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On the computation of crystalline microstructure

Published online by Cambridge University Press:  07 November 2008

Mitchell Luskin
Mitchell Luskin
Affiliation:
School of MathematicsUniversity of MinnesotaMinneapolis, MN 55455, USAE-mail:[email protected]
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Abstract

Microstructure is a feature of crystals with multiple symmetry-related energy-minimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetry-related states on a fine scale to minimize energy. This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.

Type
Research Article
Information
Acta Numerica , Volume 5 , January 1996 , pp. 191 - 257
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Abeyaratne, R. and Knowles, J. (1994), ‘Dynamics of propagating phase boundaries: thermoelastic solids with heat conduction’, Arch. Rat. Mech. Anal. 126, 203–230.CrossRefGoogle Scholar
Abeyaratne, R., Chu, C. and James, R. (1994), Kinetics and hysteresis in martensitic single crystals, in Proc. Symposium on the Mechanics of Phase Transformations and Shape Memory Alloys, ASME.Google Scholar
Adams, R. (1975), Sobolev Spaces, Academic Press, New York.Google Scholar
Alt, H.-W., Hoffmann, K.-H., Niezgódka, M. and Sprekels, J.(1985), A numerical study of structural phase transitions in shape memory alloys, Technical Report 90, Institut für Mathematik, Augsburg University.Google Scholar
Arlt, G. (1990), ‘Twinning in ferroelectric and ferroelastic ceramics: stress relief’, J. Mat. Sci. 22, 2655–2666.CrossRefGoogle Scholar
Ball, J. (1977), ‘Convexity conditions and existence theorems in nonlinear elasticity’, Arch. Ration. Mech. Anal. 63, 337–403.CrossRefGoogle Scholar
Ball, J. (1989), A version of the fundamental theorem for Young measures, in PDEs and Continuum Models of Phase Transition (Rascle, M., Serre, D. and Slemrod, M., eds), Springer, New York, pp. 207–215. Lecture Notes in Physics, vol. 344.CrossRefGoogle Scholar
Ball, J. and James, R. (1987), ‘Fine phase mixtures as minimizers of energy’, Arch. Rational Mech. Anal. 100, 13–52.CrossRefGoogle Scholar
Ball, J. and James, R. (1992), ‘Proposed experimental tests of a theory of fine microstructure and the two-well problem’, Phil. Trans. R. Soc. Lond. A 338, 389–450.CrossRefGoogle Scholar
Ball, J. and James, R. (1993), Theory for the microstructure of martensite and applications, in Proceedings of the International Conference on Martensitic Transformations (Perkins and Wayman, eds), Monterey Institute of Advanced Studies, Carmel, California, pp. 65–76.Google Scholar
Ball, J., Chu, C. and James, R. (1994), Metastability of martensite. Manuscript.Google Scholar
Ball, J., Chu, C. and James, R. (1995), Hysteresis during stress-induced variant rearrangement, in Proceedings of the International Conference on Martensitic Transformations.Google Scholar
Ball, J., Holmes, P., James, R., Pego, R. and Swart, P. (1991), ‘On the dynamics of fine structure’, J. Nonlinear Sci. 1, 17–70.CrossRefGoogle Scholar
Barsch, G., Horovitz, B. and Krumhansl, J. (1987), ‘Dynamics of twin boundaries in martensites’, Phys. Rev. Lett. 59, 1251–1254.CrossRefGoogle ScholarPubMed
Basinski, Z. S. and Christian, J. W. (1954), ‘Experiments on the martensitic transformation in single crystals of indium-thallium alloys’, Acta Metall. 2, 148–166.CrossRefGoogle Scholar
