A method for smoothing multiple yield functions

Many models of plasticity are built using multiple, simple yield surfaces. Examples include geomechanical models and crystal plasticity. This leads to numerical difficulties, most particularly during the stress update procedure, because the combined yield surface is nondifferentiable, and when employing implicit time stepping to solve numerical models, because the Jacobian is often poorly conditioned. A method is presented that produces a single C² differentiable and convex yield function from a plastic model that contains multiple yield surfaces that are individually C² differentiable and convex. C² differentiability ensures quadratic convergence of implicit stress-update procedures; convexity ensures a unique solution to the stress update problem, whereas smoothness means the Jacobian is much better conditioned. The method contains just one free parameter, and the error incurred through the smoothing procedure is quantified in terms of this parameter. The method is illustrated through three different constitutive models. The method's performance is quantified in terms of the number of iterations required during stress update as a function of the smoothing parameter. Two simple finite-element models are also solved to compare this method with existing approaches. The method has been added to the open-source “MOOSE” framework, for perfect, nonperfect, associated, and nonassociated plasticity.