Table Of ContentModelingandSimulationinScience,EngineeringandTechnology
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Advances in Multifield
Theories for Continua
with Substructure
Gianfranco Capriz
Paolo Maria Mariano
Editors
Springer Science+Business Media, LLC
Gianfranco Capriz Paolo Maria Mariano
Dipartimento cii Matematica Dipartimento cii Ingegneria Strutturale
Universita cii Pisa e Geotecnica
56127 Pisa Universita cii Roma "La Sapienza"
ltaly 00184 Roma
Italy
Library of Congress Cataloging-in-Publication Data
Advances in multified theories for continua with s.ubstructure I Gianfranco Capriz, Paolo
Maria Mariano, editors.
p. cm. - (Modeling and simulation in science, engineering and technology)
ISBN 978-1-4612-6465-1 ISBN 978-0-8176-8158-6 (eBook)
DOI 10.1007/978-0-8176-8158-6
1. Continuum mechanics. 1. Capriz, G. (Gianfranco) II. Mariano, Paolo Maria, 1966-
III. Series.
QA808.2.A385 2003
531-dc21 2003052406
CIP
AMS Subject Classifications: 74AIO, 74Al5, 74A30, 74A40, 74A35, 74A45, 74A50, 74A60,
74C15, 74N05, 74Nl5, 74N20, 74N30, 74Rl5, 74E2S, 74E30, 74H33, 76F2S, 76AOS
Printed on acid-free paper.
© 2004 Springer Science+Business Media New York
Originally published by Birkhiiuser Boston in 2004
Softcover reprint of the hardcover Ist edition 2004
All rights reserved. This work may not be translated or copied in whole or in part without the written
permis sion of the publisher (Springer Science+Business Media, LLC ), except for brief excerpts in
connection with reviews or scholarly analysis. Use in connection with any form ofinformation storage
and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by the
Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-6465-1 SPIN 10920787
9 8 7 6 5 432 1
Contents
Preface vii
1. WorkConditions and Energy Functions forIdealElastic-PlasticMaterials
M.Silhavy 1
2. GeneralizedElastic-PlasticDecompositioninDefective Crystals
G P. Parry 33
3. Cocycles, Compatibility, and Poisson Brackets forComplex Fluids
H. Cendra, J.Marsden, andT. S.Ratiu 51
4. Defect-InducedTransitions asMechanismsofPlasticity andFailure
inMultifield Continua
O.B. Naimark 75
5. Extended Thermodynamics:AMultifieldTheory ParExcellence
I.MUller 115
6. MicrostructureandTurbulenceinDilute Polymer Solutions
E. DeAngelis, C.M. Casciola, P. M.Mariano, andR.Piva 127
7. Fluxes andFlux-ConjugateStresses
R. Segev 149
8. Algebra,Geometry,and ComputationsofExact Relations forEffective
Moduli ofComposites
Y. Grabovsky 167
9. AMultifieldTheory fortheModelingoftheMacroscopicBehavior
ofShape Memory Materials
D.BernardiniandT. J.Pence 199
10. Balance ataJunction among Coherent Interfaces inMaterials
withSubstructure
G Capri;andP. M. Mariano 243
Preface
Toachievedesign,implementation,andservicingofcomplexsystemsandstruc
turesinanefficientandcost-effectiveway,adeeper knowledge andunderstandingof
thesubtle castanddetailed evolutionofmaterials isneeded.The analysis indemand
borders withthemolecularandatomicone,spanningallthewaydownfromclassical
continua.The study of the behavior of complex materials in sophisticated devices
alsoopensintricatequestionsabouttheapplicability ofprimaryaxiomsofcontinuum
mechanics suchastheultimatenatureofthematerialelementitselfandthepossibility
ofidentifyingitperfectly.Soitisnecessarytodeveloptoolsthatallowustoformulate
both theoretical models and methods ofnumerical approximationfor theanalysis of
material substructures.
Multifield theories in continuum mechanics, which bridge classical materials
science and modern continuummechanics, provide precisely these tools. Multifield
theories not only address problems of material substructures, but also encompass
well-recognized approaches to the study ofsoft condensed matter and allow one to
modeldisparateconditionsinvariousstatesofmatter.However,researchinmultifield
theories is vast, and there is little in the way of a comprehensive distillation of the
subject from anengineer's perspective.Therefore, thepapers inthepresent volume,
which grewoutofourexperienceaseditors foranengineeringjournal,1tacklesome
fundamental questions,suggestsolutions ofconcrete problems,andstrivetointerpret
a host of experimental evidence. In this spirit, each of the authors has contributed
original results having inmind their wider applicability.
