Quasistatic Contact Problems in Viscoelasticity · Quasistatic contact problems in viscoelasticity...
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Transcript of Quasistatic Contact Problems in Viscoelasticity · Quasistatic contact problems in viscoelasticity...
Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity
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AMS/IP
Studies in Advanced Mathematics Volume 30
Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity
Weimin Han and Mircea Sofonea
American Mathematical Society • International Press
P
https://doi.org/10.1090/amsip/030
Shing-Tung Yau , Genera l Edi to r
2000 Mathematics Subject Classification. P r imar y 74-02 , 74M15 , 74M10 , 74S05 , 74B05 , 74B20, 74Cxx , 74Dxx ;
Secondary 65M06 , 65M12 , 65M15 , 65M60 , 65N12 , 65N15 , 65N30 , 49J40 .
L i b r a r y o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a
Han, Weimin . Quasistatic contac t problem s i n viscoelasticit y an d viscoplasticit y / Weimi n Ha n an d Mirce a
Sofonea. p. cm . — (AMS/I P studie s i n advance d mathematics ; v . 30 )
Includes bibliographica l reference s an d index . ISBN 0-8218-3192- 5 (alk . paper ) 1. Viscoelasticity . 2 . Viscoplasticity . 3 . Contac t mechanics—Mathematica l models .
I. Sofonea , Mircea . II . Title . III . Series .
QA931 .H26 200 2 620.1/1232—dc21 200202771 6
C o p y i n g a n d r e p r i n t i n g . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .
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Dedicated t o
DAQING HAN , SUZHE N Q I N
HUIDI TANG , ELIZABET H J IN G HA N
MICHAEL Y U E HA N
and
In memor y o f SEMINICA SOFONE A
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Contents Preface x i
List o f Symbol s x v
I Nonlinea r Variationa l Problem s an d Numerica l Ap -proximation 1
1 Preliminarie s o f Functiona l Analysi s 3 1.1 Norme d Space s an d Banac h Space s 3 1.2 Linea r Operator s an d Linea r Functional s 8 1.3 Hilber t Space s 1 3 1.4 Conve x Function s 1 7 1.5 Banac h Fixed-poin t Theore m 2 1
2 Functio n Space s an d Thei r Propertie s 2 5 2.1 Th e Space s C m(n) an d I/>(Q) 2 5 2.2 Sobole v Space s 2 9 2.3 Space s o f Vector-valued Function s 3 7
3 Introductio n t o Finit e Differenc e an d Finit e Elemen t Approxi -mations 4 3 3.1 Finit e Differenc e Approximation s 4 3 3.2 Basi s o f the Finit e Elemen t Approximatio n 4 5 3.3 Finit e Elemen t Interpolatio n Erro r
Estimates 5 3 3.4 Finit e Elemen t Analysi s o f Linear Ellipti c Boundary Valu e Prob-
lems 5 4
4 Variationa l Inequalitie s 5 9 4.1 Ellipti c Variationa l Inequalitie s 5 9 4.2 Approximatio n o f Ellipti c Variationa l
Inequalities 6 5 4.3 A n Evolutionar y Variationa l Inequalit y 6 8 4.4 Semi-discret e Approximations of the Evolutionary Variational In-
equality 7 6 4.5 Full y Discret e Approximation s o f th e Evolutionar y Variationa l
Inequality 7 8
Bibliographical Notes 89
vii
viii CONTENTS
II Mathematica l Modellin g i n Contac t Mechanic s 9 1
5 Preliminarie s o f Contac t Mechanic s o f Continu a 9 3 5.1 Kinematic s o f Continu a 9 3 5.2 Dynamic s o f Continu a 9 8 5.3 Physica l Settin g o f Contac t Problem s 10 2 5.4 Contac t Boundar y Condition s an d Frictio n Law s 10 4
6 Constitutiv e Relation s i n Soli d Mechanic s 11 3 6.1 Physica l Backgroun d an d Experiment s 11 3 6.2 Constitutiv e Relation s i n Elasticit y 12 1 6.3 Constitutiv e Relation s i n Viscoelasticit y 12 5 6.4 Constitutiv e Relation s i n Viscoplasticity 13 3
7 Backgroun d o n Variationa l an d Numerica l Analysi s i n Contac t Mechanics 14 1 7.1 Functio n Space s i n Soli d Mechanic s 14 1 7.2 Semi-discret e an d Full y Discret e Approximation s 15 2 7.3 Convergenc e unde r Basi c Solutio n Regularit y 15 6 7.4 Som e Inequalitie s 16 2
8 Contac t Problem s i n Elasticit y 16 7 8.1 Frictionles s Contac t Problem s 16 7 8.2 Numerica l Analysi s o f the Frictionles s Contac t Problem s 17 3 8.3 Quasistati c Frictiona l Contac t Problem s 17 7 8.4 Numerica l Analysi s o f Quasistati c Frictiona l Contac t Problem s . 18 1 8.5 Numerica l Example s 18 4
