2D OR 3D FRICTIONAL CONTACT ALGORITHMS AND APPLICATIONS ln A LARGE DEFORMA TION CONTEXT

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1 ~~". " COMMUNICAllONS ln NUMERICAL METHODS ln ENGINEERING, Vol. Il, (1995) 2D OR 3D FRICTIONAL CONTACT ALGORITHMS AND APPLICATIONS ln A LARGE DEFORMA TION CONTEXT ZIll-QIANG FENG Polytechnic Institute of Sévenans Belfort. France SUMMARY The paper is devoted to the analysis of two- or three-dimensional contact problems with Coulomb friction and large deformation. The classical approach is based on two minimum principles or two variational inequalities: the first for unilateral contact and the second for friction. A coupled approach using only one principle or one inequality is presented. This new approach allows us to extend the notion of normality law to dissipative behaviours with a non-associated flow fuie, sncb as surface friction. Non-differentiable contact 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 contact and friction in an existing code in a very simple and neat way. Numerical examples are carried out in many difficult cases sncb as shock absorber and three-dimensional contact. The numerical results prove that 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 for large deformation and geometrically non-linear problems, which occur during most of metal forming processes. But practical applications often need unilateral contact and friction boundary conditions. ln the literature, large deformation contact problems with friction have almost exclusively been treated by penalty approximations or 'trial-and-error' methods. The augmented Lagrangian method first appeared 10 deal with constrained minirnization problems. Since friction problems are not minirnization problems, the formulation needs to be extended. Recently some extensions 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 10 the saddle-point equations of the augmented Lagrangian. One consequence of this algorithm is that the sire of the system increases because the displacements and the contact reactions (or multipliers) are both independent unknown quantities. Zero diagonal terms may appear in the stiffness matrix. ln addition, the stiffness matrix is no longer symmetric due 10 friction. This requires special solution strategies. De Saxcé and Feng proposed a theory called ISM (implicit standard materials), from which another augmented Lagrangian formulation was developed, which is essentially different from that of the first two works. ln particular, the frictional contact problem is treated in a reduced system by the formulation of De Saxcé and Feng. For the unilateral contact problems with friction, the new material law model leads 10 a single displacement variational principle and a unique inequality. ln consequence, the unilateral contact and the friction CCC /95/ Received 2 August 1995 by John Wiley & Sons, Ltd. Revised 18 August 1994

2 ~.-c " 410 z.-q. FENG are coupled. This variational approach is simpler than the classical one, which involves two variational principles and two inequalities utilized, respectively, for unilateral contact and friction.4 Recently, Heegaard and Curnier extended the augmented Lagrangian method proposed by Alart and Curnier to deal with the discrete large slip problems, but their studies were restricted to 2D frictionless contact problems. The aim of the present paper is to constitute a brief outline of previous 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 surface tractions f on part SI of S, and imposed displacements fi on part So of S. On part S2 of the boundary, contact may occur. A variational formulation for an implicit standard material behaviour on the contact surface is defined by a 'bipotential' b( -u, r). Then, the displacements vector 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 by introducing 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, the following 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 finite element method. For the sake of representation simplicity, we assume DOW that contact may occur between some points XA and XB of two bodies A and B. Consequently, the frictional contact law can be written only in terms of the relative displacements 0= A - OB and of the contact reactions r = r A = -rb. The unilateral contact conditions and Coulomb's dry friction criterion for each point 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,,u is 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)

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5 . -.. LARGE DEFORMA non CONT ACf 413 Table 1. Newton-Raphson algorithrn written in C++ for (int istep=o; istep<max_step; istep++){ //load step if (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 + Fext } } mesh->apply _boundary _conditionso; K.solve(dx,Res,I); if (contact){ contact->compute_reaction(k,dx,reac ); Res+=Reac; K.solve(dx,Res,O); } mesh->actualization(dx); if (algorithm->convergence_test(res) )break; //K dx = Res' //K dx = Res' + Reac //x = x + dx shawn in Figure 2. This example is often studied for validation of computer codes.io,11 Due to symmetry, 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 following results have been obtained:. length of non-contact area: AB = 3.75 mm. length of sliding area: BC= mm. length of sticking area: CD = 17.5 mm. Figure 3 shows the initial and deformed meshes (with an amplification factor 300). Figure 4 shows the distribution of contact reactions on AD. These results are in good agreement with other existing solutions. 10,11 F 1 1-2h., t tft t t t t t 1 G E ~ : ABC D : 1, h Figure 2. Scheme of test exarnple (h = 40 mm)

6 414 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 further explore the performance of the present method in a large de formation context and with complicated contact surfaces. The problem is displayed in Figure 5. The absorber cross-section was modelled with axisymmetric elements. The operation configuration is achieved by means of a rigid box and a rigid plate which is given a vertical motion. The rubber material behaviour was modelled by means of a Mooney-Rivlin non-linear elastic incompressible constitutive law. ln the deformed geometries (Figure 6) one can see flot only the contact between rigid bodies and deformable body, but also the contact between different parts of the SaIne deformable body. Figure 5. Initial geomeli"y

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