Use of Variational Methods for the Analysis ofContact Problems in Solid Mechanics

J. T. OdeD and N. KikuchiThe University of Texas at Austin, Austin. Texas, U.S.A.

ABSTRACT

We describe several variational formulations of the problem of contact of an incompressibleelastic body with a lubricated rigid foundation. These include variational inequalities, La-grange multiplier methods, and penalty methods. Existence theorems are presented as well asresults of finite element approximations.

1. INTRODUCTION

In this paper, we outline an analysis of a class of nonlinear boundary-value problems in thetheory of incompressible elastic bodies which arise in the study of contact problems; i.e., theproblem of the contact of an incompressible elastic body with a rigid foundation. We developvariational principles for this class of problems which embody the use of variational inequal-ities, Lagrange multipliers, and penalty methods. We study the existence and uniqueness ofsolutions to such problems, their approximation using finite elements, and we present some re-presentative numerical solutions.

Our first approach combines features of saddle point theory and the theory of variational in-equalities. In particular, much of our analysis is based on applications of the followingtheorems:

Theorem 1: Let U and V denote real reflexive Banach spaces, Ke U a nonempty closed convexsubset of U and Me V a nonempty closed convex subset of V.

Let L : K x M ~m be a real functional satisfying the following conditions:

(i) Vq E M, v -+ L (v,q) is convex and lower semicontinuousj

(ii) Vv E K, q -+ L(v,q) is concave and upper semicontinuousj

(iii) 3qO E M such that L(v,qO) -+ +.., as Ilvllll -+.., ;

(iv) limv E K (inf L(v,q» ; - Q) as Ilqllv -+.., V E K .

Then there exists a saddle point (u,p) E K x M of Lj i.e.

L(u,q) ~ L(u,p) ~ L(v,p) Vv E K, Vq EM. (1.1)

Theorem 2: Let the conditions of Theorem 1 hold except in place of (i) we have

(i') Vq E M, v -+ L (v,q) is convex and Gateaux differentiable, with G~teaux derivative

Vv E K: <aL(u,p)au

and, in addition to (ii), we have

a 11m ~ L(u + tv,p) ,t-+O+ at

(11' ) Vv E K, q ~ L (v,q) is

Vq E M: <aL(u,p)

Gateaux differentiable with derivative

a, q>V • lim at L(u,p + tq) .

t-+O+

Then there exists a saddle point (u,p) E K x M satisfying (1.1) and, moreover, (u,p) is char-acterized by

260

Contact Problems 261

<aL(u,p) u _ v> > 0Vv EK }au' 1I-

<aL(u,p) q _ p> < 0 Vq E Map' V-In (1. 2), <., ,>U and <., .>V denote duality pairing on Ll' x U and V' x V respectively.

(1. 2)

It is possible to extend Theorem 1 to cases in which K is only weakly sequentially closed; see,for example, Eke1and [1]. A proof of Theorems 1 and 2 in the forms given here can befound in Eke1and and Temam [2]. Note that (1.2) represents a pair of variational inequalities.Both reduce to variational equalities whenever u E int K and p E int M.

2. A SIGNORINI PROBLEM FOR INCO~lPRESSIBLE ELASTIC BODIES

We begin by considering the problem of the contact of an incompressible linearly elastic bodywith a rigid frictionless (lubricated) foundation. The body is characterized as the closure ofan open bounded domain n eRn, n = 1,2,3, with a smooth boundary r which consists of threeparts r • 'f

DIJ ['FUfC' where r D' rF denote the portions of the boundary on which the displace-

ments and tractions are prescribed respectively and r C is a portion of r containing the unknown

contact surface. The "classical" problem is governed by the equations

ai

.. + f aJ oJ i

a~and boundary conditions

o j Cij(v) • (vij

2µcij - POijin II (2.1)

(2.2)

and the contact conditions

on r c . (2.3)

Here the usual notations are employed; un • uini

and s a s(~) is the initial "gap" representing

the normalized distance between material particles ~ and the contact surface rc' In addition,we have the incompressibility constraint

(2.4)

Constraints (2.3) and (2.4) and homogeneous (fixed) conditions on rD characterize theconstraint set

c - {v E U : c11(v) • 0 a.e. in II ,

Vi = 0 on rD; vn - s ~ 0 a.e. on rc} ,

where 1I is an appropriate real Banach space of measurable functions on O.

