Implicit integration for a model of martensite reorientation in shape memory alloys

We present an implicit integration scheme for a constitutive model for shape memory alloys that takes into account martensite detwinning and reorientation. This latter phenomenon, which is due to variant switching in the martensite phase, is allowed to occur independently of phase transformation and without being necessarily accompanied by variation in the magnitude of inelastic strain. The model is based on the ZM model for shape memory alloys [15, 14], where the evolution of inelastic strain is shown to be governed by two distinct loading functions: • The first is a J2 plasticity-like stress-based criterion, • The second expresses a requirement on the magnitude of inelastic strain not to exceed a certain threshold that depends on the SMA considered. This last loading condition is referred to hereinafter as the saturation condition. The derivation of the state equations for the shape memory alloy is subject to this constraint, which is accounted for mathematically through the use of a scalar Lagrange multiplier and a corresponding set of Kuhn-Tucker optimality conditions in presence of unilateral constraints. The increment of inelastic strain can be determined by the use of a suitable return-mapping algorithm to enforce consistency with the above inelastic loading criteria. We show that this requires solving a set of two polynomial equations at most, where the unknowns are an inelastic multiplier, analogous to the plastic multiplier for classical elastoplastic materials, and the Lagrange multiplier; both of which are scalars. The equations can be easily solved using Newton iterations. Most importantly, the direction of the increment of inelastic strain in strain space can be found directly, without requiring iterative updating like in the majority of the available literature through the use of so-called "cutting plane" algorithms. The model is used to simulate SMA structures subjected to multiaxial nonproportional loading, including situations where reorientation takes place at saturation, where the magnitude of inelastic strain remains equal to its maximum value.