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The relaxation of a double-well energy. (English) Zbl 0825.73029

MSC:
74E05 Inhomogeneity in solid mechanics
74B05 Classical linear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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[1] Acerbi, E.; Fusco, N.: Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1984) 125-145 · Zbl 0565.49010 · doi:10.1007/BF00275731
[2] Allaire, G.; Kohn, R.: Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials, preprint. · Zbl 0805.73043
[3] Avellaneda, M.: Iterated homogenization, differential effective medium theory, and applications. Comm. Pure Appl. Math. 40 (1987) 527-554 · Zbl 0629.73010 · doi:10.1002/cpa.3160400502
[4] Avellaneda, M.: Optimal bounds and microgeometries for elastic two-phase composites. SIAM Journal Appl. Math. 47 (1987) 1216-1228 · Zbl 0632.73079 · doi:10.1137/0147082
[5] Avellaneda, M.; Cherkaev, A. V.; Lurie, K. A.; Milton, G. W.: On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality. J. Appl. Phys. 63 (1988) 4989-5003 · doi:10.1063/1.340445
[6] Avellaneda, M.; Milton, G. W.: Bounds on the effective elasticity tensor of composites based on two-point correlations, in Proc. ASME Energy-Technology Conference and Exposition, D. Hui and T. Koszic, ed., ASME (1989) · Zbl 0702.73078
[7] Ball, J. M.; Murat, F.:W 1,p -quasiconvexity and variational problems for multiple integrals. Journal Funct. Anal. 58 (1984) 225-253 · Zbl 0549.46019 · doi:10.1016/0022-1236(84)90041-7
[8] Ball, J. M.; James, R. D.: Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52 · Zbl 0629.49020 · doi:10.1007/BF00281246
[9] Ball, J. M.; James, R. D.: Proposed experimental tests of a theory of fine microstructure and the two well problem, to appear · Zbl 0758.73009
[10] Bhattacharya, K.; Firoozye, N; James, R.; Kohn, R.: in preparation
[11] Cahn, J. W.; Larche, F.: A simple model for coherent equilibrium. Acta Metall. 32 (1984) 1915-1923 · doi:10.1016/0001-6160(84)90173-1
[12] Chipot, M.; Kinderlehrer, D.: Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237-277 · Zbl 0673.73012 · doi:10.1007/BF00251759
[13] Collins, C.; Luskin, M.: The computation of the austenitic-martensitic phase transition. in Lecture Notes in Physics 344, M. Rascle et al. eds., Berlin: Springer-Verlag 1989, 34-50 · Zbl 0991.80502
[14] Collins, C.; Luskin, M.: Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem, preprint · Zbl 0735.65042
[15] Dacorogna, B.: Quasiconvexity and relaxation of nonconvex variational problems. Journal Funct. Anal. 46 (1982) 102-118 · Zbl 0547.49003 · doi:10.1016/0022-1236(82)90046-5
[16] Dacorogna, B.: Remarques sur les notions de polyconvexite, quasiconvexite, et convexite de rang 1. J. Math. Pures Appl. 64 (1985) 403-438 · Zbl 0609.49007
[17] Dacorogna, B.: Direct Methods in the Calculus of Variations. Berlin: Springer-Verlag, 1989 · Zbl 0703.49001
[18] Ericksen, J. L.: The Cauchy and Born hypotheses for crystals, in Phase Transformations and Material Instabilities in Solids, M. Gurtin, ed., Academic Press (1984) 61-78 · Zbl 0567.73112
[19] Ericksen, J. L.: Constitutive theory for some constrained elastic crystals. Int. Journal Solids Structures 22 (1986) 951-964 · Zbl 0595.73001 · doi:10.1016/0020-7683(86)90030-2
[20] J. L. Ericksen,: Stable equilibrium configurations of elastic crystals. Arch. Rat. Mech. Anal. 94 (1986) 1-14 · Zbl 0597.73006 · doi:10.1007/BF00278240
[21] Ericksen, J. L.: Twinning of crystals I, in Metastability and Incompletely Posed Problems. S. Antman et al., eds., Berlin: Springer-Verlag (1987) 77-94