Bibliographical Notes 189
III Contac t Problem s i n Viscoelasticit y 19 1
9 A Frictionles s Contac t Proble m 19 3 9.1 Proble m Statemen t 19 3 9.2 A n Existenc e an d Uniquenes s Resul t 19 5 9.3 Numerica l Approximation s 19 7 9.4 Dua l Formulatio n 20 2
10 Bilatera l Contac t wit h Sli p Dependen t Frictio n 20 7 10.1 Proble m Statemen t 20 7 10.2 A n Existenc e an d Uniquenes s Resul t 20 9 10.3 Semi-discret e Approximatio n 21 4 10.4 Full y Discret e Approximation s 21 8 10.5 Dua l Formulatio n 22 2
CONTENTS i x
11 Frictiona l Contac t wit h Norma l Complianc e 22 7 11.1 Proble m Statemen t 22 7 11.2 A n Abstrac t Proble m an d it s
Well-posedness 23 0 11.3 Semi-discret e Approximatio n 23 4 11.4 Full y Discret e Approximatio n 23 7 11.5 Application s t o th e Contac t Proble m 23 9 11.6 Continuou s Dependenc e wit h Respec t t o Contac t Condition s . . 24 3 11.7 Numerica l Example s 24 6
12 Frictiona l Contac t wit h Norma l Dampe d Respons e 25 5 12.1 Proble m Statemen t 25 5 12.2 A n Abstrac t Proble m an d it s
Well-posedness 25 7 12.3 Semi-discret e Approximatio n o f th e
Abstract Proble m 26 2 12.4 Full y Discret e Approximatio n o f the Abstrac t Proble m 26 4 12.5 Application s t o th e Contac t Proble m 26 7 12.6 Tw o Fiel d Variationa l Formulation s 27 1 12.7 Numerica l Example s 27 3
13 Othe r Viscoelasti c Contac t Problem s 28 5 13.1 Bilatera l Contac t wit h Nonloca l Coulom b Frictio n La w 28 5 13.2 Bilatera l Contac t wit h Frictio n an d Wea r 28 9 13.3 Contac t wit h Norma l Compliance , Frictio n an d Wea r 29 2 13.4 Contac t wit h Dissipativ e Frictiona l Potentia l 29 6
Bibliographical Notes 305
IV Contac t Problem s i n Viscoplasticit y 30 7
14 A Signorin i Contac t Proble m 30 9 14.1 Proble m Statemen t 30 9 14.2 Existenc e an d Uniquenes s Result s 31 1 14.3 Som e Propertie s o f the Solutio n 31 6
15 Frictionles s Contac t wit h Dissipativ e Potentia l 32 1 15.1 Proble m Statemen t an d Variationa l Analysi s 32 1 15.2 Semi-discret e Approximatio n 32 4 15.3 Full y Discret e Approximatio n 32 7 15.4 Th e Signorin i Contac t Proble m 33 0 15.5 A Frictionless Contac t Proble m wit h Norma l Complianc e . . . . 33 3 15.6 A Convergenc e Resul t 33 5 15.7 Numerica l Example s 34 0
x CONTENTS
16 Frictionles s Contac t betwee n Tw o Viscoplasti c Bodie s 34 7 16.1 Proble m Statemen t 34 7 16.2 Uniqu e Solvabilit y an d Propertie s o f the Solutio n 35 0 16.3 Semi-discret e Approximatio n 35 2 16.4 Full y Discret e Approximatio n 35 7 16.5 Numerica l Example s 36 0
17 Bilatera l Contac t wit h Tresca' s Frictio n La w 36 5 17.1 Proble m Statemen t 36 5 17.2 Existenc e an d Uniquenes s Result s 36 8 17.3 Som e Propertie s o f the Solutio n 37 6 17.4 Semi-discret e Approximatio n 37 9 17.5 Full y Discret e Approximatio n 38 4 17.6 Convergenc e o f the Full y Discret e Schem e 39 0
18 Othe r Viscoplasti c Contac t Problem s 39 7 18.1 Contac t wit h Simplifie d Coulomb' s Frictio n La w 39 7 18.2 Contac t wit h Dissipativ e Frictiona l Potentia l 39 9 18.3 Stres s Formulatio n i n Perfec t Plasticit y 40 4 18.4 A Frictionless Contac t Proble m fo r Material s with Interna l Stat e
Variable 41 4
Bibliographical Notes 1^21
Bibliography 42 3
Index 43 9
Preface Phenomena o f contac t betwee n deformabl e bodie s o r betwee n deformabl e an d rigid bodie s aboun d i n industr y an d everyda y life . Contac t o f brakin g pad s with wheels , tires with roads , pistons with skirt s are just a few simple examples . Common industria l processe s suc h a s meta l forming , meta l extrusion , involv e contact evolutions . Becaus e of the importance o f contact processe s i n structura l and mechanica l systems , a considerabl e effor t ha s bee n mad e i n it s modelin g and numerica l simulations . Th e engineerin g literatur e concernin g thi s topi c i s extensive. Owin g to thei r inheren t complexity , contac t phenomen a ar e modele d by nonlinea r evolutionar y problem s tha t ar e difficul t t o analyze .
