A Semi-implicit Algorithm for Damage-coupled Visco-plastic Fatigue Model of Solder Alloy

The paper describes a semi-implicit algorithm for a damage-coupled visco-plastic-fatigue material model. The material chosen for the investigation is a eutectic material, Sn-Pb solder, exhibiting strain-softening behavior. The stress (constitutive) Jacobian formula is derived. The semi-implicit gradient time integration procedure is implemented numerically in the commercial finite element (FE) code ABAQUS (Version 6.2) via the user-defined material subroutine UMAT. For illustration purposes, a series of numerical examples are discussed: (1) Single-element fatigue simulations to reveal the optimum combination of the user-specified tolerance and the prescribed load step size to obtain a desired accuracy at a minimum cost. (2) Two three-dimensional analyses for monotonic tensile loading and fatigue loading are conducted for a miniature specimen of solder to show the capability of the proposed procedure to deal with thermo-mechanical loading. (3) Simulation of a shear notched specimen under monotonic loading is compared with the test to illustrate the ability of this algorithm for the specimen that has a serious damage region. The numerical examples illustrate that the semi-implicit algorithm is effective for simulating cyclic thermo-viscoplastic behavior of the solder. The research can be applied to the simulation of thermo-viscoplastic behavior and fatigue life of softening materials.