The quasi-static model of contact problem with friction.

Pop, Nicolae ; Cioban, Horia ; Butnar, Lucian 等

1. INTRODUCTION

The mathematical background of the theory of contact with friction is still incomplete. The proof of the existence and uniqueness of the solution for models with normal compliance belong to the dynamic case and involve the contact bodies which are made from linear viscoelastic material. The presence of the inertial terms and viscoelastic damping is essential in order to end this proof (Pop & Cioban, 2008). If a body lies on a contact surface with friction and is subjected to increasing tangential forces, the sliding starts when the equilibrium configuration body becomes dynamically unstable. Thus, the inertial effects are essential in characterizing the dynamic transition from adhesion to the slide. Dynamic effects are used in order to prove that the resulting movements are limited in time, but have not proven to be essential in an existence theory.

The model was proposed by (Schillor, 2001) and the existence of solutions for the quasistatic problems with friction was proved by (Anderson, 1995).

2. CLASSICAL FORMULATION

We denote by [[OMEGA].sup.[alpha]] [subset] [[??].sup.d], [alpha] = 1,2, d = 2,3, the domains occupied by the two bodies in contact and by [[GAMMA].sup.[alpha]] = [partial derivative][[OMEGA].sup.[alpha]] the sufficiently regulated boundaries of the two bodies, which can be written as [[GAMMA].sup.[alpha]] = [[GAMMA].sup.[alpha].sub.D] [universal] [[GAMMA].sup.[alpha].sub.N] [universal] [[GAMMA].sup.[alpha].sub.C], [alpha] = 1,2, where [[GAMMA].sup.[alpha].sub.D], [[GAMMA].sup.[alpha].sub.N], [[GAMMA].sup.[alpha].sub.C] are open disjoint sets of [[GAMMA].sup.[alpha]]. The bodies are subject to strong volumetric density forces, [f.sup.[alpha]] in [[OMEGA].sup.[alpha]] and also to density, surface forces [T.sup.[alpha]] on [[GAMMA].sup.[alpha].sub.N]. The bodies are fixed on [[GAMMA].sup.[alpha].sub.D] [subset] [[GAMMA].sup.[alpha]] and [[GAMMA].sup.[alpha].sub.C] are the surfaces which may define the contact. The gap between the bodies, in the contact zone is defined by a function g, with g [greater than or equal to] 0. The points from [[OMEGA].sup.[alpha]] and from [[GAMMA].sup.[alpha]] will be denoted by x = ([x.sub.1], ..., [x.sub.d]), and respectively by [S.sub.i] 1 [less than or equal to] i [less than or equal to] d. The time is also denoted by t and we consider the interval t [member of] [0, T]. The quasistatic problem consists of determining the displacements [u.sup.[alpha]](x,t) = ([u.sup.[alpha].sub.1](x,t), [u.sup.[alpha].sub.2](x,t), ..., [u.sup.[alpha].sub.d] (x,t)), and also the stresses [[sigma].sup.[alpha]] = ([[sigma].sup.[alpha].sub.ij]), which satisfy the following equations, boundary conditions, contact conditions and the initial conditions:

--the equilibrium equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

--the constitutive equation

[[sigma].sup.[alpha].sub.ij] = [E.sup.[alpha].sub.ijkl] [u.sup.[alpha].sub.k.l], 1 [less than or equal to] i, j, k, l [less than or equal to] d (2)

where [E.sup.[alpha].sub.ijkl] is the tensor of elastic material coefficients, and the signs (,1) and (.) represent the partial derivatives with respect to [x.sub.l] and with respect to time, respectively. We shall denote by n the versor of the exterior normal at [[GAMMA].sup.[alpha]] and by [[sigma].sup.[alpha].sub.n] and [[sigma].sup.[alpha].sub.T] the vector of the normal stress, respectively the tangential stress on the boundary, whose values u will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly the displacement [u.sup.[alpha]] on the boundary [[GAMMA].sup.[alpha]] can be decomposed into a tangential component [u.sup.[alpha].sub.T], and respectively a normal one [u.sup.[alpha].sub.n] = [u.sup.[alpha].sub.i] [n.sup.[alpha].sub.i], [u.sup.[alpha].sub.T] = [u.sup.[alpha].sub.i] - [u.sup.[alpha].sub.n] [u.sup.[alpha].sub.i]

--the boundary conditions

[u.sup.[alpha].sub.i] = 0 on [[GAMMA].sup.[alpha].sub.D] x (0,T), 1 [less than or equal to] 1 [less than or equal to] d (3)

[[sigma].sup.[alpha].sub.ij] ([u.sup.[alpha]])[n.sup.[alpha].sub.j] = [T.sup.[alpha].sub.i] on [[GAMMA].sup.[alpha].sub.N] x (0,T), 1 < i, j [less than or equal to] d (4)

The friction contact on [[GAMMA].sup.1.sub.C] x (0,T) is supposed to be ruled by a power type law--the normal compliance law (Klarbring, 1993).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where if

[u.sup.r.sub.n] = g [??] [[sigma].sub.T] ([u.sup.[alpha]]) = 0, (6)

and if [u.sup.r.sub.n] > g it results

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [c.sub.n] = [c.sub.n] ([sigma]), [c.sub.T] = [c.sub.T] ([sigma]), [m.sub.n] and [m.sub.T] are parameters which characterize the contact interface. This parameters can be deduced experimentally. We denoted the positive part of the argument by [(*).sub.+], and [u.sup.r.sub.n] = [u.sup.1.sub.n] - [u.sup.2.sub.n].

