Incremental methods in nonlinear, three-dimensional, incompressible elasticity

M2AN. M

- M

R ^OBERT N ^ZENGWA

Incremental methods in nonlinear, three- dimensional, incompressible elasticity

M2AN. Mathematical modelling and numerical analysis - Modéli- sation mathématique et analyse numérique, tome 22, n^o2 (1988), p. 311-342

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Article numérisé dans le cadre du programme

MATHEMATICA!. MODELUNG AND NUMERICAL ANALYSIS MOOEUSATION 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 equihbrium équations of nonlinearly elastic incompressible bodies, subject to dead or live loads, with pure displacement or traction boundary conditions We establish the convergence of the methods

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

INTRODUCTION

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

pressure field, displacement field, déformation field,

unit outer normal vector to the boundary of a domain,

Euclidean inner product, Euclidean vector norm,

matrix with element A^l} (i : row index, j : column index),

(*) 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/$ 5.20 Mathematical Modelhng and Numencal Analysis © AFCET Gauthier-Villars p

li —

V =

u.v

4 = ïjf

(^Mi) (*,)

M

= u,

= («

(AtJ)

v, :

312

A:B=AtJ Bl}

adj A Ml3

skew sym

si o

T

b

T

R. NZENGWA

matrix inner product,

matrix norm associated with the matrix inner product,

adjugate of the matrix A = transpose of the cofactor 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 equals to one,

first Piola-Kirchhoff stress, second Piola-Kirchhoff stress, volume force operator surface force operator, loading operator, a dead load,

for some integer m ^ 0 and some real number

-f? sym

Mm>q Si sym

= w

-

,

= {(6,i)e ^"-

= o, Y<t(o) = Y<J?(O)

}.

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 variable

4,

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 operator

e

and y being two normed vector spaces.

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 313

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

Let fl be a bounded open connected subset of R3 with a sufficiently smooth boundary F. We assume that Ü is the référence configuration occupied by an elastic incompressible body in the absence of any applied force. The constitutive law of the material is given by

î = «(¥$)-/> adj y$

, (o.i)

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

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

d i v f + 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 i n 11}

and two nonlinear elasticity operators (^ and §2 corresponding respectively to the pure displacement and pure fraction problems. When we consider the displacement u and the pressure p as the unknowns, the first operator

§! maps %qm n Wj1* x Wm'+hq into \Vm'q for each integer m ^ 0 and real number q > 3. With the same conditions on (m, q) the second operator e2 maps S^ x Wm + l'q into a submanifold of the set of loads L. The pure displacement and traction problems respectively reduce to solving the équations

§i(B,/>) = 6 (0-6) and

h(é,p)= ( 6 , i ) . (0-7)

314 R. NZENGWA

The incrémental methods consist in approximating the above nonlinear équations by a séquence of linear problems. Unfortunately these operators are not defined between affine manifolds and moreover they do not always possess the inverse function theorem property, a crucial condition on which dépends the applicability and the convergence of the method as proved by Bernadou-Ciarlet-Hu [5]. We therefore have to modify the incrémental methods according to the problem we want to solve.

In Section 1 we study the pure displacement problem. The corresponding nonlinear operator ex defines a local diffeomorphism between the spaces

%qm n Woq x Wm + 1~q>° and VT** but since the space %qm is not an affine manifold, we define a second operator \x between ym + 2'q x wm + l>q>° and W71^ x wm + 1'q>° which are both linear spaces. This operator satisfies the crucial inverse function theorem property. We then describe the relation between both operators Gj and èa and we apply the incrémental methods to the second operator, in order to approximate the local solution for the pure displacement problem in the case of a dead load.

