On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate

• A contact problem for a thermoelastic von Kármán plate is studied.
• The dynamics is described by a hyperbolic variational inequality.
• The thermal strain is described by parabolic equation.
• The acceleration and contact force are represented by spatio-temporal measures.
• A detailed analysis of the velocity, acceleration, and reaction force of the solution is given.

We study a dynamic contact problem for a thermoelastic von Kármán plate vibrating against a rigid obstacle. The plate is subjected to a perpendicular force and to a heat source. The dynamics is described by a hyperbolic variational inequality for deflections. The parabolic equation for a 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. A detailed analysis of the velocity, acceleration, and reaction force of the solution is given. The singular nature of the dynamic contact makes it necessary to treat the acceleration and contact force as time-dependent measures with nonzero singular parts in the zones of contact. Accordingly, the velocity field over the plate suffers (global) jumps at a countable number of times with natural physical interpretations of the signs of the jumps.