Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-25T08:15:15.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Laminates and microstructure

Published online by Cambridge University Press:  26 September 2008

Pablo Pedregal
Pablo Pedregal
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with the mathematical characterization of microstructure in elastic solids. We formulate our ideas in terms of rank-one convexity and identify the set of probability measures for which Jensen's inequality for this type of functions holds. This is the set of laminates. We also introduce generalized convex hulls of sets of matrices and investigate their structure.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 4 , Issue 2 , June 1993 , pp. 121 - 149
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ball, J. M. & James, R. 1987 Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100, 15–52.CrossRefGoogle Scholar
[2]Ball, J. M. & James, R. 1992 Proposed experimental tests of a theory of fine microstructure and the two well problem. Phil. Trans. R. Soc. London A (to appear).Google Scholar
[3]Battacharya, K. 1992 Self accommodation in martensite. Arch. Rat. Mech. Anal, (in press).CrossRefGoogle Scholar
[4]Battacharya, K. 1991 Wedge-like microstructure in martensite. Acta Metal 39, 2431–2444.Google Scholar
[5]Ericksen, J. L. 1979 On the symmetry of deformable crystals. Arch. Rat. Mech. Anal. 72, 1–13.CrossRefGoogle Scholar
[6]Ericksen, J. L. 1980 Some phase transitions in crystals. Arch. Rat. Mech. Anal. 73, 99–124.CrossRefGoogle Scholar
[7]Ericksen, J. L. 1981 Changes in symmetry in elastic crystals. In IUTAM Symp. Finite Elasticity (eds Carlson, D. E. and Shield, R. T.), Nijhoff, 167–177.CrossRefGoogle Scholar
[8]Ericksen, J. L. 1983 III posed problems in thermoelasticity theory. In Systems of Nonlinear Partial Differential Equations (ed. Ball, J. M.) Reidel, 71–95.Google Scholar
[9]Ericksen, J. L. 1984 The Cauchy and Born hypotheses for crystals. In Phase Transformations and Material Instabilities in Solids (ed. Gurtin, M.) Academic Press, 61–78.Google Scholar
[10]Ericksen, J. L. 1988 Constitutive theory for some constrained elastic crystals. Int. J. Solids Structures 22, 951–964.CrossRefGoogle Scholar
[11]Ericksen, J. L. 1986 Stable equilibrium configurations of elastic crystals. Arch. Rat. Mech. Anal. 94, 1–14.Google Scholar
[12]Ericksen, J. L. 1987 Twinning of crystals I. In Metastability and Incompletely Posed Problems, IMA Vol. Math. Appl. 3 (eds Antman, S.Ericksen, L., Kinderlehrer, D. and Müller, I.) Springer-Verlag, 77–96.CrossRefGoogle Scholar
[13]Ericksen, J. L. 1989 Weak martensitic transformations in Bravais lattices. Arch. Rat. Mech. Anal. 107, 23–36.CrossRefGoogle Scholar
[14]Fonseca, I. 1987 Variational methods for elastic crystals. Arch. Rat. Mech. Anal. 97, 189–220.CrossRefGoogle Scholar
[15]Fonseca, I. 1988 The lower quasi-convex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67, 175–195.Google Scholar
[16]Gurtin, M. E. 1986 On phase transitions with bulk, interfacial, and boundary energy. Arch. Rat. Mech. Anal. 96, 243–264.CrossRefGoogle Scholar
[17]James, R. D. 1988 Microstructure and weak convergence. In Proc. Symp. Material Instabilities in Continuum Mechanics, Heriot-Watt, Edinburgh (ed. Ball, J. M.) 175–196.Google Scholar
[18]James, R. D. 1989 Personal communication.Google Scholar
[19]James, R. D. & Kinderlehrer, D. 1989 Theory of diffusionless phase transitions. In PDE's and continuum models of phase transitions, Lecture Notes in Physics, Vol. 344 (eds Rascle, M.Serre, D. and Slemrod, M.) Springer-Verlag, 51–84.Google Scholar
[20]Kinderlehrer, D. 1988 Remarks about the equilibrium configurations of crystals. In Proc. Symp. Material Instabilities in Continuum Mechanics, Heriot-Watt, Edinburgh (ed. Ball, J. M.) 217–242.Google Scholar
[21]Kohn, R. V. 1989 The relationship between linear and non-linear variational models of coherent phase transitions. In Proc. Seventh Army Conf. on Appl. Math, and Computing. West Point, CA, June.Google Scholar
[22]Lurie, K. A. & Cherkaev, A. V. 1988 On a certain variational problem of phase equilibrium. In Proc. Symp. Material Instabilities in Continuum Mechanics, Heriot-Watt, Edinburgh (ed. Ball, J. M.) 257–268.Google Scholar
[23]Matos, J. 1992 The absence of fine microstructure in α-β quartz (to appear).Google Scholar
[24]Firoozye, N. B. 1990 Optimal Translations and Relaxations of some Multiwell Energies, PhD Thesis, Courant Institute, New York University.Google Scholar
[25]Firoozye, N. B. & Kohn, R. V. 1992 Geometric parameters and the relaxation of multiwell energies. In Microstructure and Phase Transitions (eds Kinderlehrer, D. et al. ) Springer-Verlag.Google Scholar
[26]Kohn, R. V. 1991 The relaxation of a Double-Well Energy. Cont. Mech. Therm. 3, 193–236.CrossRefGoogle Scholar
