Mathematical Physics - Volume II - Numerical Methods

5.11 Non-Local properties of SPH

187

superposition of the waves. The results for the strain-softening solution show clearly the consequences of strain-softening; a displacement discontinuity develops after the superposition in the localisation zone at x = 0 and this zone localises in an area of zero width (see Figure 5.9). This discontinuity cannot propagate outside this zone, as the type of PDE in this zone has changed to elliptic and interaction with areas x̸ = 0, which are governed by hyperbolic PDEs, is not possible. Consequently, strain grows to infinity, as illustrated in Figure 5.10, and simultaneously stress in the localisation zone drops to zero, see Figure 5.11. Outside the localisation zone, the bar unloads as release waves travel to the bar ends. One can observe that the softening zone effectively acts as a free boundary.

Figure 5.9: Elastic local and nonlo cal solutions for normalised longitu dinal displacement at t = 3 L / 2 c .

Figure 5.10: Elastic local and non local solutions for normalised longi tudinal strain at t = 3 L / 2 c .

Figure 5.12: Internal energy his tory for the local and nonlocal solu tions.

Figure 5.11: Elastic local and non local solutions for normalised longi tudinal stress at t = 3 L / 2 c .

5.11.2 Main Aspects of the Smoothed Particle Hydrodynamics (SPH) Method as Nonlocal Regularisation Method In the local theory of continuum stress only depends on the deformation history at a single point x . A nonlocal theory considers additionally the influence of the deformation of surrounding points, ξ , in a representative volume element (RVE). This is done by substituting the local variable η ( x ) by a weighted average ¯ η ( x ) of the variable in the point’s spatial neighbourhood.