Bhattacharya, K. (1991), ‘Wedge-like microstructure in martensite’, Acta Metall. Mater. 39, 2431–2444.CrossRefGoogle Scholar
Bhattacharya, K. (1992), ‘Self accommodation in martensite’, Arch. Rat. Mech. Anal. 120, 201–244.CrossRefGoogle Scholar
Bhattacharya, K. (1993), ‘Comparison of the geometrically nonlinear and linear theories of martensitic transformation’, Continuum Mechanics and Thermodynamics 5, 205–242.CrossRefGoogle Scholar
Bhattacharya, K. and Kohn, R. (1995), Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials, Technical Report 1366, IMA.Google Scholar
Bhattacharya, K. and Kohn, R. (1996), ‘Symmetry, texture, and the recoverable strain of shape memory polycrystals’, Acta Metall. Mater.CrossRefGoogle Scholar
Bhattacharya, K., Firoozye, N., James, R. and Kohn, R. (1994), ‘Restrictions on microstructure’, Proc. Roy. Soc. Edinburgh A 124A, 843–878.CrossRefGoogle Scholar
Bhattacharya, K., James, R. and Swart, P. (1993), A nonlinear dynamic model for twin relaxation with applications to Au 47.5at.%Cd and other shape-memory materials, in Twinning in Advanced Materials (Yoo, M. and Wuttig, M., eds), Theoretical Materials Science.Google Scholar
Brandon, D., Lin, T. and Rogers, R. (1995), ‘Phase transitions and hysteresis in nonlocal and order-parameter models’, Meccanica 30, 541–565.CrossRefGoogle Scholar
Brighi, B. and Chipot, M. (1994), ‘Approximated convex envelope of a function’, SIAM J. Numer. Anal. 31, 128–148.CrossRefGoogle Scholar
Bruno, O. (1995), ‘Quasistatic dynamics and pseudoelasticity in polycrystalline shape memory wires’, Smart Mater. Struct. 4, 7–13.CrossRefGoogle Scholar
Bruno, O. P., Leo, P. H. and Reitich, F. (1995 a), ‘Free boundary conditions at austenite–martensite interfaces’, Physics Review Letters 74, 746–749.CrossRefGoogle ScholarPubMed
Bruno, O. P., Reitich, F. and Leo, P. H. (1995 b), The overall elastic energy of polycrystalline martensitic solids. Manuscript.CrossRefGoogle Scholar
Burkart, M. W. and Read, T. A. (1953), ‘Diffusionless phase changes in the indium-thallium system’, Trans. AIME J. Metals 197, 1516–1524.Google Scholar
Carstensen, C. and Plecháč, P. (1995), Numerical solution of the scalar double-well problem allowing microstructure, Technical Report 1752, Technische Hoch-schule Darmstadt.Google Scholar
Chipot, M. (1991), ‘Numerical analysis of oscillations in nonconvex problems’, Numer. Math. 59, 747–767.CrossRefGoogle Scholar
Chipot, M. and Collins, C. (1992), ‘Numerical approximations in variational problems with potential wells’, SIAM J. Numer. Anal. 29, 1002–1019.CrossRefGoogle Scholar
Chipot, M. and Kinderlehrer, D. (1988), ‘Equilibrium configurations of crystals’, Arch. Rat. Mech. Anal. 103, 237–277.CrossRefGoogle Scholar
Chipot, M., Collins, C. and Kinderlehrer, D. (1995), ‘Numerical analysis of oscillations in multiple well problems’, Numer. Math. 70, 259–282.CrossRefGoogle Scholar
Chu, C. and James, R. (1995), Analysis of microstructures in Cu-14% Al-3.9% Ni by energy minimization, in Proceedings of the International Conference on Martensitic Transformations.CrossRefGoogle Scholar
Ciarlet, P. (1978), The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.Google Scholar
