Weakly nonlinear oscillations superimposed on finite circumferential shearof a compressible, nonlinear, viscoelastic, isotropic material

A study is presented of the weakly nonlinear oscillatory response of a comp ressible, nonlinear, viscoelastic, isotropic material. In its reference con figuration, the material takes the form of a hollow concentric circular cyl inder 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 st ate 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 w ith a generalized Blatz-Ko model. The parameters in this model are chosen s o that this static deformation consists of circumferential shear, without r adial displacement, in which case the normal stresses are self-equilibratin g and no local volume change is induced. The outer shell is then subjected to a sinusoidal rotational disturbance, and the subsequent motion of the cy linder is studied. It is found that due to dynamic effects the normal stres ses are no longer self-equilibrated and radial motion is induced. The metho d 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 nume rical results presented in this paper are for steady-state solutions to the problem of primary resonance of the first made; however, the method is rea dily extendible. Results are given for both the purely elastic case and the case with linear damping. Many of the phenomena often associated with nonl inear oscillations occur, including mode saturation, the possibility of jum p phenomena, and quadratic shift; however, the interpretation of these phen omena is complicated by the interaction of the spatial and temporal distrib utions 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 r esult that even a slight amount of compressibility has a significant impact on the motion.