Dual formulation of a quasistatic viscoelastic contact problem with tresca's friction law

We consider quasistatic evolution of a viscoelastic boby which is in bilateral frictional contact with a rigid foundation. We derive two variational formulations for the problem:the primal formulation in terms of the displacements and the dual formulation in terms of the stress field. We prove the existence of a unique solution to each one and establish the equivalence between the two variational formulations. We also prove the continuous dependence of the solution on the friction yield limit.


INTRODUCTION
We investigate a model for the process of quasistatic frictional contact between a viscoelastic body and a rigid foundation.Processes of frictional contact are very common in industry and everyday life.Contact without lubrication can be found for example in breaking systems and in train wheels and tracks.Often, in practice, the main interest lies in the contact stress, since the behavior of the system and especially the surface integrity and wear depend on it.For this reason we obtain and analyze a formulation of the problem in terms of the stress, the so-called "dual formulation".
This work is a continuation of [15,16] where related problems were investigated, but there only the primal formulations, in terms of the displacements, were considered.
We study the contact problem for a general viscoelastic material with constitutive law cr = A(s(u)) + g(s(u)), (1.1) where a denotes the stress tensor, u the displacements and s = s(u) the linearized strain tensor.A and Q are general nonlinear constitutive functions.Here and below, a dot above a variable denotes the time derivative.Such general constitutive laws were used recently in [15,16].
We assume that there is no loss of contact between the body and the foundation, i.e. the contact is bilateral.The friction process 1s described by Tresca's law Jerri < g, } Jar! < g =* Ur = 0, larl = g =* there exists A > 0 such that ar = -A.ut',(1.2) which takes place on the contact surface r c, where ur denotes the tangential velocity, a -c represents the tangential traction and g is the friction yield limit.The strong inequality holds in the stick zone and the equality in the slip zone.
The classical formulation of the model consists of a system of evolution equations with a frictional boundary condition on the contact surface.Since, generally, such problems do not have classical solutions, we reformulate the model as a variational inequality for the displacements.This is the primal formulation.Our interest lies in obtaining and analyzing a dual formulation in terms of the stress.Indeed, as noted above, in most engineering applications the distribution of the contact stress is of greater importance than the displacements.Moreover, even when the displacements are reasonably accurate, in the numerical solutions of contact problems, the stress which is obtained by numerical differentiation is considerably less accurate.
In this paper we prove the existence of a unique weak solution for each one of the formulations and establish their equivalence.The existence and uniqueness result is accomplished in a number of steps, where fixed point arguments are used.From the equivalence we find that the stress in the primal problem is the solution of the dual one.
We prove the continuous dependence of the solutions on the friction yield limit g.This is important in applications since it indicates that small inaccuracies in g lead to small variations in the solution.
A number of quasistatic contact problems with friction have been investigated recently in the literature.Constitutive laws of linear elasticity and normal compliance contact condition were used in [4,12] while Sigorini's condition in (5-7].Bilateral contact with more general constitutive laws can be found in [1][2][3]16], and viscoelastic contact problems with normal compliance and friction were studied in [15].A quasistatic problem which takes thermal effects into consideration, especially the frictional heat generation, was analyzed in [14].Except for [12], these papers deal with the primal formulation of the problem.In [12] only the dual problem was considered.Here, we deal with both formulations and their equivalence. The rest of the paper is organized as follows.Section 2 contains notation and preliminary material.In Section 3 we describe the classical model for the process and formulate it as two equivalent variational inequalities, problems P 1 and P 2 .Problem Ph the primal one, is an evolution variational inequality in which the unknown is the displacement field u.Problem P 2 , the dual problem, is an inequality in terms of the stress field a.We establish the existence and uniqueness of the solution for each of the problems in Section 4 by using arguments from the theory of elliptic variational inequalities and a fixed point theorem.In Section 5 we prove the equivalence of the formulations in the sense that if u is the unique solution of P 1 and stress field a is the solution of P 2 , then u and a are related by the viscoelastic constitutive law (1.1).Finally, in Section 6 we show the continuous dependence of the solutions of problems P 1 and P 2 on the friction yield limit g.