Colli, P. (1995), ‘Global existence for the three-dimensional Frémond model of shape memory alloys’, Nonlinear analysis, theory, methods, and applications 24, 1565–1579.Google Scholar
Colli, P., Frémond, M. and Visintin, A. (1990), ‘Thermo-mechanical evolution of shape memory alloys’, Quart. Appl. Math. 48, 31–47.CrossRefGoogle Scholar
Collins, C. (1993 a), Computation of twinning, in Microstructure and Phase Transitions (Ericksen, J., James, R., Kinderlehrer, D. and Luskin, M., eds), Springer, New York, pp. 39–50. IMA Volumes in Mathematics and Its Applications, vol. 54.CrossRefGoogle Scholar
Collins, C. (1993 b), Computations of twinning in shape-memory materials, in Smart Structures and Materials 1993: Mathematics in Smart Structures (Banks, H. T., ed.), Proc. SPIE 1919, pp. 30–37.CrossRefGoogle Scholar
Collins, C. (1994), Comparison of computational results for twinning in the two-well problem, in Proceedings of the 2nd International Conference on Intelligent Materials (Rogers, C. and Wallace, G., eds), Technomic, Lancaster, PA, pp. 391–401.Google Scholar
Collins, C. and Luskin, M. (1989), The computation of the austenitic–martensitic phase transition, in Partial Differential Equations and Continuum Models of Phase Transitions (Rascle, M., Serre, D. and Slemrod, M., eds), Springer, New York, pp. 34–50. Lecture Notes in Physics, vol. 344.Google Scholar
Collins, C. and Luskin, M. (1990), Numerical modeling of the microstructure of crystals with symmetry-related variants, in Proceedings of the US–Japan Workshop on Smart/Intelligent Materials and Systems (Ahmad, I.Aizawa, M., Crowson, A. and Rogers, C., eds), Technomic, Lancaster, PA, pp. 309–318.Google Scholar
Collins, C. and Luskin, M. (1991 a), Numerical analysis of microstructure for crystals with a nonconvex energy density, in Progress in Partial Differential Equations: the Metz Surveys (Chipot, M. and Paulin, J. S. J., eds), Longman, Harlow, UK, pp. 156–165. Pitman Research Notes in Mathematics Series, vol. 249.Google Scholar
Collins, C. and Luskin, M. (1991 b), ‘Optimal order estimates for the finite element approximation of the solution of a nonconvex variational problem’, Math. Comp. 57, 621–637.CrossRefGoogle Scholar
Collins, C., Kinderlehrer, D. and Luskin, M. (1991 a), ‘Numerical approximation of the solution of a variational problem with a double well potential’, SIAM J. Numer. Anal. 28, 321–332.CrossRefGoogle Scholar
Collins, C., Luskin, M. and Riordan, J. (1991 b), Computational images of crystalline microstructure, in Computing Optimal Geometries (Taylor, J., ed.), Amer. Math. Soc., Providence, pp. 16–18. AMS Special Lectures in Mathematics and AMS Videotape Library.Google Scholar
Collins, C., Luskin, M. and Riordan, J. (1993), Computational results for a two-dimensional model of crystalline microstructure, in Microstructure and Phase Transitions (Ericksen, J., James, R., Kinderlehrer, D. and Luskin, M., eds), Springer, New York, pp. 51–56. IMA Volumes in Mathematics and Its Applications, vol. 54.CrossRefGoogle Scholar
Dacorogna, B. (1989), Direct methods in the calculus of variations, Springer, Berlin.CrossRefGoogle Scholar
DeSimone, A. (1993), ‘Energy minimizers for large ferromagnetic bodies’, Arch. Rat. Mech. Anal. 125, 99–143.CrossRefGoogle Scholar
Dolzmann, G. and Müller, S. (1995), ‘Microstructures with finite surface energy: the two-well problem’, Arch. Rat. Mech. Anal. 132, 101–141.CrossRefGoogle Scholar
