# Stability of microstructure for tetragonal to monoclinic martensitic transformations

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 3, page 663-685
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBelik, Pavel, and Luskin, Mitchell. "Stability of microstructure for tetragonal to monoclinic martensitic transformations." ESAIM: Mathematical Modelling and Numerical Analysis 34.3 (2010): 663-685. <http://eudml.org/doc/197438>.

@article{Belik2010,

abstract = {
We give an analysis of the stability and uniqueness of the simply
laminated microstructure for all three tetragonal to monoclinic
martensitic transformations. The energy density for tetragonal to
monoclinic transformations has four rotationally invariant wells since
the transformation has four variants. One of these tetragonal to
monoclinic martensitic transformations corresponds to the shearing of
the rectangular side, one corresponds to the shearing of the square
base, and one corresponds to the shearing of the plane orthogonal to a
diagonal in the square base. We show that the simply laminated
microstructure is stable except for a class of special material
parameters. In each case that the microstructure is stable, we derive
error estimates for the finite element approximation.
},

author = {Belik, Pavel, Luskin, Mitchell},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Martensitic transformation; microstructure; nonconvex
variational problem; simple laminate; tetragonal; monoclinic; volume
fraction; Young measure; finite element; error estimate; stability; uniqueness; simply laminated microstructure; tetragonal to monoclinic martensitic transformations; energy density; rotationally invariant wells; error estimates; finite element approximation},

language = {eng},

month = {3},

number = {3},

pages = {663-685},

publisher = {EDP Sciences},

title = {Stability of microstructure for tetragonal to monoclinic martensitic transformations},

url = {http://eudml.org/doc/197438},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Belik, Pavel

AU - Luskin, Mitchell

TI - Stability of microstructure for tetragonal to monoclinic martensitic transformations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 3

SP - 663

EP - 685

AB -
We give an analysis of the stability and uniqueness of the simply
laminated microstructure for all three tetragonal to monoclinic
martensitic transformations. The energy density for tetragonal to
monoclinic transformations has four rotationally invariant wells since
the transformation has four variants. One of these tetragonal to
monoclinic martensitic transformations corresponds to the shearing of
the rectangular side, one corresponds to the shearing of the square
base, and one corresponds to the shearing of the plane orthogonal to a
diagonal in the square base. We show that the simply laminated
microstructure is stable except for a class of special material
parameters. In each case that the microstructure is stable, we derive
error estimates for the finite element approximation.

LA - eng

KW - Martensitic transformation; microstructure; nonconvex
variational problem; simple laminate; tetragonal; monoclinic; volume
fraction; Young measure; finite element; error estimate; stability; uniqueness; simply laminated microstructure; tetragonal to monoclinic martensitic transformations; energy density; rotationally invariant wells; error estimates; finite element approximation