(2.5)

It is easily shown that solutions of the problem (2.1) - (2.4) can be characterized by the fol-lowing variational equality:

Find u E C such that

(2.6)

Moreover, (2.6) is equivalent to the constrained minimization problem

(2.7)}fividx - f gividsfF

2L (rF), then F is a functional on the Sobo1ev space

R 1,2,3.

u E C: F(u) ~ F(v) Vv E c

F(v) = f µcij (v)cij (v)dx - fII II

~ 2Note that if µ E L (0), fi E L (0), gi E1 n ( 1 n(H (0» and we may take U = H (n» , n

262 J. T. Oden and N. Kikuchi

3. A RESOLUTION BY LAGRANGE MULTIPLIEP-S

We reformulate (2.7) by relaxing the incompressibility constraint (2.4). Then, instead of theset C of (2.5) we have

K = {v E U: vn - s ~ 0 a.e. on rC; vi = 0 a.e. on rn} .

Next we introduce the Lagrangian,

(3.1)

L(v,q) = F(v) - f qcii(v)dx . (3.2)o

Since cii(v) E L2(0) , the Lagrange multiplier q E L2(0); i.e. in this case the constraint set

M in Theorem 1 is a subspace of a complete normed linear space.

Clearly, conditions (i) and (ii) of TIleorem 1 (indeed i') and (ii') of Theorem 2) are satisfied.In fact, L(.,q) is 8tpict~y convex for every q if mes (fn) > O. Condition (iii) also holds by

1( nKorn's inequality. To verify (iv) we note that for Uo = (HO 0» , infvcKL(v,q) 2 infU L(v,q),

since Uo C K (because of s ~ 0). Then it suffices to show that infvcUOL(v,q)-+ - '" as Ilql Ict~.From the characterization of a minimizer v of L(.,q) on U ,q 0

~ qcii(v)dx ; l fividx - ~ 2µcij (vq)cij (v)dx

2 (21111llo,,,,lle(vq)llo + Ilfllo)llvl11 •

Here Ilvll = ess. sup E ,.,Ivl,and Ilello2

= ti . lile 1102 We recall the lemma proved0,'" x .. oJ= ijby Tartar [3],

Lemma 1: For q E L2(Q) , there exists a v E Uo such that div v = q and IIvl 11 ~ C I Iql 10, C > ~

provided the compatibility condition f q dx a 0 is satisfied.o

Let the set M be defined by

M = {q E L2(n): f q dx = O} • (3.3)o

Then, for every q E M, it can be shown that Ile(v )110 * + '"if I Iql 10 -+ + "'. On the other~~, q

provided there exists a positive number m such that

µ(x) ~ m a.e. in 0 . (3.4)

Therefore, ifisfied.

infv E u L(v,q) -+ - '"

oThus all conditions of Theorem 1 are sat-

In conclusion, we have

Theorem 3: Let mes fD > 0 and (3.4) hold. Then there exists a unique saddle point (u,p) of

the functional L of (3.2) and (u,p) E K x M. ~loreover, u is a solution of the variational in-equality (2.6) and of the minimization problem (2.7).

4. A RESOLUTION BY PENALTY METHODS

An alternate formulation of the Signorini problenl described in Section 2 can be constructedusing penalty methods. That is, instead of satisfying the constraint cii(u) = 0 a.e. in 0, thepenalty functional

1 f 2p(v) = 2 (cii(v» dx (4.1)o

is appended to the functional F of (2.7). It is important to note that p(.) is weakly lowersemicontinuous on U, p(v) ~ 0 for every v E U, and P(v) ; 0 iff cii(v) = 0 a.e. in Q. The pen-alized problem is

uA E K: E(UA,A) ~ E(V,A) V v E K

E(v,A) = F(v) + AP(V) .(4.2)

Contact Problems 263

For every ~ > 0, E(.,~) is strictly convex, Gateaux differentiable, and coercive on K if mes(fn) > O. Then the penalized problem has a unique solution uA for each~~ O. Moreover, u~ is

uniformly bounded for every A > O. Since the Sobolev space (H1(Q»n is a reflexive Banachspace there exists a subsequence uA which converges weakly to some element u in (Hl(O»n. ~:ore-over, if the min1mizer of F on C is uo'

P(u~) ~ (F(uO) - F(u~»/~ .

Sinc~ F(uO) and F(u~) are bounded, p(u) = 0 as A -+"', i.e. £ii(u~) converges strongly to £itu)

in L (n). This means that £ii(u) • 0 a.e. in the domain n. Since

F(UA) ~ F(u~) + AP(U~) ~ F(uO) ,

we obtain F(u) ~ F(uO) by taking the limit A ~ + "'.Since u E C, the limit u has to be a min-

imizer of F on C. Because of the uniqueness of minimizers of F on C, u is exactly the same asuo. Similar arguments can be given to show that, for every convergent subsequence of u~, the

limit is unique. Thus, the original sequence uA converges weakly to the minimizer Uo of F on C.

