Regularization of material instabilities by meshfree approximations with intrinsic length scales

Meshfree approximation, such as Moving Least Square (MLS) and Reproducing Kernel (RK) approximations, possess intrinsic non-local properties. These non-local properties of meshfree approximations are exploited to incorporate an intrinsic length scale which regularizes problems with material instabilities. The discrete equilibrium equation is obtained by employing an assumed strain method in the Galerkin approximation. This proposed method is essentially uniformly non-local, but in contrast to non-local "nite elements, no kinematic modes are observed. Gradient-type regularization can also be modelled by this method without the additional boundary conditions and other complications of the conventional gradient methods. Numerical examples show that the displacement-based MLS/RK formulation (1-level regularization) is su$cient to remedy mesh-sensitivity in damage-induced strain localization. For strain localization associated with plasticity, a two-level MLS/RK regularization in displacement and strain shown to be e!ective.