Secondary Deformations Due to Axial Shearing of the Annular Region Between Two Eccentrically Placed Cylinders

Fosdick and Kao 1] extended a conjecture of Ericksen's 2] for non-linear fluids, to non-linear elastic solids, and showed that unless the material moduli of an isotropic elastic material satisfied certain special relations, axial shearing of cylinders would be necessarily accompanied by secondary deformations if the cross-section were not a circle or the annular region between two concentric circles. Further, they used the driving force as the small parameter for a perturbation analysis and showed that the secondary deformation will occur at fourth order, much in common with what is known for non-linear fluids. Here, we show that if on the other hand the driving force is not small (of O(1)), but the departure of the cylinder from circular symmetry is small, then secondary deformations appear at first order, the parameter for perturbance being the divergence from circular symmetry. This revised version was published online in August 2006 with corrections to the Cover Date.