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

Robert V. Kohn; Barbara Niethammer

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 2, page 377-398
  • ISSN: 0764-583X

Abstract

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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 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.

How to cite

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Kohn, Robert V., and Niethammer, Barbara. "Geometrically nonlinear shape-memory polycrystals made from a two-variant material." ESAIM: Mathematical Modelling and Numerical Analysis 34.2 (2010): 377-398. <http://eudml.org/doc/197448>.

@article{Kohn2010,
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. },
author = {Kohn, Robert V., Niethammer, Barbara},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Shape memory polycrystals; recoverable strain; nonlinear homogenization.; geometrically nonlinear shape-memory polycrystals; two-variant elastic material; symmetry},
language = {eng},
month = {3},
number = {2},
pages = {377-398},
publisher = {EDP Sciences},
title = {Geometrically nonlinear shape-memory polycrystals made from a two-variant material},
url = {http://eudml.org/doc/197448},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Kohn, Robert V.
AU - Niethammer, Barbara
TI - Geometrically nonlinear shape-memory polycrystals made from a two-variant material
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 2
SP - 377
EP - 398
AB - 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.
LA - eng
KW - Shape memory polycrystals; recoverable strain; nonlinear homogenization.; geometrically nonlinear shape-memory polycrystals; two-variant elastic material; symmetry
UR - http://eudml.org/doc/197448
ER -

References

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  11. F. John, Uniqueness of Non-Linear Elastic Equilibrium for Prescribed Boundary Displacements and Sufficiently Small Strains. Comm. Pure Appl. Math.25 (1972) 617-635.  
  12. R.V. Kohn, The relaxation of a double-well energy. Continuum Mech. Thermodyn.3 (1991) 193-236  
  13. R.V. Kohn and V. Lods, in preparation (1999).  
  14. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems II. Comm. Pure Appl. Math.34 (1986) 139-182.  
  15. S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal.99 (1987) 189-212.  
  16. Y.C. Shu and K. Bhattacharya, The influence of texture on the shape-memory effect in polycrystals. Acta Mater.46 (1998) 5457-5473.  

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