Numerical Analysis of Elastic Elliptic Membrane Shells in Frictional Contact with Wear

Abstract

[Abstract] We consider a family of linearly elastic and elliptic membrane shells, all sharing the same middle surface, with thickness 2ε, clamped along their entire lateral face, which upon deformation may enter in frictional contact with a moving foundation along its lower face. As a result of friction, material might be removed from the interface, thus causing wear. In this paper, we focus in the numerical analysis of our model and provide a priori error estimates for displacements, stress and wear fields. We have implemented the fully discrete scheme with the FreeFEM++ programming environment, by using an intrinsic (basis free) formulation, reduction of order, penalization and Newton method. We carried on a series of numerical simulations to demonstrate the capabilities of our code, providing experimental evidence of our theoretical results regarding the convergence with respect to the thickness parameter ε and mesh size h. As a practical application, we have also included the results of a simulation of the wear that undergoes the tyre of a vehicle braking on a road.