Instabilities due to strain-softening solved using the SPH method

The local continuum damage models used with the quasi brittle materials can lead to strain softening and an ill-posed boundary value problem, when the character of governing partial differtial equations changes locally, leading to a mesh sensitive numerical instability. This work primarily considered the strain softening effects in the SPH spatial discretization, combined with a local continuum damage model, which had been observed to lead to the instabilities in the classic FEM [1]. Simulation programme presented in here conists of analysis of an one dimensional wave propagation problem that was analytically solved by Bažant in [2] and a cube high velocity impact on a flat quasi brittle panel. The first set of results demonstrate that width of the strain softening region in the SPH is controlled by the smoothing length rather than discretisation density, which means that the SPH method is inherently non-local and suggests that the SPH smoothing length should be linked to the material characteristic length scale in solid mechanics simulations. The second set of results demonstrates that the SPH provides stable and satisfactory solutions for a high velocity impact case, which will be used for further validation of the numerical tools developed within the project EXTREME.