MPM stress update schemes#

Comparison of USF, USL, MUSL and USAVG. $x$ is the position vector, $v$ is the velocity, $a$ is the acceleration, $\epsilon$ is the strain tensor and $\sigma$ is the stress tensor. Subscripts, $p$ and $I$ represents values at material points and background nodes. $\mathrm{\Delta }{t}^{\ast }=\mathrm{\Delta }t$ is the time step size for USF, USL and MUSL. For USAVG, $\mathrm{\Delta }{t}^{\ast }=\frac{1}{2}\mathrm{\Delta }t$. (reproduced after Kularathna., 2018).

[1] Nairn, J. A. (2003). Material point method calculations with explicit cracks. Computer Modeling in Engineering and Sciences, 4(6), 649-664.

[2] Kularathna, S. (2018). Splitting solution scheme for material point method (Doctoral dissertation, University of Cambridge).