The initial conditions are given by

u(x,0) = [u.sub.0](x) on [[OMEGA].sup.1] [universal] [[OMEGA].sup.2] (8)

[??](x,0) = [u.sub.1](x) on [[OMEGA].sup.1] [universal] [[OMEGA].sup.2]. (9)

3. VARIATIONAL FORMULATION AND INCREMENTAL METHOD

Further on we will introduce the space of admissible displacements, V = {[v.sup.[alpha]] [member of] [H.sup.[alpha]] ([[OMEGA].sup.[alpha]]).sup.d] : [v.sup.[alpha]] = 0 a.e. on [[GAMMA].sup.[alpha].sub.D]} and the constraints set K, respectively

K = {v [member of] V : [v.sup.1.sub.n] - [v.sup.2.sub.n] [less than or equal to] g a.e. on [[GAMMA].sup.1.sub.C]}. (10)

Now, let us suppose that

[F.sub.i] [member of] [L.sup.2] ([[OMEGA].sup.[alpha]]), [T.sub.i] [member of] [L.sup.2]([[GAMMA].sup.[alpha].sub.N]), g [member of] [H.sup.1/2] ([[GAMMA].sup.1.sub.C]), (11)

and the elastic coefficients satisfy the following conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where [[alpha].sub.1] is a positive constant. Furthermore in order to have well defined integrals on the boundary, [[GAMMA].sup.1.sub.C], it is necessary for the following relations to hold

[c.sub.n] (s), [C.sub.T] (s) [member of] [L.sup.[infinity]] ([[GAMMA].sup.1.sub.C]), 1 [less than or equal to] [m.sub.n], [m.sub.T] < [infinity] if d = 2, (13)

1 [less than or equal to] [m.sub.n], [m.sub.T] [less than or equal to] 3 if d = 3. (14)

Those restrictions on [m.sub.n] and [m.sub.T] result from the embediness theorem, which establishes that for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with 2 [less than or equal to] q [less than or equal to] [epsilon] for d = 2 and with 2 [less than or equal to] q [less than or equal to] 4 for d = 3, where we denoted by [gamma] the trace operator.

The weak variational form of the quas-istatic contact problem is given by the next problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[for all]v [member of] V, with the initial conditions u(x,0) = [u.sub.0](x), [??](x,0) = [u.sub.1](x), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

The equivalence of the variational problem (15) with (1)-(9) is established using standard methods. (Fortin et al., 2009). Through the temporal discretization of the quasi-static problem one obtains the so-called incremental problem which is equivalent to a sequence of static problems.

The incremental form is obtained by the variational formulation of the problem (15) through the approximation of the temporal derivatives of the displacements with finite differences.

Although at each increment the dependence on the loading path is neglected, the consideration of a sequence constructed as above depends on how the applied forces variate. From a mechanical point of view and also numerically speaking the problem resulting at each step is similar to a static problem.

Thus, in order to obtain incremental formulations of the quasistatic contact problem, we introduce a partition, ([t.sup.0], [t.sup.1], ..., [t.sup.n]) of the time interval [0,T] and the following notations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

we observe that at each time [t.sup.k] we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Discredized variational formulation for the quasi-static model of the contact problem with friction is given by the below proposition:

Proposition 3.1. Find [DELTA][u.sup.k] [member of] [K.sup.k] that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

Note that incremental inequality (20) is formally similar to a quasi-variational inequality for a static sequence contact problem.

4. CONCLUSION

The quasistatic problem will be resolved step by step, such that at each step we shall calculated small strains and small displacements and we will add to the previously calculated result, for small changes of the applied forces. Obviously, both the contact area and the contact state are changing (open contact, fixed contact and sliding contact).

This algorithm and the research described in this paper are leading us to get a much faster algorithm by using the sequential quadratic programming methods. This is esential to be able to consider realistic large scale industrial problems which could imply million of degrees of freedom.

5. REFERENCES

Anderson, L.E. (1995). A global existence results for a quasistatic contact problem with friction, Advances in Mathematical Sciences and Application, 5(1), 249-286, ISSN 1343-4373

Fortin, M. et al. (2009). Frictional contact in solid mechanics, 6th International Congress on Industrial and Applied Mathematics, Jeltsch, R., Wanner, G.,(Eds.), 131-154, European Mathematical Society Publishing House, ISBN 978-3-03719-056-2, Zurich

Klarbring, A. (1993). Mathematical programing in contact problems, Computational Methods in Contact Mechanics, Computational Mech. Publ., Southampton, 233-263, ISBN 1-85312-872-4

Pop, N. & Cioban, H. (2008). Genralized Jacobians and Newton Method for Solving the Frictional Contact Problems, Annals for DAAAM for 2008 & Proceedings of the 19th International DAAAM Symposium, Katalinic, B., pp.1093-1094, ISSN 1726-9679, Trnava, Slovakia, Oct.2008, DAAAM International, Vienna

Shillor, A. (2001). Quasistatic Problems in Contact Mechanics, Int. J. Applied Mathematics in Computer Sciences, 11(1), 189-204, ISSN 1641-876X