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nonlinear homogenization.

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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Geometrically nonlinear shape-memory polycrystals made from a two-variant material
Robert V. Kohn, Barbara Niethammer
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 34 / Issue 2 / March 2000

Published online by Cambridge University Press:

15 April 2002, pp. 377-398

Print publication:

March 2000
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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.

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

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