologyand Economics Budapest Hungary ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid MechanicsandIts Applications ISBN978-3-319-56933-8 ISBN978-3-319-56934-5 (eBook) DOI 10.1007/978-3-319-56934-5 LibraryofCongressControlNumber:2017936900 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of 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 transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Thepredictionoftheresponseofamaterial(orastructure)toexternalloadingisan ordinaryengineeringproblem.Itisanecessarystepinthedesignofanyman-made tools, devices, and constructions. Increasing complexity of advanced electronics, modern machinery and equipment, and of corresponding materials demands more and more sophisticated methods for their proper handling. The accuracy in the predictionofthematerialbehaviordependsonthechosentheoreticaldescription.It iswellunderstoodthatidealizedclassicaltheories(likelinearelasticityandFourier’s heat conduction) work well if a material can be considered as homogeneous and a loading is not extremal. Such theories are not sufficient for inhomogeneous microstructured materials. At the same time, the complete atomistic (or even quantum mechanics) exposition is possible only in principle and in any case it is difficult for using in practice. The compromise between the full accuracy and a practical treatment can be achieved in various ways. One of such possibilities is presentedin this book. The considered approach supposes the introduction of internal variables to characterize the influence of a microstructure on the global behavior of a material. This idea is not new and has been exploited at least for 50 years. It was broadly appliedinrheology,plasticity,andphase-fieldtheory.However,itsfullpowerwas uncovered only recently. The use of the internal variable concept in a more extendedcontext,i.e.,theintroductionofadualinternalvariable,providesaunified treatment both internal variables of state and dynamic degrees of freedom. This extension covers both parabolic evolution equations for dissipative internal vari- ables andhyperbolicevolutionequationsintheabsence ofdissipation.Bothforms of evolution equations follow from the dissipation inequality and, therefore, are thermodynamicallyconsistent.Thestructureofwell-knownevolutionequationsfor theCosseratmicrorotationandforthemicromorphicmicrodeformationisrecovered in the framework of the proposed approach. In the case of heat conduction, a hyperbolic evolution equation for the microtemperature is obtained by keeping the coupled parabolic equation for the global temperature. Thus, the framework of the construction of advanced continuum theories is described and illustrated in the book. The coupling between mechanical and v vi Preface thermal effects is treated in the dynamic context. Static problems are omitted due to their relative simplicity. The three-dimensional theory is complemented by examples in the one-dimensional case. An implementation of the theory into numerical algorithm is provided as well. The book summarizes results obtained during the collaboration between co-authors in the framework of joint Estonian–Hungarian research projects under the agreement of scientific cooperation between the Estonian and Hungarian Academies of Sciences. Itisourpleasantdutytothankthosewhohelpedustoclarify ourownviewsin debates over matters of principle, particularly Gérard Maugin (Paris), Wolfgang Muschik (Berlin), Jüri Engelbrecht (Tallinn), Tamás Fülöp, and Csaba Asszonyi (Budapest), to each of whom we owe more than we can express. Mihhail Berezovski(DaytonaBeach)improved numerical schemes andtheiradaptationfor particular problems. A careful reading of a draft by Jüri Engelbrecht enabled us to eliminate numerous misprints and other slips. This book is based on the results of our research supported by various funding sources. These include the grants from the Estonian Science Foundation 7037 and 8702, the grant from the Estonian Research Council PUT434 (A.B.), OTKA and NKFIA grants K104260, K116197, K116375 (P.V.), and we are very grateful for their support. Thisbookisintendedforgraduateandpostgraduatestudentsandscientistsinthe area of applied mathematics, mechanics, and engineering sciences, who are acquainted with the basics of mechanics of continuous media. Tallinn, Estonia Arkadi Berezovski Budapest, Hungary Peter Ván February 2017 Contents 1 Instead of Introduction.. ..... .... .... .... .... .... ..... .... 1 Part I Internal Variables in Thermomechanics 2 Introduction... .... .... ..... .... .... .... .... .... ..... .... 21 3 Thermomechanical Single Internal Variable Theory ... ..... .... 35 4 Dual Internal Variables . ..... .... .... .... .... .... ..... .... 59 Part II Dispersive Elastic Waves in One Dimension 5 Internal Variables and Microinertia .... .... .... .... ..... .... 75 6 Dispersive Elastic Waves. ..... .... .... .... .... .... ..... .... 85 7 One-Dimensional Microelasticity ... .... .... .... .... ..... .... 99 8 Influence of Nonlinearity. ..... .... .... .... .... .... ..... .... 113 Part III Thermal Effects 9 The Role of Heterogeneity in Heat Pulse Propagation in a Solid with Inner Structure .... .... .... .... .... ..... .... 123 10 Heat Conduction in Microstructured Solids .. .... .... ..... .... 131 11 One-Dimensional Thermoelasticity with Dual Internal Variables.. .... ..... .... .... .... .... .... ..... .... 147 12 Influence of Microstructure on Thermoelastic Wave Propagation.. .... ..... .... .... .... .... .... ..... .... 163 vii viii Contents Part IV Weakly Nonlocal Thermoelasticity for Microstructured Solids 13 Microdeformation and Microtemperature.... .... .... ..... .... 175 Summary. .... .... .... .... ..... .... .... .... .... .... ..... .... 191 Appendix A: Sketch of Thermostatics... .... .... .... .... ..... .... 193 Appendix B: Finite-Volume Numerical Algorithm. .... .... ..... .... 201 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 219 Chapter 1 Instead of Introduction Abstract Wavedispersionisacharacteristicfeatureofsolidswithmicrostructure. The development of models of linear wave dispersion effects in isotropic solids is used as an example of the construction of more and more sophisticated dispersive waveequations.Theconsideredmodelsaredistinctinfoundationsandderivations. 1.1 One-DimensionalElasticWavesinHeterogeneous Solids In the classical continuum mechanics, materials are treated as homogeneous. This idealization simplifies the consideration substantially. Wave propagation in homo- geneous media is the subject of classical acoustics, optics, and elasticity. The cor- respondingwaveequationisthestandardexampleofhyperbolicpartialdifferential equationsintextbooks.Theclassicalwaveequationisapplicableforthenumerical simulationofwavepropagationinmaterialscomposedbydistinctbuthomogeneous pieces(likealaminate). It is instructive to demonstrate what happens if a wave propagates through a heterogeneous solid with a known composition and material parameters. An one- dimensional elastic pulse propagation in inhomogeneous solids is considered as a benchmark. In this case the computational domain corresponds to a homogeneous solid,exceptforaregionoflengthd,wherecertaininhomogeneityisinserted(see Fig.1.1). Thepulsepropagationisdescribedbythestandardwaveequation u =c2u , (1.1) tt xx whereuisthedisplacement,cistheelasticwavespeed,subscriptsdenotederivatives. Itshouldbenotedthatthevalueoftheelasticwavespeedcisprescribedaccording tothechosensubstructure.Insimulationsconsideredbelow,thevaluesofthedensity ©SpringerInternationalPublishingAG2017 1 A.BerezovskiandP.Ván,InternalVariablesinThermoelasticity, SolidMechanicsandItsApplications243,DOI10.1007/978-3-319-56934-5_1
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