A Total-Lagrangian Material Point Method for solid mechanics problems involving large deformations

Abstract

The material point method (MPM) has found successful applications in many engineering problems involving large displacement, large deformation and contacts. The standard MPM formulation, which adopts piece-wise linear basis functions, suffers from the so-called cell-crossing instability, low order of convergence and numerical fracture. Modifications have been made to this standard MPM to mitigate these issues: B-spline MPM (BSMPM), the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) all decrease cell-crossing instabilities and increase the order of convergence, but only CPDI effectively suppresses numerical fracture. However, these methods, CPDI in particular, significantly increase the method’s implementation and computational complexity. This paper presents a total Lagrangian MPM, dubbed TLMPM, that overcomes the issues of the conventional MPM while being more efficient and easier to implement than CPDI. The method is used for impact analyses of a cylinder bar made of steel and necking and fracture of cylinder alloy specimens. The numerical solutions are in satisfactory agreement with the experiment data. No numerical fracture occurred for simulations involving very large tensile deformation without special treatment such as done in the CPDI. Convergence analyses using the method of manufactured solutions show that the TLMPM is second-order accurate for problems of which boundary is axis-aligned. For the challenging generalized vortex problem, it also converges quadratically for relatively coarse meshes. Moreover, the model is able to simulate physically based fracture using continuum damage mechanics.