Geometrically nonlinear shape-memory polycrystals made from a two-variant material

¹ Courant Institute, 251 Mercer Street, New York University, New York, NY 10012. (kohn@cims.nyu.edu)
² Inst. für Angew. Math., Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany. (igel@iam.uni-bonn.de)

Received: 11 August 1999

Abstract

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.