zbMATH — the first resource for mathematics

Fine phase mixtures as minimizers of energy. (English) Zbl 0629.49020
Solid-solid phase transformations lead to certain characteristic microstructural features involving fine mixtures of phases. In this paper the martensitic transformation is investigated by theoretical methods. The main idea that the austenit/finely twinned martensite interface is modelled by certain minimizing sequences for the total free energy. It is assumed that the body prefers to be deformed in three states specified by three constant deformation gradients E, \(F^+\), \(F^-\). Here E is the unit tensor. The authors show that it is possible to arrange a very fine mixture of layers \(F^+/F^-/F^+/..\). on one side of an appropriately oriented interface so that the average deformation gradient of these layers does approximately satisfy conditions of compatibility with E. These configurations are minimizers of the total free energy. The finely twinned configurations of martensites are described as the approximations of generalized curves in Sobolev space. It is shown that the failure of lower semicontinuity of the free energy functional is a typical property for solids which change phase.
The paper gives some different examples of fineness in energy minimizers. These examples are as follows: 1. Fine twins in minimization problems with no absolute minimum; 2. Strongly elliptic energy with minimizers having fine boundary wrinkles; 3. Minimizers of energy having a finer and finer mixture of phases as an interface is approached from one side.
Reviewer: I.Ecsedi

49S05 Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)
49J20 Existence theories for optimal control problems involving partial differential equations
74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
74E30 Composite and mixture properties
Full Text: DOI
[1] R. A. Adams, Sobolev Spaces. New York: Academic Press, 1975.
[2] S. Agmon, A. Douglis & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II. Comm. Pure Appl. Math. 17 (1964), 35-92. · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337-403. · Zbl 0368.73040 · doi:10.1007/BF00279992
[4] J. M. Ball, Strict convexity, strong ellipticity, and regularity in the calculus of variations. Math. Proc. Camb. Phil. Soc. 87 (1980), 501-513. · Zbl 0451.35028 · doi:10.1017/S0305004100056930
[5] J. M. Ball, J. C. Currie & P. J. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order. J. Functional Anal. 41 (1981), 135-174. · Zbl 0459.35020 · doi:10.1016/0022-1236(81)90085-9
[6] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London A 306 (1982), 557-611. · Zbl 0513.73020 · doi:10.1098/rsta.1982.0095
[7] J. M. Ball & J. E. Marsden, Quasiconvexity, positivity of the second variation, and elastic stability. Arch. Rational Mech. Anal. 86 (1984), 251-277. · Zbl 0552.73006 · doi:10.1007/BF00281558
[8] J. M. Ball & F. Murat, W 1, P -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984), 225-253. · Zbl 0549.46019 · doi:10.1016/0022-1236(84)90041-7
[9] J. M. Ball & G. Knowles, Liapunov functions for thermomechanics with spatially varying boundary temperatures. Arch. Rational Mech. Anal. 92 (1986), 193-204. · Zbl 0624.73006 · doi:10.1007/BF00254826
[10] Z. S. Basinski & J. W. Christian, Crystallography of deformation by twin boundary movements in indium-thallium alloys. Acta Met. 2 (1954), 101-116; also, Experiments on the martensitic transformation in single crystals of indium-thallium alloys. Acta Met. 2 (1954), 148-166. · doi:10.1016/0001-6160(54)90100-5
[11] J. S. Bowles & J. K. MacKenzie, The crystallography of martensitic transformations I and II. Acta Met. 2 (1954), 129-137, 138-147. · doi:10.1016/0001-6160(54)90102-9
[12] D. A. G. Bruggeman, Berechnung verschiedener physikalischer Konstanten, von Heterogenen Substanzen. Ann. Phys. 5 (1935), 636-664. · doi:10.1002/andp.19354160705
[13] M. W. Burkart & T. A. Read, Diffusionless phase change in the indium-thallium system. Trans. AIME J. Metals 197 (1953), 1516-1524.
[14] M. Chipot & D. Kinderlehrer, Equilibrium configurations of crystals, to appear. · Zbl 0673.73012
[15] J. W. Christian, The Theory of Transformations in Metals and Alloys. Pergamon Press, 1975.
[16] J. L. Ericksen, Some phase transitions in crystals. Arch. Rational Mech. Anal. 73 (1980), 99-124. · Zbl 0429.73007 · doi:10.1007/BF00258233
[17] J. L. Ericksen, Constitutive theory for some constrained elastic crystals. IMA Preprint # 123, Institute for Mathematics and its Applications, University of Minnesota. · Zbl 0595.73001
[18] J. L. Ericksen, Some surface defects in unstressed thermoelastic solids. Arch. Rational Mech. Anal. 88 (1985), 337-345. · Zbl 0588.73188 · doi:10.1007/BF00250870
[19] J. L. Ericksen, Some Constrained Elastic Crystals, in Material Instabilities in Continuum Mechanics, (ed. J. M. Ball). Oxford University Press, to appear. · Zbl 0655.73022
[20] I. Fonseca, Variational methods for elastic crystals. Arch. Rational Mech. Anal., 97 (1987), 189-220. · Zbl 0611.73023 · doi:10.1007/BF00250808
[21] J. W. Gibbs, On the equilibrium of heterogeneous substances, in The Scientific Papers of J. Willard Gibbs, Vol. 1. Dover Publications, New York, 1961. · Zbl 0098.20905
[22] M. E. Gurtin, Two-phase deformations of elastic solids. Arch. Rational Mech. Anal. 84 (1983), 1-29. · Zbl 0525.73054 · doi:10.1007/BF00251547
[23] R. D. James, Finite deformation by mechanical twinning. Arch. Rational Mech. Anal. 77 (1981), 143-176. · Zbl 0537.73031 · doi:10.1007/BF00250621
[24] R. D. James, The stability and metastability of quartz, in Metastability and Incompletely Posed Problems, IMA, Vol. 3 (ed. S. Antman, J. L. Ericksen, D. Kinderlehrer & I. Müller) Springer-Verlag, 1987, 147-176.