~IDreover, for the minimizer uA' we have

f ~£1i(u,)viidx = f fividx - f 2µ£ij(u~)vi jdxo 1\, 0 0 '

for every v E UO' since Uo C K. Then

Ilgrad (A£ii(uA» I 1-1 ;;sup Elf A£i'(u )vi idxllllvl11v -uo 0 ~ A ,

~ Ilfll_l + 211µ110)lu~"1 .

-1 nThus, grad (~eii(uA» is uniformly bounded in (H (0» for each A ~ O. Since the image of

2 -1 n 2grad: L (0) (H (n» is isomorphic to L (O)/N, N = {u • constant} (see Tarter [4], Lemma 8,p. 30), ~£1i (uA) is bounded in L2(Q)/N. TIlenthere exists a subsequence (still denoted

2~£ii(uA»' which converges weakly to some l1mit -p in L (0). We note that the limit -p is uni-

que within constants. Since £ii(uA) converges strongly to £ii(u), and since F(.) is weakly

lower semicontinuous. we have

Theorem 4: Let mes (rD) > 0, and (3.4) hold. Then the sequence u~ of solutions of the penal-

ized problem (4.2) converges weakly to the solution of the constrained problem (2.7) as A ~ + ~Moreover, the sequence A£ii(uA) also converges to -p, which is identified with the pressure, in

2L (0) modulo constants.

5. FINITE ELEMENT APPROXIMATIONS

The use of penalty methods as a basis for finite element approximations of constrained problemshas been described by several authors; see, for example, Hughes, Liu. and Brooks [5]. Reddy [6],and the references therein. We briefly outline results of examples solved numerically by fin-ite element methods.

We consider the plane-strain rigid punch problem indicated in Fig. 1 and the half-cylinderHertz problem shown in Fig. 2. Eight-node isoparametric elements are used. Our formulationemploys the Lagrange multiplier method plus a penalty term for the incompressibility constraint:

Some remarks on these results are listed asNumericalfollows:

L(v,q) = F(v) + AP(V) - f q(v-rc n

results are indicated in the figures.

s)ds • (5.1)

(1) When 3 x 3 Gaussian numerical integration is taken for both the strain energy and the pen-alty functional, convergence of deformations as A -+ + ~(i.e. Poisson's ratio v -+ 0.5 forplane strain problems) is not obtained. Indeed, for the Hertz problem, adequate resultsare obtained only for Poisson's ratio v < 0.495. If 3 x 3 Gaussian quadrature is usedfor the strain energy, and reduced 2 x 2-Gaussian quadrature is used for the penalty func-tional, then rapid convergence of deformstions as A -+ + '" is observed. The pressureA£ii(u

A) seems to converge at the Gauss points for 2 x 2 Gaussian quadrature integrations.

264 J. T. Oden and N. Kikuchi

T1.5 •

10.5

Rigid Foundation (Unilateral Condition)

'1.;;: '/.::71~II~I/~r~iiJll~

Young's Modulus 1000.

Poisson's Ratio 0.499999

9-Node Isopara

X B-Node Isopara

r .1..LO.

100. t200.300. r :&'400. t..·.1t------6.--- ..q- /

500 '1Contact Pressure

Fig. 1. A rigid punch problem on a rigid foundation for incompressible body.

T4.

1~.

/.

..0.0'.0-1200.

P .~ 100.

9-Node Isoparametric F.E,

3 x 3 & 2 x 2 Integration

E.1000. ,,·0.49999

8. -- !

(2) If the resulting equations ~re solved byUzawa's method, then the iteration factorcan be chosen independently of the penaltyparameter A.

Space does not permit the inclusion of manyimportant details. These will be discussed ina forthcoming companion paper.

ACKNOWLEDGMENT

The assistance of Mr. Y. J, Song in performingthe numerical experiments is gratefully acknowl-edged. This work was supported by the U.S. AirForce Office of Scientific Research under Con-tract F-49620-78-C-00B3.

REFERENCES

9-Node Isoparametric F.E.

3 x 3 Integration

Fig. 2, A Hertz problem of anincompressible body.

1. Ekeland, 1., "On the Variational Principle,"J. Math. Anal. Appl. 47 (1974) 324-353.

2. Ekeland, I. and Temam, R., Analyse Convexeet PPoblemes Variationnels, Dunod, Paris(1974) .

3. Tartar, L., "Nonlinear Partial DifferentialEquations Using Compactness Method," MRCTechnical Summary Report No. 1584, Universi-ty of Wisconsin (1976).

4. Hughes, T. J. R., Liu, W. K., and Brooks, A.,"Finite Element Analysis of IncompressibleViscous Flows by the Penalty Function Formu-lation," J. Compo Phys. 30 (1979) 1-60.

5. Reddy, J. N., "Penalty Methods for FiniteElement Approximations of Steady StokeseanFlows," to appear in Int, J. Engng. Sci.