Solvability of a dynamic rational contact with limited interpenetration for viscoelastic plates

Abstract

The solvability of the rational contact with limited interpenetration of different kind of viscolastic plates is proved. The biharmonic plates, von Kármán plates, Reissner-Mindlin plates and full von Kármán systems are treated. The viscoelasticity can have the classical ("short memory") form or the form of a certain singular memory. For all models some convergence of the solutions to the solutions of the Signorini contact is proved provided the thickness of the interpenetration tends to zero.These are the premises of the rational contact model which was introduced by [7] and [8], where the solvability of its static version has been proved. The first dynamic (frictionless) rational contact has been investigated in [9]. It concerns a boundary contact of a body with a foundation.Since 2006 a series of papers about the solvability of dynamic Signorini contact problems for different models of plates [1]-[4] was published. The purpose of this paper is to extend these results to the rational contact with limited interpenetration. Unlike [9] we face here a domain contact.2 Abstract formulation of the problem for the clamped or simply supported viscoelastic plate and the scheme of its solutionLet Ω ⊂ R 2 be a bounded domain with a sufficiently smooth boundary Γ . Let X be a Sobolev-type Hilbert space defined on Ω, let Y be the space of traces of elements from X on Γ . Let A, B : X → X * be two linear symmetric strongly elliptic operators in the form D * aD, D * bD, respectively, where D is a differential operator and a, b are positively definite matrices or tensors of time constant but possibly space-dependent elements. Let I ≡ [0, T ] be a time interval. Here the dual space X * is defined via the suitable generalization of the L 2 (Ω) scalar product. Let X ≡ L 2 (I; X). We introduce the bilinear forms A :We shall denote the elements of v ∈ X or v : I → X such that v ∈ X as displacements, and their first time derivatives (denoted by dots) as velocities. Let γ be a negative real number. Let p : R →R ≡ R ∪ {+∞} be a nonincreasing function such that p(x) = 0 for x ≥ 0, p(x) ∈ R for x > γ, and lim xցγ p(x) = +∞, where γ ∈ R is a given bound of the interpenetration. Our problem is to find u ∈ X such thatu ∈ X for which the following set of relations holds

Contacts with limited interpenetration

“…This depth does not influence the estimates of different norms to the solutions, because the normal stress was treated there as defined by the Green formula and not in the form of its expression by the constitutive relation. The proof, less straightforward than those made in papers [1] , [2] , [3] , [4] , may be seen as an alternative proof of the existence of solution to the quasistatic Signorini contact with friction. There are some differences with respect of Andersson's proof which is based on the appropriate Yosida approximations.…”

“…The further papers dealt with the dynamic frictionless contact for viscoelastic material. In [3] the boundary contact of a body and in [4] the domain contact of different models of plates have been analyzed.…”

“…The common feature of refs. [1] , [2] , [3] , [4] , [5] is the proof of the existence of solution to problems posed. Of course, this has no straightforward impact to technical practice.…”

“…Summing up: the presented and other results published (Refs. [1] , [2] , [3] , [4] ) create a reliable basis of the numerical analysis of the problems; the method is ready to be used in solving a wide class of contact problems arising in technical practice. …”

“…the presented and other results published (Refs. [1] , [2] , [3] , [4] ) create a reliable basis of the numerical analysis of the problems;…”