Implementation of the Large Time Increment Method for the Simulation of Pseudoelastic Shape Memory Alloys

Abstract

The paper presents a numerical implementation of the Large Time Increment (LaTIn) method for the integration of the ZM model [1] for SMAs in the pseudoelastic range. LaTIn was initially proposed as an alternative to the conventional incremental approach for the integration of nonlinear constitutive models [2]. It is adapted here for the simulation of pseudoelastic SMA behavior and is shown to be especially useful in situations where the phase transformation process presents little to no hardening. In these situations, a slight stress variation during a load increment can result in large variation of the volume fraction of martensite within a representative volume element of the SMA. This can lead to difficulty in numerical convergence if the incremental method is used. LaTIn involves two stages: in the first stage a solution satisfying the conditions of static equilibrium is obtained for each load increment without considering the consistency with the phase transformation conditions, then in the second stage consistent increments of the local state variables are determined for the entire loading path. The two stages take place sequentially, in contrast to the incremental method that requires satisfying the global equilibrium and local consistency conditions simultaneously at a given load increment before proceeding to the next. The numerical integration algorithm consists of the following steps: 1. Division of the loading path into a finite number of increments, 2. Solution for all the load increments of the static equilibrium problem in which the local consistency conditions are relaxed, 3. Update of the state variables in accordance with the consistency conditions for all the load increments. Steps 2 and 3 are repeated until a solution is reached that satisfies simultaneously the equilibrium and consistency requirements. An algorithm is presented for the implicit integration of the time-discrete equations. The algorithm is used for finite element simulations using Abaqus, in which the model is implemented by means of a user material subroutine. The simulation results are discussed in comparison with those obtained using conventional step-by-step incremental integration.