Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack

Abstract

This work studies and simulates the dynamics of a Gao beam that may come in contact with a reactive foundation and which has an evolving symmetric edge crack. The problem is set as a variational inequality coupled with a differential equation for the evolution of the damage variable, which measures the severity of the crack. The existence of a local weak solution for the model is proved using approximations, result for abstract problems in Hilbert spaces, and a priori estimates. A finite-element scheme for the problem is proposed, and representative simulations presented. In particular, the dependence of the beam's vibration frequency on the crack's position and size is depicted. © The author 2011.