Smoothed polygonal finite element method for generalized elastic solids subjected to torsion

Abstract

Orthopaedic implants made of titanium alloy such as Ti-30Nb-10Ta-5Zr (TNTZ-30) are biocompatible and exhibit nonlinear elastic behavior in the ‘small’ strain regime (Hao et al., 2005). Conventional material modeling approach based on Cauchy or Green elasticity, upon linearization of the strain, inexorably leads to Hooke’s law which is incapable of describing the said nonlinear response. Recently, Rajagopal introduced a generalization of the theory of elastic materials (Rajagopal, 2003, 2014), wherein the linearized strain can be expressed as a nonlinear function of stress. Consequently, Devendiran et al. (2016) developed a thermodynamically consistent constitutive equation for the generalized elastic solid, in order to capture the response of materials showing nonlinear behavior in the small strain regime. In this paper, we study the response of a long cylinder made of TNTZ-30 with non-circular cross section subjected to end torsion. An explicit form of the constitutive equation derived in Devendiran et al. (2016) is used to study the response of the cylinder. The cross-section is discretized with quadratic serendipity polygonal elements. A novel one point integration rule is presented to compute the corrected derivatives, which are then used to compute the terms in the stiffness matrix. Unlike the conventional Hooke’s law, the results computed using the new constitutive equation show stress softening behavior even in the small strain regime.