ROBERT NZENGWA Incrementalmethodsinnonlinear,three- … · divergence of a tensor field T: O, cz...

$: ROBERT NZENGWA Incrementalmethodsinnonlinear,three- … · divergence of a tensor field T: O, cz IR3- Ml3, first and second Fréchet derivatives of an operator e^ .. \r J\. v> I ,$
RAIROMODÉLISATION MATHÉMATIQUE

ET ANALYSE NUMÉRIQUE

ROBERT NZENGWAIncremental methods in nonlinear, three-dimensional, incompressible elasticityRAIRO – Modélisation mathématique et analyse numérique,tome 22, no 2 (1988), p. 311-342.<http://www.numdam.org/item?id=M2AN_1988__22_2_311_0>

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MATHEMATICA!. MODELUNG AND NUMERICAL ANALYSISMOOEUSATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

(Vol. 22, n° 2, 1988, p. 311 à 342)

INCREMENTAL METHODS IN NONLINEAR, THREE-DIMENSIONAL,INCOMPRESSIBLE ELASTICITY (*)

by Robert NZENGWA (*)

Communiqué par P G. ClARLET

Abstract — In this paper, we apply the incrémental methods to approximate the equihbriuméquations of nonlinearly elastic incompressible bodies, subject to dead or live loads, with puredisplacement or traction boundary conditions We establish the convergence of the methods

Résume — Dans cet article, nous appliquons des méthodes incrémentales pour approcher leséquations d'équilibre de corps élastiques non linéaires incompressibles, soumis à des forcesmortes ou a des forces vives, avec des conditions aux limites de déplacement pur ou de tractionpure Nous établissons la convergence de ces méthodes

INTRODUCTION

We first specify the notation we shall use, concerning notably vectorfields, matrices, function spaces, derivatives and norms. In what follows,Latin indices take their values in the set {1, 2, 3} and the repeated indexconvention is used.

pressure field,displacement field,déformation field,unit outer normal vector to the boundary of adomain,Euclidean inner product,Euclidean vector norm,matrix with element Al} (i : row index, j : columnindex),

(*) Reçu en octobre 1986(*) Laboratoire d'Analyse Numérique, Tour 55-65, Université Pierre et Marie Curie, 4 place

Jussieu, 75005 Pans

M2 AN Modélisation mathématique et Analyse numérique 0399-0516/88/02/311/32/$ 5.20Mathematical Modelhng and Numencal Analysis © AFCET Gauthier-Villars

pli —

V =

u.v

ïjf4 =

(Mi)

(*,)

M= u,

= («(AtJ)

v, :

312

A:B=AtJ Bl}

adj A

Ml3

skewsymsio3

T

bT

R. NZENGWA

matrix inner product,matrix norm associated with the matrix innerproduct,adjugate of the matrix A = transpose of thecofactor matrix,set of all matrices of order 3,set of skew symmetrie matrices of order 3,set of symmetrie matrices of order 3,set of symmetrie positive definite matrices,set of orthogonal matrices,set of orthogonal matrices, with determinant equalsto one,first Piola-Kirchhoff stress,second Piola-Kirchhoff stress,volume force operatorsurface force operator,loading operator,a dead load,for some integer m ^ 0 and some real number

-f? sym

Mm>q

Si sym

= wm-2,

= {(6,i)e ^"-= o, Y<t(o) = Y<J?(O)T}.

1"1/", f b+ C T = O

4

¥<t>div

— » _

9

= (ô(<t>;)efy3

T = (9,7;,) G O?3

usual partial derivatives,

partial derivative with regard to the matrix variable4,gradient of a mapping c[> : fi <= [R3 -• IR3,divergence of a tensor field T: O, cz IR3 -• Ml3,first and second Fréchet derivatives of an operatore . \r vr Ck* {u\ en T tV V^ Û" (n\ ei T f Y V \ Y^ . J\. —> I , ö yU ) c JU yA., I ) , U \ll ) fc 1^2 V-"- ï * ) j ^

and y being two normed vector spaces.

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numerical Analysis

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 313

We dénote by C1; C2? etc., constants which are independent of thevarious functions found in a given inequality.

Let fl be a bounded open connected subset of R3 with a sufficientlysmooth boundary F. We assume that Ü is the référence configurationoccupied by an elastic incompressible body in the absence of any appliedforce. The constitutive law of the material is given by

î = «(¥$)-/> adj y$T , (o.i)

for a given C00 matrix-valued function a : (x, A) e Ü x M3 -• M3, f beingthe first Piola-Kirchhoff tensor. For details about the constitutive law, theinterested reader should refer to [14].

The equilibrium équations for such a body subject to body forceb are of the form

divf + è = 0 in H , (0.2)det V£ - 1 = 0 in CL , (0.3)

the déformation <J> and the pressure p being the unknowns. The équations(0.2) and (0.3) together with the boundary condition

$ = id on T (0.4)

constitute the pure displacement problem while the boundary condition

{«(¥$) ~/>adj V$T}v = T onT (0.5)

together with the équations (0.2)-(0.3) constitute the pure traction problem.In what follows the real number q > 3. Taking into account équation (0.3)

leads to the définition of the Sobolev submanifold

22, = { £ e Wm + 2 '*,det V$ - l in 11}

and two nonlinear elasticity operators ( and §2 corresponding respectivelyto the pure displacement and pure fraction problems. When we consider thedisplacement u and the pressure p as the unknowns, the first operator§! maps %q

m n Wj1* x Wm'+hq into \Vm'q for each integer m ^ 0 and realnumber q > 3. With the same conditions on (m, q) the second operatore2 maps S^ x Wm + l'q into a submanifold of the set of loads L. The puredisplacement and traction problems respectively reduce to solving theéquations

§i(B,/>) = 6 (0-6)

and

h(é,p)= ( 6 , i ) . (0-7)

vol. 22, n° 2, 1988

314 R. NZENGWA

The incrémental methods consist in approximating the above nonlinearéquations by a séquence of linear problems. Unfortunately these operatorsare not defined between affine manifolds and moreover they do not alwayspossess the inverse function theorem property, a crucial condition on whichdépends the applicability and the convergence of the method as proved byBernadou-Ciarlet-Hu [5]. We therefore have to modify the incrémentalmethods according to the problem we want to solve.

