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Stability of microstructure for tetragonal to monoclinic martensitic transformations

Published online by Cambridge University Press:  15 April 2002

Pavel Belik  and
Mitchell Luskin
Pavel Belik
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA; (belik@math.umn.edu)
Mitchell Luskin
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA; (luskin@math.umn.edu)
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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.

Keywords

Martensitic transformationmicrostructurenonconvex variational problemsimple laminatetetragonalmonoclinicvolume fractionYoung measurefinite elementerror estimate
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 3 , May 2000 , pp. 663 - 685
Copyright
© EDP Sciences, SMAI, 2000

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References

R. Adams. Sobolev Spaces. Academic Press, New York (1975).
Ball, J. and James, R., Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52. CrossRef
Ball, J. and James, R., Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A 338 (1992) 389-450. CrossRef
Bhattacharya, K., Self accomodation in martensite. Arch. Rat. Mech. Anal. 120 (1992) 201-244. CrossRef
K. Bhattacharya and G. Dolzmann, Relaxation of some multiwell problems, in Proc. R. Soc. Edinburgh: Section A, to appear.
Bhattacharya, K., Li, B. and Luskin, M., The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation. Arch. Rat. Mech. Anal. 149 (2000) 123-154. CrossRef
Brighi, B. and Chipot, M., Approximation of infima in the calculus of variations. J. Comput. Appl. Math. 98 (1998) 273-287. CrossRef
Carstensen, C. and Plechác, P., Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. , 66 (1997) 997-1026. CrossRef
Carstensen, C. and Plechác, P., Adaptive algorithms for scalar non-convex variational problems. Appl. Numer. Math. 26 (1998) 203-216. CrossRef
Chipot, M., Numerical analysis of oscillations in nonconvex problems. Numer. Math. 59 (1991) 747-767. CrossRef
Chipot, M. and Collins, C., Numerical approximations in variational problems with potential wells. SIAM J. Numer. Anal. 29 (1992) 1002-1019. CrossRef
Chipot, M., Collins, C., and Kinderlehrer, D., Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282 . CrossRef
Chipot, M. and Kinderlehrer, D., Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237-277. CrossRef
M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of nonconvex problems. (preprint, 1997).
Collins, C., Kinderlehrer, D., and Luskin, M., Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal. 28 (1991) 321-332. CrossRef
Collins, C. and Luskin, M., Optimal order estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp. 57 (1991) 621-637. CrossRef
B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin, (1989).
Dolzmann, G., Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621-1635. CrossRef
French, D., On the convergence of finite element approximations of a relaxed variational problem. SIAM J. Numer. Anal. 28 (1991) 419-436.
Jian, L. and James, R., Prediction of microstructure in monoclinic LaNbO4 by energy minimization. Acta Mater. 45 (1997) 4271-4281. CrossRef
Kinderlehrer, D. and Pedregal, P., Characterizations of gradient Young measures. Arch. Rat. Mech. Anal. 115 (1991) 329-365. CrossRef
Kruzík, M., Numerical approach to double well problems. SIAM J. Numer. Anal. 35 (1998) 1833-1849. CrossRef
Li, B. and Luskin, M., Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35 (1998) 376-392. CrossRef
B. Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructure. Math. Comp. 67(223) (1998) 917-946.
Li, B. and Luskin, M., Approximation of a martensitic laminate with varying volume fractions. Math. Model. Numer. Anal. 33 (1999) 67-87. CrossRef
Simultaneous, Z. Li numerical approximation of microstructures and relaxed minimizers. Numer. Math. 78 (1997) 21-38.
Luskin, M., Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75 (1996) 205-221. CrossRef
M. Luskin, On the computation of crystalline microstructure. Acta Numer. (1996) 191-257.
M. Luskin and L. Ma, Analysis of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal. 29 320-331.
Nicolaides, R. and Walkington, N., Strong convergence of numerical solutions to degenerate variational problems. Math. Comp. 64 (1995) 117-127. CrossRef
Pedregal, P., Numerical approximation of parametrized measures. Num. Funct. Anal. Opt. 16 (1995) 1049-1066. CrossRef
Pedregal, P., On the numerical analysis of non-convex variational problems. Numer. Math. 74 (1996) 325-336. CrossRef
Roubícek, T., Numerical approximation of relaxed variational problems. J. Convex Anal. 3 (1996) 329-347.
Simha, N., Crystallography of the tetragonal → monoclinic transformation in zirconia. J. Phys. IV Colloq. France 5 (1995) C81121-C81126. CrossRef
Simha, N., Twin and habit plane microstructures due to the tetragonal to monoclinic transformation of zirconia. J. Mech. Phys. Solids 45 (1997) 261-292. CrossRef
V. Sverák, Lower-semicontinuity of variational integrals and compensated compactness, in Proceedings ICM 94, Zürich (1995). Birkhäuser.
L. Tartar, Compensated compactness and applications to partial differential equations, in: Nonlinear analysis and mechanics, R. Knops, Ed., Pitman Research Notes in Mathematics, London 39 (1978) 136-212.
Zanzotto, G., 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.