Numerical Continuum Mechanics

442 Pages·2012·7.446 MB·English

by  Vladimir N Kukudzhanov, Alexei Zhurov

De Gruyter Studies in Mathematical Physics 15
Editors
Michael Efroimsky, Bethesda, USA
Leonard Gamberg, Reading, USA
Dmitry Gitman, São Paulo, Brasil
Alexander Lazarian, Madison, USA
Boris Smirnov, Moscow, Russia

Vladimir N. Kukudzhanov
Numerical Continuum
Mechanics
De Gruyter
Physics and Astronomy Classification 2010: 46.15.-x, 02.70.Bf, 02.60.Lj, 46.05.+b, 46.50.+a,
46.35.+z, 46.70.De, 46.70.Hg, 83.60.Df, 62.20.-x, 61.72.Lk, 81.40.Cd
ISBN 978-3-11-027322-9
e-ISBN 978-3-11-027338-0
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detailed bibliographic data are available in the Internet at http://dnb.dnb.de.
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Preface
Thebookisdevotedtocomputationalmethodsincontinuumthermomechanicsandis
basedonthelecturecoursetheauthordeliveredfor15yearstostudentsoftheMoscow
InstituteofPhysicsandTechnologyandtheTsiolkovskiiRussianStateTechnological
University.
The book has a dual purpose: educational, to introduce the reader to advanced
foundations of computational mechanics, and scientific, to acquaint the reader, by
morecomplexexamples,withthestate-of-the-artdirectionsofresearchandleadthem
toanindependentresearchworkinthisareaofcontinuummechanics.
The presentation of the materialis intended for engineering physicists rather than
computational mathematicians. The book does not aim to give rigorous substantia-
tion of methods. Proofs of theorems are often avoided. Presentation of ideas and
qualitativeconsiderations,illustratedbyspecificexamplesrelatedtoimportantappli-
cations, is preferred to rigorous exposition of theorems with detailed statements of
all conditions imposed on the functions involved. At the same time, the conditions
and limitations that are essentialfor practicalapplications of methods are discussed
quitethoroughly. Insufficientattentiontosuchconditionscanresultinseriouserrors.
This style of presentation is justified by the fact that the book is primarily intended
forengineersandphysicists,whoaremoreinterestedintheessenceoftheproblemin
questionratherthanaformallyrigorousapproachtoitssolution. Alongwiththesolu-
tion oftypicalexamples,illustratingtheapplicationofmethods, conditions thatmay
lead to inefficiency of the methods are discussed and possible ways of overcoming
drawbacksaresuggested.
The book is organized into parts, chapters and sections, which are numbered se-
quentially. The formulas and figures have double numbering, with the first number
indicating the section and the second showing the number within the section. The
facultative information – suchas proofs of theorems, subtleties of the application of
methods,andsoon–isgiveninsmallertypeandmaybeskippedatfirstreading.
PartI,consistingofChapter1,coversissuesandstatementsofproblemsthatdiffer
from traditional approaches in the modern nonlinear continuum mechanics and are
rarely included in the educational literature or differ in the formulation of equations
speciallyadaptedforefficientnumericalanalysis. DiscussedinChapter1areintegral
anddivergenceformsofconservationlaws,variationalprinciplesandgeneralizedso-
lutionsofcontinuummechanics,thethermodynamictheoryofcontinuousmediawith
internalvariablesallowingthedescriptionofmaterialstructure,constitutiveequations
of composite media with complex rheology, elastoviscoplastic media with damage,
vi Preface
and the theory of large elastoplastic deformations. Also discussed are nonclassical
methodsofdescribingthemotionofcontinua,includingmixedLagrangian–Eulerian
methodsanddescriptioninarbitraryadaptablemovingcoordinates. Thepresentation
isperformedinsuchawaythatthereaderdoesnothavetouseadditionalliteratureto
learnthematerial. Thechaptercanbeusefultomoreadvancedreadersaswell.
PartII,comprisingChapters2to4,outlinesthebasicsofnumericalmethodsforthe
solution of finite differenceequations. This partisclose tothe contentof traditional
courses on numerical methods with focus on efficient methods for nonlinear prob-
lems of continuum mechanics. Solution methods for stiff and singularly perturbed
boundary-value problems andnonlinear waveunsteadyproblems arediscussed. Sta-
bilityanalysismethodsusingdifferentialapproximationsareoutlined.
PartIII(Chapters5to8)givesthe descriptionanddevelopmentof specialnumer-
icalmethods of continuum mechanicsandalsodiscussestheir applicationtosolving
certainclassesofone-dimensionalandmultidimensionalunsteadydynamicproblems
and generalization to two- or three-dimensional problems for elastic and elastovis-
coplasticmedia. Finitedifferenceschemesforunsteadyproblemswithdiscontinuous
solutions are analyzed by the method of differential approximations. The methods
of splitting in directions and physicalprocessesfor media with complex rheological
relationsaredeveloped;thesemethodsallowonetoreducecomplexproblemstosuc-
cessivesolution ofproblemsforsimpler media. Efficientnumerical-analyticalmeth-
ods for elastoplastic and elastoviscoplastic problems in two or more dimensions are
suggested. Specialmethods are consideredthat allow one to solve problems involv-
inglargeorverylargedeformationsofelastoplasticsolidsunderextremethermome-
chanicalloads. ThesemethodsarebasedonnonclassicalmixedLagrangian–Eulerian
approaches to the description of the motion of continuous media, adaptable moving
gridtechniques,andtheparticle-in-celltechniqueanditsmodifications. Solutions of
several problems are given: penetration of a rigid indenter into an elastoplastic ma-
terial with fracture, formation of a cumulative jet under the action of a detonation
wave,indentationofasine-shapedrigidstampintoanelastoplasticmaterial,fracture
of anelasticlayer(glass)whenimpactedbyasteelcylinder,impactofadeformable
cylindrical projectile on a deformable slab at a supersonic speed and their fracture,
andsomeothers.
Chapter 8 deals with damage and continuum fracture of elastoplastic and elasto-
viscoplasticmediawithdefects,underquasistaticanddynamicthermomechanicalac-
tions.
AlsoincludedinChapters5to8arenewresults,whichonlyappearinjournalpub-
lications and arenot included inthe educationalliterature. The newmethods canbe
usefulto the advancedreaderswho specializeinnumericalsimulation of continuum
mechanicalproblemsaswellastopostgraduateandPhDstudents.
AttheendofChapters2to7,therearenumerousexercisesdesignedtosupplement
thetextandconsolidatetheconceptsdiscussed. Theyservetostimulatethereaderto
furtherstudyandtoreinforceanddeveloppracticalskills.
Preface vii
The book has an extended table of contents to help the reader quickly locate the
desired information. The brief list of notations includes symbols and terms most
frequently used in computational mathematics and mechanics. The bibliography in-
cludesreferencescitedinthetexttoindicatetheauthorswhocontributedtotheresults
andrefertheinterestedreadertomoredetailedinformationandothereducationallit-
erature.
Thebookisintendedforgraduateandpostgraduatestudentsintheareaofapplied
mathematics, mechanics, and engineering sciences,who areacquainted withthe ba-
sics of mechanics of continuous media and main concepts of computational mathe-
matics. The first two parts of the book aim at extending the reader’s knowledge in
these disciplines. The book may also be helpful for a wide range of engineers, sci-
entists, university teachers,and PhD students engagedin the fields of computational
mathematicsandmechanicsofcontinuousmedia.
I am very grateful to my colleagues and pupils Nikolai Burago, Alexander Lev-
itin, and Sergei Lychev for their valuable comments and fruitful discussions, which
helpedimprovethebook. IwouldalsoliketothankAlexeiZhurovfortranslatingthe
manuscriptintoEnglishthoroughlyandconscientiously.
Moscow,October2012 VladimirKukudzhanov