Ekeland, I. and Temam, R. (1974), Analyse convexe et problèmes variationnels, Dunod, Paris.Google Scholar
Ericksen, J. (1980), ‘Some phase transitions in crystals’, Arch. Rat. Mech. Anal. 73, 99–124.CrossRefGoogle Scholar
Ericksen, J. (1986), ‘Constitutive theory for some constrained elastic crystals’, J. Solids and Structures 22, 951–964.CrossRefGoogle Scholar
Ericksen, J. (1987 a), Some constrained elastic crystals, in Material Instabilities in Continuum Mechanics and Related Problems (Ball, J., ed.), Oxford University Press, pp. 119–137.Google Scholar
Ericksen, J. (1987 b), Twinning of crystals I, in Metastability and Incompletely Posed Problems (Antman, S., Ericksen, J., Kinderlehrer, D. and Müller, I., eds), Springer, New York, pp. 77–96. IMA Volumes in Mathematics and Its Applications, vol. 3.CrossRefGoogle Scholar
Eshelby, J. D. (1961), Elastic inclusions and inhomogeneities, in Progress in Solid Mechanics, vol. 2 (Sneddon, I. N. and Hill, R., eds), pp. 87–140.Google Scholar
Firoozye, N. (1993), ‘Geometric parameters and the relaxation of multiwell energies’, pp. 85–110. IMA Volumes in Mathematics and Its Applications, vol. 54.CrossRefGoogle Scholar
Fonseca, I. (1987), ‘Variational methods for elastic crystals’, Arch. Rational Mech. Anal. 97, 189–220.CrossRefGoogle Scholar
Frémond, M. (1990), Shape memory alloys, a thermomechanical model, in Free bounday problems: Theory and applications, vol. I (Hoffman, K.-H. and Sprekels, J., eds), Longman, Harlow, UK, pp. 295–306.Google Scholar
French, D. (1991), ‘On the convergence of finite element approximations of a relaxed variational problem’, SIAM J. Numer. Anal. 28, 419–436.Google Scholar
French, D. and Jensen, S. (1991), ‘Behavior in the large of numerical solutions to one-dimensional nonlinear viscoelasticity by continuous time Galerkin methods’, Comp. Meth. Appl. Mech. Eng. 86, 105–124.CrossRefGoogle Scholar
French, D. and Walhbin, L. (1993), ‘On the numerical approximation of an evolution problem in nonlinear viscoelasticity’, Computer Methods in Applied Mechanics and Engineering 107, 101–116.CrossRefGoogle Scholar
Friesecke, G. (1994), ‘A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems’, Proc. Roy. Soc. Edinb. 124A, 437–471.Google Scholar
Friesecke, G. and Dolzmann, G. (1996), ‘Time discretization and global existence for a quasi-linear evolution equation with nonconvex energy’, SIAM J. Math. Anal.Google Scholar
Friesecke, G. and McLeod, J. B. (submitted), ‘Dynamic stability of nonminimizing phase mixtures’, Proc. Roy. Soc. London.Google Scholar
Friesecke, G. and McLeod, J. B. (To appear), ‘Dynamics as a mechanism preventing the formation of finer and finer microstructure’, Arch. Rat. Mech. Anal.Google Scholar
Glowinski, R. (1984), Numerical Methods for Nonlinear Variational Problems, Springer, New York.CrossRefGoogle Scholar
Goodman, J., Kohn, R. and Reyna, L. (1986), ‘Numerical study of a relaxed variational problem from optimal design’, Comp. Meth. in Appl. Mech. and Eng. 57, 107–127.CrossRefGoogle Scholar
Gremaud, P. (1993), ‘On an elliptic-parabolic problem related to phase transitions in shape memory alloys’, Numer. Funct. Anal, and Optim. 14, 355–370.CrossRefGoogle Scholar