In Section 1 we study the pure displacement problem. The correspondingnonlinear operator ex defines a local diffeomorphism between the spaces%qm n Woq x Wm + 1~q>° and VT** but since the space %q

m is not an affinemanifold, we define a second operator \x between ym + 2'q x wm + l>q>° andW71^ x wm + 1'q>° which are both linear spaces. This operator satisfies thecrucial inverse function theorem property. We then describe the relationbetween both operators Gj and èa and we apply the incrémental methods tothe second operator, in order to approximate the local solution for the puredisplacement problem in the case of a dead load.

In Section 2 we consider the pure traction problem. The associatedoperator 02 is defined on a non affine manifold. As in Section 1 we define asecond operator §2 between the spaces Wm + 2>q x Wm + hq and L x Wm + hq.Using results due to Chillingworth-Marsden-Wan [6] and Wan, Y. H. [21],we prove that this operator defines a local diffeomorphism between an openneighborhood of (id, 0) in the linear space Çsym x Wm + lyq and a non affineSobolev submanifold A^ofLx Wm + 1'q. From the décomposition of the loadspace as L = Le © skew, we show that the submanifold TV is the graph of afunction G defined from Le x Wm + 1'q into Skew. Let £ be a dead load inLe with no axis of equiiibrium ; then iocaîiy there exists a unique rotation Qsuch that the element (Ql,0) e N. Consequently we obtain the localunique solution corresponding to the load £. The solution corresponding tothe dead load Ql is obtained from the solution of an appropriate differentialéquation between the above considered spaces, with Q£ as a parameter inthe vector field. The approximation comprises two steps : we first approachthe parameter Q | , then we apply Euler's method to the differentialéquation in which the approximate value of g£ is a parameter.

Finally we consider a class of live loads. Under some additionalassumptions on the loading operator f we first prove the existence and localuniqueness of a solution via the fixed point theorem, using in a crucial waythe polar décomposition of invertible matrices. The approximation of thesolution consists in approaching a finite séquence of solutions for tractionproblems of the dead load type.



1. APPROXIMATION OF A PURE DISPLACEMENT BOUNDARY-VALUE PROBLEMFOR INCOMPRESSIBLE MATERIALS

1.1. Existence

Let ft be an open bounded and connected subset of R3 with a sufficientlysmooth boundary F. We assume that ft is the référence configuration of anincompressible body in the absence of any applied forces. The equilibriuméquations of the body, when it is subjected to a body force b, are :

m il , U--1-1Jdet V$ = 1 inft , (1.1-2)

$ = id onT . (1-1-3)

Given a constitutive law for the first Piola-Kirchhoff stress f as defined in[14]

<J?T , (1.1-4)

where p is the pressure and the constitutive law is a matrix-valuedC00 mapping a : ft x M3 -> M3, the problem then consists in solving thefollowing boundary-value problem for the displacement field u — (ut) andthe pressure p :

-djv {«(/ + Vw)-/?ad^ (ƒ + Vw)T} =b in ft , (1-1-5)

d e t ( / + Vw) = l i n f t , (1-1-6)w = 0 o n T . (1-1-7)

The dependence on x in the constitutive law has been omitted in theformulas. The second Piola-Kirchhoff stress

CT= (7 + Vw)-1 f (1.1-8)

could be equivalently used in these équations as in [5],In what follows the different Sobolev spaces are defined for integers m

and real numbers q satisfying the condition

mmO and q>3 . (1.1-9)

Considering the équations (l.l-6)-(l.l-7) and the condition (1.1-9) wedefine the set of admissible displacements

%qm= { w e V " " + 2 ' * , d e t ( / + V u ) = l i n ft}. (1.1-10)

vol. 22, n° 2, 1988

316 R NZENGWA

This set has a C -submanifold structure because of the condition (1.1-9)as shown in [14], The pure displacement problem reduces to solving theéquations

(u,p)el%xWm + l>*9 (1.1-11)

- div [q (I + Vu) - p adj (/ + Vuf) = b in XI . (1.1-12)

In order to analyse the properties of the left hand side of the équation(1.1-12) which will be useful in the sequel, we gather here all the propertiesof the constitutive law a under our hypotheses :

• the matrix-valued function a : Ù x Ml3 -> Ml3 is of class C™,(1.1-13)

• the référence configuration is a natural state, i.e. a(I) = 0,(1.1-14)

• the prindple of material frame-indifference is satisfied, i. e.

a(QF) = Qa(F), forallQinO\ (1.1-15)

• letting Ç = 9^a(/), there exists a real number (3 ;> 0 such that

Cl]kîel]ekî^^el}el] (1.1-16)

where e = (etJ) is a symmetrie matrix.From the above hypotheses we deduce that the équations (1.1-11) and

(1.1-12) are equivalent to solving

where the nonlinear operator Gj is defined by

- - d i v {q(I + Vw)-/?adj (ƒ +Vw)T} e W™'*. (1.1-18)

This operator is C °° and locally bounded as a composition of the mappingsu i-> adj (ƒ -+- Vw), (u,p) \-^p sxQ (I + Vw)T and the Nemytsky operator$ >-> «(Y^) which are all C00 and locally bounded ([13], [20]). It is clearfrom (1.1-14) that 9^0, 0) = 0.

In order to apply the inverse function theorem we must compute§{(0,0) and verify that it is an isomorphism between TQ %q

m x Wm + 1'q>° andW71^, where

TQ H= {u e Ym + 2>v, div u = 0} (1.1-19)

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is the tangent space to %qm at the origüi O and

eWm + 1-*9 f /> = o l . (1.1-20)Ja

The équation

is equivalent to the linear pure displacement boundary-value problem

- d i v {C G (M) -pi} = b in XI, (1.1-22)divu = g inft, (1.1-23)

u = A o n T , (1.1-24)

with

0 = 0 and A = 0 . (1.1-25)

The tensor e (M) is the linearized strain tensor defined by

e ( M ) = (e„(M)) , with 2 6 ^ ( ^ = ^ + 3 ^ . (1.1-26)

Clearly the operator defined by (1.1-21) is continuous. Under thehypotheses (Ll-13)-(1.1-16) and the additional hypothesis :

• The complementarity condition of Agmon-Douglis-Nirenberg ([3], [4])holds for the équations (l.l-22)-(l.l-24), (1.1-27)

it follows ([14]) that the operator §i (Q, 0) is an isomorphism between thespaces TQXq

m x wm + 1'q>° and Wm>q. We keep these various hypothesesthroughout ; we can now easily deduce existence results :