InChapter 1,M.Silhavy deals with theenergetics ofidealelastic-plasticcrys
talline materials under large deformations, showing that the convexification of a
dissipation function (introduced to describe the set of possible free energies) pro
vides the optimal lower bound of the plastic work. New subtle consequencesensue
from thecelebratedIl'yushincondition: forexample,theexistenceofanew kind of
energy function and the validity ofacondition stronger thanthecommon normality
condition.
InChapter 2,G.Parry alsoinvestigates crystal plasticity:When defective crys
tals are described by lattice vectors, one is able to deduce elastic invariants under
deformation paths. An unusual but physically significant decomposition of the de
formation gradient into elastic and plastic parts is obtained by purely geometrical
IG.CaprizandP.M.Mariano,eds.,Multifield Theories,Internat.J.SolidsStructures,38-6-7(2001).
viii Preface
arguments, without resorting to the notion of stress; thus, issues in kinematics are
discerned fromissuesassociated withinteractions.Thelatteronesare,infact,power
conjugatedwiththeformer and are,inthissense,subsequent tothem.
Motivated by questions that arise in the mechanics of complex fluids, such
as spin glasses,H.Cendra, J.E. Marsden, and T.S. Ratiu introduce, in Chapter 3,
inthe general context, related Poisson brackets withcocycles. Possible connections
with the variational multisymplectic approach are suggested, with implications for
further research. Thus the subtle and geometrical nature of mechanics, inparticular
mechanics ofcomplex bodies, isemphasized.
InChapter 4,O.Nairmarkdiscusses physical interpretationsofawidenumber
ofdetailedexperimentalresultsondefect-inducedtransitionsandstudiestheirconse
quences forthemacroscopic irreversiblemechanisms ofmetalplasticity andfailure.
Dynamic phenomenainvolving solitary wavesarecharacterized bytheirconnection
with the essential features of the defect substructure of bodies under experimental
examination. Here as well, the data and the qualitative behavior lead to additional
theoretical questions.
Evensimplesubstances,suchasmonoatomicgases,sometimesdisplaycomplex
behavior. When a monoatomic gas is rarefied, the Navier-Stokes-Fourier scheme
does notholdsatisfactorily.Extended thermodynamics becomeanecessary tool,and
hundreds or thousands of moments of the distribution function enter the picture. In
Chapter5,I.Mullerexamines thisissue,providesananalysisofthescattering oflight
inagas, and investigates theshock structure inashock tube.
When turbulence ispresent, drag reduction is observed in polymeric fluidsas
a consequence of their macromolecularsubstructure.When the polymeric flow be
comes semidense,amultifieldapproach canbeusedtoshowthatthetermsinvolving
spatialgradients (inparticularLaplacians),requiredinthebalanceequations,arecon
sequences of microstress,that is,of contact interactions between neighboring fami
liesofpolymeric chains.This matter isdiscussed byE.DeAngelis,C.M. Casciola,
P.M. Mariano,andR.Piva inChapter 6.
The choice of the manifold M of substructural states can be, at times, quite
unusual. Nevertheless the material substructure can always be characterized by an
abstract,extensiveproperty.Thenotionofstressstillmakessenseandcanbeachieved
without the notion of "velocity." The material elements are considered as integral
manifolds of a flux vector field.Such a point of view is discussed in Chapter 7 by
R. Segev, and the results are illustrated by an application to the case in which the
extensiveproperty istheelectric charge.
Composites are natural examplesofcomplex materials,and theymay becon
structed in a number of different ways. For polycrystals, one possible method of
analysis is todeduce "effective" elastic constants by homogenization. However, in
general, there is no guarantee that the tensors of material properties be symmetric.
Nevertheless, exact relations among material constants can be obtained which are
preserved by the homogenization procedure. They give us information on the re
sulting composite, regardless ofthecomplexityofthe material texture.This issue is
discussedbyY.Grabovsky inChapter 8.