In Section 2 we consider the pure traction problem. The associated operator 02 is defined on a non affine manifold. As in Section 1 we define a second 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 open neighborhood of (id, 0) in the linear space Çsym x Wm + lyq and a non affine Sobolev submanifold A^ofLx Wm + 1'q. From the décomposition of the load space as L = Le © skew, we show that the submanifold TV is the graph of a function G defined from Le x Wm + 1'q into Skew. Let £ be a dead load in Le with no axis of equiiibrium ; then iocaîiy there exists a unique rotation Q such that the element (Ql,0) e N. Consequently we obtain the local unique solution corresponding to the load £. The solution corresponding to the dead load Ql is obtained from the solution of an appropriate differential équation between the above considered spaces, with Q£ as a parameter in the vector field. The approximation comprises two steps : we first approach the parameter Q | , then we apply Euler's method to the differential équation in which the approximate value of g£ is a parameter.

Finally we consider a class of live loads. Under some additional assumptions on the loading operator f we first prove the existence and local uniqueness of a solution via the fixed point theorem, using in a crucial way the polar décomposition of invertible matrices. The approximation of the solution consists in approaching a finite séquence of solutions for traction problems of the dead load type.

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 315

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

1.1. Existence

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

m il , U--1-1J det 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-valued C00 mapping a : ft x M3 -> M3, the problem then consists in solving the following boundary-value problem for the displacement field u — (u^t) and the pressure p :

- d j v {«(/ + 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 the formulas. 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) we define the set of admissible displacements

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

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 the équations

(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 properties of 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 mappings u 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 clear from (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 %qm x Wm + 1'q>° and W71^, where

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

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 317 is the tangent space to %qm at the origüi O and

eW

-*

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 the hypotheses (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 the spaces T^QX^qm x w^{m + 1}'^q>° and W^m>^q. We keep these various hypotheses throughout ; 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 neighborhood

U x P of (0,0) in %^qm x W^{m + 1}><i>^ö such that, for each load b in B, the pure displacement boundary-value problem (l.l-5)-(l.l-7) has exactly one solution in UxP. D Such a solution therefore satisfies the équation (1,1-17) ; however the incrémental method cannot be directly described. Indeed in the description of the method [5], we compute the approximate solution (u^{n + 1},p^{n + 1}) at step 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 the pressure respectively. Since %^qm is not a linear space u^{n + a} = uⁿ + Suⁿ does not

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 V^{m + 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 w^{m + hq}>° . (1.1-31) We may write

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

^ w T ) / ; . (1.1-32) We find that the intégral of the above expression vanishes using Green's formula, the Piola identity and the boundary condition. Clearly this operator is also Cœ and locally bounded with all its derivatives. It is easy to conclude that

§ J ( Q > O ) ( M , / O = (b,g) (1.1-33) defines an isomorpbism between r + M y ^ + M.o

I f , «x^ « + U,oi Indeed the équation (1.1-33) is equivalent to the équations (l.l-22)-(l. 1-24) for the particular case in which h = 0. The operator §î(Q, 0) is also continuous and one-to-one. lts image is the subset

\(b,g,h)e Wm^q x Wm + 1'? ) Ox wm + 2-i/q,q ^ f g _ f § . v = o l ,

l *~ J a JT ~ J

(1.1-34) which contains W™** x Wm + 1'9'0 x {0} , ([14]). We easily conclude using the 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), there exist a neighborhood B x G of ( 0 , 0 ) in W™^ x w^{m + hq}>° and a neighbor- hood UxP of ( 0 , 0 ) in ym + 2>q x wm + 1>q>° such that \x defines a diffeomorphism between the two neighborhoods. D

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 319 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 to approximate the solution of the équation

§i(«,/>) = ( 6 . 0 ) (1.1-35) which is exactly the solution for the pure displacement boundary-value problem (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. We shall 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 not consider the case of a gradient vector field b e Cl(Cl, U3) for which a trivial solution 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 the neighborhood 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 < . . . < k^N = 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 un, b pn) e ym + 2>q x Wm + 1>q>°. (1.2-6)

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 and pressure 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 relevant results proved so far :

We showed that the operator hx is C °° and has locally bounded derivatives of 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 a neighborhood UxP of ( 0 , 0 ) i n r + 2 l ? x r + 1'?' ° a n d a neighborhood B 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 both Banach spaces. We now describe these neighborhoods in the following theorem.