Gremaud, P. (1994), ‘Numerical analysis of a nonconvex variational problem related to solid–solid phase transitions’, SIAM J. Numer. Anal. 31, 111–127.CrossRefGoogle Scholar
Gremaud, P. (1995), ‘Numerical optimization and quasiconvexity’, Euro. J. of Applied Mathematics 6, 69–82.CrossRefGoogle Scholar
Gurtin, M. (1981), Topics in Finite Elasticity, SIAM, Philadelphia.CrossRefGoogle Scholar
Herstein, I. N. (1975), Topics in Algebra, 2nd edn, Wiley, New York.Google Scholar
Hoffmann, K.-H. and Niezgódka, M. (1990), ‘Existence and uniqueness of global solutions to an extended model of the dynamical developments in shape memory alloys’, Nonlinear Analysis 15, 977–990.CrossRefGoogle Scholar
Hoffmann, K.-H. and Zheng, S. (1988), ‘Uniqueness for structured phase transitions in shape memory alloys’, Math. Methods Appl. Sciences 10, 145–151.CrossRefGoogle Scholar
Hoffmann, K.-H. and Zou, J. (1995), ‘Finite element approximations of Landau–Ginzburg's equation model for structural phase transitions in shape memory alloys’, Mathematical Modelling and Numerical Analysis 29, 629–655.CrossRefGoogle Scholar
Horovitz, B., Barsch, G. and Krumhansl, J. (1991), ‘Twin bands in martensites: statics and dynamics’, Phys. Rev. B 43, 1021–1033.CrossRefGoogle ScholarPubMed
James, R. (1986), ‘Displacive phase transformations in solids’, Journal of the Mechanics and Physics of Solids 34, 359–394.CrossRefGoogle Scholar
James, R. (1987 a), Microstructure and weak convergence, in Material Instabilities in Continuum Mechanics and Related Mathematical Problems (Ball, J., ed.), Oxford University Press, pp. 175–196.Google Scholar
James, R. (1987 b), The stability and metastability of quartz, in Metastability and Incompletely Posed Problems (Antman, S. and Ericksen, J., eds), Springer, New York, pp. 147–176.CrossRefGoogle Scholar
James, R. (1989), Minimizing sequences and the microstructure of crystals, in Proceedings of the Society of Metals Conference on Phase Transformations, Cambridge University Press.Google Scholar
James, R. and Kinderlehrer, D. (1989), Theory of diffusionless phase transitions, in PDE's and continuum models of phase transitions (Rascle, M., Serre, D. and Slemrod, M., eds), Springer, pp. 51–84. Lecture Notes in Physics, vol. 344.CrossRefGoogle Scholar
Kartha, S., Castán, T., Krumhansl, J. and Sethna, J. (1994), ‘The spin-glass nature of tweed precursors in martensitic transformations’, Phys. Rev. Lett. 67, 3630.CrossRefGoogle Scholar
Kartha, S., Krumhansl, J. A., Sethna, J. P. and Wickham, L. K. (1995), ‘Disorder-driven pretransitional tweed in martensitic transformations’, Phys. Rev. B 52, 803.CrossRefGoogle ScholarPubMed
Khachaturyan, A. G. (1967), ‘Some questions concerning the theory of phase transformations in solids’, Soviet Phys. Solid State 8, 2163–2168.Google Scholar
Khachaturyan, A. G. (1983), Theory of structural transformations in solids, Wiley, New York.Google Scholar
Khachaturyan, A. G. and Shatalov, G. (1969), ‘Theory of macroscopic periodicity for a phase transition in the solid state’, Soviet Phys. JETP 29, 557–561.Google Scholar
Kinderlehrer, D. (1987), ‘Twinning in crystals II’, pp. 185–212. IMA Volumes in Mathematics and Its Applications, vol. 3.CrossRefGoogle Scholar
Kinderlehrer, D. and Ma, L. (1994 a), ‘Computational hysteresis in modeling magnetic systems’, IEEE. Trans. Magn. 30.6, 4380–4382.CrossRefGoogle Scholar
