### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 377-398 |

Number of pages | 22 |

Journal | Mathematical Modelling and Numerical Analysis |

Volume | 34 |

Issue number | 2 |

State | Published - Mar 2000 |

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### Keywords

- Nonlinear homogsnization
- Recoverable strain
- Shape memory polycrystals

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics
- Modeling and Simulation

### Cite this

*Mathematical Modelling and Numerical Analysis*,

*34*(2), 377-398.

**Geometrically nonlinear shape-memory polycrystals made from a two-variant material.** / Kohn, Robert; Niethammer, Barbara.

Research output: Contribution to journal › Article

*Mathematical Modelling and Numerical Analysis*, vol. 34, no. 2, pp. 377-398.

}

TY - JOUR

T1 - Geometrically nonlinear shape-memory polycrystals made from a two-variant material

AU - Kohn, Robert

AU - Niethammer, Barbara

PY - 2000/3

Y1 - 2000/3

N2 - 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.

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.

KW - Nonlinear homogsnization

KW - Recoverable strain

KW - Shape memory polycrystals

UR - http://www.scopus.com/inward/record.url?scp=0034401616&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034401616&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034401616

VL - 34

SP - 377

EP - 398

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 2

ER -