Hysteretic models with stiffness and strength degradation in a mathematical programming format

In this paper, we present several hysteretic models formulated using an energy approach. In each case, the behavior of the model is completely described by specifying two scalar-valued functions—a stored energy function and a dissipation potential. Consequently, different types of mathematical programs arise in incremental non-linear analyses involving these models. It is relatively well-known how classical plasticity models can be described using an energy approach, and lead to mathematical programming problems. However, in this paper, we demonstrate that plasticity models with non-associated flow rules, softening plasticity or strength degradation models, and damage or stiffness degradation models can be represented in this framework as well. The energy approach serves to unify formulation and implementation of a broad class of hysteretic models. In addition, it helps motivate regularization strategies needed in optimization and inverse problems. The types of models considered in this paper are ones commonly applied in earthquake engineering. MATLAB implementations are included as online supplemental data with this paper to illustrate the conceptual simplicity of implementing models formulated using this approach.