In earl y mathematica l publication s i t wa s invariabl y assume d tha t th e de -formable bodie s wer e linearl y elasti c an d th e processe s wer e static . However , a long tim e ag o i t wa s recognize d th e nee d t o conside r contac t problem s involv -ing viscoelasti c an d viscoplasti c material s a s wel l a s a larg e variet y o f contac t and frictiona l boundar y conditions , whic h lea d t o tim e dependen t models . Th e mathematical theor y o f contac t problems , tha t ca n predic t reliabl y th e evolu -tion of the contact proces s in different situation s and under various conditions, i s emerging currently . I t deal s with rigorou s modelin g o f the contac t phenomena , based on the fundamental physica l principles, a s well as with the variational an d numerical analysi s of the models . A thorough treatmen t o f contact problem s re-quire knowledge from functiona l analysis , moder n partia l differentia l equations , numerical approximation s an d erro r analysis .
The purpos e o f thi s boo k i s to introduc e th e reade r t o a mathematica l the -ory o f contac t problem s involvin g deformabl e bodies . Th e content s cove r th e mechanical modeling , mathematica l formulations , variationa l analysis , an d th e numerical solutio n o f th e associate d formulations . Ou r intentio n i s t o giv e a complete treatmen t o f some contac t problem s b y presentin g argument s an d re -sults i n modeling , analysis , an d numerica l simulations .
In the boo k we treat quasistati c contac t processe s i n the infinitesima l strai n theory. Quasistati c processe s aris e when th e applie d force s var y slowl y i n time , and therefor e th e syste m respons e i s relativel y slo w s o tha t th e inertia l term s in th e equation s o f motio n ca n b e neglected . W e mode l th e materia l behavio r with elastic , viscoelasti c o r viscoplasti c constitutiv e laws ; som e o f ou r result s are extende d t o material s wit h interna l stat e variable s an d t o perfectl y plasti c materials. Th e contac t i s modeled wit h variou s conditions , includin g Signorin i nonpenetration condition , norma l compliance and normal damped response con-ditions. Th e friction i s modeled with versions of Coulomb's an d Tresca' s frictio n laws o r wit h law s involvin g a dissipativ e frictiona l potential . W e als o conside r problems wit h frictio n an d wea r an d w e us e a versio n o f th e Archar d la w t o model th e evolutio n o f wear .
XI
Xll PREFACE
Variational analysi s o f th e model s include s existenc e an d uniquenes s result s of wea k solution s a s wel l a s result s o f continuou s dependenc e o f th e solutio n on th e da t a an d parameters . Link s betwee n differen t mechanica l model s ar e discussed; fo r example , elasticit y a s a limitin g cas e o f viscoelasticity , perfec t plasticity a s a limitin g cas e of viscoplasticity, Signorin i nonpenetrat io n conditio n as a limitin g cas e o f th e norma l complianc e contac t condition . I n carryin g ou t the variationa l analysi s w e systematically us e result s o n ellipti c an d evolutionar y variational inequalities , conve x analysis , nonlinea r equation s wit h monoton e operators an d fixed point s o f operators .
Two kind s o f approximatio n scheme s ar e introduce d an d analyzed . Whe n only th e spatia l variable s ar e discretized , w e obtai n semi-discret e schemes . I f both th e spatia l an d tempora l variable s ar e discretized , w e arrive a t full y discret e schemes. Fo r bo t h kind s o f scheme s w e prov e existenc e an d uniquenes s results . We sho w convergenc e o f the discret e solution s unde r th e basi c solutio n regularit y available fro m th e well-posednes s result s o f th e variationa l problems . W e als o present optima l orde r erro r estimate s unde r additiona l regularit y assumptio n o n the solution .
To demonstrat e th e performanc e o f th e numerica l schemes , a numbe r o f nu -merical simulation s ar e discussed . Th e tes t problem s rang e fro m on e t o thre e dimensional geometries . Th e finite elemen t metho d i s used t o discretiz e th e spa -tial domai n an d finite difference s ar e use d fo r th e tim e derivatives . W e describ e in th e boo k numerica l result s i n th e stud y o f som e mode l contac t problem s fo r elastic, viscoelasti c an d viscoplasti c materials .