Kinderlehrer, D. and Ma, L. (1994 b), The simulation of hysteresis in nonlinear systems, in Mathematics in Smart Structures and Materials (Banks, H. T., ed.), SPIE, pp. 78–87. Vol. 2192.Google Scholar
Kinderlehrer, D. and Pedregal, P. (1991), ‘Characterizations of gradient Young measures’, Arch. Rat. Mech. Anal. 115, 329–365.CrossRefGoogle Scholar
Kirkpatrick, S., Gelatt, C. D. Jr, and Vecchi, M. P. (1983), ‘Optimization by simulated annealing’, Science 220, 671–680.CrossRefGoogle ScholarPubMed
Klouček, P. and Luskin, M. (1994 a), ‘The computation of the dynamics of martensitic microstructure’, Continuum Mech. Thermodyn. 6, 209–240.CrossRefGoogle Scholar
Klouček, P. and Luskin, M. (1994 b), ‘Computational modeling of the martensitic transformation with surface energy’, Mathematical and Computer Modelling 20, 101–121.CrossRefGoogle Scholar
Klouček, P., Li, B. and Luskin, M. (1996), ‘Analysis of a class of nonconforming finite elements for crystalline microstructures’, Math. Comput. To appear.Google Scholar
Kohn, R. (1991), ‘Relaxation of a double-well energy’, Continuum Mechanics and Thermodynamics 3, 193–236.CrossRefGoogle Scholar
Kohn, R. and Müller, S. (1992 a), ‘Branching of twins near an austenite/twinned-martensite interface’, Philosophical Magazine 66A, 697–715.CrossRefGoogle Scholar
Kohn, R. and Müller, S. (1992 b), ‘Relaxation and regularization of nonconvex variational problems’, Rend. Sem. Mat. Fis. Univ. Milano 62, 89–113.CrossRefGoogle Scholar
Kohn, R. and Müller, S. (1994), ‘Surface energy and microstructure in coherent phase transitions’, Comm. Pure Appl. Math. 47, 405–435.CrossRefGoogle Scholar
Kohn, R. and Strang, G. (1983), ‘Explicit relaxation of a variational problem in optimal design’, Bull. A.M.S. 9, 211–214.CrossRefGoogle Scholar
Kohn, R. and Strang, G. (1986), ‘Optimal design and relaxation of variational problems I, II, and III’, Commun. Pure Appl. Math. 39, 113–137, 139–182, and 353–377.CrossRefGoogle Scholar
Kružík, M. (1995), Numerical approach to double well problems. Manuscript.Google Scholar
Leo, P. H., Shield, T. W. and Bruno, O. P. (1993), ‘Transient heat transfer effects on the pseudoelastic hysteresis of shape memory wires’, Acta metallurgica 41, 2477–2485.CrossRefGoogle Scholar
Li, B. and Luskin, M. (1996), Finite element analysis of microstructure for the cubic to tetragonal transformation, Technical Report 1373, IMA.Google Scholar
Luskin, M. (1991), Numerical analysis of microstructure for crystals with a nonconvex energy density, in Progress in Partial differential equations: the Metz Surveys (Chipot, M. and Paulin, J. S. J., eds), Longman, Harlow, UK, pp. 156–165.Google Scholar
Luskin, M. (1996 a), Approximation of a laminated microstructure for a rotationally invariant, double well energy density, Technical report. To appear.CrossRefGoogle Scholar
Luskin, M. (1996 b), ‘Numerical analysis of a microstructure for a rotationally invariant, double well energy’, Zeitschrift für Angewandte Mathematik und Mechanik.Google Scholar
Luskin, M. and Ma, L. (1992), ‘Analysis of the finite element approximation of microstructure in micromagnetics’, SIAM J. Numer. Anal. 29, 320–331.CrossRefGoogle Scholar
Luskin, M. and Ma, L. (1993), Numerical optimization of the micromagnetics energy, in Mathematics in Smart Materials, SPIE, pp. 19–29.Google Scholar
