Nonlinear oscillations of a rod with residual stress in the presence of contact dry friction

Abstract

Wave processes are studied in an elastic rod interacting with its environment according to the law of dry friction. Oscillations of the system at residual stresses are emerged under the influence of instantaneous unloading. Analytical solution of the partial differential equation of hyperbolic type with a nonlinear mechanism of energy dissipation is obtained. Thus, it makes it possible to describe the oscillatory process at various loading parameters of the system for an arbitrary section of the rod. The property, that for the class of problems under consideration dry friction does not change the natural frequency of the system, is shown additionally. The problem can be considered as a model for describing dynamic processes in rocks under the effect of internal stresses. Results of the modeling problem allow us to conclude: if we consider that rocks resist differently to tension and compression, such as resistance to tension is less than compression’ resistance, it is obvious that in the case of relatively rapid unloading of the system, the residual compressions will create stresses that correspond to the first maximal displacements of the end section, and that exceed the limit of tensile strength as well. In this case, the edge sections undergo the brittle fracture that can be sustained by the tensile stresses of the subsequent maximum displacements.