Contents
Preface v
I Basicequationsofcontinuummechanics
1 Basicequationsofcontinuousmedia 3
1.1 Methodsofdescribingmotionofcontinuousmedia ............... 3
1.1.1 Coordinatesystemsandmethodsofdescribingmotionof
continuousmedia ................................... 3
1.1.2 Euleriandescription ................................. 4
1.1.3 Lagrangiandescription ............................... 5
1.1.4 Differentiationofbases ............................... 5
1.1.5 Descriptionofdeformationsandratesofdeformationofa
continuousmedium .................................. 7
1.2 Conservationlaws. Integralanddifferentialforms ................ 9
1.2.1 Integralformofconservationlaws ...................... 9
1.2.2 Differentialformofconservationlaws ................... 11
1.2.3 Conservationlawsatsolutiondiscontinuities .............. 13
1.2.4 Conclusions ........................................ 14
1.3 Thermodynamics .......................................... 15
1.3.1 Firstlawofthermodynamics ........................... 15
1.3.2 Secondlawofthermodynamics ........................ 16
1.3.3 Conclusions ........................................ 18
1.4 Constitutiveequations ...................................... 18
1.4.1 Generalformofconstitutiveequations. Internalvariables .... 18
1.4.2 Equationsofviscouscompressibleheat-conductinggases .... 21
1.4.3 Thermoelasticisotropicmedia ......................... 21
1.4.4 Combinedmedia .................................... 22
1.4.5 Rigid-plasticmediawithtranslationallyisotropichardening .. 24
1.4.6 Elastoplasticmodel .................................. 25
1.5 Theoryofplasticflow. Theoryofinternalvariables ............... 26
1.5.1 Statementoftheproblem. Equationsofan
elastoplasticmedium ................................. 26
1.5.2 Equationsofanelastoviscoplasticmedium................ 30