Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction

Abstract

The paper studies the evolution of the thermomechanical and electric state of a thermoviscoelastic thermistor that is in frictional contact with a reactive foundation. The mechanical process is dynamic, while the electric process is quasistatic. Friction is modeled with a nonmonotone relation between the tangential traction and tangential velocity. Frictional heat generation is taken into account and so is the strong dependence of the electric conductivity on the temperature. The mathematical model for the process is in the form of a system that consists of dynamic hyperbolic subdifferential inclusion for the mechanical state coupled with a nonlinear parabolic equation for the temperature and an elliptic equation for the electric potential. The paper establishes the existence of a weak solution to the problem by using time delays, a priori estimates and a convergence method.

Keywords:

• Thermoviscoelastic thermistor
• temperature dependent electric conductivity
• evolution hemivariational inequality
• frictional contact
• time delay
• existence of a weak solution

AMS Subject Classifications:

• 35J25
• 35K05
• 47J20
• 49J52
• 49J53
• 74F05
• 74H10
• 74H20

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme [grant agreement number 295118]; and the National Science Center of Poland under the Maestro Advanced [project number DEC-2012/06/A/ST1/00262].