THEOREM 1 : For each pair of numbers (m, q) satisfying condition (1.1-9), there exist a neighborhood B of 0 in Wm>q and a neighborhoodU x P of (0,0) in %q

m x Wm + 1><i>ö such that, for each load b in B, the puredisplacement boundary-value problem (l.l-5)-(l.l-7) has exactly one solutionin UxP. D

Such a solution therefore satisfies the équation (1,1-17) ; however theincrémental method cannot be directly described. Indeed in the descriptionof the method [5], we compute the approximate solution (un + 1,pn + 1) atstep n + 1 from (un,pn) by a formula of the form

un + 1 = un + §un and pn + 1 = pn -h bpn (1.1-28)

where bun and §pn are « small » incréments of the displacement and thepressure respectively. Since %q

m is not a linear space un + a = un + Sun does not

vol. 22, n° 2, 1988

318 R NZENGWA

belong to %qm in général We overcome this difficulty by defining a second

operator èx between linear spaces such that

9i (« , />)= ( è , 0 ) (Ll-29)

be equivalent to (1.1-17) for the load b. A natural définition for

§i is

§i". («,/>) e Vm + 2*q x w m + 1'*-°->

-+ (§i (">/>)> det (/ + Vw) - 1) e Wm>* x Wm + 1 '* '° . (1.1-30)

We must therefore verify that

det (ƒ + VM) - 1 e wm + hq>° . (1.1-31)

We may write

det (/ + Ç M ) - 1 = d ivw+ {3I + 2M, + 2 ^ +1", + i -

^ w T ) / ; . (1.1-32)

We find that the intégral of the above expression vanishes using Green'sformula, the Piola identity and the boundary condition. Clearly thisoperator is also Cœ and locally bounded with all its derivatives. It is easy toconclude that

§J (Q>O)(M, /O= (b,g) (1.1-33)

defines an isomorpbism between r + M y ^ + M.oI f , « x ^ « + U,oi Indeed the équation (1.1-33) is equivalent to theéquations (l.l-22)-(l. 1-24) for the particular case in which h = 0. Theoperator §î(Q, 0) is also continuous and one-to-one. lts image is the subset

\(b,g,h)e Wm^q x W m + 1 ' ? ) O x wm + 2-i/q,q ^ f g _ f § . v = o l ,l *~ Ja JT ~ J

(1.1-34)

which contains W™** x Wm + 1 ' 9 ' 0 x {0} , ([14]). We easily conclude usingthe Banach theorem.

We can now deduce a theorem similar to theorem 1.

THEOREM 2 : For each pair of numbers (m,q) satisfying (1.1-9), thereexist a neighborhood B x G of (0 ,0) in W™^ x wm + hq>° and a neighbor-hood UxP of (0 ,0 ) in ym + 2>q x wm + 1>q>° such that \x defines adiffeomorphism between the two neighborhoods. D

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelhng and Numencal Analysis


Clearly the operator bx satisfies équation (1.1-29) in these neighborhoods.For a given dead load b in B, we shall use the incrémental method toapproximate the solution of the équation

§i(«,/>) = (6.0) (1.1-35)

which is exactly the solution for the pure displacement boundary-valueproblem (l.l-5)-(l.l-7).

1.2. Description and convergence of the method.

We begin by specifying some subséquent notations. We dénote by

'O | a | gm

the norm of the Sobolev spaces W"1'*, by

the norm of the product space Wm + 2yq x Wm + hq and its subspaces. Weshall dénote by the same symbol ||.| | the norms of the spaces L(X, Y),L(Y,X) and L2(X, Y) for the normed spaces X and Y. We shall notconsider the case of a gradient vector field b e Cl(Cl, U3) for which a trivialsolution is given by the pair (0,p),

= f £Jo

.xdt . (1-2-1)

We can now describe the approximation of the boundary-value problem(l.l-5)-(l.l-7) by an incrémental method. Given a dead load b in theneighborhood B, as defined in theorem 2, we approximate the solution(U>P)> (QiiUiP) = (6>0)) as follows : let there be given any partition

0 = \° < X1 < . . . < kN = 1 (1.2-2)

of the interval [0, 1]. We let

Abn = (X" + 1 -\n)b, O ^ n ^ i V - 1 , (1.2-3)

u° = 0, p° = 0, (1.2-4)

then assuming un and pn are known, we solve the linear problem

&(%">Pn)(&ê«9 Spn) = (Abn, 0) , (1.2-5)

( b u n , b p n ) e y m + 2>q x Wm + 1>q>°. (1.2-6)

vol. 22, n° 2, 1988

320 R. NZENGWA

We define the (n + 1 )-st approximate displacement and pressure

un + l =un + hun and pn + 1 = pn + 8j?n . (1.2-7)

We end this algorithm by computing the N-th approximate Piola-Kirchhoff second stress tensor from the AT-th approximate displacement andpressure by

?" = à(uN,pN)= (I + VuNrlx

VuN) -pNadi (I + ¥uN)T} . (1.2-8)

Bef ore proving the convergence of this method, we recall some relevantresults proved so far :

We showed that the operator hx is C °° and has locally bounded derivativesof all orders. We deduced (theorem 2) from the hypotheses (1.1-13)-(1.1-16) and (1.1-27) that ^ defines a local diffeomorphism between aneighborhood UxP of ( 0 , 0 ) i n r + 2 l ? x r + 1 ' ? ' °anda neighborhoodB x G of (Q, 0) in Wm*q x wm + 1>q>°. For each element (u,p) in U x P,Q[(u,p) is an isomorphism between the above spaces which are bothBanach spaces. We now describe these neighborhoods in the followingtheorem.

