  We propose a novel projection method to treat near-incompressibility and
volumetric locking in small- and large-deformation elasticity and plasticity
within the context of higher order material point methods. The material point
method is well known to exhibit volumetric locking due to the presence of large
numbers of material points per element that are used to decrease the quadrature
error. Although there has been considerable research on the treatment of
near-incompressibility in the traditional material point method, the issue has
not been studied in depth for higher order material point methods. Using the
Bbar and Fbar methods as our point of departure we develop an appropriate
projection technique for material point methods that use higher order shape
functions for the background discretization. The approach is based on the
projection of the dilatational part of the appropriate strain rate measure onto
a lower dimensional approximation space, according to the traditional Bbar and
Fbar techniques, but tailored to the material point method. The presented
numerical examples exhibit reduced stress oscillations and are free of
volumetric locking and hourglassing phenomena.
