# 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

## Access Full Article

top## Abstract

top## How to cite

topKohn, 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

top- J.M. Ball and R.D. James, Proposed experimental test of a theory of fine microstructures and the two-well problem. Phil. Trans. R. Soc. Lond. A338 (1992) 389-450. Zbl0758.73009
- J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225-253. Zbl0549.46019
- K. Bhattacharya, Theory of martensitic microstructure and the shape-memory effect, unpublished lecture notes.
- K. Bhattacharya and R.V. Kohn, Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials. Arch. Rat. Mech. Anal.139 (1998) 99-180. Zbl0894.73225
- A. Braides, Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci. Detta XL, V. Ser., Mem. Mat.9 (1985) 313-322. Zbl0582.49014
- O.P. Bruno and G.H. Goldsztein, A fast algorithm for the simulation of polycrystalline misfits: martensitic transformations in two space dimensions, Proc. Roy. Soc. Lond. Ser. A (to appear). Zbl0968.74063
- O.P. Bruno and G.H. Goldsztein, Numerical simulation of martensitic transformations in two- and three-dimensional polycrystals, J. Mech. Phys. Solids (to appear). Zbl0966.74078
- F. John and L. Nirenberg, On functions of bounded mean oscillation. Comm. Pure Appl. Math.14 (1961) 415-426. Zbl0102.04302
- F. John, Rotation and strain. Comm. Pure Appl. Math.14 (1961) 391-413. Zbl0102.17404
- F. John, Bounds for deformations in terms of average strains, in Inequalities III, O. Shisha Ed., Academic Press (1972) 129-143.
- F. John, Uniqueness of Non-Linear Elastic Equilibrium for Prescribed Boundary Displacements and Sufficiently Small Strains. Comm. Pure Appl. Math.25 (1972) 617-635. Zbl0287.73009
- R.V. Kohn, The relaxation of a double-well energy. Continuum Mech. Thermodyn.3 (1991) 193-236 Zbl0825.73029
- R.V. Kohn and V. Lods, in preparation (1999).
- R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems II. Comm. Pure Appl. Math.34 (1986) 139-182. Zbl0621.49008
- S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal.99 (1987) 189-212. Zbl0629.73009
- Y.C. Shu and K. Bhattacharya, The influence of texture on the shape-memory effect in polycrystals. Acta Mater.46 (1998) 5457-5473.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.