Implicit integration and consistent tangent modulus of a time-dependent non-unified constitutive model

This paper describes the implicit integration and consistent tangent modulus of an inelastic constitutive model with transient and steady strain rates, both of which are time- and temperature-dependent; the transient rate is in uenced by the evolution of back stress decomposed into parts, while the steady rate depends only on applied stress and temperature. Such a non-uni ed model is useful for high-temperature structural analysis and is practical owing to the ease in determining material constants. The implicit integration is shown to result in two scalar-valued coupled equations, and the consistent tangent modulus is derived in a quite versatile form by introducing a set of fourth-rank constitutive parameters into the discretized evolution rule of back stress. The constitutive model is, then, implemented in a nite element program and applied to a lead-free solder joint analysis. It is demonstrated that the implicit integration is very accurate if the multilinear kinematic hardening model of Ohno and Wang is employed, and that the consistent tangent modulus certainly a ords quadratic convergence to the Newton–Raphson iteration in solving nodal force equilibrium equations