THEOREM 3 (Bernadou-Ciariet-Hu 1983 theorem 2) : For each pair ofnumbers Cm, q ) satisfying (1.1-9), there exists a real number p0 > 0 dependingon m and q such that if

O ^ P ^ P o , (1.2-9)

then

Q[(u,p) is an isomorphism for each (u,p) e Upx Pp , (1.2-10)

where

t / p x P p = {(u,p)eYm + 2>qxWn + 1>(i>\ \\(u,p)\\^p} , (1.2-11)

7 P = sup | | { 8 i ( « > J P ) } " 1 | | < + o o , (1.2-12)

llt^ifei^i)}"1- {i(ulPl){uT£)eüxP Ij (Ki _ fe pi

+ oo . D (1.2-13)


INCREMENTAL METHODS IN NONLÏNEAR ELASTICITY 321

We deduce from this theorem and a classical resuit in differentialéquations (e.g. Crouzeix-Mignot [9]) :

THEOREM 4 (Bernadou-Ciarlet-Hu 1983 theorem 3) : For each realnumber p which satisfies the inequality (1.2-9) and for each load (b, g) whichsatisfies

P7p1 , (1.2-14)

the differential équation defined in the closed bail U? x P 9 by

g(O) = Q anrf £(0) = 0 (1.2-16)

/iay a unique solution which satisfies

h(uÇK)9p(\)) = ^{b,g) , O ^ X ^ 1 . D (1.2-17)

Consequently, one has

^ Ci max {\n + 1 - \rt} , (1.2-18)

and, as a by-product,

m a x {^" + 1 - ^ " } . (1-2-19)^n^N-1

m a x {^ + 1 - n > (1-2-20)

where C j and C2 are constants which depend only on p, Lp and

2. APPROXIMATION OF A PURE TRACTION BOUNDARY-VALUE PROBLEM FORINCOMPRESSIBLE MATERIALS

2.1. Preliminary

To begin with, let us recall some définitions that we shall use in thesequel. The mapping,

fc:(£,T)eL- f 6 ® J C + f T0JCGM 3 (2.1-1)Ja Jr

vol. 22, n° 2, 1988

322 R. NZENGWA

is called the astatic load. A load | = (§, T) is equilibrated relative to thenatural state if and only if

(2-1-2)

We define two supplementary subsets of the load space L,

Le= {t e L,k(£) e sym} , (2.1-3)

skew = {(e L, k(t) e skew} . (2.1-4)

For a given load f in L, skew (£) will dénote the component ofjP in the subspace skew. A load £ in Le is without axis of equilibrium if thefollowing equivalent conditions are satisfied :

det ( & ( £ ) - t r f c ( | ) / ) # O (2.1-5)

or if the mapping

w e skew -• k {{) w + wk (jP ) e skew (2.1-6)

is an isomorphism. We define two canonical projections

Ae : g = (L, skew (£)) e L -+ i9 E L€, (2.1-7)

A : i = (2e, skew ( f ) ) e L - > skew (£) e skew . (2.1-8)

A loading operator is a continuous mapping

i : <$> e r + 2 ' ? - £(*) = (fe(*), ï (*)) e 4 • (2-1-9)

A loading operator is said to be a dead foad if it is constant(£(<!>)= (ê> l))> otherwise it is called a live load. One should refer toChillingworth-Marsden-Wan [6] for details on the above définition andassumptions.

We can now describe the pure traction boundary-value problem. Weconsider the constitutive law of an incompressible material as defined inSection 1 by the relation (1.1-4) and we keep the hypotheses (1.1-13)-(1.1-16). The problem consists in finding a déformation $ and a pressure/» thatsatisfy

q , (2.1-10)

} - § in H , (2.1-11)

j , } v = T on T, (2.1-12)det V$ - 1 = 0 in H . (2.1-13)

Green's formula implies that the load (b, T) must be of total force zéro.



Similarly as in Section 1 these équations can be put in the form of anoperator

§2= (*,jP)e Wm + 2>qx Wm + 1>q-> (-div { « ( ¥ $ ) - jV} v , d e t y ^ - l ) e L x r + l l ?

5 (2.1-14)

which is also C°° and has locally bounded derivatives of all orders. Clearlyone has

§ 2 ( & 0 ) = (0 ,0) . (2.145)

In order to eliminate the indétermination due to rigid body motions weshall restrict this operator to the set of Çsym x Wm +1} q. In this way we define

§2 : (?,/>) e Çsym x Wm + 1-*-> h(<?,P) e L x Wm + 1-* . (2.1-16)

The équation

^ ) = (è,J»0) (2.1-17)

is equivalent to

-div {Ç

{Ce

e (v) -pi} =(v) —pi} v =

div v =

fe in ft ,T on T,g in ft .

(2.1-18)(2.1-19)(2.1-20)

In addition to the hypotheses (1.1-13)-(1.1-16) we assume• the complementarity condition of Agmon-Douglis-Nirenberg ([3], [4])

holds for the équations (2.1-18)-(2.1-20). • (2.1-21)

THEOREM 5 : Under the hypotheses (1.1-13)-1.1-16) and (2.1-21), thelinear operator 9 (id, 0 ) de fines an isomorphism between the spacesÇsym x Wm + hq and L€ x Wm + hq.

Proof: Clearly ^(id, 0) is a continuous one-to-one mapping between thespaces Çsym x Wm + x'q and LexWm +1;q. Let voeWm + hq dénote the rightinverse of g (see [14]) for the divergence operator. One has

divgo = 0 . (2.1-22)

We let

Si = Eo-2o(O). (2.1-23)

vol. 22, n° 2, 1988

324 R- NZENGWA

We still have

d i v 2 1 = g , but ?h(0) = 0 . (2.1-24)

The linear mapping

w : x s O, -+ (skew A) x e U3 , (2.1-25)

where the matrix

4 = Y2i(0), (2.1-26)

satisfies the condition

divw = 0 and w(0) = 0 . (2.1-27)

It follows that the vector field

22 = Ei - w (2.1-28)

is the right inverse of g in the space Csvm. The change of variable in theéquations (2.1-18)-(2.1-20) leads to an equivalent system

-d iy{C e (u)-ql} = b' in ft , (2.1-30){Ç e (u)~ql}v = T' o n T , (2.1-31)

diyu = 0 in ft, (2.1-32)

where

b' = b + diy {Ç e (»)} , (2.1-33)T' = T - Ç G (g)v . (2.1-34)

Evidently the load (è' , T ') belongs to Le. For m => 3/2 — 1 it is known([21]) that this system possesses a unique solution in H^*2 x Hm + 1. Thehypothesis (2.1-21) guarantees that the regularity results hold for m = 0 andq > 3 ; e/. Geymonat [10]. D

T H E O REM 6 : There exists a neighborhood U x P of (id, 0) i'w

Csym x ^ m + 1'9 whose image under the operator 02 is a C °° submanifold N in

L x Wm + 1'«.