Ma, L. (1993), Computation of magnetostrictive materials, in Mathematics in Smart Materials, SPIE, pp. 47–54.Google Scholar
Ma, L. and Walkington, N. (1995), ‘On algorithms for non-convex optimization’, SIAM J. Numer. Anal. 32, 900–923.CrossRefGoogle Scholar
Müller, S. (1993), ‘Singular perturbations as a selection criterion for periodic minimizing sequences’, Calc. Var. 1, 169–204.CrossRefGoogle Scholar
Müller, S. and Šverák, V. (1995), Attainment results for the two-well problem by convex integration, Technical Report SFB 256, Universität Bonn.Google Scholar
Nicolaides, R. A. and Walkington, N. (1993), ‘Computation of microstructure utilizing Young measure representations’, J. Intelligent Material Systems and Structures 4, 457–462.CrossRefGoogle Scholar
Niezgódka, M. and Sprekels, J. (1991), ‘Convergent numerical approximations of the thermomechanical phase transitions in shape memory alloys’, Numer. Math. 58, 759–778.CrossRefGoogle Scholar
Ortiz, M. and Giola, G. (1994), ‘The morphology and folding patterns of buckling-driven thin-film blisters’, J. Math. Phys. Solids 42, 531–559.CrossRefGoogle Scholar
Pedregal, P. (1993), ‘Laminates and microstructure’, Europ. J. Appl. Math. 4, 121–149.CrossRefGoogle Scholar
Pedregal, P. (1995), On the numerical analysis of non-convex variational problems. Manuscript.Google Scholar
Pedregal, P. (1996), ‘Numerical approximation of parametrized measures’, Num. Funct. Anal. Opt. 16, 1049–1066.CrossRefGoogle Scholar
Pego, R. (1987), ‘Phase transitions in one-dimensional nonlinear viscoelasticity’, Arch. Rat. Mech. Anal. 97, 353–394.CrossRefGoogle Scholar
Pitteri, M. (1984), ‘Reconciliation of local and global symmetries of crystals’, J. Elasticity 14, 175–190.CrossRefGoogle Scholar
Pitteri, M. and Zanzotto, G. (1996 a), Continuum models for twinning and phase transitions in crystals, Chapman and Hall, London.Google Scholar
Pitteri, M. and Zanzotto, G. (1996 b), Twinning in symmetry-breaking phase transitions. Manuscript.Google Scholar
Polak, E. (1971), Computational Methods in Optimization, Academic Press.Google Scholar
Quarteroni, A. and Valli, A. (1994), Numerical Approximation of Partial Differential Equations, Springer, Berlin.CrossRefGoogle Scholar
Rannacher, R. and Turek, S. (1992), ‘Simple nonconforming quadrilaterial Stokes element’, Numer. Meth. for PDEs 8, 97–111.CrossRefGoogle Scholar
Roitburd, A. (1969), ‘The domain structure of crystals formed in the solid phase’, Soviet Phys. Solid State 10, 2870–2876.Google Scholar
Roitburd, A. (1978), ‘Martensitic transformation as a typical phase transition in solids’, Solid State Physics 34, 317–390.CrossRefGoogle Scholar
Roubíček, T. (1994), ‘Finite element approximation of a microstructure evolution’, Math. Methods in the Applied Sciences 17, 377–393.CrossRefGoogle Scholar
Roubíček, T. (1996 a), ‘Numerical approximation of relaxed variational problems’, J. Convex Analysis.Google Scholar
Roubíček, T. (1996 b), Relaxation in Optimization Theory and Variational Calculus, Walter de Gruyter, Berlin.Google Scholar
Rudin, W. (1987), Real and Complex Analysis, 3rd edn, McGraw-Hill, New York.Google Scholar
Rybka, P. (1992), ‘Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions’, Proc. Royal Soc. Edinburgh 120A, 101–138.CrossRefGoogle Scholar