THEOREM 3 (Bernadou-Ciariet-Hu 1983 theorem 2) : For each pair of numbers Cm, q ) satisfying (1.1-9), there exists a real number p0 > 0 depending on 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 U^px P^p , (1.2-10) where

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

7P= s u p | | { 8 i ( « >JP ) } "1| | < + o o , (1.2-12)

llt^ifei^i)}"

- {i

(u

uT£)eüxP Ij

_

+ oo . D (1.2-13)

INCREMENTAL METHODS IN NONLÏNEAR ELASTICITY 321 We deduce from this theorem and a classical resuit in differential équations (e.g. Crouzeix-Mignot [9]) :

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

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 FOR INCOMPRESSIBLE MATERIALS

2.1. Preliminary

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

fc:(£,T)eL- f 6 ® J C + f T 0 J C G M³ (2.1-1) Ja Jr

322 R. NZENGWA

is called the astatic load. A load | = (§, T) is equilibrated relative to the natural 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 of jP in the subspace skew. A load £ in Le is without axis of equilibrium if the following 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

'

- £(*) = (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 to Chillingworth-Marsden-Wan [6] for details on the above définition and assumptions.

We can now describe the pure traction boundary-value problem. We consider the constitutive law of an incompressible material as defined in Section 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/» that satisfy

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.

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 323

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

§2= (*,jP)e Wm + 2>qx Wm + 1>q-> (-div { « ( ¥ $ ) - j

V} 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. Clearly one has

§2( & 0 ) = ( 0 , 0 ) . (2.145) In order to eliminate the indétermination due to rigid body motions we shall 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

- d i v {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), the linear operator 9^ (id, 0 ) de fines an isomorphism between the spaces Çsym x Wm + hq and L€ x Wm + hq.

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

divg^o = 0 . (2.1-22)

We let

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

324 R- NZENGWA

We still have

d i v 21 =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 the équations (2.1-18)-(2.1-20) leads to an equivalent system

- d i y { 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. The hypothesis (2.1-21) guarantees that the regularity results hold for m = 0 and q > 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 projections Ae and A of the load space L onto its supplementary subspaces Le and skew, by letting

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

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 325 For convenience we let

*e(<g,p) = ( Aeè2(î,Jp )?d e t V c p - l ) (2.1-36) and rewrite

&(?,P) = (h(<£>P)> A§2(<P,/>)) • (2.1-37) Clearly 0g(id,0) is an isomorphism between Çs y mx Wm + 1'9 and Le xWm + l'q and be defines a local diffeomorphism. For a given load 2, we shall write

L = Ae£ . (2.1-38)

The mapping

X: ( < f , p ) € Çs y r ax W " + 1 ' ^

1 Ç^symxW^{m + 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) in U x P, we have

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

&(?,/>)= ( L A & l ê e } "

^ ^ ) ^ ) , (2.1-41)

where

L ) . (2.1-42) We deduce immediately that the image N of the neighborhood UxP under 62 is the graph of the function

G : (L, g)eL

xW

>«^Ah

{h

y

(f

, 0) e skew , (2.1-43)

G ( 0 , 0 ) = ( Q , 0 ) and G' (Q, 0)(fe, öf) = 0

for ail (fe, ^ ) G Le x Wm + 1>? . (2.1-44) D We have thus shown that Af is a submanifold of class C °°.

326 R. NZENGWA

In the sequel, we choose the neighborhood U x P to be the closed bail defined 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 for the 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 that for each load ( | , g) e N

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

C

= sup IH&CîP^)}"

!!- (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 dead load

We first consider the case of a dead load that belongs to the submanifold N (the results obtained in this case will be used later). Let there be given a load ( |50 ) 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 of the 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 in Section 1 : we let

(p° = id and />° = 0 , (2.2-4)

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 327 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 (<p^N,p^N).