Proof: First, we redefine the operator 02, using the projectionsAe and A of the load space L onto its supplementary subspacesLe and skew, by letting

?det V<p-1)G Le x skew x wm + hq . (2.1-35)



For convenience we let

*e(<g,p) = (Aeè2( î , Jp)?detVcp-l) (2.1-36)

and rewrite

&(?,P) = (h(<£>P)> A§2(<P,/>)) • (2.1-37)

Clearly 0g(id,0) is an isomorphism between Ç s y mxWm + 1 '9 andLe xWm + l'q and be defines a local diffeomorphism. For a given load2, we shall write

L = A e £ . (2.1-38)

The mapping

X:(<f,p)€Ç sy raxW" + 1 ' ^

1 ÇsymxWm + 1-« (2.1-39)

is of class C °° and also defines a local diffeomorphism in a neighborhood of(id, 0) in ÇsymxWm + hq that we dénote UxP. For ail (<£,p) inU x P, we have

i ( î ^ ) = §;(id )0)X($ Î Jp). (2.1-40)

We may then write

&(?,/>)= (LA&lêe}" 1 ^^)^ ) , (2.1-41)

where

L ) . (2.1-42)

We deduce immediately that the image N of the neighborhoodUxP under 62 is the graph of the function

G : (L, g)eLexWm + 1>«^Ah2{hey1 (fe, 0) e skew , (2.1-43)

which is also C°° and satisfies

G(0 ,0 )=(Q,0) and G' (Q, 0)(fe, öf) = 0for ail (fe, ) G Le x Wm + 1>? . (2.1-44)

D We have thus shown that Af is a submanifold of class C °°.

vol. 22, n° 2, 1988

326 R. NZENGWA

In the sequel, we choose the neighborhood U x P to be the closed baildefined by

UxP = 5(id, Po/2) x B(0, Po/2) . (2.1-45)

Let a load ( | , 0 ) be in N, then we infer from theorem 5 that a solution forthe pure traction boundary-value problem (2.1-10)-(2.1-13) is

(?,/>)= {h}~\l,0). (2.1-46)

Arguing as in Section 1, we find that there exists a constant Co such thatfor each load ( | , g) e N

|| || C Q - ' P O , (2.1-47)

where

C o = sup IH&CîP^)}"1!!- (2-1-48)

Thus the projection of TV onto LexWm + 1'q is contained in the bail

2.2. Approximation of the pure traction boundary-value problem with deadload

We first consider the case of a dead load that belongs to the submanifoldN (the results obtained in this case will be used later). Let there be given aload ( | 5 0) in N ; then the solution of the problem is such that

1(<?,P)= (L,0) . (2.2-1)

We can now proceed as in Section 1. We consider a regular partition ofthe interval [0, 1],

0 = k° < X1 < . . . < \N = 1 (2.2-2)

such that

A\n = kn + 1 - \ n = j-9 O^n^N -1 , (2.2-3)

and we let A approach infinity. The incrémental method is defined as inSection 1 : we let

(p° = id and />° = 0 , (2.2-4)



and, assuming <p" and pn are known, we solve the linear problem

(2.2-5)

(2.2-6)

where

§é" = \eWl^Pn) • (2.2-7)

Then we define the (n + 1 )-st approximate solution

(p = <p -+- oip y p = p -h op , (2.2-ö)

and in particular we obtain the N-th approximate solution (<pN,pN).

The convergence of this method also follows from classical result aboutordinary differential équation. It suffices to note that the solution of theéquation

9* (?,ƒ>) = ( L O ) (2.2-9)

is the value at the point 1 of the solution of the differential équation

1 (|e, 0) , (2.2-10)dt v~

£(0) = id and q (0) = 0 . (2.2-11)

The vector field defined in the bail ï?(id, p0/2) x B(0, p0/2) by

2i(£,/>> l) = {&($,P)}~\L, 0) (2.2-12)

for f in N, with

-ó"1Po)> (2.2-13)

is of class C00 with respect to ail its arguments. Moreover it is locallybounded because of the local boundedness of the different operators it iscomposed of. The equilibrated part of the load, le (or f, evidently) appearsas a parameter in the vector field (2.2-12) and the solution of the équations(2.2-12)-(2.2-13) dépends continuously on this parameter. More precisely,there exists a constant C depending on the vector field vx such that for givenparameters fle and £2e> tf (<PIJ.PI)

an<3 {WiiPi) a r e t n e solutions of theéquations (2.2-10) and (2.2-11), we have

IKSi - 92^1 -Pi)\\ S C(2 l)| |2ie " Czell • (2.2-14)

vol. 22, n° 2, 1988

328 R. NZENGWA

We shall now use these results to treat the case of an equilibrated load| , (f e Le). We begin by recalling essential results due to Chillingworth-Marsden-Wan [6, Sections 4, 5], which will be necessary for constructing theincrémental method. We adapt them to the case of incompressible materialsin the following form :

THEOREM 7 : Let £0 be a load in Le without axis of equilibrium. Thenthere exist a real number T) > 0, a neighborhood O0 of ]P0 in Le and aneighborhood VI of I in O3, such that:

• each load f e O0 is without axis of equilibrium, (2.2-15)• for each pair ( | , X) e O0 x [0, t\], there exists a unique Q e Vl such

that

\(Ql,0)eN , (2.2-16)

• Q is differentiable with respect to £ and Q is the unique solution of theimplicit équation (2.2-17)

QeO3, (2.2-18)

, Ö ) = 0 (2-2-19)

where H is the mapping defined by

H: {K L Q) e R x Le x O3-> AQl - ~ G(\Ae Qg, 0) 6 skew . (2.2-20)A.

Proof: It follows from the décomposition of L onto Le ® skew that

= skew ( ö O - (2.2-21)

The results then follow from [6, theorem 5.1]. DFrom the above theorem, we conclude that if a load l in the neighborhood

OQ is sufficiently small, we can define in O3 a curve Q(tl) that is defined fort = 1. This is equivalent to saying that we can choose the pair (O0, [0, 1])such that the condition (2.2-16) is satisfied for each element. We cantherefore define a mapping

O3 (2.2-22)

such that

Q(t)leN . (2.2-23)

It is clear that the neighborhood O0 is contained in the closed bail



B(Q, CQ1 p0) (2.1-48). For each load £ in O0 there exists a unique solution

(<P,p) e î?(id, Po/2) x B(Q, p0/2) of the équation

U<2>P)= ( g (DLO) . (2.2-24)

This solution dépends continuously on the load f. Letting

?, (2.2-25)

we deduce from the principe of material frame-indifference and from therelation

det V$ = det V<p , (2.2-26)

that

§2(*,P)= (LO) . (2.2-27)

We have thus shown that the pair (<&,p) is a solution of the pure tractionboundary-value problem. We obtain the same results for the gênerai case ofa load (£, g), with g e Wm + 1'q not necessary zero.

