Mathematics

While in bulk shape memory materials interfaces separate austenite and a very fine mixture of variants of martensite, in thin films the formation of a wide variety of exact austenite-martensite interfaces is possible. These are used to construct some deformations, tents and tunnels, which are energy minimizing and could be used as the basis of simple actuators. In the proposed model, a deformable film subject to hydrostatic pressure on its lower surface is considered. Following an approach first introduced by Bhattacharya and James, a direct derivation of a theory of a very thin film from three dimensional nonlinear elasticity is given. The theory is then specialized to a film in the form of a long strip made of a material which can undergo a martensitic phase transformation. It is found that, if the pressure is less than a temperature-dependent critical value, the film bulges up in a tunnel-like configuration and the resulting microstructure is a fine mixture of austenite and martensite. If the pressure is increased above the critical value, only martensite is present. In both cases, the cross-section of the tunnel takes the form of a circular arch.