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

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 locally bounded because of the local boundedness of the different operators it is composed of. The equilibrated part of the load, le (or f, evidently) appears as 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 v^x such that for given parameters f^le and £²e> tf (<PIJ.PI) an<3 {WiiPi) a r e t n e solutions of the équations (2.2-10) and (2.2-11), we have

IKSi - 9 2 ^ 1 -Pi)\\ S C(^{2 l})||2ie " Czell • (2.2-14)

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 the incrémental method. We adapt them to the case of incompressible materials in the following form :

THEOREM 7 : Let £⁰ be a load in L^e without axis of equilibrium. Then there exist a real number T) > 0, a neighborhood O⁰ of ]P⁰ in L^e and a neighborhood 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 V^l such that

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

• Q is differentiable with respect to £ and Q is the unique solution of the implicit 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]. D From the above theorem, we conclude that if a load l in the neighborhood O^Q is sufficiently small, we can define in O³ a curve Q(tl) that is defined for t = 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 can therefore 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

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 329 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 ( D L O ) . (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 the relation

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

§ 2 ( * , P ) = ( L O ) . (2.2-27) We have thus shown that the pair (<&,p) is a solution of the pure traction boundary-value problem. We obtain the same results for the gênerai case of a load (£, g), with g e W^{m + 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 restrict ourself to the neighborhood Vj, the set of such rotations reduces to the single rotation that we have denoted Ö(£). In the construction of an incrémental algorithm, we must successively approximate Q(l), then the solution (<p,p) and finally $. We approximate Q(i) by an incrémental method. 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 V^o of the matrix 0. It is no loss of generality to assume that the neighborhood Vt defined in (2.2-16) satisfies the condition

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

330 R. NZENGWA

Then the implicit équations (2.2-18)-(2.2-20) are equivalent to the équations 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 )

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

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

We can choose the neighborhood Vo (equivalently Vt) and O0 in such a way 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 its arguments and its derivatives are also locally bounded. The load f then appears 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 vector field v² such that if | i and |² belong to 0^O, 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

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 3 3 1 instead consider an approximation obtained from a truncated Taylor expansion 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 to conclude 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)(A^c 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 V^o and O⁰ such that H and H² satisfy the condition (2.2-36). We therefore approximate the curve w(\, l) by a curve w2(K, i) which is the solution of the équations

| J i r < ^x - - « > ' ^{OâXS1 (2} - ² " ⁴⁶⁾

and

w²(0) = 0 . (2.2-47)

Indeed, the operator H² (unlike H) can be computed, since its expression is deduced from linearized incompressible elasticity. Moreover we have the following 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 C independent 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 value theorem and the condition (2.2-36), we obtain

W

(K O - iv(x, f)|| ^ x

J ^ (x, w, o j * r(

, f)

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 ) | | ë C | | ! | |4. (2.2-53)

Proof: Recall that (<p,p) and (<p,p) are equal to (^(1), ^(1)) for the parameters l^e 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 the estimâtes (2.2-14) and (2.2-48). We have

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

which yields

II(«P - f,p-p)\\ S C(

)||{|| \\w(l, i) - w

(l, | ) | | . (2.2-55)

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

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 333 w² by Euler's method. Let w^ dénote the M-th approximation of w2( l , {)- We approximate the solution of the équations (2.2-10)-(2.2-ll), in which the parameter is computed from w^. We then define the method as follows. Let M and N be two positive integers. We define two regular partitions 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 = wm+ ô>vm. (2.2-61) We compute the approximate parameter from the M-th approximation of

Ff = A

expw?t. (2.2-62)

<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 incrémental method. Then there exists a constant C such that

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

334 R. NZENGWA

Proof : From the proof of convergence of Euler's method we know that there 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 in Section 1, we have

,v

)> - f)\\ â C ( M -f «ru ) . (2.2-69)

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

||(exp w(1)

- (exp w?)

