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

Gradient-dependent damage formulations incorporate higher-order derivatives of state variables in the consti- tutive equations. Di erent formulations have been derived for this gradient enhancement, comparison of which is di cult in a nite 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 di erent gradient enhanced models with the non-local damage model. The formulation of the additional boundary conditions is addressed. Nu- merical examples show the objectivity with respect to the discretization and the di erences between various gradient formulations with second- and fourth-order derivatives. It is shown that with the same underlying internal length scale, very di erent results can be obtained.