Meshless discretisation of nonlocal damage theories

Abstract

In this chapter, meshless discretisation methods are explored in the implementation of nonlocal continuum damage theories. Integral-type and gradient-type nonlocality are both considered. The main advantage of using a meshless implementation (compared to more established discretisation methods such as the finite element method) is that the higher-order continuity requirements imposed by gradient-type nonlocality can be accomodated straightforwardly Thus, meshless methods are particularly suited as an implementational framework to test and compare various nonlocal theories. Here, the element-free Galerkin (EFG) method is used. In particular, second-order and fourth order gradient damage models are compared to integral-type damage models whereby the integral nonlocal operator acts on the equivalent strain or on the displacements. No signficant differences in response are found, which implies that the inclusion of a fourth-order term in the gradient-type nonlocality is of lesser importance. Finally, the mathematical non-locality of EFG interpolation functions is tested to ascertain whether it provides a mechanical nonlocality to the description. It is shown that this is not the case. However, despite this lack of intrinsic mechanical nonlocality, the EFG method is an excellent tool for the numerical implementation of a nonlocal continuum theory.