SPH elastic dynamics

The standard smoothed particle hydrodynamics (SPH) formulation of fluid dynamics can exhibit an instability called the tensile instability. This instability may occur with both positive and negative pressure. Usually the effects are small, but in the case of elastic or brittle solids the effects may be severe. Under tension, a brittle solid can fracture, but it is difficult to disentangle the physical fracture and fragmentation from the nonphysical clumping of SPH particles due to the tensile instability. Recently, one of us (JJM) has shown how this instability can be removed by an artificial stress which introduces negligible errors in long-wavelength modes. In this paper we show how the algorithm can be improved by basing the artificial stress on the signs of the principal stresses. We determine the parameters of the artificial stress from the dispersion relation for elastic waves in a uniform material. We apply the algorithm to oscillating beams, colliding rings and brittle solids. The results are in very good agreement with theory, and with other high-accuracy methods.