Rybka, P. (1995), Viscous damping prevents propagation of singularities in the system of viscoelasticity, Technical Report 184, Departament of Mathematics, Centro de Investigación y Estudios Avanzados del IPN, Mexico.Google Scholar
Schryvers, D. (1993), ‘Microtwin sequences in thermoelastic NixAl100–x martensite studied by conventional and high-resolution transmission electron microscopy’, Phil. Mag. A68, 1017–1032.CrossRefGoogle Scholar
Sethna, J. P., Kartha, S., Castan, T. and Krumhansl, J. A. (1992), ‘Tweed in martensites: A potential new spin glass’, Physica Scripta T42, 214.CrossRefGoogle Scholar
Shearer, M. (1986), ‘Nonuniqueness of admissible solutions of Riemann initial value problems’, Arch. Rat. Mech. Anal. 93, 45–59.CrossRefGoogle Scholar
Silling, S. (1989), ‘Phase changes induced by deformation in isothermal elastic crystals’, J. of the Mech. and Phys. of Solids 37, 293–316.CrossRefGoogle Scholar
Silling, S. (1992), ‘Dynamic growth of martensitic plates in an elastic material’, Journal of Elasticity 28, 143–164.CrossRefGoogle Scholar
Slemrod, M. (1983), ‘Admissibility criteria for propagating phase boundaries in a van der Waals fluid’, Arch. Rat. Mech. Anal. 81, 301–315.CrossRefGoogle Scholar
Swart, P. and Holmes, P. (1992), ‘Energy minimization and the formation of microstructure in dynamic anti-plane shear’, Arch. Rational Mech. Anal. 121, 37–85.CrossRefGoogle Scholar
Tartar, L. (1984), étude des oscillations dans les équations aux dérivées partielles nonlinéaires, in Lecture Notes in Physics vol. 195, Springer, pp. 384–412.Google Scholar
Tartar, L. (1990), ‘H-measures, a new approach for studying homogenization, oscillations, and concentration effects in partial differential equations’, Proc. Roy. Soc. Edinburgh 115A, 193–230.Google Scholar
Truskinovsky, L. (1985), ‘The structure of isothermal phase shock’, (Soviet Physics Doklady ) Dokl. Acad. Nauk SSSR 285 (2), 309–315.Google Scholar
Truskinovsky, L. (1987), ‘Dynamics of non-equilibrium phase boundaries in the heat-conductive nonlinear elastic medium’, J. Appl. Math. & Mech. (PMM) 51 (6), 777–784.CrossRefGoogle Scholar
Truskinovsky, L. (1994), ‘Transition to ‘detonation’ in dynamic phase changes’, Arch, for Rational Mech. Anal. 125, 375–397.CrossRefGoogle Scholar
Truskinovsky, L. and Zanzotto, G. (1995), ‘Finite-scale microstructures and metastability in one-dimensional elasticity’, Meccanica 30, 557–589.CrossRefGoogle Scholar
Truskinovsky, L. and Zanzotto, G. (1996), ‘Ericksen's bar revisited’, Mech. Phys. Solids.CrossRefGoogle Scholar
Sverák, V. (1992), ‘Rank-one convexity does not imply quasiconvexity’, Proc. Royal Soc. Edinburgh 120A, 185–189.CrossRefGoogle Scholar
Wang, Y., Chen, L.-Q. and Khachaturyan, A. G. (1994), Computer simulation of microstructure evolution in coherent solids, in Solid–Solid Phase Transformations (Johnson, W. C., Howe, J. M., Laughlin, D. E. and Soffa, W. A., eds), The Minerals, Metals & Materials Society, pp. 245–265.Google Scholar
Wen, S., Khachaturyan, A. G. and Morris, J. W. Jr (1981), ‘Computer simulation of a ‘Tweed-Transformation’ in an idealized elastic crystal’, Metallurgical Trans. A 12A, 581–587.CrossRefGoogle Scholar
Wloka, J. (1987), Partial Differential Equations, Cambridge University Press.CrossRefGoogle Scholar
Zanzotto, G. (1996), Weak phase transitions in simple lattices. Manuscript.Google Scholar