[25] R. D. James, Displacive phase transformations in solids. J. Mech. Phys. Solids 34 (1986), 359-394. · Zbl 0585.73198 · doi:10.1016/0022-5096(86)90008-6
[26] D. Kinderlehrer, Remarks about equilibrium configurations of crystals, in Material Instabilities in Continuum Mechanics, (ed. J. M. Ball). Oxford University Press, to appear. · Zbl 0850.73037
[27] B. Klosowicz & K. A. Lurie, On the optimal nonhomogeneity of a torsional elastic bar. Arch. of Mech. 24 (1971), 239-249. · Zbl 0252.73032
[28] R. V. Kohn & G. Strang, Explicit relaxation of a variational problem in optimal design. Bull. Amer. Math. Soc. 9 (1983), 211-214. · Zbl 0527.49002 · doi:10.1090/S0273-0979-1983-15158-3
[29] N. A. Lavrov, K. A. Lurie & A. V. Cherkaev, Nonuniform rod of extremal torsional stiffness. Mech. of Sol. 15 (1980), 74-80.
[30] K. A. Lurie, A. V. Cherkaev & A. V. Fedorov, Regularization of optimal design problems for bars and plates I, II. J. Opt. Th. Appl. 37 (1982), 499-522 and 523-543; also, On the existence of solutions to some problems of optimal design for bars and plates. J. Opt. Th. Appl. 42 (1984), 247-281. · Zbl 0464.73109 · doi:10.1007/BF00934953
[31] M. Marcus & V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45 (1972), 294-320. · Zbl 0236.46033 · doi:10.1007/BF00251378
[32] G. W. Milton, Modelling the properties of composites by laminates, in Proc. Workshop on Homogenization and Effective Moduli of Materials and Media, 1984, to appear. · Zbl 0631.73011
[33] F. Murat & L. Tartar, Calcul des variations et homogénéisation, in Ecole d’été d’homogénéisation, Bureau sans Nappe, Eyrolles, Paris, July 1983, to appear.
[34] N. Nakanishi, Characteristics of stress-strain behaviour associated with thermoelastic martensitic transformation. Arch. Mech. 35 (1983), 37-62.
[35] Z. Nishiyama, Martensitic Transformation. Academic Press, 1978.
[36] W. Noll, A general framework for problems in the statics of finite elasticity, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations (ed. G. M. de la Penha & L. A. Medeiros), North-Holland, 1978. · Zbl 0415.73046
[37] K. Otsuka & K. Shimizu, Morphology and crystallography of thermoelastic ?? Cu-Al-Ni martensite. Japanese J. Appl. Phys. 8 (1969), 1196-1204. · doi:10.1143/JJAP.8.1196
[38] A. C. Pipkin, Some examples of crinkles, in Homogenization and Effective Moduli of Materials and Media. IMA, Vol. 1 (ed. J. L. Ericksen, D. Kinderlehrer, R. Kohn & J. L. Lions), Springer-Verlag, 1986. · Zbl 0648.73020
[39] U. E. Raitum, The extension of extremal problems connected with a linear elliptic equation. Soviet Math. Dokl. 19 (1978), 1342-1345. · Zbl 0428.49002
[40] U. E. Raitum, On optimal control problems for linear elliptic equations. Soviet Math. Dokl. 20 (1979), 129-132.
[41] Yu. G. Reshetnyak, On the stability of conformal mappings in multidimensional spaces. Siberian Math. J. 8 (1967), 69-85. · Zbl 0172.37801 · doi:10.1007/BF01040573
[42] Yu. G. Reshetnyak, Liouville’s theorem on conformal mappings under minimal regularity assumptions. Siberian Math. J. 8 (1967), 631-653. · Zbl 0167.36102
[43] H. C. Simpson & S. J. Spector, On failure of the complementing condition and nonuniqueness in linear elastostatics. J. Elasticity 15 (1985), 229-231. · Zbl 0576.73012 · doi:10.1007/BF00041996
[44] H. C. Simpson & S. J. Spector, On the sign of the second variation in finite elasticity, Arch. Rational Mech. Anal. 98 (1987), 1-30. · Zbl 0657.73027 · doi:10.1007/BF00279960
[45] L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of nonlinear partial differential equations (ed. J. M. Ball). Reidel, 1983. · Zbl 0536.35003
[46] L. Tartar, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires, in Lecture Notes in Physics 195. Springer-Verlag, 1984, 384-412.
[47] C. Truesdell, Some challenges offered to analysis by rational thermomechanics in Contemporary Developments in Continuum Mechanics and Partial Differential Equations (ed. G. M. de la Penha & L. A. J. Medeiros), North-Holland, 1978.
[48] G. van Tendeloo, J. van Landuyt & S. Amelinckx, The ?-? phase transition in quartz and AlPO4 as studied by electron microscopy and diffraction. Phys. Stat. Sol. a 33 (1976), 723-735. · doi:10.1002/pssa.2210330233
[49] C. M. Wayman, Introduction to the Crystallography of Martensitic Transformations. MacMillan, 1964.
[50] M. S. Wechsler, D. S. Lieberman & T. A. Read, On the theory of the formation of martensite. Trans. AIME J. Metals, 197 (1953), 1503-1515.
[51] H. Weyl, The Classical Groups. Princeton, 1946. · Zbl 1024.20502
[52] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory. Chelsea, 1980.
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.