Let £ be in Le. The orbit of £, viz.,

*i= {QlQeO3}, (2.2-28)

which we identify with the set

{{Qt,0),QeO3}, (2.2-29)

intersects the manifold N for at least four rotations [6], but if we restrictourself to the neighborhood Vj, the set of such rotations reduces to thesingle rotation that we have denoted Ö(£). In the construction of anincrémental algorithm, we must successively approximate Q(l), then thesolution (<p,p) and finally $. We approximate Q(i) by an incrémentalmethod. For this purpose we here recall the following well known results(Lang, S [12]).

The exponential mapping

exp : w e skew -> exp (w) e O3 (2.2-30)

defines a diffeomorphism in a neighborhood Vo of the matrix 0. It is no lossof generality to assume that the neighborhood Vt defined in (2.2-16)satisfies the condition

V, = exp(V0). (2.2-31)

vol. 22, n" 2, 1988

330 R. NZENGWA

Then the implicit équations (2.2-18)-(2.2-20) are equivalent to theéquations w € skew,

H{K w, | ) = 0 ? O i X ^ l , (2.2-33)

where the function H is defined by

H(K wj) = H(K exp w, {) . (2.2-34)

Clearly we have

n Jjr n TT

^ ( 0 , 0 , C ) W = |g(0, / ,C)w. (2.2-35)

We deduce from Theorem 5.1 of [6] that the linear mapping (2.2-35) is anisomorphism in the space skew for each load £ in O0.

We can choose the neighborhood Vo (equivalently Vt) and O0 in such away that :

( > w> i) is an isomorphism for (X, w, f) € [0, 1] x VQ x Oö .ow "" ""

(2.2-36)

Then for a fixed load 2 in O0, the solution of the équation (2.2-32)-(2.2-34) is a curve which is the solution of the differential équation

l d i fr."(M>.O, 0 S X S 1 , (2.2-37)

together with the initial condition

w(0) = 0 . (2.2-38)

The vector field v2 of this équation is differentiable with respect to itsarguments and its derivatives are also locally bounded. The load f thenappears to be a parameter in this vector field, consequently the mapping

ö : ! e O0 -+ exp w (1, f) e O3 (2.2-39)

is Lipschitzian. Therefore there exists a constant C depending on the vectorfield v2 such that if | i and | 2 belong to 0O, we have

|| exp H>(1, &) - exp W ( l , I J ) || â C (22)112! — €2 M - (2-2-40)

The f uil expression of the vector field v2 is quite involved. We shall



instead consider an approximation obtained from a truncated Taylorexpansion of the function H. More precisely we write (see [12])

H(\, w(l, t), t) = A exp wt - ~ G"(0, 0)(Ae exp wt, 0)2 + \ 2 T(w, t) ,

(2.2-41)

since

G(Q, 0) = 0 and G'(0,0) = 0 . (2.2-42)

We use the fact that G and ail its derivatives are locally bounded toconclude that there exists a constant C such that

|| TXw, f ) H S C H ! | | 3 . (2.2-43)

We let

H2(K w, t) = A exp wt - | G"(0, 0)(Ac exp w£, 0)2 . (2.2-44)

We still have

^(0,0,0-^(0,0,0. (2.2-45)

So, we can again consider the neighborhoods Vo and O0 such thatH and H2 satisfy the condition (2.2-36). We therefore approximate the curvew(\, l) by a curve w2(K, i) which is the solution of the équations

| J i r < x - - « > ' OâXS1 (2-2"46)

andw2(0) = 0 . (2.2-47)

Indeed, the operator H2 (unlike H) can be computed, since its expressionis deduced from linearized incompressible elasticity. Moreover we have thefollowing estimate :

LEMME 1 : Let w and w2 be the exact solutions o f the équations (2.2-37)-(2.2-38), and (2.2-46)-(2.2-47), respectively. Then there exists a constant Cindependent of V such that :

Sup || W(X, £) - w2{\9 t)\\ ^ C \\i\\3 . (2.2-48)

\ e [0,1]

vol. 22, n° 2, 1988

332 R. NZENGWA

Proof : We write

H{K w, f) = H2(K w, i) + X2 T(w, i). (2.2-49)

Using équations (2.2-37)-(2.2-38) and (2.2-46)-(2.2-47), the mean valuetheorem and the condition (2.2-36), we obtain

W2(K O - iv(x, f)|| x2 J (x, w, o j * r(w , f), (2.2-50)

where w belongs to the interval [wXi w]. The conclusion follows from the

local boundedness of the derivatives of H2. •Notice the effect on the estimâtes of the remaining term T in the Taylor's

expansion of H.We now establish some estimâtes we shall use in the sequel.

LEMME 2 : Let (<p,/>) be the solution of the équation

(l, i) i, 0) (2.2-51)

and let (<p,p) dénote the exact value of the solution of the differential

équations (2.2-10)-(2.2-ll) at the point 1, with Fe as the parameter, where

Ze = Aeexp w2(l, 0 i • (2.2-52)

Then there exists a constant C independent of £ such that

| | (<P-f , />- /7) | |ëC| | ! | |4 . (2.2-53)

Proof: Recall that (<p,p) and (<p,p) are equal to (^(1), ^(1)) for the

parameters le and \e respectively, where (ty(t), q(t)) is the solution of the

differential équations (2.2-10)-(2.2-ll). The result then follows from theestimâtes (2.2-14) and (2.2-48). We have

| | y \\ || - L||> (2.2-54)

which yields

II(«P - f,p-p)\\ S C(gl)||{|| \\w(l, i) - wz(l, | ) | | . (2.2-55)

Then we conclude from lemma 2. •We can now describe the method. We first start by approximating

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numencal Analysis


w2 by Euler's method. Let w^ dénote the M-th approximation ofw 2 ( l , {)- We approximate the solution of the équations (2.2-10)-(2.2-ll), inwhich the parameter is computed from w^. We then define the method asfollows. Let M and N be two positive integers. We define two regularpartitions of the interval [O, 1],

0 = \x° < ja,1 < . . . < jmM = 1 , (2.2-56)0 = \ ° < X1 < . • • < XN = 1 . (2.2-57)

We let

w£ = 0 , (2.2-58)

then assuming that w™ is known, we solve the linear problem

(|xm, w™, V) hwm — — Ajxm (M*"1»

w™<> | ) » (2.2-59)

Ôwm G skew , (2.2-60)

and we define the (m + 1 )-st approximation

w>m + 1 = w m + ô>vm. (2.2-61)

We compute the approximate parameter from the M-th approximation of

Ff = Aeexpw?t. (2.2-62)

Applying the incrémental method (2.2-4)-(2.2-8), we compute the iV-thapproximation (<pN,pN). Finally we let

<p = (exp w2M)T 9* . (2.2-63)

We next establish the convergence of the method.

