A sequentially updated projection method for integrating the rate form of plasticity

Abstract We propose a novel fully implicit algorithm named the Implicit Sequential Projection Method (ISPM) for integrating the rate form of plasticity. ISPM is inspired by the implicit construction of the Closest Point Projection Method (CPPM) and the Cutting Plane Method’s (CPM) treatment of state variables as functions of the plastic multiplier. Though all the state variables are treated as functions of the plastic multiplier, the evaluation of the derivatives of the state variables with respect to the plastic multiplier at the unknown solution makes ISPM a fully implicit algorithm similar to CPPM. ISPM iteratively updates the solution by projecting the state at the end of prior iteration to the yield surface, producing its characteristic sequential projection of the solution, hence the name. Furthermore, ISPM provides a straightforward to derive consistent tangent operator (CTO) with global convergence on par with CPPM. The mathematical formulation, main attributes, and geometric details of the return mapping are discussed in relation to CPPM and CPM, with an emphasis on showing the distinctness of the algorithm. Further, we establish the relationship between ISPM and CPPM. The versatility and accuracy of ISPM and its CTO are explored through a series of numerical tests conducted with three plasticity models. Tests that focus on material-level iterations show that the stress remapping of ISPM is as accurate as CPPM’s. Further, the mixed stress–strain loading history test conducted with non-linear elastic Drucker–Prager plastic material shows that ISPM can provide a runtime advantage over CPPM for some complex plasticity models involving all three stress invariants.