NOTATION AND PRELIMINARIES
In this short section we present the notations and some preliminary material we shall use, and for further details we refer the reader to [8,10,11] or [13].We denote by SM the space of second order symmetric tensors on ~M (M = 2, 3 ), while "•" and I • I represent the inner product and the Euclidean norm on SM and ~M, respectively.Let n c ~M be a bounded domain with a Lipschitz boundary r and let v denote the unit outer normal on r.We let where and below i,j = 1, ... , M, summation over repeated indices is implied and the index that follows a comma indicates a partial derivative.H, 1-l, H 1 and 11. 1 are real Hilbert spaces endowed with the inner products given by and (u, r)?t = L uqrij dx, (u, v) H 1 = (u, v) H+(e(u), (e( v)}7-l' (cr, -rh-£ 1 = (cr, r)7-l+(div a, div r) H' respectively.Here s: H 1 ---+ 1t and div: 1t 1 ---+ H are the strain and the divergence operators, respectively, defined by The associated norms on the space H, 1t, H 1 and 1t 1 are denoted by I • IH, I • 17-£, I • IH 1 and I • 17-£ 1 , respectively.Let Hr = H 1 1 2 (r)M and let y : Ht ---+ Hr be the trace map.For every element v e H 1 we use, when no confusion is likely, the notation v for the trace yv of v on r and we denote by Vv = v • v and vr = v -Vv • v the normal and the tangential components of v on r, respectively.
Let H~ be the dual of Hr and let (•, •) denote the duality pairing between H~ and H 1 .For every a E 1t 1 , let av be the element of H~ given by We also denote by a v and a r the normal and tangential traces of a (see, e.g., [13]).We recall that if a is a regular function, then for all v E H1, where da is the surface measure element, and Finally, let (X, I • lx) be a real normed space, then C(O, T; X) and C 1 (0, T; X) denote the spaces of continuous and continuously respectively.

THE PROBLEM AND VARIATIONAL FORMULATION
The physical setting we consider is as follows.A viscoelastic body occupies the domain Q and has surface r that is partitioned into three disjoint measurable parts r v, r N and r c such that meas r D > 0. The body is clamped on r D X (0, T) and the displacements vanish there.Surface tractions fN act on r D x (0, T).The solid is in bilateral frictional contact with a rigid foundation on r c x (0, T), which means that the body and foundation have a compliant shape on r c and there is no loss of contact.A volume force of density f 0 is applied in Q x (0, T).We are interested in the evolution of the frictional process on the time interval [0, T], for T> 0. Assuming the constitutive law (1.1 ), Tresca's law of friction (1.2) and slow evolution of f 0 and f N, the classical formulation of the mechanical problem is the following.
Problem P: Find a displacement field u: n X [O,T] ~ ~M and a stress field a : n X [0, T] ~ s M such that Ia.I < g ::::} u. = o, Jar I = g ::::} there exists A. > 0 such that a. = -A.u., Here (3 .2) is the quasistatic equation of motion, since the inertial terms have been omitted, and u 0 is the initial displacement field.
To describe the two varational formulations of problem (3.1)--(3.6),we need additional notation.Let V denote the closed subspace of H 1 given by v = { v E Hl I v == 0 on r D, Vv = 0 on r c}.
Since meas r D > 0, Korn's inequality holds (see, e.g., [9] p. 79), thus Here and below, C denotes a positive generic constant which may depend on n, r, A, Q and T but does not depend on time or on the input data f 0 , fN, u 0 and g, and whose value may vary form place to place.On V we use the inner product and it follows from (3.7) that I•IH 1 and l•lv are equivalent norms on V. Therefore, (V, I • I v) is a real Hilbert space.
In the study of the mechanical problem (3.1)-(3.6)we assume that

Also and
(a) there exists L > 0 such _that We obtain from (3.9) that for r E 1-l the function x ~ A(x; r(x)) belongs to 1t and hence we may consider A as an operator on 1t with range in 1-t.Moreover, A: 1t ~ 1t is a strongly monotone Lipschitz continuous operator and, therefore, A is invertible and its inverse A-1 : 1t ~ 1t is also a strongly monotone Lipschitz continuous operator.Similar arguments allow us to consider g: 1-l ~ 1t as a Lipschitz continuous operator, too.We assume in addition that g E L 00 (r c) and g(x) > 0 a.e. on r c, Let f(t) be the element of V given by Let j be a continuous seminorm on V given by j(v) = f glv-rl da "'v E V.

Jrc
(3.17)By (3.13) the integral is well defined and, moreover, Finally, for each t E [0, T ], the set l:(t) is given by Now, we obtain from (3.15)   Problem P 2 : Find the stress field a ; [0, T] ~ 7t such that a(t) E :E(t), (B(a, e(uo))(t), ra(t))7-l> 0 Vr e l:(t), t e [0, T]. (3.36)We conclude that if {u, a} is a regular solution of the mechanical problem P then u is a solution of problem P 1 and a is a solution of problem P2.For this reason we refer to problems P 1 and P 2 as variational formulations of problem P. We will show in Section 5 that these formulations are equivalent.

