A model for shape memory alloys with the possibility of voids

Abstract

The paper is devoted to the study of a mathematical model for the thermomechanical evolution of metallic shape memory alloys. The main novelty of our approach consists in the fact that we include the possibility for these materials to exhibit voids during the phase change process. Indeed, in the engineering paper (60) has been recently proved that voids may appear when the mixture is produced by the aggregations of powder. Hence, the composition of the mixture varies (under either thermal or mechanical actions) in this way: the martensites and the austenite transform into one another whereas the voids volume fraction evolves. The first goal of this contribution is hence to state a PDE system capturing all these modelling aspects in order then to establish the well-posedness of the associated initial-boundary value problem. 1. Introduction. Shape memory alloys are mixtures of many martensites variants and of austenite. They exhibit an unusual behavior: even if they are permanently deformed, they can totally recover their initial shape just by thermal or mechanical means. There may be voids in the mixture, which may appear when the mixture is produced by the aggregations of powders, as it has been recently proved in the engineering paper (60). Of course, the voids are filled either with gas or air when appearing or when aggregating powders. We do not take into account the gas phase mechanical properties (which are mainly described in terms of pressure and temperature) because we focus on the mechanical behaviour of the solid mixture, i.e., we assume the volume fraction of voids is small. The composition of the mixture varies: the martensites and the austenite transform into one another whereas the voids volume fraction evolves. These phase changes can be produced either by thermal actions or by mechanical actions. The striking properties of shape memory alloys result from interactions between mechanical and thermal actions (cf., e.g., (9, 42)). We assume that the phases can coexist at each point and we suppose that, besides austenite, only two martensitic variants are present. However, this choice provides a sufficiently good description of the phenomenon, as we want describe a macroscopic predictive theory which can be used for engineering purposes. The