Table Of ContentSolid Mechanics and Its Applications
Arkadi Berezovski
Peter Ván
Internal
Variables in
Thermoelasticity
Solid Mechanics and Its Applications
Volume 243
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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
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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
