Weakly Nonlinear Oscillations Superimposed on Finite Circumferential Shear of a Compressible, Nonlinear, Viscoelastic, Isotropic Material

Abstract

A study is presented of the weakly nonlinear oscillatory response of a compressible, nonlinear, viscoelastic, isotropic material. In its reference configuration, the material takes the form of a hollow concentric circular cylinder bonded to a fixed support at its inner boundary and to a rigid shell at its outer boundary. The cylinder is brought to a deformed equilibrium state in which the outer shell has been rotated through a finite angle about the axis of the cylinder. The material model used combines linear damping with a generalized Blatz-Ko model. The parameters in this model are chosen so that this static deformation consists of circumferential shear, without radial displacement, in which case the normal stresses are self-equilibrating and no local volume change is induced. The outer shell is then subjected to a sinusoidal rotational disturbance, and the subsequent motion of the cylinder is studied. It is found that due to dynamic effects the normal stresses are no longer self-equilibrated and radial motion is induced. The method of multiple scales is used to analyze the case where the motion is small but finite. This problem has unusual features for which special techniques for using the method of multiple scales were developed. The particular numerical results presented in this paper are for steady-state solutions to the problem of primary resonance of the first mode; however, the method is readily extendible. Results are given for both the purely elastic case and the case with linear damping. Many of the phenomena often associated with nonlinear oscillations occur, including mode saturation, the possibility of jump phenomena, and quadratic shift; however, the interpretation of these phenomena is complicated by the interaction of the spatial and temporal distributions of the deformation. These phenomena are examined by considering the influence of various system parameters on the moment that must be placed on the outer cylinder to maintain the motion. Of particular interest is the result that even a slight amount of compressibility has a significant impact on the motion.