Geometrically nonlinear shape-memory polycrystals made from a two-variant material
Courant Institute, 251 Mercer Street, New York University,
New York, NY 10012. (firstname.lastname@example.org)
2 Inst. für Angew. Math., Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany. (email@example.com)
Bhattacharya and Kohn have used small-strain (geometrically linear) elasticity to analyze the recoverable strains of shape-memory polycrystals. The adequacy of small-strain theory is open to question, however, since some shape-memory materials recover as much as 10 percent strain. This paper provides the first progress toward an analogous geometrically nonlinear theory. We consider a model problem, involving polycrystals made from a two-variant elastic material in two space dimensions. The linear theory predicts that a polycrystal with sufficient symmetry can have no recoverable strain. The nonlinear theory corrects this to the statement that a polycrystal with sufficient symmetry can have recoverable strain no larger than the 3/2 power of the transformation strain. This result is in a certain sense optimal. Our analysis makes use of Fritz John's theory of deformations with uniformly small strain.
Mathematics Subject Classification: 74B20 / 74Q20
Key words: Shape memory polycrystals / recoverable strain / nonlinear homogenization.
© EDP Sciences, SMAI, 2000