Mathematical Physics - Volume II - Numerical Methods

5.11 Non-Local properties of SPH

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Experiment 3 - Influence of constant smoothing length, ∆ p = variable, λ =variable, h = 2 . 5 [ mm ] Experiment 3 demonstrates the behaviour of SPH for different interparticle distances, subjected to a fixed smoothing length size, h = 2 . 5 [ mm ] . The particle density, i.e. the number of neighbouring particles, was changed, by varying the interparticle distance used in the model. Figure 5.34 shows the stress-strain curves for the three particle densities considered. It is clear that all bars underwent linear strain-softening behaviour. Damage distribution at response time t = 3 / 2 · L / c e is shown in Figure 5.35 for the central particle of the bar. The results indicate that the size of the damaged zone did not depend on the interparticle distance ∆ p , i.e. particle density. However, the damage peak value was dependant on particle density (the lowest value obtained for the highest particle density).

Figure 5.34: Longitudinal stress vs. longitu dinal strain curves for the central particle obtained with different values of ∆ p ; SPH-experiment 3.

Figure 5.35: : Damage distribution obtained with different particle densities at response time t = 3 / 2 · L / c e ; SPH-experiment 3.

Figure 5.36: Analytical solution and the nu merical results for longitudinal displacement at re sponse time t = 3 / 2 · L / c e ; SPH-experiment 3.

Figure 5.37: : Analytical solution and the numerical results for longitudinal strain at response time t = 3 / 2 · L / c e ; SPH-experiment 3.

The distribution of longitudinal displacement, longitudinal strain and longitudinal stress shown in Figure 5.36, Figure 5.37 and Figure 5.38, respectively, corroborate the above statement and are independent of the particle density. The effects of damage are smoothed out over an increasing number of particles with increasing interparticle distance.