Dispersion and isogeometric analyses of second-order and fourth-order implicit gradient-enhanced plasticity models

Implicit gradient plasticity models incorporate higher-order spatial gradients via an additional Helmholtz type equation for the plastic multiplier. So far, the enrichment has been limited to second-order spatial gradients, resulting in a formulation that can be discretised using C0-continuous finite elements. Herein, an implicit gradient plasticity model is formulated that includes a fourth-order gradient term as well. A comparison between the localisation properties of both the implicit gradient plasticity formulations and the explicit second-order gradient plasticity model is made using a dispersion analysis. The higher-order continuity requirement for the fourth-order implicit gradient plasticity model has been met by exploiting the higher-order continuity property of isogeometric analysis, which uses nonuniform rational B-splines as shape functions instead of Lagrange polynomials. The discretised variables, displacements, and plastic multiplier may require different orders of interpolation, an issue that is also addressed. Numerical results show that both formulations can be used as a localisation limiter, but that quantitative differences occur, and a different evolution of the localisation band is obtained for 2-dimensional problems.