Preface ix
Shape memory behavior arises in the study of materials undergoing solid-to
solidtransformationsbetween austeniteandmartensite phases.Theirexotic behavior
is a natural proving ground for the potential of multifield theories. In Chapter 9,
D. Bernardini and T.J. Pence consider three morphological descriptors of the sub
structural state:Twoareofascalarnature,andthethirdisasecond-ordertensor.This
approach accounts notonlyforpseudoelasticityandshapememory behavior,butalso
for thehightemperaturereorientation ofmartensitic variants.
Interfaces andjunctionsarecommon incomplexmaterials.Ofteninterfaces are
characterized by their own surface energy, and an additional energy accrues along
junctions. The surface energy must be positive, while, at the junction, there may
occurachange ofsign.Themodelingofsharpinterfacesandjunctionsbecomesmore
intricate when one views themasstructured surfaces orlines.Surface microstresses
and self-forces, as wellas linestresses andself-forces, enter the picture and need to
bebalanced.The matter isdiscussedinChapter 10byG Capriz and P.M.Mariano.
This collection of invited papers had its origins atTaormina,Italy,during the
2001 International Congress of the Italian Association for Theoretical and Applied
Mechanics (AIMETA); it was there that many of the authors discussed with us the
basic organization of this text.Wearegrateful totheorganizers oftheCongress and
to the Italian National Group of Mathematical Physics for the grants that made our
workpossible.Furthermore,weexpress ourappreciation toTomGrasso andhisstaff
atBirkhauserBoston forproviding kindsupport. Last butnotleast,wethankNicola
Bellomo foracceptingour book intohisseries Modeling andSimulationinScience,
Engineering andTechnology.
Gianfranco Capri:
Paolo Maria Mariano
October 2003
Advances in Multifield Theories
for Continua with Substructure
1. Work Conditions and Energy Functions for
Ideal Elastic-Plastic Materials
M. Silhavy"
Abstract. The paper deals withthe cyclic second law andI1'yushin'scondition
forisothermal,ideal,isotropic,elastic-plasticmaterialsatlargedeformations.The
second lawisequivalenttotheexistenceoftheelasticpotential andthenonnega
tivity ofplastic power.The material admits infinitely manyfreeenergies: The set
ofallenergy functions isdescribedintermsofadissipationfunction. Itsconvex
ification provides theoptimal lower bound for plastic work;italso figuresinthe
maximalandminimalenergies.I1'yushin'sconditionisequivalenttotheexistence
oftheelasticpotential and anewconditionthatisstrongerthan thenormality of
theplastic stretchingand theconvexityofthestress range. I1'yushin'scondition
is also equivalentto the existenceof anew kind of energy function called "the
initial and finalextendedenergy functions." Materials of type C are introduced
forwhichtheinitialextendedenergyfunctionhasadditional convexityproperties.
Itcan beviewed asastored energy ofaHencky hyperelastic material associated
withtheelastic-plasticmaterial.
1.1. Introduction
This paperdeals withthecyclic second law andIl'yushin'sconditionforisothermal,
ideal,isotropic elastic-plasticmaterials atlargedeformations. The ideal natureofthe
material excludes hardening. Both the isothermal cyclic second law and Il'yushin's
conditionassertthenonnegativityoftheworkofexternalforcesonforcertainclasses
ofcyclic processes.Theformer speaksofcyclesinthestatespace;thelatterofcycles
inthespace oftotal deformations,
Sections 1.3-1.5 analyze the material from the point of view of the general
theory ofactions andpotentialsonthennodynarnicalsystems byColeman andOwen
[5]. That theory provides an unbiased derivation of potentials like the free energy;
moreover, itenables onetodiscuss theiruniqueness/nonuniquenessandtodetermine
the maximal and minimal potentials. Formaterials treated here,the plastic flowrule
can be "nonassociated," and the restrictions equivalent to the second law are the
existenceoftheelastic potential fortheelastic stress function and the nonnegativity
"Mathematical Institute oftheAV CR,Zitna25, 11567 Prague, CzechRepublic, silhavy@math.
cas.cz andDepartment ofMathematics, University ofPisa,ViaF.Buonarroti 2,56127Pisa,Italy,
[email protected].
G. Capriz et al. (eds.), Advances in Multifield Theories for Continua with Substructure
© Birkhäuser Boston 2004
