MPM stress update schemes

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