The boo k i s intende d t o b e self-containe d an d accessibl e t o a larg e numbe r of readers . I t i s divide d int o fou r parts , a s describe d i n th e following .
Part I i s devote d t o th e basi c notion s an d result s whic h ar e fundamen -tal t o th e developmen t late r i n thi s book . W e revie w her e th e backgroun d on functiona l analysis , functio n spaces , finite differenc e approximation s an d fi-nite elemen t method . The n w e appl y thes e result s i n th e stud y o f ellipti c an d evolutionary variationa l inequalities . Th e materia l presente d i n thi s par t i s self -contained. I t doe s no t nee d an y knowledg e i n Contac t Mechanic s an d coul d b e aimed a t graduat e student s an d researcher s intereste d i n a genera l t reatmen t o f variational problem s an d thei r numerica l approximations .
Part II present s preliminar y materia l i n Contac t Mechanics . W e summariz e here basi c notion s an d genera l principle s o f Mechanic s o f Continua . The n w e introduce contac t boundar y condition s wit h o r withou t frictio n a s wel l a s consti -tutive law s whic h ar e use d i n th e res t o f th e book . W e als o presen t preliminar y results o n variationa l an d numerica l analysi s i n contac t problem s an d w e appl y these result s i n th e stud y o f model s involvin g elasti c bodies . Th e materia l i n Par t I I i s aime d a t thos e reader s wh o ar e intereste d i n th e mechanica l back -ground o f contac t problem s an d mathematica l theor y o f som e contac t problem s in elasticity .
Parts III and IV represen t th e mai n par t s o f th e book . The y dea l wit h the stud y o f quasistati c problem s fo r Kelvin-Voig t viscoelasti c material s an d rate-type viscoplasti c materials , respectively . Thes e par t s ar e wri t te n base d o n our origina l research . W e conside r her e a numbe r o f problem s wit h variou s
PREFACE xm
contact an d frictiona l o r frictionles s boundar y conditions , fo r whic h w e provid e variational analysi s an d numerica l approximations . Fo r som e o f th e contac t problems, w e include result s o f numerica l simulation s t o sho w th e performanc e of th e numerica l schemes . Thes e tw o part s o f th e boo k woul d interes t mainl y researchers fo r a n in-dept h knowledg e of the mathematica l theor y o f quasistati c contact problems .
' Th e lis t o f the reference s a t th e en d o f the boo k i s by n o mean s exhaustive . It onl y include s paper s o r book s tha t wer e use d fo r o r ar e directl y connecte d with th e subject s treate d i n thi s book . Eac h par t i s conclude d wit h a sectio n entitled Bibliographical Notes tha t discusse s reference s o n th e principa l result s treated, a s well as information o n importan t topic s relate d to , bu t no t include d in, th e bod y o f the text .
Each o f the fou r part s o f the boo k i s divide d int o severa l chapters . Al l th e chapters ar e numbere d consecutively . Mathematica l relation s (equalitie s o r in -equalities) ar e numbered b y chapter an d thei r orde r of occurrence. Fo r example , (5.3) i s th e thir d numbere d mathematica l relatio n i n Chapte r 5 . Definitions , examples, problems , theorems , lemmas , corollaries , proposition s an d remark s are numbere d consecutivel y withi n eac h chapter . Fo r example , i n Chapte r 10 , Problem 10. 1 is followed b y Theore m 10.2 .
The presen t boo k i s the resul t o f three years o f cooperation betwee n th e tw o authors. I n writin g i t w e hav e draw n o n th e result s o f ou r join t collaboratio n with numerou s colleague s an d friend s t o who m w e addres s ou r thanks . W e express ou r gratitud e t o Professo r Mei r Shillo r fo r ou r beneficia l cooperatio n as wel l a s fo r th e interestin g discussion s o n th e model s treate d i n th e book . We particularl y than k Dr . J.R . Fernandez-Garcf a wh o provide d th e numerica l simulations include d i n Part s II I an d I V o f the book . W e extend ou r thank s t o Dr. M . Barboteu wh o realized th e numerica l result s i n Par t I I o f the book . W e especially than k Professo r Shing-Tun g Ya u fo r hi s suppor t an d encouragemen t of our work .
The wor k o f W.H . wa s supporte d b y NSF/DARP A unde r Gran t DMS -9874015, Jame s Va n Alle n Natura l Scienc e Fellowship , Facult y Developmen t Award a t th e Universit y o f Iowa, an d K.C . Won g Educatio n Foundation . Par t of th e boo k manuscrip t wa s prepare d whe n W.H . wa s visitin g th e Stat e Ke y Laboratory o f Scientifi c an d Engineerin g Computing , Chines e Academ y o f Sci -ences, durin g th e summe r o f 200 0 throug h th e suppor t o f th e K.C . Won g Ed -ucation Foundation . H e particularl y thank s Professo r Lie-hen g Wan g fo r th e warm hospitality .