[22] Firoozye, N.: Optimal Translations and Relaxations of Some Multiwell Energies, Ph. D. thesis, New York University, 1990
[23] Firoozye, N.: Optimal use of the translation method and relaxations of variational problems. Comm. Pure Appl. Math., to appear. · Zbl 0733.49018
[24] Fonseca, I.: Variational methods for elastic crystals. Arch. Rat. Mech. Anal. 97 (1987) 189-220 · Zbl 0611.73023 · doi:10.1007/BF00250808
[25] Fonseca, I.: Stability of elastic crystals, in Non-Classical Continuum Mechanics, R. Knops and A. Lacey eds., Cambridge: University Press (1987) 187-196
[26] Fonseca, I.: The lower quasiconvex envelope of the stored energy function for an elastic crystal. Journal Math. Pures et Appl. 67 (1988) 175-195 · Zbl 0718.73075
[27] Fonseca, I.; Tartar, L.: The displacement problem for elastic crystals. Proc. Roy. Soc. Edinburgh 113 AA (1989) 159-180 · Zbl 0691.73016
[28] Francfort, G. A.; Murat, F.: Homogenization and optimal bounds in linear elasticity. Arch. Rat. Mech. Anal. 94 (1986) 307-334 · Zbl 0604.73013 · doi:10.1007/BF00280908
[29] Gerard, P.: Moyennisation et regularite 2-microlocale, Annales Scientifiques de l’Ecole Normale Superieure to appear
[30] Gerard, P.: Microlocal defect measures, preprint
[31] Grinfel’d, M. A.: Conditions for thermodynamic phase equilibrium in a nonlinear elastic material. Doklady Akad. Nauk SSSR Geophysics 251 (1980) 824-828
[32] Grinfel’d, M. A.: Stability of interphase boundaries in solid elastic media. Prikl. Matem. Mekhan. 51 (1987) 489-496 · Zbl 0667.73002
[33] Grinfel’d, M.: Continuum methods in the theory of phase transitions in solids. Phys. Earth and Planetary Interiors 50 (1988) 99-109 · doi:10.1016/0031-9201(88)90099-4
[34] Grindfel’d, M. A.; Langman, S. L.: Average thermoelastic moduli of two-phase media. Izvestia Earth Physics 21 (1985) 594-602
[35] Hong, M.; Wedge, D. E.; Morris, J. W.: The state and habit of the Fe16N2 precipitate in b.c.c. iron: elastic theory. Acta Metallurgica 32 (1984) 279-288 · doi:10.1016/0001-6160(84)90056-7
[36] James, R. D.: The arrangement of coherent phases in a loaded body, in Phase Transformations and Material Instabilities in Solids, M. Gurtin, ed., Academic Press (1984) 79-98 · Zbl 0594.73114
[37] James, R. D.: Displacive phase transformations in solids. Journal Mech. Phys. Solids 34 (1986) 359-394 · Zbl 0585.73198 · doi:10.1016/0022-5096(86)90008-6
[38] James, R. D.: The stability and metastability of quartz, in Metastability and Incompletely Posed Problems, S. Antman et al. eds., Berlin: Springer-Verlag (1987) 147-176
[39] James, R. D.; Kinderlehrer, D.: Theory of diffusionless phase transitions, in Lecture Notes in Physics 344, M. Rascle et al., eds., Berlin: Springer-Verlag (1989) 51-84 · Zbl 0991.74504
[40] Johnson, W. C.; Voorhees, P. W.: Phase equilibrium in two-phase coherent solids. Metall. Trans. 18 A (1987) 1213-1228 · doi:10.1007/BF02647191
[41] Kaganova, I. M.; Roitburd, A. L.: An anisotropic crystalline inclusion in an anisotropic matrix. Sov. Phys. Crystallogr. 34 (1989) 650-653
[42] Kaganova, I. M.; Roitburd, A. L.: Equilibrium between elasticially-interacting phases. Sov. Phys. JETP 67 (1988) 1173-1183
[43] Khachaturyan, A. G.: Some questions concerning the theory of phase transformations in solids. Soviet Physics-Solid State 8 (1967) 2163-2168
[44] Khachaturyan, A. G.: Theory of Structural Transformations in Solids. John Wiley and Sons (1983)
[45] Khachaturyan, A. G.; Shatalov, G. A.: Theory of macroscopic periodicity for a phase transition in the solid state. Soviet Physics JETP 29 (1969) 557-561
