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.

翻译：本文提出一种新颖的投影方法，用于处理高阶物质点法框架下小变形与大变形弹塑性分析中的近不可压缩性与体积锁死问题。物质点法因采用大量物质点以降低积分误差，易产生体积锁死现象。尽管传统物质点法中近不可压缩性处理已有较多研究，但高阶物质点法中的该问题尚未得到深入探讨。以Bbar与Fbar方法为出发点，我们针对背景离散采用高阶形函数的物质点法，发展了一种适用的投影技术。该方法依据传统Bbar与Fbar技术，将相应应变率度量的膨胀部分投影至低维近似空间，并针对物质点法特点进行专门设计。数值算例表明，所提方法能有效减小应力振荡，并完全消除体积锁死与沙漏现象。