Small amplitude sinusoidal disturbances superimposed on finite circular shear of a compressible, non-linearly elastic material

Abstract

Consider a hollow concentric circular cylinder of a non-linear compressible isotropic elastic material. In its reference configuration, the cylinder is 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. Deformation consists of the rotation of the cylindrical material surfaces, producing circumferential shear, and possibly their radial expansion or contraction. The latter effect arises from volume changes induced by the normal stresses associated with shear. The strain energy function of the material is assumed to be such that the deformation consists of circumferential shear, without radial displacement. In this 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 inertia causes the shearing motion to induce a radial motion. When the amplitude of the shell rotation is very small, the radial and shearing displacements of the cylinder vary sinusoidally with the same period as the disturbance. The depndence of this motion on the period of oscillation and the initial shell displacement is discussed.