Jiří Jarušek

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

A coupled thermoviscoelastic frictional contact problem is investigated. The contact is modelled by the Signorini condition for the displacement velocities and the friction by the Coulomb law. The heat generated by friction is described by a non-linear boundary condition with at most linear growth. The weak formulation of the problem consists of a variational inequality for the elasticity part and a variational equation for the heat conduction part. In order to prove the existence of a solution to this problem we first use an approximation of the Signorini condition by the penalty method. The existence of a solution for the approximate problem is shown using the fixed-point theorem of Schauder. This theorem is applied to the composition of the solution operator for the contact problem with given temperature field and the solution operator for the heat equation problem with known displacement field. To obtain this proof, the unique solvability of both problems is necessary. Due to this reason it is necessary to introduce the penalty method. While the penalized contact problem has a unique solution, this is not clear for the original contact problem. The solvability of the original frictional contact problem is verified by an investigation of the limit for vanishing penalty parameter.

Conical differentiability for evolution variational inequalities

Abstract The conical differentiability of solutions to the parabolic variational inequality with respect to the right-hand side is proved in the paper. From one side the result is based on the Lipschitz continuity in H 1 2 ,1 (Q) of solutions to the variational inequality with respect to the right-hand side. On the other side, in view of the polyhedricity of the convex cone K={v∈ H ;v |Σ c ⩾0,v |Σ d =0}, we prove new results on sensitivity analysis of parabolic variational inequalities. Therefore, we have a positive answer to the question raised by Fulbert Mignot (J. Funct. Anal. 22 (1976) 25–32).

SOLVABILITY OF DYNAMIC CONTACT PROBLEMS FOR ELASTIC VON KÁRMÁN PLATES

The existence of solutions is proved for unilateral dynamic contact problems of elastic von Karman plates. Boundary conditions for a free and clamped plate are considered.

Solvability of a rational contact model with limited interpenetration in viscoelastodynamics

A rational model of a dynamical contact of a viscoelastic body with its support is presented. It is assumed that the contact is frictionless and permits a limited interpenetration which is prescribed. A weak formulation of the problem is given and the existence of its solutions is proved. It is also shown that if the depth of the interpenetration tends to zero, the solutions converge to a solution of the Signorini contact (without interpenetration).

A vibrating thermoelastic plate in a contact with an obstacle

Abstract We deal with a dynamic contact problem for a thermoelastic plate vibrating against a rigid obstacle. Dynamics is described by a hyperbolic variational inequality for deflections. The plate is subjected to a perpendicular force and to a heat source. The parabolic equation for the thermal strain resultant contains the time derivative of the deflection. We formulate a weak solution of the system and verify its existence using the penalization method.

Dynamic Contact Problem for Viscoelastic von Kármán-Donnell Shells

We deal with initial-boundary value problems describing vertical vibrations of viscoelastic von Karman-Donnell shells with a rigid inner obstacle. The short memory (Kelvin-Voigt) material is considered. A weak formulation of the problem is in the form of the hyperbolic variational inequality. We solve the problem using the penalization method.

On hyperbolic contact problems

Abstract We deal with hyperbolic variational inequalities modeling vibrations of two-dimensional structures with an obstacle. We focus on the plates with moderately large deflections. The nonlinear strain-displacements relations imply nonlinear elliptic parts of differential operators in considered problems.We distinguish two types of problems. In the first case only the deflections are considered with accelerations and the plane displacements are expressed using the Airy stress function. In the case of plane accelerations the full von K´arm´an system consisting of two equations and one variational inequality is considered. The existence of solutions is derived using the penalization method.