Adaptivity in linear elastic fracture mechanics based on shape sensitivity analysis

If crack growth of an elastic body is viewed as a shape change we can use the well known concept of shape sensitivity analysis to compute the energy release rate. To do this, we adopt as cost function the total potential energy and as state equation the equilibrium equation. The shape derivative of the total potential energy stored in the cracked body Π˙ depends on the displacement field u and on the shape change velocity field V   which characterize the crack growth. Following this procedure the present paper deals with the derivation of a novel a posteriori error estimator which is an upper bound of the global error |Π˙-Π˙h|. This error estimator has been specifically designed to evaluate the energy release rate in mesh refinement or re-meshing procedures so as to obtain improved meshes for which the optimal rate of convergence is recovered even in a case of singularities. This novel estimator is capable to capture all source of errors for the energy release rate Π˙: the ones from stress concentration and the errors from the sensitivity of the solution to shape changes due to crack growth. Finally, well known three-dimensional examples of un-cracked and cracked body are considered in order to illustrate the potentiality of the proposed methodology.