EXISTENCE AND UNIQUENESS
In this section we state and prove the existence and uniqueness of solutions of the variational problems P 1 and P 2 .Our first result is: The proof of the theorem is based on fixed point arguments and will be carried out in several steps.Similar ideas were used in [ 1,15] but there the setting was different and so was the choice of the operators.To simplify the notation we shall not indicate explicitly the dependence on t.
Let 17 E C(O, T; 1-l ), and consider the following variational problem.
Problem P 1 ry: Find v 11 : [0, T] ~ V such that (A(s(v 11 (t))), s(w)-s(vry(t)))1i+(rJ(t), s(w)-s(vry(t)))7-i (4.1) and let a= s(f), and set a 11 = a 11 -a.From ( for each t E [0, T] and using (3.16) we deduce Therefore, by (4.7)-(4.9) it is straightforward to show that a 11 is a solution for ( 4.6), such that a 11 E C(O, T; 1{), if and only if a 71 E C(O, T; 7-l) and      In this section we study the link between the solutions u and a of the problems P 1 and P 2 , respectively.The main result of this section is that problems P 1 and P 2 are equivalent formulations of the mechanical problem P. then a is a solution of P 2 and a E C(O, T; 1-ft).ii) Conversely, let a be the solution of Problem P 2 • Then, there exists a unique function u E C 1 (0, T; V) such that (5.We now prove that u is a solution of Problem P (5.11) (5.12) and since the converse inequality follows from the fact that cr(t) E ~(t), for each t E [0, T], we obtain (5.3) which, in turn, implies (5.2).The inequality (3.34) follows now from (5.1) and (5.2).This concludes the proof of Theorem 4.4.D

CONTINUOUS DEPENDENCE ON THE FRICTION YIELD LIMIT
We show that the solution of the variational problems P 1 and P2 depend continuously on the friction yield limit g.The main results in this section is the following.THEOREM 6.1 Assume that (3.9)-(3.12)and (3.14) hold.Let ui, cri be the solutions of the variational problems P 1 and P 2 , respectively, with gi which satisfy (3.13), i= 1, 2. Then, there exists C > 0, which depends on n, r, A, g and T, such that In addition to the mathematical interest in this result, Theorem 6.1 is important in mechanical applications since it shows that small inaccuracies or vanatlons in the friction yield limit lead to small changes in the solutions for both problems P 1 and P 2 •  As an application to Theorem 6.1 we consider a physical setting where a viscoelastic part or component of a system is being acted upon by tractions on a part of its surface and is in bilateral frictional contact with a moving harder element.If we assume that the friction yield limit g is a Lipschitz continuous function with respect to v*, the velocity of the element, then Theorem 6.1 guarantees that the solution of the problem depends on v* Lipschitz continuously.
have r E b(t){:::::}r -a E bQ, r-a 17 (t))H (4.10) 'Vr E I:o, for each t E [0, T].Using standard arguments from the theory of elliptic variational inequalities we find that there exists an unique element a 71 (t) E 1{ which is a solution of ( 4.1 0).

( 4 . 16 )
The existence part in Theorem 4.4 follows now from ( 4.6) and ( 4.16), while the uniqueness part results from the uniqueness of the fixed point of the operator e.

THEOREM 5 . 1
Let condition (3.9)-(3.14)hold.i) If u is the solution of Problem P 1 and a is the function given by a= A(e(u)) + Q(e(u)), (5.1)
E V which is a solution of (4.1).Now, let t 1 , t2 E [0, T ], and for the sakeof simplicity we denote v 11 (ti) == vi, rJ(ti) == 1Ji, f(ti) = fi for i == 1, 2. = uo + l v~(s)ds t E [O,T] The operator A has a unique fixed point r,* E C(O, T; 1t ).Proof of Theorem 4.1 Let 71* E C(O, T; 'Jt) be the fixed point of A and let v 77 • be the solution of problem P 1 ry for 71 = r,*.We show that the function u 71 ., given in (4.3), is a solution of P 1 • Indeed, we get from (4.3) that The proof of Theorem 4.4 is similar to that of Theorem 4.1.Let 17 E C(O, T; 7-l ), and consider the following auxiliary problem.Problem P2ry Find a 11 : [O,T] -+ 1i1 such that, for each t E [0, Jl, Problem P 217 has a unique solution a 71 E C(O,T; 1i1).