A quasi-static contact problem in thermoviscoelastic diffusion theory

• The quasi-static contact problem of thermoviscoelastic diffusion rod is analyzed.
• The existence and uniqueness of solution is proved.
• The exponential decay of the weak solution is proved.
• Some numerical results are given.

The problem of thermoviscoelastic quasi-static contact between a rod and a rigid obstacle, when the diffusion effect is taken into account, is modeled and analyzed. The contact is modeled by the Signorini's condition and the stress–strain constitutive equation is of the Kelvin–Voigt type. In the quasi-static case, the governing equations correspond to the coupling of an elliptic and two parabolic equations. It poses some new mathematical difficulties due to the nonlinear boundary conditions. The existence of solutions is proved as the limit of solutions to a penalized problem. Moreover, we show that the weak solution converges to zero exponentially as time goes to infinity. Finally, we give some computational results where the influence of diffusion and viscosity are illustrated in contact.