W.H. Iowa Cit y
M.S. Perpignan
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List o f Symbol s
Sets
N: th e se t o f positive integer s
Z+: th e se t o f non-negative integer s
R: th e rea l lin e
R = R U {±oc}: extende d rea l lin e
R+ : th e se t o f non-negative number s
Rd: th e d-dimensiona l Euclidea n spac e
§d: th e spac e o f second orde r symmetri c tensor s o n R d
Q+: th e se t o f orthogonal matrice s o f orde r d with determinan t 1
Sd' th e uni t spher e i n R d
ft: a n open , bounded , connecte d se t i n R d wit h a Lipschit z boundar y F
T: th e boundary o f the domain f2 , that i s decomposed a s T = T i u r2 U ^ with Ti , T 2 and T 3 relatively ope n wit h respec t t o T
Fi: th e par t o f the boundar y wher e displacemen t conditio n i s specified; meas (Ti) > 0 is assumed throughou t th e boo k
T2: th e par t o f the boundar y wher e tractio n conditio n i s specifie d
T3: th e par t o f the boundar y fo r contac t
/ = [0,T] : tim e interva l o f interes t
/ = (0,T )
xv
XVI LIST OF SYMBOLS
Operators
e: deformatio n operator , i.e . e(u) = (sij(u)), Sij(u)=±(uij +u jti) (pag e 97 )
Div: divergenc e operator , i.e . Dive r = (cr^j ) (pag e 100 )
II^: finite elemen t interpolatio n operato r (pag e 52 )
VK' projectio n operato r ont o a se t K (pag e 16 )
Vh: finite elemen t projectio n operato r (page s 77 , 181 )
VQH: finite elemen t projectio n o n Q h (pag e 154 )
Function space s
Lp(fl): th e Lebesgue space of p-integrable functions , wit h the usua l mod -ification i f p = o o (pag e 28 )
Cm(fl): th e spac e o f function s whos e derivative s u p t o an d includin g order m ar e continuou s u p t o th e boundar y F (pag e 26 )
CQ°(Q): th e space of infinitely differentiabl e function s wit h compact sup -port i n ft (pag e 26 )
Wk,p(il): th e Sobolev space of functions whos e weak derivatives of orders less than o r equa l t o k ar e p-integrable o n ft (pag e 30 )
Hk(Sl) = W k>2(Sl) (page 30 )
W^p(n): the closure of Cg°(fi) in Wk*(Q) (page 30)
Hk(Q) = W^ 2(Ct) (page 30)
H'1^): th e dua l o f H%(Q) (page 34 )
# 2 ( r ) : a Sobole v spac e on T , defined a s the rang e o f the trac e operato r o n t f 1 ^ ) (pag e 36 )
# - 5 ( r ) : th e dua l o f H*(T) (pag e 36 )
Hr = Hi{T) d (pag e 143 )
H^: dua l o f H r (pag e 144 )
H = {v = (v u . . . ,v d)T : Vi e L 2(ft), 1 < i < d} = L 2(Q)d,
inner produc t (U,V)H = jQ ui(x) v i(x) dx (pag e 142 )
Q = {r = {r i3) : r tj = r3% G L2(tt), 1 < i , j < d} = L 2(ft)fxd, inner produc t ((T,T)Q — J fi cr^(x ) r^(x) dx (pag e 142 )
Q1 = { r € Q : D i v r e # } , inner produc t (cr,r)g 1 = (CT,T)Q -f - (Diver , D i v T )# (pag e 145 )
V - { v e H 1 ^ : v = 0 a.e . o n I \ } , inner produc t (ix , v)y = (e(ix),e(i;)) Q (pag e 144 )
LIST OF SYMBOLS xvn
Vi — {v G V : Vy = 0 a.e . o n ^ } wit h th e inne r produc t (u,v)y (page 144 )
y2 = { v G " K : v v < Q a.e. o n ^ } wit h th e inne r produc t (u,v)v (page 144 )
X: a Hilber t spac e o r it s subse t wit h inne r produc t (• , -)x, o r a Banac h space o r it s subse t wit h nor m | | • ||x
Cm(7;X) = {ve C(7;X ) : v& G C(7;X), j = 1 , . . . ,m } (pag e 37 )
LP(I;X) = {v : I -^ X measurabl e : | |V||LP(J ;X) < o o } (pag e 38 )
Wk*(I;X) = {ve L"(0,T;X ) : \\v^\\ LP{I.x) < oc Vj < k} (pag e 39 )
Hk(I-X) = W f c '2(/;X) (pag e 40 )
Other symbol s
d: a positive intege r takin g th e value s 1,2, 3 i n application s
c: a generi c positiv e constan t
r+ = max{0,r} : positiv e par t o f r
v. uni t outwar d norma l o n th e boundar y o f Q
(•, •): th e dualit y pairin g betwee n H*(T) an d H~* (V) (pag e 36 )
(•, -)r- th e dualit y pairin g betwee n H? an d H^ (pag e 144 )
V: fo r an y
3: ther e exist(s )
A: closur e o f the se t A o
int A o r A: interio r o f the se t A
dA: boundar y o f the se t A
diji th e Kronecke r delt a
s.t.: suc h tha t
a.e.: almos t everywher e
iff: i f and onl y i f
O(h): fo r som e constan t c > 0 independent o f h suc h tha t |0(/i) | < ch
Awn = wn — wn-\\ backwar d differenc e
&nWn = Aw n/kn: backwar d divide d differenc e
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BIBLIOGRAPHY 42 3
Bibliography
[1] R.A . Adams , Sobolev Spaces, Academi c Press , Ne w York, 1975 .
