Many crystalline materials exhibit solid-to-solid martensitic phase transformations in response to certain changes in temperature or applied load. These martensitic transformations result from a change in the stability of the material's crystal structure. It is, therefore, desirable to have a detailed understanding of the possible modes through which a crystal structure may become unstable. The current work establishes the connections between three crystalline stability criteria: phonon-stability, homogenized-continuum- stability, and the presently introduced Cauchy-Born-stability criterion. Stability with respect to phonon perturbations, which probe all bounded perturbations of a uniformly deformed specimen under "hard-device" loading (i.e., all around displacement type boundary conditions) is hereby called "constrained material stability". A more general "material stability" criterion, motivated by considering "soft" loading devices, is also introduced. This criterion considers, in addition to all bounded perturbations, all "quasi-uniform" perturbations (i.e., uniform deformations and internal atomic shifts) of a uniformly deformed specimen, and it is recommend as the relevant crystal stability criterion.
Bibliographical noteFunding Information:
This work was supported by the National Science Foundation Grant CMS 0409084 (Dr. Ken Chong, Program Director), the Department of Energy Computational Science Graduate Fellowship Program of the Office of Scientific Computing and Office of Defense Programs in the Department of Energy under contract DE-FG02-97ER25308 (for R. Elliott), a CAREER grant from the National Science Foundation (for J. Shaw).
- Asymptotic analysis
- Finite strain
- Phase transformation
- Stability and bifurcation