THEOREM 8 : Let £ be a dead load in O0. Let (<p,p) be the solution of

7)= (£,0) (2.2-64)

and let (<$>N,pN) be the approximate solution computed by the incrémentalmethod. Then there exists a constant C such that

| | ( ( ) ) . (2.2-65)

vol. 22, n° 2, 1988

334 R. NZENGWA

Proof : From the proof of convergence of Euler's method we know thatthere exists a constant C such that

I K 1 | § (2-2-66)

We may write

$ - $" = (exp w{l)T - exp w2(l)T + exp w2(l)T) -

- (exp (w2M)T) <p + (exw?f($ -f). (2.2-67)

Now we can also write

¥ ~ ? N = S - S + ÎP - ?N- (2.2-68)

Using the estimate (2.2-53) and the convergence resuit established inSection 1, we have

,v2M)> - f)\\ â C ( M -f «ru ) . (2.2-69)

From (2.2-48) and (2.2-67) we deduce the estimate

||(exp w(1)T - (exp w?)r) <p || S C ( M i + || T|| ) . (2.2-70)

We combine both inequalities to obtain the desired resuit. A similarestimate holds for \\p — pN ||. D

To sum sup, our approach is based on an essential resuit on compressiblematerials due to Chillingworth-Marsden-Wan [6] : given a sufficient« small » dead load £0 in Le without axis of equilibrium, there exists aneighborhood O0 of £0 in Le, in which each load is without axis ofequilibrium and has an orbit which intersects TV. We have adapted this resultto incompressible materials and finally we have constructed an incrémentalmethod for each load £ in O0. We shall also use this result to treat live loadsin the next paragraph.

2.3. Approximation of the pure traction boundary-value problem with liveloads

In this paragraph we keep all the hypotheses made on the nonlinearoperator 62. Let us consider a loading operator £ as define by (2.1-7). Theproblem consists in solving the équation

h(&P)= (£(*), 0) (2.3-1)



where

?(*)=(*W,î(*))H- (2.3-2)

In order to guarantee the existence of a solution for this problem, we firstassume that

h0 = ?_(&) e Le (2.3-3)

satisfies the condition

det k(h0) >0 . (2.3-4)

Then we establish a lemma which will be usefull in the sequel.

LEMMA 3 : Let h be a dead load in Le which satisfies the inequality (2.3-4).Then RT h is without axis ofequilibrium, where R is the polar décompositionrotation of k(h).

Proof: Clearly RT k(h) belongs to the set of symmetrie, positive definitematrices 5>. It suffices to consider a diagonalizing orthonormal matrix P ofRTk(h). We have

2,a3), ax 9 a2 , ax > 0 . (2.3-5)

Consequently we have

det (cr RTk(h) I - RTk(h)) = det ((04 + a2 + a3) ƒ - L>) 0 (2.3-6)

which is an equivalent définition of a load without axis of equilibrium. DWe easily conclude that there exists a neighborhood U of h0 in

L in which each load satisfies the inequality (2.3-4). Consequently for eachload h in £/, we can define the polar décomposition R(h) of the matrixk(h), This leads to the définition of the mapping

Ç:/je U^R(h)T h e Le (2.3-7)

since the orthogonal matrix R(h) satisfies the équation

{h)T k(h)]m = k(h) . (2.3-8)

From lemma 3 we infer that the loads Ç (h ) is without axis of equilibrium

for each load h in the neighborhood U. If we choose the load h0 to be in the

closed bail B(0, CQ1 pm) (2.2-13), then we can choose a closed neighbor-

hood Ue of h0 contained in B(0, CôVo) s u c n t n a t

1 ^ ) . (2.3-9)

vol. 22, n° 2, 1988

336 R- NZENGWA

We deduce from Theorem 7 that for each load h in U there exists arotation Q(h) such that

Q ( h ) R ( h ) T heN . (2.3-10)

Consequently there exists a pair (<p,p) such that

h,0). (2.3-11)

Let4>=R(h)Q(h)Tv, (2.3-12)

then, because of the material frame indifférence, we have

U&P)= (6 ,0 ) . (2.3-13)

We have thus shown that there exists a local unique solution for the puretraction problem for each dead load in the neighborhood U. In the previousparagraph we established such an existence resuit and the possibility ofapproximating the solution only for équilibrâted loads. We have establisheda stronger resuit, which is essential because in the case of the live loadproblem we shall show that the solution is nothing but that corresponding toa particular dead load in U. This will be possible if at least locally the imageof the live loading operator £ is contained in the neighborhood U. Wetherefore need an additional assumption on the loading operator £.

We begin by considering the set of déformations

{ > * = * ( * ) QihY f,heU,<pe £( id, P o /2 )} , (2.3-14)

where (<$>,p) is the solution for the load Q{h) R(h) hT as shown inparagraph (2.1). Using relation (2.1-45) and the fact that the mappings Qand R are all locally bounded, we clearly see that the set of déformations(2.3-14) is bounded. We consider a closed bail B(id, 8) which contains thisset and we assume that the loading operator restricted to this bail isLipschitzian with a Lipschitz constant d such that

<fô<r0, ro = C0-1 Po/2. (2.3-15)

Consequently we have the following inclusions :

£(£(id, 8)) c U^l-\Ue) . (2.3-16)

Using this inclusion, we define a mapping

XiheU^ f ( $ ) e U, (2.3-17)



where

$ = R(h)Q(h)T<S and &($,/>) = {h, 0) . (2.3-18)

We easily show that this mapping is also Lipschitzian.More precisely, for given loads hx and h2 in U, we have

with

7 = C(C(/?) + C ( Ö ) C « ) + C a ) C ( g 1 ) ) . (2.3-20)

We observe that the constants C (R) and C(£) can be computedindependently of the nature of the material, while the constants C ( g ) andC(vx) depend only on the constitutive law. Hence the constant 7 can becomputed if the constitutive law is known. Let

p = max (7, ô/r0) , (2.3-21)

and assume finally that the constant d is such that

dp < 1 . (2.3-22)

We can now establish an existence resuit.