UR - http://eudml.org/doc/197438

ER -

## References

top- R. Adams. Sobolev Spaces. Academic Press, New York (1975). Zbl0314.46030
- J. Ball and R. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal.100 (1987) 13-52. Zbl0629.49020
- J. Ball and R. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A338 (1992) 389-450. Zbl0758.73009
- K. Bhattacharya, Self accomodation in martensite. Arch. Rat. Mech. Anal.120 (1992) 201-244. Zbl0771.73007
- K. Bhattacharya and G. Dolzmann, Relaxation of some multiwell problems, in Proc. R. Soc. Edinburgh: Section A, to appear. Zbl0977.74029
- K. Bhattacharya, B. Li and M. Luskin, The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation. Arch. Rat. Mech. Anal.149 (2000) 123-154. Zbl0942.74056
- B. Brighi and M. Chipot, Approximation of infima in the calculus of variations. J. Comput. Appl. Math.98 (1998) 273-287. Zbl0937.65071
- C. Carstensen and P. Plechác, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp., 66 (1997) 997-1026. Zbl0870.65055
- C. Carstensen and P. Plechác, Adaptive algorithms for scalar non-convex variational problems. Appl. Numer. Math.26 (1998) 203-216. Zbl0894.65029
- M. Chipot, Numerical analysis of oscillations in nonconvex problems. Numer. Math.59 (1991) 747-767. Zbl0712.65063
- M. Chipot and C. Collins, Numerical approximations in variational problems with potential wells. SIAM J. Numer. Anal.29 (1992) 1002-1019. Zbl0763.65049
- M. Chipot, C. Collins, and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math.70 (1995) 259-282 . Zbl0824.65045
- M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rat. Mech. Anal.103 (1988) 237-277. Zbl0673.73012
- M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of nonconvex problems. (preprint, 1997).
- C. Collins, D. Kinderlehrer, and M. Luskin, Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal.28 (1991) 321-332. Zbl0725.65067
- C. Collins and M. Luskin, Optimal order estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp.57 (1991) 621-637. Zbl0735.65042
- B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin, (1989). Zbl0703.49001
- G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal.36 (1999) 1621-1635. Zbl0941.65062
- D. French, On the convergence of finite element approximations of a relaxed variational problem. SIAM J. Numer. Anal.28 (1991) 419-436. Zbl0696.65070
- L. Jian and R. James, Prediction of microstructure in monoclinic LaNbO4 by energy minimization. Acta Mater.45 (1997) 4271-4281.
- D. Kinderlehrer and P. Pedregal, Characterizations of gradient Young measures. Arch. Rat. Mech. Anal.115 (1991) 329-365. Zbl0754.49020
- M. Kruzík, Numerical approach to double well problems. SIAM J. Numer. Anal.35 (1998) 1833-1849. Zbl0929.49016
- B. Li and M. Luskin, Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal.35 (1998) 376-392. Zbl0919.49020
- B. Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructure. Math. Comp.67(223) (1998) 917-946. Zbl0901.73076
- B. Li and M. Luskin, Approximation of a martensitic laminate with varying volume fractions. Math. Model. Numer. Anal.33 (1999) 67-87. Zbl0928.74012
- Z. Li, Simultaneous numerical approximation of microstructures and relaxed minimizers. Numer. Math.78 (1997) 21-38. Zbl0890.65067
- M. Luskin, Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math.75 (1996) 205-221. Zbl0874.73060
- M. Luskin, On the computation of crystalline microstructure. Acta Numer. (1996) 191-257. Zbl0867.65033
- M. Luskin and L. Ma, Analysis of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal.29 320-331. Zbl0760.65113
- R. Nicolaides and N. Walkington, Strong convergence of numerical solutions to degenerate variational problems. Math. Comp.64 (1995) 117-127. Zbl0821.65040
- P. Pedregal, Numerical approximation of parametrized measures. Num. Funct. Anal. Opt.16 (1995) 1049-1066. Zbl0848.65049
- P. Pedregal, On the numerical analysis of non-convex variational problems. Numer. Math.74 (1996) 325-336. Zbl0858.65059
- T. Roubícek, Numerical approximation of relaxed variational problems. J. Convex Anal.3 (1996) 329-347. Zbl0881.65058
- N. Simha, Crystallography of the tetragonal → monoclinic transformation in zirconia. J. Phys. IV Colloq. France5 (1995) C81121-C81126.
- N. Simha, Twin and habit plane microstructures due to the tetragonal to monoclinic transformation of zirconia. J. Mech. Phys. Solids45 (1997) 261-292.
- V. Sverák, Lower-semicontinuity of variational integrals and compensated compactness, in Proceedings ICM 94, Zürich (1995). Birkhäuser. Zbl0852.49010
- L. Tartar, Compensated compactness and applications to partial differential equations, in: Nonlinear analysis and mechanics, R. Knops, Ed., Pitman Research Notes in Mathematics, London39 (1978) 136-212.
- G. Zanzotto, Twinning in minerals and metals: remarks on the comparison of a thermoelasticity theory with some available experimental results. Atti Acc. Lincei Rend. Fis.82 (1988) 725-756. Zbl0737.73013

## Citations in EuDML Documents

top## NotesEmbed ?

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