[27]Dacorogna, B. 1985 Remarques sur les notions de polyconvexité, quasi-convexité et convexit´e de rang I. J. Math. Pures Appl. 64, 403–438.Google Scholar
[28]Sverak, V. 1992 Rank-one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. 120A, 185–189.Google Scholar
[29]Chipot, M. & Kinderlehrer, D. 1988 Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103, 237–277.Google Scholar
[30]Dacorogna, B. 1989 Direct Methods in the Calculus of Variations. Springer-Verlag.Google Scholar
[31]Kohn, R. V. 1990 Personal communication.Google Scholar
[32]Tartar, L. 1991 Microstructure and phase transitions (eds Kinderlehrer, D. et al. ) Springer-Verlag.Google Scholar
[33]Milton, G. & Nesi, V. 1992 Polycrystalline configurations that maximize electrical resistivity. J. Mech. Phys. Solids (to appear).Google Scholar
[34]Ball, J. M. 1990 Sets of gradients with no rank one connections. J. Math. Pures et Appl. 69, 241–259.Google Scholar
[35]Matos, J. 1990 Some Mathematical Methods in Elasticity. Ph.D. thesis, University of Minnesota.Google Scholar
[36]Tartar, L. 1982 The compensated-compactness method applied to systems of conservation laws. In Systems of Nonlinear Partial Diff. Eq. (ed. Ball, J. M.) NATO ASI Series, Vol. C111, Reidel, 263–285.Google Scholar
[37]Kinderlehrer, D. & Pedregal, P. 1991 Characterizations of gradient Young measures. Arch. Rat. Mech. Anal. 115, 329–365.CrossRefGoogle Scholar
[38]Kinderlehrer, D. & Pedregal, P. 1992 Remarks about Young measures supported on two wells (to appear).Google Scholar
[39]Kinderlehrer, D. & Pedregal, P. 1992 Weak convergence of integrands and the Young measure representation. SIAM J. Math. Anal. 23, 1–19.Google Scholar
[40]Collins, C. & Luskin, M. 1989 The computation of the austenitic-martensitic phase transition. In PDE'S and continuum models of phase transitions, Lecture Notes in Physics vol. 344 (eds Rascle, M.Serre, D. and Slemrod, M.) Springer-Verlag, 34–50.Google Scholar
[41]Collins, C. & Luskin, D. 1990 Numerical modelling of the microstructure of crystals with symmetry-related variants. Proc. ARO US-Japan Workshop on Smart/Intelligent Materials and Systems, Technomic.Google Scholar
[42]Collins, C. & Luskin, M. 1992 Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem (Math. Comput. to appear).Google Scholar
[43]Chipot, M. & Collins, C. 1992 Numerical approximation in variational problems with potential wells (to appear).CrossRefGoogle Scholar
[44]Collins, C., Kinderlehrer, D. & Luskin, M. 1991 Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal. 28, 321–333.Google Scholar
[45]Nicolaides, R. A. & Walkington, N. 1992 Computation of microstructure utilizing Young measure representation. In Proc. Recent Adv. Adaptive Sensory Materials and their Appl., Blacksburg, VA.Google Scholar
[46]Ball, J. M. & Murat, F. 1990 Remarks on rank-one convexity and quasiconvexity. Proc. Dundee Conference on Differential Equations.Google Scholar
[47]Dacorogna, B. & Marcellini, P. 1988 A counterexample in the vectorial calculus of variations. In Proc. Symp. Material Instabilities in Continuum Mechanics, Heriot-Watt, Edinburgh (ed. Ball, J. M.) 77–83.Google Scholar
[48]Kohn, R. V. & Strang, G. 1987 Optimal design and relaxation of variational problems, I-III. Comm. Pure Appl. Math. 34, 113–137, 139–182, 353–377.Google Scholar
[49]Marcellini, P. 1984 Quasiconvex quadratic forms in two dimensions. Appl. Math. Optim. 11, 183–189.Google Scholar
[50]Serre, D. 1983 Formes quadratiques in the calculus of variations. J. de Math. Pures et Appl. 62, 177–196.Google Scholar
[51]Sverak, V. 1990 Examples of rank one convex functions. Proc. R. Soc. Edinb. 114A, 237–242.Google Scholar
[52]Terpstra, F. J. 1938 Die darstellung biquadratischer formen als summen von quadraten mit anwendung auf die variationsrechnung. Math. Ann. 116, 166–180.CrossRefGoogle Scholar
[53]Firoozye, N. B. 1992 Optimal use of the translation method and relaxations of variational problems (to appear).Google Scholar
[54]Pipkin, A. C. 1989 Elastic material with two preferred states, Preprint, August.Google Scholar
[55]Milton, G. 1986 Modelling properties of composites by laminates. In Homogeneization of Effective Moduli of Materials and Media (eds Ericksen, J. et al. ) Springer-Verlag.Google Scholar
[56]Zhang, K. 1992 Rank-one connections and the three well problem (preprint).Google Scholar
[57]Sverak, V. 1992 On regularity for the Monge-Ampère equation without convexity assumptions (to appear).Google Scholar
[58]Young, L. C. 1969 Lectures on the Calculus of Variations and Optimal Control Theory. Saunders.Google Scholar
[59]Ball, J. M. 1989 A version of the fundamental theorem for Young measures. In PDE's and continuum models of phase transitions, Lecture Notes in Physics vol. 344 (eds Rascle, M.Serre, D. and Slemrod, M.), Springer-Verlag, 207–215.CrossRefGoogle Scholar