Nonlinear thermoviscoelastic Timoshenko beams with dynamic frictional contact

ABSTRACT

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.

Keywords:

• Nonlinear Timoshenko beam
• Kirchhoff type
• Coulomb's law of friction
• discontinuous frictional coefficient
• regularized variational formulation

2010 Mathematics Subject Classifications:

• 74M10
• 35M13
• 65M12

Disclosure statement

No potential conflict of interest was reported by the author(s).