An F‐Bar B‐Spline Material Point Method for Capturing Size‐Dependent Strain Localization With Finite Strain Gradient Plasticity

The classical material point method (MPM) is particularly suitable for large deformation problems with surface contact, but its capability to capture size effects remains relatively limited. In order to capture the size effect widely observed in solid materials as well as to eliminate mesh dependency, we present an F‐bar B‐spline MPM for solving finite strain‐gradient plasticity in the micromorphic approach. A multiplicative finite strain thermoplasticity model is first regularized in a micromorphic approach, where a global internal variable is introduced to take into account the size effect. An implicit B‐spline MPM is developed to solve the coupled problem in a monolithic manner with consistent linearization. This implementation provides a general framework to incorporate phenomenological elasto‐plasticity models while preserving the prescribed size effect in the MPM. Although B‐spline basis function is effective in suppressing the volumetric‐locking deformation pattern when volume‐preserving plasticity model is used, severe stress oscillation may manifest when the deformation mode approaches the incompressible limit. Therefore, we propose an F‐bar method for the implicit B‐spline MPM to suppress the spurious stress oscillation. The exact linearization of the F‐bar method and its coupling with the balance equation of micromorphic momentum are derived in closed form. Three representative numerical examples are presented to validate our implementation and demonstrate the advantages of our method. Results show that the proposed method is effective in capturing the size effect at extreme conditions with large distortion and contact, and the F‐bar method suppresses the spurious stress oscillations associated with volume‐preserving plastic flow. One limitation is that the convergence behavior is less satisfactory when strain‐softening model is used due to the inherent limitation of the implicit MPM.