Gradient Elasto-plasticity with the Generalised Interpolation Material Point Method

The modelling of geomechanics problems can require a method that allows large deformations and non-linear material behaviour, in this respect the Generalised Material Point Method (GIMPM) is ideal. A fully implicit version of GIMPM has recently been developed for geomechanics problems and some aspects of its implementation are described here. An area that has received less attention in material point methods is that conventional analysis techniques constructed in terms of stress and strain are unable to resolve structural instabilities such as shear banding. This is because they do not contain any measure of the length of the microstructure of the material analysed, such as molecule size or grain structure. Gradient theories provide extensions of the classical equations with additional higher-order terms. The use of length scales makes it possible to model a finite thickness shear band which is not possible with traditional methods. Much work has been done on using gradient theories to include the effect of microstructure in the finite element method (and other numerical analysis techniques) however this yet to be combined with material point methods. In this paper the key equations that are required to extend the implicit GIMPM method to include gradient elasto-plasticity are detailed