THEOREM 9 : Under the additional condition (2.3-22) on the loadingoperator £, the live load pure traction boundary-value problem possesses alocal unique solution.

Proof : Clearly the condition (2.3-22) implies that the mapping x admits aunique fixed point. If h dénotes the fixed point then we have

&(*,/>) = ik, 0) = (i(*), 0) . • (2.3-23)

We have thus shown that the solution of the live load problem is nothingbut that of the dead load problem in which the fixed point h is the load. Wealso note that the polar décomposition is used in a crucial way in theconstruction of the mapping x- This was possible because we assumedcondition (2.3-4).

It is now easy to describe an approximation scheme of the solution basedon the dead load case. Let us consider the séquence of dead loadsrecursively defined in U as follows :

ho,hi + 1 = X(hj). (2.3-24)

vol. 22, n° 2, 1988

338 R NZENGWA

This séquence converges to the fixed point h and consequently thecorresponding séquence of solutions

(*,,/>,) with &(*,,/>,)= (d, ,0) , (2.3-25)

converges to the exact solution of (2.3-23). The incrémental method willconsist in approximating the séquence (2.3-25) of traction problems withdead load, and in using the methods described in paragraph 2.2. We shallconsider a finite number of terms hp j = 0, 1, . . . , / and estimate thedistance between the exact solution of the problem and the approximatesolution corresponding to hj.

The method is described as follows. We let

then assuming h} is known, we set

f = R{h]f W . (2.3-27)

The load f is in Le. We therefore use results of paragraph 2.2 to

approximate the corresponding solution, Le. we apply the method (2.2-57)-(2.2-64) to approximate the problem

§2 (*,,/>,)= ( f , 0 ) . (2.3-28)

We let

£ Y , (2.3-29)

where (fyN,pN) is the approximate solution of (2.3-28). We compute the

(/ + 1 )-st approximate load

hJ + 1 =l(V)- (2.3-30)

In this fashion we obtain the J-th approximate déformation and pressure(W'P*)- Upper indices are used for denoting approximate terms.

We now establish convergence results. We let

[T] = Sup || r | | (2.3-31)

dénote the bound of the remainder in the Taylor's expansion of the functionH, when restricted to the closed bail U.



THEOREM 10 : Let (<&,p) be the local exact solution obtained in theorem

9, and let (§ J , pJ) be the approximate solution. Then there exists a constant C

such that

{ l l } ^ . (2.3-32)

Proof; Consider again the séquence (hj) j ^ 0 defined in (2.3-24). Leth be the fixed point of the contracting mapping x- Clearly we have

||(Ô,-Ô)||S ^ï-Wh-hoW^Cidy)'. (2.3-33)

Let (<&,p) and (,<£>,,Pj) be such that

h($,P) = (Ô»0) and Wk,,P,)=(hp0). (2.3-34)

By construction we have

*; = K(Ô,)ô(!,)TiP, and 4> = R(h)Qa)T<£, (2.3-35)

where

{, = R{hj)T h, and £ = R(h)T (h,) . (2.3-36)

Hence we deduce from (2.3-19) and inequality (2.2-14) that

; - 4>,P, -P)\\ ^y\\h, -h\\ SC(dyy . (2.3-37)

We want to estimate \\(4>J — §,pJ — P)\\- We may write

* \\(&-*J,P'-PJ)

(2.3-38)

It therefore suffices to estimate the first term in the right hand side. Weshow by induction that for each integer j we have

l + l + [T]} . (2.3-39)

For ; = 0 , the resuit holds (c/. § 2.2). Suppose that it is true for somejmO. Consider ty + 1 , q} + x) and ($, + 1,pJ + i) satisfying

U% + u q J + i ) = ( 6 ; + 1 > 0 ) . (2.3-40)

vol 22, n° 2, 1988

340 R- NZENGWA

Arguing as in (2.3-19) and (2.3-37) we also obtain

||(*; + l - * ; + l ,«, + l - P ; + l ) | |S 'Y| |6 ' + 1 - 6 / + 1 | | . (2.3-41)

But

/ ? J + 1 = £ ( ^ ) and Ô/ + 1 = £ (* , ) , (2-3-42)

hence we have

\\h> + 1-h, + 1\\Sd\\&-4>,\\, (2.3-43)

and by induction the foUowing inequality holds

(2.3-44)

since dy <: 1. Finally we write

I ! (x\t K n n ^ 11 < 11 CÂJ + 1 — «tf rtf + ! ^ ^ I !

(2.3-45)

We then apply the estimate established in the incrémental method fordead load to the first term in the right hand side. This yields

. (2.3-46)

Hence we conclude that the result also holds at the (ƒ + 1 )-st step and theproof is complete. D

We infer from the above theorem that for a given s > 0, we may choose Jsuch that

C(dy)J^B/3, (2.3-47)

then we choose M and N such that

{ ^ M } (2-3"48)

and we obtain

\ \ ® ? \ \ ^ (2.3-49)

hence the smaller is [T], the better is the approximation.



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solutions of elliptic partial differential équations sàtisfying gênerai boundaryconditions I, Comm. Pure Appl. Math XII (1959), 623-727.

[4] S. AGMON, A. DOUGLIS and L. NIRENBERG, Estimâtes near the boundary forsolutions of elliptic partial differential équations sàtisfying gênerai boundaryconditions II, Comm. Pure Appl. Math XVII (1964), 35-92.

[5] M. BERNADOU, P. G. CIARLET and J. Hu, On the convergence o f the semi-discrète incrémental method in nonlinear, three-dimensional, elasticity, Journalof Elasticity 14 (1984), 425-440.

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M2AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numencal Analysis

ROBERT NZENGWA Incrementalmethodsinnonlinear,three- … · divergence of a tensor field T: O, cz...

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