Coupled torsion and inflation of functionally graded and residually stressed Mooney–Rivlin hollow cylinders

Abstract

Functionally graded structures have material properties continuously varying in one or more directions. Examples include human teeth, sea shells, bamboo stems, and mammal organs in which the varying volume fraction and orientation of fibers optimize their functionalities. Here we analytically study large deformations due to combined torsion and inflation of residually stressed and radially graded Mooney–Rivlin hollow circular cylinders to provide insights into how grading their material moduli according to a power law function of the radius can be advantageously used. We simulate residual stresses in a thin wall hollow cylinder by inverting it inside out and in a thick wall cylinder by assuming that a longitudinal wedge opening parallel to the cylinder axis is closed by deforming it axisymmetrically. It is found that positive integer values of the power law exponent are generally advantageous but its negative values can have deleterious effects on stress distributions. Furthermore, a hollow cylinder with residual stresses generated by one of the foregoing methods in the reference configuration should be modeled as orthotropic rather than transversely isotropic for subsequent deformations. The analytical solutions provided here should help numerical analysts verify their algorithms for large deformations of rubber-like materials that are modeled by the Mooney–Rivlin relation, and design engineers for exploiting the radial gradation of the two material moduli to reduce structure’s mass without sacrificing its performance.