[46] Kinderlehrer, D.: Twinning of crystals II, in Metastability and Incompletely Posed Problems, S. Antman et al., eds., Berlin: Springer-Verlag (1987) 135-146 · Zbl 0638.73007
[47] Kohn, R. V.: The relationship between linear and nonlinear variational models of coherent phase transitions, in Trans. 7th Army Conf. on Appl. Math. and Computing, ARO Report No. 90-1 (1990) · Zbl 0692.73016
[48] Kohn, R. V.: Relaxation of the elastic energy for a system of two coherent phases with well-ordered elastic moduli, in preparation
[49] Kohn, R. V.; Lipton, R.: Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials. Arch. Rat. Mech. Anal. 102 (1988) 331-350 · Zbl 0662.73012 · doi:10.1007/BF00251534
[50] Kohn, R. V.; Milton, G. W.: On bounding the effective conductivity of anisotropic composites, in Homogenization and Effective Moduli of Materials and Media. J. Erickensen et al., eds., Berlin: Springer-Verlag (1986) 97-125 · Zbl 0631.73012
[51] Kohn, R. V.; Muller, S.: in preparation
[52] Kohn, R. V.; Sternberg, P.: Local minimisers and singular perturbations. Proc. Roy. Soc. Edinburgh 111 A (1989) 69-84 · Zbl 0676.49011
[53] Kohn, R. V.; Strang, F.: Optimal design and relaxation of variational problems, I?III. Comm. Pure Appl. Math. 34 (1987) 113-137, 139-182 and 353-377 · Zbl 0609.49008
[54] Kohn, R. V.; Vogelius, M.: Relaxation of a variational method for impedance computed tomography. Comm. Pure Appl. Math. 40 (1987) 745-777 · Zbl 0659.49009 · doi:10.1002/cpa.3160400605
[55] Kostlan, E.; Morris, J. W.: The preferred habit of a coherent thin-plate inclusion in an anisotropic elastic solid. Acta Metallurgica 35 (1987) 2167-2175 · doi:10.1016/0001-6160(87)90046-0
[56] Larche, F.; Cahn, J. W.: A linear theory of thermochemical equilibrium of solids under stress. Acta Metallurgica 21 (1973) 1051-1063 · doi:10.1016/0001-6160(73)90021-7
[57] Larche, F.; Cahn, J. W.: A nonlinear theory of thermochemical equilibrium of solids under stress. Acta Metallurgica 26 (1978) 53-60 · doi:10.1016/0001-6160(78)90201-8
[58] Larche, F. C.; J. Cahn, W.: Thermomechanical equilibrium of multiphase solids under stress. Acta Metallurgica 26 (1978) 1579-1589 · doi:10.1016/0001-6160(78)90067-6
[59] Lee, J. K.; Barnett, D. M.; Aaronson, H. I.: The elastic strain energy of coherent ellipsoidal precipitates in anisotropic crystalline solids. Metallurgical Trans. 8 A (1977) 963-970 · doi:10.1007/BF02661580
[60] Lurie, K. A.; Cherkaev, A. V.: Exact estimates of the conductivity of a binary mixture of isotropic components. Proc. Roy. Soc. Edinburgh 104 A (1986) 21-38 · Zbl 0623.73011
[61] Lurie, K. A.; Cherkaev, A. V.: On a certain variational problem of phase equilibrium. in Material Instabilities in Continuum Mechanics, J. M. Ball, ed., Oxford: University Press (1988) 257-268 · Zbl 0850.93486
[62] Mayo, W. E.; Tsakalakos, T.: The influence of elastic strain energy on the formation of coherent hexagonal phases. Metallurgical Trans. 11 A (1980) 1637-1644 · doi:10.1007/BF02660518
[63] Milton, G.: Modelling the properties of composites by laminates, in Homogenization and Effective Moduli of Materials and Media, J. Ericksen etal., eds., Berlin: Springer-Verlag (1986) 150-175
[64] Milton, G. W.: On characterizing the set of possible effective tensors of composites: the variational method and the translation method. Comm. Pure Appl. Math. 43 (1990) 63-125 · Zbl 0751.73041 · doi:10.1002/cpa.3160430104
[65] Milton, G. W.; Kohn, R. V.: Variational bounds on the effective moduli of anisotropic composites. Journal Mech. Phys. Solids 36 (1988) 597-629 · Zbl 0672.73012 · doi:10.1016/0022-5096(88)90001-4
[66] Murat, F.: Tartar, L.; personal communication (1989) see also ref. 80.)
