<p>The Material Point Method (MPM) is a numerical method commonly used for solid mechanics simulations involving large deformations, history-dependent materials, or complex interactions such as contact and fracture. MPM can be treated as a finite element method where the integration points are allowed to move independently of the mesh. Matrix-free implementations of finite element operators provide higher performance on modern supercomputing hardware due to the higher arithmetic intensity, as measured by FLOPs per byte of memory transferred from memory, and the lower memory bandwidth requirements per degree of freedom when compared to assembled sparse matrices. In this paper, we derive a mathematical formulation for evaluation of finite element bases at arbitrary quadrature points per element and demonstrate the performance of matrix-free implementation of these MPM operators on GPU hardware. The measured performance compares favorably with matrix-free finite element operators on elements without a tensor product structure.</p>