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

Published online by Cambridge University Press:  15 April 2002

Robert V. Kohn and
Barbara Niethammer
Robert V. Kohn
Affiliation:
Courant Institute, 251 Mercer Street, New York University, New York, NY 10012. (kohn@cims.nyu.edu)
Barbara Niethammer
Affiliation:
Inst. für Angew. Math., Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany. (igel@iam.uni-bonn.de)
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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 someshape-memory materials recover as much as 10 percent strain. This paperprovides the first progress toward an analogous geometrically nonlineartheory. We consider a model problem, involving polycrystals madefrom a two-variant elastic material in two space dimensions. The lineartheory predicts that a polycrystal with sufficient symmetry can have norecoverable strain. The nonlinear theory corrects this to the statement thata polycrystal with sufficient symmetry can have recoverable strain nolarger than the 3/2 power of the transformation strain. This result isin a certain sense optimal. Our analysis makes use of Fritz John'stheory of deformations with uniformly small strain.

Keywords

Shape memory polycrystalsrecoverable strainnonlinear homogenization.

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