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.COMMUNICAllONS lN NUMERICAL METHODS lN ENGINEERING, Vol. Il,409-416 (1995)

2D OR 3D FRICTIONAL CONTACT ALGORITHMS ANDAPPLICATIONS lN A LARGE DEFORMA TION CONTEXT

ZIll-QIANG FENGPolytechnic Institute of Sévenans. 90010 Belfort. France

SUMMARYThe paper is devoted to the analysis of two- or three-dimensional contact problems with Coulomb frictionand large deformation. The classical approach is based on two minimum principles or two variationalinequalities: the first for unilateral contact and the second for friction. A coupled approach using only oneprinciple or one inequality is presented. This new approach allows us to extend the notion of normalitylaw to dissipative behaviours with a non-associated flow fUIe, sncb as surface friction. Non-differentiablecontact potentials are regularized by means of the augmented Lagrangian method. Using the C++language, an object-oriented finite element database is created, which allows us to implement the contactand friction in an existing code in a very simple and neat way. Numerical examples are carried out inmany difficult cases sncb as shock absorber and three-dimensional contact. The numerical results provethat this approach is robust and efficient conceming numerical stability.

KEY WORDS contact; friction; finite element method; large deformation; augmented Lagrangian method

1. INTRODUCflON

ln recent years, considerable progress bas been made in the development of solution methods forlarge deformation and geometrically non-linear problems, which occur during most of metalforming processes. But practical applications often need unilateral contact and friction boundaryconditions. ln the literature, large deformation contact problems with friction have almostexclusively been treated by penalty approximations or 'trial-and-error' methods. The augmentedLagrangian method first appeared 10 deal with constrained minirnization problems. Since frictionproblems are not minirnization problems, the formulation needs to be extended. Recently someextensions have been obtained in mutually independent works by Alart and Cumier, 1 Simo and

Laursen,2 and De Saxcé and Feng.3 The first two works consist of applying Newton's method 10the saddle-point equations of the augmented Lagrangian. One consequence of this algorithm isthat the sire of the system increases because the displacements and the contact reactions (ormultipliers) are both independent unknown quantities. Zero diagonal terms may appear in thestiffness matrix. ln addition, the stiffness matrix is no longer symmetric due 10 friction. Thisrequires special solution strategies. De Saxcé and Feng proposed a theory called ISM (implicitstandard materials), from which another augmented Lagrangian formulation was developed,which is essentially different from that of the first two works. ln particular, the frictional contactproblem is treated in a reduced system by the formulation of De Saxcé and Feng. For the unilateralcontact problems with friction, the new material law model leads 10 a single displacementvariational principle and a unique inequality. ln consequence, the unilateral contact and the friction

CCC 1069-8299/95/050409-08 Received 2 August 1994@ 1995 by John Wiley & Sons, Ltd. Revised 18 August 1994

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410 z.-Q. FENG

are coupled. This variational approach is simpler than the classical one, which involves twovariational principles and two inequalities utilized, respectively, for unilateral contact and friction.4Recently, Heegaard and Curnier extended the augmented Lagrangian method proposed by Alartand Curnier to deal with the discrete large slip problems, but their studies were restricted to 2Dfrictionless contact problems. The aim of the present paper is to constitute a brief outline ofprevious work,3.6-8 and to extend it to generallarge deformation contact problems with friction.

2. V ARIATIONAL PRINCIPLE AND INEQUALUY

Let .Q be a structure with boundary S, subjected to imposed body forces f, imposed surfacetractions f on part SI of S, and imposed displacements fi on part So of S. On part S2 of theboundary, contact may occur. A variational formulation for an implicit standard materialbehaviour on the contact surface is defined by a 'bipotential' b( -u, r). Then, the displacementsvector ° and the contact reactions vector r are linked by an implicit subdifferentiallaw:

r e a_ub( -0, r), -0 e arb( -0, r) (1)

For the sake of generality, an implicit standard behaviour of the contact bodies is assumed byintroducing a bipotential P(t, a) such that

a e a,p(t, a) te aop(t, a) (2)

where t and a denote, respectively, the generalized strain and stress fields. On this basis, thefollowing functional, called 'bifunctional', is defined:

B(o, a) = l P(t(o), a) da + l b(o, r(a» dS - I f. ° da - l t. ° dS - l r(a). fi dS (3)a S2 a SI Sa

An exact solution (0, a) is simultaneously a solution of the following variational principles:

ln! B(Ok, a) and ln! B(o, aS) (4)° k kinematically admissible as statically admissible

Usually, only the first variational principle is used which cornes to the displacement finiteelement method.