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Index absolute continuity , 4 0 acceleration, 9 5
balance la w o f angula r momentum , 100
balance la w of mass, 9 9 balance la w of momentum, 10 0 Banach fixed-point theorem , 2 1 Banach space , 7 bidual, 1 2 biliear form , 6 3
bounded, 6 3 continuous, 6 3 elliptic, 6 3 symmetric, 6 3
Bochner integrable , 3 8 bounded operator , 9 bulk modulus , 12 3 Burgers model , 12 8
Cea's inequality , 5 6 generalization of , 66 , 67
Cartesian produc t space , 4 Cauchy sequence , 6 Cauchy stres s tensor , 10 1 Cauchy-Lipschitz theorem , 20 2 Cauchy-Schwartz inequality , 14 , 28 closed set , 6
weakly, 1 2 closure, 7 compact support , 2 6 compliance tensor , 12 2 conjugate exponent , 1 1 constitutive law , 10 4 contact condition , 10 6
bilateral, 10 6 normal compliance , 10 7 normal dampe d response , 10 7 Signorini, 10 6
contact proble m i n elasticit y bilateral wit h Tresca' s friction ,
177
frictionless wit h deformable sup -port, 17 2
Signorini frictionless , 16 7 simplified Coulomb' s friction, 17 9
contact proble m i n perfec t plastic -ity, 40 3
bilateral wit h Tresca' s friction , 412
frictionless wit h normal dampe d response, 41 2
normal dampe d respons e wit h Tresca's friction , 41 3
contact proble m i n viscoelasticit y bilateral with friction an d wear ,
289 bilateral with nonlocal Coulomb
friction, 28 5 bilateral with slip dependent fric -
tion, 20 7 bilateral wit h Tresca' s friction ,
302 bilateral wit h viscou s friction ,
302 frictional wit h norma l dampe d
response, 25 5 frictional wit h norma l compli -
ance, 22 7 normal dampe d respons e wit h
power-law friction , 30 4 normal dampe d respons e wit h
Tresca's friction , 30 4 Signorini frictionless , 19 3 viscous contac t wit h power-la w
friction, 30 3 viscous contact with Tresca's fric -
tion, 30 3 with dissipative frictional poten -
tial, 29 6 with norma l compliance , fric -
tion an d wear , 29 2 contact proble m i n viscoplasticit y
bilateral wit h Tresca' s friction ,
440 INDEX
365 frictionless fo r materials with in-
ternal stat e variable , 41 4 frictionless wit h dissipativ e po -
tential, 32 1 frictionless wit h normal compli-
ance, 33 3 Signorini frictionless , 309 , 330 unilateral frictionless betwee n two
bodies, 34 7 with dissipative frictional poten -
tial, 39 9 with simplifie d Coulomb' s fric -
tion, 39 7 continuous embedding , 3 3 continuous medium , 9 3 continuous operator , 9 continuum, 9 3 convergence, 6 , 5 7
strong, 6 weak, 1 1 weak * , 12
convex combination , 8 convex function , 1 7
strictly, 1 7 convex set , 8 creep, 11 7 current configuration , 9 4
deformation gradient , 9 5 deformed configuration , 9 4 dense subset , 8 density o f smooth functions , 14 8 derivative, 3 7
strong, 3 7 weak, 3 9
difference backward, 4 3 centered, 4 3 forward, 4 3
displacement, 9 5 normal, 10 5 tangential, 10 5
dual formulation , 202 , 222 dual space , 1 1 dynamic processes , 10 2
effective domain , 1 8 elastic deformation , 11 6 elasticity operator , 13 2 elasticity tensor , 12 2 elastoplastic model , 11 7 elliptic variationa l inequality , 5 9
first kind , 5 9 second kind , 5 9
epigraph, 1 8 equation o f equilibrium, 10 2 equation o f motion , 10 2 error estimate , 5 7 Euclidean norm , 4 Euclidean space , 4
finite elemen t interpolant , 5 2 finite elemen t projection , 7 7 finite elemen t space , 5 0 finite element s
