SPH as a nonlocal regularisation method: Solution for instabilities due to strain-softening

Within the framework of continuum damage mechanics (CDM), mechanical loading leads to material damage and consequent degradation of material properties. This can result in strain-softening behaviour, which when implemented as a local model in the finite element (FE) method, leads to an ill-posed boundary value problem, resulting in significant mesh sensitivity of the solution. It is well-known that the addition of a characteristic length scale to CDM models, a non-local approach, maintains the character of the governing equations. In this paper, the similarities between the Smooth Particle Hydrodynamic (SPH) method and non-local integral regularisation methods are discussed. A 1D dynamic strain-softening problem is used as the test problem for a series of numerical experiments, to investigate the behaviour of SPH. An analytical solution for the test problem is derived, following the solution for a 1D stress state derived by Bazˇant and Belytschko in 1985. An equivalent SPH model of the problem is developed, using the stable Total-Lagrange form of the method, combined with a local bi-linear elastic-damage strength model. A series of numerical experiments, using both SPH and FE solvers, demonstrate that the width of the strain-softening region is controlled by the element size in FE, but in SPH it is controlled by the smoothing length rather than the inter-particle distance, which is the analogous to the element size in the FE method. This investigation indicates that the SPH method is inherently non-local numerical method and suggests that the SPH smoothing length should be linked to the material characteristic length scale in solid mechanics simulations.