On the implementation of gradient plasticity with the material point method

Abstract

The Material Point Method (MPM) is a computational method which allows solid mechanics problems to be modelled using material points which move through a fixed background grid. State variables are stored at these material points and tracked throughout the simulation. The MPM is ideal for modelling geomechanics problems that require the ability to capture large deformations and non-linear material behaviour. A well documented grid crossing error exists in the MPM which occurs when material points move between grid elements. The Generalised Interpolation Material Point (GIMP) method was proposed to alleviate this problem [2]. An implicit implementation of the GIMP method is used in this work. Conventional analysis techniques constructed in terms of stress and strain are able to handle large deformations well, however they are unable to deal with structural instabilities such as shear banding. Because there is no measure relating to the microstructure of the analysed material, the width of a shear band is highly mesh dependent. Gradient theories enrich these conventional theories with the addition of higher-order terms to include a length scale. Using gradient methods it is possible to model a shear band with a finite thickness without it being mesh dependent. Although there has been lots of work on gradient theories within the Finite Element Method (FEM) it is an area which has received less attention in the MPM. In this work an existing gradient elasto-plasticity theory [4], used with the FEM, is applied to the GIMP method. The MPM and GIMP method are first introduced and the key equations that are required to include gradient elasto-plasticity are detailed. The effect of introducing a length scale is then demonstrated in a numerical example.