affine-equivalent, 4 9 finite strai n tensor , 9 6 first Piola-Kirchhof f stres s tensor, 10 0 first Piola-Kirchof f stres s vector, 10 0 fixed point , 2 1 friction condition , 10 6
Coulomb, 11 0 generalized Coulomb , 11 0 nonlocal Coulomb , 11 1 slip dependen t Tresca , 10 9 Tresca, 10 9 viscous, 11 1
friction force , 10 6 frictionless condition , 10 8 Frobenius norm , 4 fully discret e approximation , 7 8 functional, 9
linear continuous , 1 1
Gateaux derivative , 2 0 generalized variationa l lemma , 2 9 generalized Voig t model , 12 8 global interpolation , 5 2 Green-Saint Venan t tensor , 9 6
Holder inequality , 2 8 Hencky material , 12 4
INDEX 441
Hilbert space , 1 4 Hooke's law , 12 1
indicator function , 1 9 inelastic deformation , 11 6 infinitesimal deformation , 9 7 infinitesimal strai n tensor , 9 7 inner product , 1 3 inner produc t space , 1 3 internal stat e variable , 13 8 interpolation erro r estimate , 5 3
Jacobian, 9 4
Kelvin-Voigt model , 12 6 kinematics o f continua , 9 3 Korn's inequality , 14 4
Lame coefficients , 12 3 Lax-Milgram Lemma , 6 4 Lebesgue's theorem , 2 9 limit, 6
weak, 1 1 weak * , 1 2
linear space , 3 linearized strai n tensor , 9 7 Lipschitz continuou s function , 2 7 Lipschitz continuou s operator , 9 Lipschitz domain , 3 1 local interpolation , 5 1 locally p-integrable , 2 9 lower semicontinuou s (l.s.c. ) func -
tion, 1 8
mass density , 9 8 material
anisotropic, 12 3 homogeneous, 12 3 isotropic, 12 3 nonhomogeneous, 12 3
material point , 9 4 Maxwell model , 12 6
generalized, 12 6 mesh, 4 6 mesh parameter , 4 7 motion, 9 4 multi-index, 2 5
norm, 3 equivalent, 5 Euclidean, 4 Frobenius, 4 operator, 1 0
normed space , 4 complete, 7 reflexive, 1 2 separable, 1 3
operator, 8 bounded, 9 compact, 1 3 continuous, 9 domain, 8 linear, 9 Lipschitz continuous , 9 maximal monotone , 19 5 monotone, 15 , 19 5 range, 9 strictly monotone , 1 5 strongly monotone , 1 5
orthogonal, 1 5
parti t ion, 4 6 perfectly plasti c model , 134 , 13 8 Perzyna's law , 13 6 Poisson's ratio , 12 3 Prandt l -Reuss flo w rule , 13 8 primal formulation , 202 , 22 2 projection, 1 6 projection operator , 1 6 proper, 1 7 proximal element , 6 1 proximity operator , 6 1
quasistatic processes , 10 2 quasivariational inequality , 27 2
reference configuration , 9 4 reference element , 4 6 reflexive norme d space , 1 2 relaxation, 11 8 Riesz representatio n theorem , 1 5 right Cauchy-Gree n tensor , 9 5 rigid bod y motion , 9 5
442 INDEX
rigid-plastic model , 11 7
scheme backward, 4 4 Crank-Nicolson, 4 4 forward, 4 4 generalized mid-point , 4 5
semi-discrete approximation , 7 6 semi-norm, 5 separable norme d space , 1 3 small strai n tensor , 9 7 Sobolev space , 2 9
embedding, 3 2 of integer order , 3 0 of non-intege r order , 3 5
Sokolowski model , 13 4 standard viscoelasti c model , 12 6 step-size, 4 3 strain
elastic, 11 6 plastic, 11 6
strength limit , 11 6 stress
normal, 10 6 tangential, 10 6
stress tensor , 10 2 stress vector , 10 2 stress-strain diagram , 11 3 subdifferentiable function , 1 9 subdifferential, 1 9 subgradient, 1 9 support, 2 6 support functional , 2 0
trace, 33 , 143 , 146 trace operator , 34 , 143 , 146 triangulation, 4 6
regular, 4 7 tribology, 10 4
undeformed configuration , 9 4
velocity, 9 5 viscoelastic constitutiv e law , 13 2 viscoelastic phenomenon , 11 8 viscoplastic constitutive law, 133 , 135
viscosity operator , 132 , 19 4 von Mise s function , 13 7
