Stability of microstructure for tetragonal to monoclinic martensitic transformations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 34 (2000) no. 3, p. 663-685
@article{M2AN_2000__34_3_663_0,
     author = {B\v el\'\i k, Pavel and Luskin, Mitchell},
     title = {Stability of microstructure for tetragonal to monoclinic martensitic transformations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {34},
     number = {3},
     year = {2000},
     pages = {663-685},
     zbl = {0981.74042},
     mrnumber = {1763530},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2000__34_3_663_0}
}
Bělík, Pavel; Luskin, Mitchell. Stability of microstructure for tetragonal to monoclinic martensitic transformations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 34 (2000) no. 3, pp. 663-685. http://www.numdam.org/item/M2AN_2000__34_3_663_0/

[1] R. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[2] J. Ball and R. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52. | MR 906132 | Zbl 0629.49020

[3] J. Ball and R. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A 338 (1992) 389-450. | Zbl 0758.73009

[4] K. Bhattacharya, Self accomodation in martensite. Arch. Rat. Mech. Anal. 120 (1992) 201-244. | MR 1183551 | Zbl 0771.73007

[5] K. Bhattacharya and G. Dolzmann, Relaxation of some multiwell problems, in Proc. R. Soc. Edinburgh: Section A, to appear. | Zbl 0977.74029

[6] 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. | MR 1719149 | Zbl 0942.74056

[7] B. Brighi and M. Chipot, Approximation of infima in the calculus of variations. J. Comput. Appl. Math. 98 (1998) 273-287. | MR 1656994 | Zbl 0937.65071

[8] C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp., 66 (1997) 997-1026. | MR 1415798 | Zbl 0870.65055

[9] C. Carstensen and P. Plecháč, Adaptive algorithms for scalar non-convex variational problems. Appl. Numer. Math. 26 (1998) 203-216. | MR 1602868 | Zbl 0894.65029

[10] M. Chipot, Numerical analysis of oscillations in nonconvex problems. Numer. Math. 59 (1991) 747-767. | MR 1128031 | Zbl 0712.65063

[11] M. Chipot and C. Collins, Numerical approximations in variational problems with potential wells. SIAM J. Numer. Anal. 29 (1992) 1002-1019. | MR 1173182 | Zbl 0763.65049

[12] M. Chipot, C. Collins, and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282. | MR 1330864 | Zbl 0824.65045

[13] M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237-277. | MR 955934 | Zbl 0673.73012

[14] M. Chipot and S. Muller, Sharp energy estimates for finite element approximations of nonconvex problems. (preprint, 1997).

[15] 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. | MR 1087507 | Zbl 0725.65067

[16] 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. | MR 1094944 | Zbl 0735.65042

[17] B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin, (1989). | MR 990890 | Zbl 0703.49001

[18] G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM. J. Numer. Anal. 36 (1999) 1621-1635. | MR 1706747 | Zbl 0941.65062

[19] D. French, On the convergence of finite element approximations of a relaxed variational problem. SIAM J. Numer Anal. 28 (1991) 419-436. | MR 1043613 | Zbl 0696.65070

[20] L. Jian and R. James, Prediction of microstructure in monoclinic LaNbO4 by energy minimization. Acta Mater. 45 (1997) 4271-4281.

[21] D. Kinderlehrer and P. Pedregal, Characterizations of gradient Young measures. Arch. Rat. Mech. Anal. 115 (1991) 29-365. | MR 1139835 | Zbl 0754.49020

[22] M. Kruzík, Numerical approach to double well problems. SIAM. J. Numer Anal. 35 (1998) 1833-1849. | MR 1639950 | Zbl 0929.49016

[23] B. Li and M. Luskin, Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35 (1998) 376-392. | MR 1618484 | Zbl 0919.49020

[24] B. Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructure. Math. Comp. 67(223) (1998) 917-946. | MR 1459391 | Zbl 0901.73076

[25] B. Li and M. Luskin, Approximation of a martensitic laminate with varying volume fractions. Math. Model. Numer. Anal. 33 (1999) 67-87. | Numdam | MR 1685744 | Zbl 0928.74012

[26] Z. Li, Simultaneous numerical approximation of microstructures and relaxed minimizers. Numer. Math. 78 (1997) 21-38. | MR 1483567 | Zbl 0890.65067

[27] M. Luskin, Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75 (1996) 205-221. | MR 1421987 | Zbl 0874.73060

[28] M. Luskin, On the computation of crystalline microstructure. Acta. Numer. (1996) 191-257. | MR 1624603 | Zbl 0867.65033

[29] M. Luskin and L. Ma, Analysis of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal. 29 320-331. | MR 1154269 | Zbl 0760.65113

[30] R. Nicolaides and N. Walkington, Strong convergence of numerical solutions to degenerate variational problems. Math. Comp. 64 (1995) 117-127. | MR 1262281 | Zbl 0821.65040

[31] P. Pedregal, Numerical approximation of parametrized measures. Num. Funct. Anal. Opt. 16 (1995) 1049-1066. | MR 1355286 | Zbl 0848.65049

[32] P. Pedregal, On the numerical analysis of non-convex variational problems. Numer. Math. 74 (1996) 325-336. | MR 1408606 | Zbl 0858.65059

[33] T. Roubíček, Numerical approximation of relaxed variational problems. J. Convex. Anal. 3 (1996) 329-347. | MR 1448060 | Zbl 0881.65058

[34] N. Simha, Crystallography of the tetragonal → monoclinic transformation in zirconia. J. Phys. IV Colloq. France 5 (1995). C81121-C81126.

[35] N. Simha, Twin and habit plane microstructures due to the tetragonal to monoclinic transformation of zircoma. J. Mech. Phys. Solids 45 (1997) 261-292.

[36] V. Šverák, Lower-semicontinuity of variational integrals and compensated compactness, in Proceedings ICM 94, Zürich (1995) Birkhäuser. | MR 1404015 | Zbl 0852.49010

[37] 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. | MR 584398 | Zbl 0437.35004

[38] 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. | Zbl 0737.73012