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

top
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

top

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

top
  1. 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
  2. J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225-253.  Zbl0549.46019
  3. K. Bhattacharya, Theory of martensitic microstructure and the shape-memory effect, unpublished lecture notes.  
  4. 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
  5. 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
  6. 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
  7. 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
  8. F. John and L. Nirenberg, On functions of bounded mean oscillation. Comm. Pure Appl. Math.14 (1961) 415-426.  Zbl0102.04302
  9. F. John, Rotation and strain. Comm. Pure Appl. Math.14 (1961) 391-413.  Zbl0102.17404
  10. F. John, Bounds for deformations in terms of average strains, in Inequalities III, O. Shisha Ed., Academic Press (1972) 129-143.  
  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.  Zbl0287.73009
  12. R.V. Kohn, The relaxation of a double-well energy. Continuum Mech. Thermodyn.3 (1991) 193-236  Zbl0825.73029
  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.  Zbl0621.49008
  15. S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal.99 (1987) 189-212.  Zbl0629.73009
  16. Y.C. Shu and K. Bhattacharya, The influence of texture on the shape-memory effect in polycrystals. Acta Mater.46 (1998) 5457-5473.  

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.