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A global existence result for a quasistatic contact problem with friction. (English) Zbl 0832.35056

The purpose of this important paper is to study of a question concerning to a quasistatic contact problem for a linearly elastic nonviscous system. Friction is modelled by a so-called normal compliance power law. The problem is formulated mathematically as a variational inequality in a Sobolev space. The author consider the following family of PDEs for \(0\leq t\leq T\), \[ \text{div } \sigma(u(t))=-f(t)\quad\text{in }\Omega,\quad u(t)\bigl|_{S_u}= 0,\quad \sigma(u(t))\cdot n= q\quad\text{on }S_t,\;u(0)= 0, \] \(\Omega\subset \mathbb{R}^d\) \((d= 2\) or 3) is a bounded open set with Lipschitz boundary, \(u\) is the displacement field, \(f\) are the volume forces of density, \(q\) are the surface forces of density, \(n\) is the outward unit normal vector to \(\partial\Omega\), \(\sigma\) denotes the stress tensor.
A theorem of existence is given, without any restriction on the magnitude of the applied forces or on the coefficient of friction. The question of uniqueness for the quasistatic problem in this paper is open. A time- dependent displacement field which solves the problem may be a discontinuous function of time if the forces are large.

MSC:

35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
74A55 Theories of friction (tribology)
74M15 Contact in solid mechanics
49J40 Variational inequalities
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