Frictional Wear of a Thermoelastic Beam

We present two versions of a mathematical model for the evolution of wear in a thermoelastic beam resulting from frictional contact with a rigid moving surface. One version is quasistatic and the other dynamic. We show that the quasistatic problem allows for the decoupling of the mechanical and thermal aspects of the process. The problem reduces to that of the heat equation with nonlinear and history-dependent boundary conditions. Then the displacements, shear stresses, and wear can be obtained by quadrature. We establish the existence of a local weak solution for the problem and a partial uniqueness result and obtain conditions for the solution’s further regularity. The dynamic problem consists of the heat equation coupled with the equation of motion and the Archard wear equation. We prove the existence and uniqueness of the local weak solution of this problem too