Deformation of Shape-Memory Materials

Abstract

AbstractWe present an overview of a theory of martensitic transformations developed by J. M. Ball and the author, and additional related results by Bhattacharya, Chu and Collins and Luskin and Kinderlehrer. This theory is in a form that is amenable to detailed numerical computations of microstructure. We describe the energy minimizing microstructures of a single crystal under applied displacements and under detailed dead loading. The relation of the theory to the crystallographic theory of martensite is presented.A consequence of the theory is the sensitivity of the patterns of microstructure to the precise lattice parameters. In the case of the wedge-like microstructure studied by Bhattacharya, it is found that this microstructure is only possible as a coherent, energy minimizing microstructure if very restrictive conditions on the lattice parameters are satisfied. Materials that exhibit the wedge satisfy this relation closely. The analysis suggests that optimal shape-memory behavior may be related to relations of this type.The theory has a relation to the work of Khachaturyan, Roitburd and Shatalov. We compare and contrast the two approaches. We present results that suggest that any approach based on the kinematics of linear elasticity will make serious quantitative errors in the prediction of microstructure.