For the sake of representation simplicity, we assume DOW that contact may occur betweensome points XA and XB of two bodies A and B. Consequently, the frictional contact law can bewritten only in terms of the relative displacements 0= °A - OB and of the contact reactionsr = r A = -rB. The unilateral contact conditions and Coulomb's dry friction criterion for eachpoint are stated as

un+g>O rn >0 IIr,II",urn (5)

where n and t denote, respectively, the normal and tangential directions to the contact surface, ,uis the friction coefficient and g the initial gap.

Let "/l be the convex Coulomb friction set (Figure 1) defined by

"/l={(r"rJsuchthatllr,II",urn} (6)

With the usual notation of an indicator function and denoting the time derivative by an overdot,the contact with friction can be represented by the following bipotential:

b(-un, -0" r n' r,) =,ur nll ~ Il + 'l'R+(Un+ g) +'1' K/l(rl, rJ (7)

~.-c

-...

LARGE DEFORMA nON CONT ACf 413

Table 1. Newton-Raphson algorithrn written in C++

for (int istep=O; istep<max_step; istep++){ //load stepif (contact) contact->contact_detectionO; //local frame, gap vector ...for (int iter=O; iter<max_iter; iter++){ //N-R equilibrium iteration

mesh- >compute stiffness>matrix (K);mesh->compute=residual_vector(Res); //Res' = -Fint + Fextmesh->apply _boundary _conditionsO;K.solve(dx,Res,I); //K dx = Res'if (contact){

contact->compute_reaction(K,dx,Reac );Res+=Reac; K.solve(dx,Res,O); //K dx = Res' + Reac

}mesh->actualization(dx); //x = x + dxif (algorithm->convergence_test(Res) )break;

}}

shawn in Figure 2. This example is often studied for validation of computer codes.IO,11 Due tosymmetry, only half of the slab is modelled. The characteristics of this example are:

. Young's modulus: E= 13,000 daN/mm2

. Poisson'sratio: v=0.2

. friction coefficient: .u = 1.0

. boundary conditions: Ux = 0 on ED, UX = Uy = 0 at point D

. loadings: f= -5 daN/mm2 on GE, F= -10 daN/mm2 on GA.

The half-slab is modelled by 512 linear quadrilateral plane strain elements. The followingresults have been obtained:

. length of non-contact area: AB = 3.75 mm. length of sliding area: BC= 18.75 mm

. length of sticking area: CD = 17.5 mm.

Figure 3 shows the initial and deformed meshes (with an amplification factor 300). Figure 4shows the distribution of contact reactions on AD. These results are in good agreement withother existing solutions. 10,11

1 2h - .,t tft t 1 t t t t

G E ~: 1

, F 1 h

: : ABC D :

Figure 2. Scheme of test exarnple (h = 40 mm)

414 Z.-Q. FENG

Figure 3. Initial and defonned meshes

Figure 4. Contact reactions on AD

The second ex ample simulates a rubber shock absorber. ln doing so, we wish to furtherexplore the performance of the present method in a large de formation context and withcomplicated contact surfaces. The problem is displayed in Figure 5. The absorber cross-sectionwas modelled with axisymmetric elements. The operation configuration is achieved by meansof a rigid box and a rigid plate which is given a vertical motion. The rubber material behaviourwas modelled by means of a Mooney-Rivlin non-linear elastic incompressible constitutivelaw. ln the deformed geometries (Figure 6) one can see flot only the contact between rigidbodies and deformable body, but also the contact between different parts of the SaInedeformable body.

Figure 5. Initial geomelI"y