The main objective of this proposal is the development of mathematical tools for modeling the shape-memory effect and pseudo-elasticity in shape-memory polycrystals. For this purpose, the investigator makes use of suitable generalizations of the linear comparison homogenization methods for these material systems, incorporating the effects both of crystallographic texture and of morphological texture (as determined by the two-point correlations of the microstructures). The main mathematical challenge arises as a consequence of the lack of convexity (or, more precisely, quasi-convexity) of the relevant energy functions. It is known that this breakdown of convexity leads to the development of additional microstructures at the single crystal level, which lies at the heart of the shape-memory effect. Because of this, linear comparison estimates are also developed for the "relaxation" (or quasi-convexification) of single-crystal shape-memory alloys (SMAs), exploiting a recently uncovered connection between homogenization at the polycrystal level and relaxation at the single-crystal level. The project results in novel and highly efficient multi-scale modeling techniques capable of handling complex microstructures, coupled strongly nonlinear response, microstructure evolution, and the possible development of instabilities. The mathematical techniques developed in this work are of broad application to large classes of polymeric, metallic, biological, and geological material systems, including multi-functional materials, and lend themselves to numerical implementation in constitutive subroutines for use with standard numerical packages.
The shape-memory effect is the ability of certain materials to recover, upon heating, apparently permanent deformation sustained below a certain critical temperature. This effect is usually accompanied by pseudo-elasticity whereby single-crystal samples of these materials are observed to undergo fairly large strains (in the order of 10%) at nearly constant stress, which is fully recovered upon unloading. These two properties make shape-memory alloys (SMAs) very attractive as low-frequency, robust actuators and sensing devices for a variety of novel technological applications. However, SMAs are normally used in polycrystalline form, typically as wires, or thin strips and films, where the shape-memory effect is much reduced. For this reason, it is essential to understand and model the relations between the single-crystal behavior, the microstructure, and the macroscopic behavior of the polycrystals. Improvements in the modeling of SMA polycrystals should result in improved performance for these materials, which are being used extensively in industry, as actuators, sensors, couplings, and electrical connectors, as well as in an ever increasing number of medical and robotics applications.