Shape sensitivity analysis and the energy momentum tensor for the kinematic and static models of torsion

Abstract

The aim of this paper is to bridge shape sensitivity analysis and configurational mechanics by means of a widespread use of the shape derivative concept. This technique will be applied as a systematic procedure to obtain the Eshelby’s energy momentum tensor associated to the problem under consideration. In order to highlight special features of this procedure and without loss of generality, we focus our attention in the application of shape sensitivity analysis to the problem of twisted straight bars within the framework of linear elasticity. Kinematic and static variational formulations as well as the direct method of sensitivity analysis are used to perform shape derivatives of both models. Integral expressions of first and second order shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross-section shape change, are achieved. These integral expressions put in evidence the relationship between shape sensitivity analysis and the first and second order Eshelby’s energy momentum tensors. Also, the null divergence property of these tensors is easily proved by comparing, in each case, the domain and boundary integral shape derivative arrived at. Finally, an example with a known exact solution, corresponding to an elastic bar with elliptical transverse cross-section submitted to twist, is presented in order to illustrate the usefulness of these tensors to compute the corresponding shape derivatives.