[67] Pipkin, A. C.: Elastic materials with two preferred states. Quart. J. Mech. Appl. Math. 44 (1991) 1-15 · Zbl 0735.73032 · doi:10.1093/qjmam/44.1.1
[68] Roitburd, A. L.: Kristallografiya 12 (1967) 567ff. (In Russian)
[69] Roitburd, A. L.: The domain structure of crystals formed in the solid phase. Sov. Phys. Solid State 10 (1969) 2870-2876
[70] Roitburd, A. L.: Domain structure caused by internal stresses in heterophase solids. Phys. Stat. Sol. (a) 16 (1973) 329-339 · doi:10.1002/pssa.2210160136
[71] Roitburd, A. L.: Martensitic transformation as a typical phase transformation in solids, in Solids State Physics 33 Academic Press (1978) 317-390
[72] Roitburd, A. L.: Equilibrium and phase diagrams of coherent phases in solids. Sov. Phys. Solid State 26 (1984) 1229-1233
[73] Roitburd, A. L.: Thermodynamics of solid solution precipitation. Sov. Phys. Solid State 27 (1985) 598-603
[74] Roitburd, A. L.: Phase equilibrium in solids. Sov. Phys. Solid State 28 (1986) 1716-1718
[75] Roitburd, A. L.; Kosenko, N. S.: Orientational dependence of the elastic energy of a plane interlayer in a system of coherent phases. Phys. Stat. Sol. (a) 35 (1976) 735-746 · doi:10.1002/pssa.2210350239
[76] Rosakis, P.: Compact zones of shear transformation in an anisotropic solid, preprint · Zbl 0763.73005
[77] Rybka, P.: Dynamical Modelling of Phase Transitions in Solids by Means of Viscoelasticity in Many Dimensions. Ph. D. Thesis NYU 1990 · Zbl 0758.73004
[78] Schneck, R.; Rokhlin, S. I.; Dariel, M. P.: Criterion for predicting the morphology of crystalline cubic precipitates in a cubic matrix. Metallurgical Trans. 16 A (1985) 197-202
[79] Tartar, L.: Estimations fines de coefficients homogeneises, in Ennio de Giorgi’s Colloquium, P. Kree, ed., Pitman (1985) 168-187
[80] Tartar, L.: H-measures: a new approach for studying homogenization, oscillations, and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh 115A (1990) 193-230 · Zbl 0774.35008
[81] Tartar, L.: H-measures and small amplitude homogenization, in Random Media and Composites, R. Kohn and G. Milton, eds., SIAM (1989) 89-99 · Zbl 0790.73009
[82] Tsakalakos, T.: On the strain energy of transformation inhomogeneities in solids, in Micromechanics and Inhomogeneity ? The Toshio Mura 65th Anniversary Volume. G. Weng et al., eds., Berlin: Springer-Verlag (1990) 469-496
[83] Wayman, C. M.: Introduction to the Crystallography of Martensitic Transformations, MacMillan (1964)
[84] Wen, S. H.; Khatchaturyan, A. G.; Morris, J. W.: Computer simulation of a ?tweedtransformation? in an idealized elastic crystal. Metall. Trans. 12 A (1981) 581-587 · doi:10.1007/BF02649732
[85] Wen, S. H.; Kostlan, E.; Hong, M.; Khachaturyan, A. G.; Morris, Jr., J. W.: The preferred habit of a tetragonal inclusion in a cubic matrix. Acta Metallurgica 29 (1981) 1247-1254 · doi:10.1016/0001-6160(81)90015-8
[86] Wert, J. A.: The strain energy of a disc-shaped GP zone. Acta Metallurgica 24 (1976) 65-71 · doi:10.1016/0001-6160(76)90148-6
[87] Williams, R. O.: Long-period superlattices in the copper-gold system as two-phase mixtures. Metall. Trans. 11 A (1980) 247-253 · doi:10.1007/BF02660629
[88] Williams, R. O.: The calculation of coherent phase equilibria. CALPHAD 8 (1984) 1-14 · doi:10.1016/0364-5916(84)90024-5
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