A simple model for shape memory alloys under multiaxial non-proportional loading

The paper presents a mathematical description of shape memory alloys within the framework of continuum mechanics, where the spatially multidimensional case is considered. A new simple model is introduced, which correctly describes not only all well-studied shape memory effects (e.g. one-way-effect, pseudoelasticity) but also the more complex behavior (e.g. reorientation of stress-induced martensite). The key idea is a new set of internal state variables, which are averaged values for a representative volume element of the polycrystalline material. A tensor-valued variable describes the state of orientation of martensite. The relative volume fraction of stress induced martensite is defined by taking a certain tensor norm of this internal variable. A free energy is chosen and thermodynamical forces are derived. These forces are sufficient to define the onset of the phase transitions, so that we do not need to introduce transition surfaces explicitly within the evolution equations. Finite element discretizations for the approximation of the field variables (strain, stress) and finite difference approximations for the time integration of local variables (internal variables) are explained.