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Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. (English) Zbl 0672.73079

It is well-known that no engineering surfaces are perfectly flat, no matter how precise the machining process used to produce an apparently flat finish. Under magnification one observes that all polished surfaces have undulations that form hills and valleys, the dimensions of which are large in comparison with molecular dimensions. Furthermore, the surface layers (contaminants, adsorbed materials, oxides, work-hardened layers) which cover most exposed metallic surfaces and which meet in actual contact processes, do not have the same mechanical properties as the underlying bulk materials. It is, therefore, natural in developing continuum mechanics models for contact problems, to assign to the interface a separate structure characterized by phenomenological laws independent of the constitutive equations that characterize the parent bulk materials.
However, the use of a separate characterization of the interface has not been frequent in continuum mechanics formulations of problems involving the dry contact between solid bodies. Usually, unilateral contact conditions are adopted which simply assert that, when two deformable bodies are pressed together, no mutual penetration of the bodies occurs. In other words, the compressed interface is assumed to have no normal compliance. This approach has led to serious mathematical difficulties, particularly in the formulation of dynamic contact problems. To date, no general theory of existence is available for these problems even in the frictionless case.
Many of the unresolved mathematical difficulties can be traced to the requirement of an unilateral (noncompliant) contact constraint. In fact, these unilateral dynamic contact problems are a particular case of general classes of evolution problems governed by second-order (in time) partial differential equations and subjected to unilateral constraints on the unknown displacement field itself as opposed to constraints on the time derivative of the displacement. Regularization (penalization) techniques and monotonicity arguments, used successfully in the case of constraints on the time derivative of the unknown function, do not, in general yield the desired results when the constraint is on the function itself because, in this case, the corresponding multivalued operator is not monotone.
Departing from analysis that attempt to model ideal noncompliant interfaces, in (*) Comput. Methods. Appl. Mech. Eng. 52, 527-634 (1985; Zbl 0544.73146) the authors advocated the use of simple phenomenological laws for the interface in continuum mechanics models of dynamic contact problems. These constitutive models were designed to simulate behavior observed in an extensive volume of experimental work.
In (*), we were thus trying to incorporate in a simple (certainly still too simple) continuum mechanics model some of the properties identified by numerous tribologists on the complex behavior of metallic interfaces, an approach used earlier by others in engineering computations of static contact problems. Perhaps not surprisingly, we shall show that, by taking into account the normal commpliance of the interface, we were also removing the essential difficulty in the formulation of physically realistic and mathematically well posed dynamic contact problems.
It is the objective of this paper to present existence and uniqueness results for these classes of problems. We present in Section 2 formal statements of the problems to be studied and the particular interface laws adopted on the basis of the study done in (*). In Section 3, we establish the variational statements which govern a class of dynamic frictionlss contact problems involving linearly elastic bodies and a class of dynamic frictional contact problems involving linearly viscoelastic bodies. In Section 4, we prove the existence and uniqueness of solutions for these classes of problems.
The techniques used in the proofs are now classical: Faedo-Galerkin approximations, regularization techniques, compactness and monotonicity arguments. Indeed, in the frictionless case, we encounter a second order hyperbolic semilinear differential equation, the essential distinguishing feature relative to other equations treated in the literature being the fact that the nonlinearity arises on the boundary. The existence proof given here employs essentially the strategy of the proof of theorem 1.1 in G. Duvaut and J. L. Lions [Inequalities in mechanics and physics (1976; Zbl 0331.35002)]. When friction is taken into account, we are led to a variational inequality which is similar in several respects to those studied earlier by J. L. Lions’ book on “Quelques méthodes de résolution des problèmes aux limites non linéaires”; pp. 8-14 (1969; Zbl 0189.406). Here we extend their results to a case in which the normal and frictional stresses on the contact boundary depend nonlinearly on the normal interface deformation.

MSC:

74A55 Theories of friction (tribology)
74M15 Contact in solid mechanics
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators
74S30 Other numerical methods in solid mechanics (MSC2010)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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