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This paper focuses on mathematical and numerical analyses of Kirchhoff type nonlinear thermoviscoelastic Timoshenko beams with Coulomb friction law. The dynamic friction is described by a slip rate dependent coefficient function and frictional heat generation. The friction function is discontinuous and understood as a set-valued function. Due to nonsmooth frictional conditions, we employ a regularization technique to establish a nonlinear approximate variational problem. Then we use time discretizations to set up the corresponding numerical formulations. The convergence of numerical trajectories is proved by using necessary a priori estimates of the regularized problem which allow us to vanish time step sizes in the limits. The uniqueness of weak solutions is also proved. A finite element method is combined to propose the fully discrete numerical schemes. The Newton–Raphson method is used to find a fully discrete approximation at each time step. Several groups of data are selected to present numerical simulations.

Nonlinear thermoviscoelastic Timoshenko beams with dynamic frictional contact