zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.