weak * convergence, 1 2 weak * limit, 1 2 weak convergence , 1 1 weak derivative , 2 9 weak formulation , 5 5 weak limit , 1 1 weakly close d set , 1 2 weakly lower semicontinuous (w.l.s.c. )
function, 1 8
yield condition , 13 7 yield function , 13 7 yield limit , 11 6 Young's modulus , 121 , 123
Titles i n Thi s Serie s
30 Weima n Ha n an d Mirce a Sofonea , Quasistati c Contac t Problem s i n Viscoelasticit y
and Viscoplasticity , 200 2
29 Shuxin g Che n an d S.-T . Yau , Editors , Geometr y an d Nonlinea r Partia l Differentia l
Equations, 200 2
28 Valenti n Afraimovic h an d Sze-B i Hsu , Lecture s o n Chaoti c Dynamica l Systems , 200 2
27 M . R a m Murty , Introductio n t o p-adi c Analyti c Numbe r Theory , 200 2
26 Raymon d Chan , Yue-Kue n Kwok , Davi d Yao , an d Qian g Zhang , Editors ,
Applied Probability , 200 2
25 Dongga o Deng , Dare n Huang , Rong-Qin g Jia , We i Lin , an d Jia n Zhon g Wong ,
Editors, Wavele t Analysi s an d Applications , 200 2
24 Jan e Gilman , Wil l ia m W . Menasco , an d Xiao-Son g Lin , Editors , Knots , Braids ,
and Mappin g Clas s Groups—Paper s Dedicate d t o Joa n S . Birman , 200 1
23 Cumru n Vaf a an d S.-T . Yau , Editors , Winte r Schoo l o n Mirro r Symmetry , Vecto r
Bundles an d Lagrangia n Submanifolds , 200 1
22 Carlo s Berenstein , Der-Che n Chang , an d Jingzh i Tie , Laguerr e Calculu s an d It s
Applications o n th e Heisenber g Group , 200 1
21 Jiirge n Jost , Bosoni c Strings : A Mathematica l Treatment , 200 1
20 L o Yan g an d S.-T . Yau , Editors , Firs t Internationa l Congres s o f Chines e
Mathematicians, 200 1
19 So-Chi n Che n an d Mei-Ch i Shaw , Partia l Differentia l Equation s i n Severa l Comple x
Variables, 200 1
18 Fangyan g Zheng , Comple x Differentia l Geometry , 200 0
17 Le i Gu o an d Stephe n S.-T . Yau , Editors , Lecture s o n Systems , Control , an d
Information, 200 0
16 Rud i Weikar d an d Gilber t Weinstein , Editors , Differentia l Equation s an d
Mathematical Physics , 200 0
15 Lin g Hsia o an d Zhoupin g Xin , Editors , Som e Curren t Topic s o n Nonlinea r
Conservation Laws , 200 0
14 Jun-ich i Igusa , A n Introductio n t o th e Theor y o f Loca l Zet a Functions , 200 0
13 Vasilio s Alexiade s an d Georg e Siopsis , Editors , Trend s i n Mathematica l Physics ,
1999
12 Shen g Gong , Th e Bieberbac h Conjecture , 199 9
11 Shinich i Mochizuki , Foundation s o f p-adic Teichmulle r Theory , 199 9
10 Duon g H . Phong , Lu c Vinet , an d Shing-Tun g Yau , Editors , Mirro r Symmetr y III ,
1999
9 Shing-Tun g Yau , Editor , Mirro r Symmetr y I , 199 8
8 Jiirge n Jost , Wilfri d Kendall , U m b e r t o Mosco , Michae l Rockner ,
and Karl-Theodo r Sturm , Ne w Direction s i n Dirichle t Forms , 199 8
7 D . A . Buel l an d J . T . Teite lbaum , Editors , Computationa l Perspective s o n Numbe r
Theory, 199 8
6 Harol d Levine , Partia l Differentia l Equations , 199 7
5 Qi-ken g Lu , Stephe n S.-T . Yau , an d Anato l y Libgober , Editors , Singularitie s an d Complex Geometry , 199 7
4 Vyjayanth i Char i an d Iva n B . Penkov , Editors , Modula r Interfaces : Modula r Li e Algebras, Quantu m Groups , an d Li e Superalgebras , 199 7
3 Xia-X i Din g an d Tai-Pin g Liu , Editors , Nonlinea r Evolutionar y Partia l Differentia l Equations, 199 7
2.2 Wil l ia m H . Kazez , Editor , Geometri c Topology , 199 7 2.1 Wil l ia m H . Kazez , Editor , Geometri c Topology , 199 7
TITLES I N THI S SERIE S
1 B . Green e an d S.-T . Yau , Editors , Mirro r Symmetr y II , 199 7