) <p || S C ( M i + || T|| ) . (2.2-70)

estimate holds for \\p — pN ||. D

To sum sup, our approach is based on an essential resuit on compressible materials due to Chillingworth-Marsden-Wan [6] : given a sufficient

« small » dead load £0 in Le without axis of equilibrium, there exists a neighborhood O0 of £0 in Le, in which each load is without axis of equilibrium and has an orbit which intersects TV. We have adapted this result to incompressible materials and finally we have constructed an incrémental method for each load £ in O0. We shall also use this result to treat live loads in the next paragraph.

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

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

h(&P)= (£(*), 0) (2.3-1)

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 3 3 5 where

? ( * ) = ( * W , î ( * ) ) H - (2.3-2) In order to guarantee the existence of a solution for this problem, we first assume 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 R^T h is without axis ofequilibrium, where R is the polar décomposition rotation of k(h).

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

2,a³), a^{x 9} a² , a^x > 0 . (2.3-5) Consequently we have

det (cr R^Tk(h) I - R^Tk(h)) = det ((04 + a² + a³) ƒ - L>) ^ 0 (2.3-6) which is an equivalent définition of a load without axis of equilibrium. D We easily conclude that there exists a neighborhood U of h0 in L in which each load satisfies the inequality (2.3-4). Consequently for each load h in £/, we can define the polar décomposition R(h) of the matrix k(h), This leads to the définition of the mapping

Ç:/je U^R(h)^T h e L^e (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 U^e of h⁰ contained in B(0, CôVo) s u c n t n a t

1 ^ ) . (2.3-9)

336 R- NZENGWA

We deduce from Theorem 7 that for each load h in U there exists a rotation 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) Let

4>=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 pure traction problem for each dead load in the neighborhood U. In the previous paragraph we established such an existence resuit and the possibility of approximating the solution only for équilibrâted loads. We have established a stronger resuit, which is essential because in the case of the live load problem we shall show that the solution is nothing but that corresponding to a particular dead load in U. This will be possible if at least locally the image of the live loading operator £ is contained in the neighborhood U. We therefore need an additional assumption on the loading operator £.

We begin by considering the set of déformations

{ > * = * ( * ) QihY f,heU,<pe £ ( i d , ^{P o}/ 2 ) } , (2.3-14) where (<$>,p) is the solution for the load Q{h) R(h) hT as shown in paragraph (2.1). Using relation (2.1-45) and the fact that the mappings Q and 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 this set and we assume that the loading operator restricted to this bail is Lipschitzian with a Lipschitz constant d such that

<fô<r⁰, r^o = 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)

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 337 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 ( g1) ) . (2.3-20) We observe that the constants C (R) and C(£) can be computed independently of the nature of the material, while the constants C ( g ) and C(vx) depend only on the constitutive law. Hence the constant 7 can be computed 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 loading operator £, the live load pure traction boundary-value problem possesses a local unique solution.

Proof : Clearly the condition (2.3-22) implies that the mapping x admits a unique 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 nothing but that of the dead load problem in which the fixed point h is the load. We also note that the polar décomposition is used in a crucial way in the construction of the mapping x- This was possible because we assumed condition (2.3-4).

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

ho,h^{i + 1} = ^X(hj). (2.3-24)

vol. 22, n° 2, 1988

338 R NZENGWA

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

(*,,/>,) with &(*,,/>,)= ( d , , 0 ) , (2.3-25) converges to the exact solution of (2.3-23). The incrémental method will consist in approximating the séquence (2.3-25) of traction problems with dead load, and in using the methods described in paragraph 2.2. We shall consider a finite number of terms hp j = 0, 1, . . . , / and estimate the distance between the exact solution of the problem and the approximate solution 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)

(/ + 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 function H, when restricted to the closed bail U.

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 3 3 9 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). Let h 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) b e 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 — §,p^J — P)\\- We may write

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

(2.3-38) It therefore suffices to estimate the first term in the right hand side. We show 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 some jmO. Consider ty + 1, q} + x) and ($, + 1,pJ + i) satisfying

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

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 for dead 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 the proof is complete. D We infer from the above theorem that for a given s > 0, we may choose J such 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.

INCREMENTAL METHODS IN NONLINEAR ELASTICITY 341

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