Dispersion analysis and element-free Galerkin solutions of second- and fourth-order gradient-enhanced damage models

Abstract

Gradient-dependent damage formulations incorporate higher-order derivatives of state variables in the constitutive equations. Different formulations have been derived for this gradient enhancement, comparison of which is difficult in a finite element context due to higher-order continuity requirements for certain formulations. On the other hand, the higher-order continuity requirements are met naturally by element-free Galerkin (EFG) shape functions. Thus, the EFG method provides a suitable tool for the assessment of gradient enhanced continuum models. Dispersion analyses have been carried out to compare different gradient enhanced models with the non-local damage model. The formulation of the additional boundary conditions is addressed. Numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second- and fourth-order derivatives. It is shown that with the same underlying internal length scale, very different results can be obtained.