Implicit Integration by Linearization for High-Temperature Inelastic Constitutive Models

Abstract

In this study, a linearization approach is used to develop an implicit integration scheme for the high-temperature inelastic constitutive models based on non-linear kinematic hardening. A non-unified model is first considered in which inelastic strain rate is divided into the transient and steady parts driven, respectively, by effective stress and applied stress. By discretizing the constitutive relations using the backward Euler method, and by linearizing the resulting discretized relations, a tensor equation is derived to iteratively achieve the implicit integration of constitutive variables. The implicit integration scheme developed is shown to be applicable to a unified constitutive model in which back stress evolves due to static and dynamic recoveries in addition to strain hardening. The integration scheme is then programmed as a subroutine in a finite element code and applied to a lead-free solder joint analysis. It is thus demonstrated that the integration scheme affords the quadratic convergence of iteration even for considerably large increments, and that the non-unified and unified models give almost the same results as each other in the joint analysis.