@article{M2AN_2000__34_3_663_0, author = {B\v{e}l{\'\i}k, Pavel and Luskin, Mitchell}, title = {Stability of microstructure for tetragonal to monoclinic martensitic transformations}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {663--685}, publisher = {Dunod}, address = {Paris}, volume = {34}, number = {3}, year = {2000}, mrnumber = {1763530}, zbl = {0981.74042}, language = {en}, url = {http://archive.numdam.org/item/M2AN_2000__34_3_663_0/} }
TY - JOUR AU - Bělík, Pavel AU - Luskin, Mitchell TI - Stability of microstructure for tetragonal to monoclinic martensitic transformations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2000 SP - 663 EP - 685 VL - 34 IS - 3 PB - Dunod PP - Paris UR - http://archive.numdam.org/item/M2AN_2000__34_3_663_0/ LA - en ID - M2AN_2000__34_3_663_0 ER -
%0 Journal Article %A Bělík, Pavel %A Luskin, Mitchell %T Stability of microstructure for tetragonal to monoclinic martensitic transformations %J ESAIM: Modélisation mathématique et analyse numérique %D 2000 %P 663-685 %V 34 %N 3 %I Dunod %C Paris %U http://archive.numdam.org/item/M2AN_2000__34_3_663_0/ %G en %F M2AN_2000__34_3_663_0
Bělík, Pavel; Luskin, Mitchell. Stability of microstructure for tetragonal to monoclinic martensitic transformations. ESAIM: Modélisation mathématique et analyse numérique, Tome 34 (2000) no. 3, pp. 663-685. http://archive.numdam.org/item/M2AN_2000__34_3_663_0/
[1] Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl
,[2] Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52. | MR | Zbl
and ,[3] 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
and ,[4] Self accomodation in martensite. Arch. Rat. Mech. Anal. 120 (1992) 201-244. | MR | Zbl
,[5] Relaxation of some multiwell problems, in Proc. R. Soc. Edinburgh: Section A, to appear. | Zbl
and ,[6] The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation. Arch. Rat. Mech. Anal. 149 (2000) 123-154. | MR | Zbl
, and ,[7] Approximation of infima in the calculus of variations. J. Comput. Appl. Math. 98 (1998) 273-287. | MR | Zbl
and ,[8] Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp., 66 (1997) 997-1026. | MR | Zbl
and ,[9] Adaptive algorithms for scalar non-convex variational problems. Appl. Numer. Math. 26 (1998) 203-216. | MR | Zbl
and ,[10] Numerical analysis of oscillations in nonconvex problems. Numer. Math. 59 (1991) 747-767. | MR | Zbl
,[11] Numerical approximations in variational problems with potential wells. SIAM J. Numer. Anal. 29 (1992) 1002-1019. | MR | Zbl
and ,[12] Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282. | MR | Zbl
, , and ,[13] Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237-277. | MR | Zbl
and ,[14] Sharp energy estimates for finite element approximations of nonconvex problems. (preprint, 1997).
and ,[15] Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal. 28 (1991) 321-332. | MR | Zbl
, , and ,[16] Optimal order estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp. 57 (1991) 621-637. | MR | Zbl
and ,[17] Direct methods in the calculus of variations. Springer-Verlag, Berlin, (1989). | MR | Zbl
,[18] Numerical computation of rank-one convex envelopes. SIAM. J. Numer. Anal. 36 (1999) 1621-1635. | MR | Zbl
,[19] On the convergence of finite element approximations of a relaxed variational problem. SIAM J. Numer Anal. 28 (1991) 419-436. | MR | Zbl
,[20] Prediction of microstructure in monoclinic LaNbO4 by energy minimization. Acta Mater. 45 (1997) 4271-4281.
and ,[21] Characterizations of gradient Young measures. Arch. Rat. Mech. Anal. 115 (1991) 29-365. | MR | Zbl
and ,[22] Numerical approach to double well problems. SIAM. J. Numer Anal. 35 (1998) 1833-1849. | MR | Zbl
,[23] Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35 (1998) 376-392. | MR | Zbl
and ,[24] Nonconforming finite element approximation of crystalline microstructure. Math. Comp. 67(223) (1998) 917-946. | MR | Zbl
and ,[25] Approximation of a martensitic laminate with varying volume fractions. Math. Model. Numer. Anal. 33 (1999) 67-87. | Numdam | MR | Zbl
and ,[26] Simultaneous numerical approximation of microstructures and relaxed minimizers. Numer. Math. 78 (1997) 21-38. | MR | Zbl
,[27] Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75 (1996) 205-221. | MR | Zbl
,[28] On the computation of crystalline microstructure. Acta. Numer. (1996) 191-257. | MR | Zbl
,[29] Analysis of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal. 29 320-331. | MR | Zbl
and ,[30] Strong convergence of numerical solutions to degenerate variational problems. Math. Comp. 64 (1995) 117-127. | MR | Zbl
and ,[31] Numerical approximation of parametrized measures. Num. Funct. Anal. Opt. 16 (1995) 1049-1066. | MR | Zbl
,[32] On the numerical analysis of non-convex variational problems. Numer. Math. 74 (1996) 325-336. | MR | Zbl
,[33] Numerical approximation of relaxed variational problems. J. Convex. Anal. 3 (1996) 329-347. | MR | Zbl
,[34] Crystallography of the tetragonal → monoclinic transformation in zirconia. J. Phys. IV Colloq. France 5 (1995). C81121-C81126.
,[35] Twin and habit plane microstructures due to the tetragonal to monoclinic transformation of zircoma. J. Mech. Phys. Solids 45 (1997) 261-292.
,[36] Lower-semicontinuity of variational integrals and compensated compactness, in Proceedings ICM 94, Zürich (1995) Birkhäuser. | MR | Zbl
,[37] 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 | Zbl
,[38] 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
,