Finite Coupled Torsion and Inflation of Functionally Graded Mooney-Rivlin Cylinders with and without Residual Stresses

Abstract

Functionally graded structures have material properties that continuously vary in one or more directions. Examples include human teeth, seashells, bamboo stems and human organs, where the varying volume fraction of fibers and their orientations optimize functionality. Deformations of such structures typically involve bending, stretching, and shearing. An everyday example of shearing deformation is the twisting of wet fabrics to extract water. In this study, we analytically examine the large deformations of functionally graded Mooney-Rivlin circular cylinders, focusing on how radial grading of material moduli can be beneficially utilized. We investigate the finite deformations caused by pressures applied to the bounding surfaces and axial loads or twisting moments on the end faces. We also simulate residual stresses in a hollow cylinder either by inverting it inside out or by closing a longitudinal wedge opening parallel to the cylinder axis through axisymmetric deformation before other loads are applied. It is observed that the maximum shear stress in an initially stress-free Mooney-Rivlin cylinder can occur at an interior point. In the absence of axial forces on the end faces, the cylinder elongates when twisted, with the degree of elongation depending on the grading of the material moduli. These findings should aid numerical analysts in verifying their algorithms for simulating large deformations of rubber-like materials modeled by the Mooney-Rivlin relation.