Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T21:59:56.320Z Has data issue: false hasContentIssue false
  • English
  • Français

A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials

Published online by Cambridge University Press:  14 February 2007

Vincenzo Nesi  and
Enrico Rogora
Vincenzo Nesi
Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza”, Italy;  nesi@mat.uniroma1.it
Enrico Rogora
Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza”, Italy;  nesi@mat.uniroma1.it
Get access

Abstract

The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-r convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-r convex forms arises. In the present paper, we define the concept of extremal 2-forms  and characterize them in the rotationally invariant jointly rank-r convex case.

Keywords

Compensated compactnessrank-r convexityeffective conductivityquadratic forms.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 1 , January 2007 , pp. 1 - 34
Copyright
© EDP Sciences, SMAI, 2007

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

Allaire, G. and Francfort, G., Existence of minimizers for non-quasiconvex functionals arising in optimal design. Ann. Inst. H. Poincaré Anal. non Linéaire 15 (1998) 301–339. CrossRef
G. Allaire and R.V. Kohn, Optimal lower bounds on the elastic energy of a composite made from two non-well ordered isotropic materials. Quart. Appl. Math. LII (1994) 311–333.
Allaire, G. and Lods, V., Minimizer for a double-well problem with affine boundary conditions. Proc. Roy. Soc. Edinburgh Sec. A 129 (1999) 439–466. CrossRef
G. Allaire and H. Maillot, H-measures and bounds on the effective properties of composite materials. Port. Math. (N.S.) 60 (2003) 161–192.
Avellaneda, M., Cherkaev, A.V., Lurie, K.A. and Milton, G.W., On the effective conductivity of polycrystals and a three dimensional phase interchange inequality. J. Appl. Phys. 63 (1988) 4989–5003. CrossRef
M.J. Beran, Nuovo Cimento 38 (1965) 771–782. CrossRef
Bergman, D.J., The dielectric constant of a composite material: a problem in classical physics. Phys. Rep. 43 (1978) 377-407. CrossRef
Bergman, D.J., Rigorous bounds for the complex dielectric constant of a two-component composite. Ann. Physics 138 (1982) 78–114. CrossRef
R. Bhatia, Matrix Analysis. Graduate texts in Mathematics, Springer-Verlag, New York (1997).
Berryman, J.G. and Milton, G.W., Microgeometry of random composites and porous media. J. Phys. D: Appl. Phys. 21 (1988) 87–94. CrossRef
A. Cherkaev, Variational methods for structural optimization. Applied Mathematical Sciences 140, Springer-Verlag, Berlin (2000).
Cherkaev, A.V. and Gibiansky, L.V., The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites. Proc. Roy. Soc. Edinburgh Sect. A 122 (1992) 93–125. CrossRef
K. Clark and G. Milton, Optimal bounds correlating electric, magnetic and thermal properties of two phases, two dimensional composites. Proc. R. Soc. Lond. A, 448 (1995) 161–190.
G. Dal Maso, An introduction to $\Gamma$ -convergence. Progress in Nonlinear Differential Equations and their Applications 8, Birkhauser Boston, Inc., Boston, MA (1993).
De Giorgi, E. and Spagnolo, S., Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine. Bull. Un. Mat. Ital (4) 8 (1973) 391–411.
Dell'Antonio, G. and Nesi, V., A scalar inequality which bounds the effective conductivity of composites. Proc. Royal Soc. London A 431 (1990) 519–530. CrossRef
Dykhne, A.M., Conductivity of a two-dimensional two-phase system. Soviet Physiscs JETP 32 (1971) 63–65.
Fonseca, I. and Müller, S., A-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30 (1999) 1355–1390. CrossRef
L.V. Gibiansky, Effective properties of a plane two-phase elastic composites: coupled bulk-shear moduli bounds, in Homogenization, Ser. Adv. Math, Appl. Sci. 50, World Sci. Publishing, River Edge, NJ (1999) 214–258.
L.V. Gibiansky and A.V. Cherkaev, Design of composite plates of extremal rigidity and/or Microstructures of composites of extremal rigidity and exact bounds on the associated energy density, in Topics in the mathematical modelling of composite materials, A. Cherkaev and R. Kohn Eds., Progr. Nonlinear Differential Equations Appl. 31, Birkhäuser Boston, Inc., Boston, MA, (1997).
Gibiansky, L.V. and Cherkaev, A.V., Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite. J. Mech. Phys. Solids 41 (1993) 937–980.
Gibiansky, L.V. and Torquato, S., Link between the conductivity and elastic moduli of composite materials. Phys. Rev. Lett. 71 (1993) 2927–2930. CrossRef
Gibiansky, L.V. and Torquato, S., Connection between the conductivity and bulk modulus of Isotropic composite materials. Proc. Roy. Soc. London A 452 (1996) 253–283. CrossRef
Gibiansky, L.V. and Torquato, S., Phase-interchange relations for the elastic moduli of two-phase composites. Internat. J. Engrg. Sci. 34 (1996) 739–760. CrossRef
Goldsztein, G.H., Rigid-pefectly-plastic two-dimensional polycrystals. Proc. Roy. Soc. Lond. A 457 (2003) 1949–1968. CrossRef
Hashin, Z. and Shtrikman, S., A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys. 33 (1962) 3125–3131. CrossRef
V.V. Jikov, S.M. Kozlov and O. A. Oleĭnik, Homogenization of differential operators and integral functionals. Translated from the Russian by G.A. Yosifian, Springer-Verlag, Berlin (1994).
Keller, J.B., A theorem on the conductivity of a composite medium. J. Math. Phys. 5 (1964) 548–549. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems I. Comm. Pure Appl. Math. 39 (1986) 113–137. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems II. Comm. Pure Appl. Math. 39 (1986) 139–182. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems III. Comm. Pure Appl. Math. 39 (1986) 353–377. CrossRef
Lurie, K.A. and Cherkaev, A.V., Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportions. Proc. Roy. Soc. Edinburgh Sect. A 99 (1984) 71–87. CrossRef
M. Milgrom and M.M. Shtrickman, Linear response of two-phase composites with cross moduli: Exact universal relations. Physical Review A (Atomic, Molecular and Optical Physics) 40 (1989) 1568–1575. CrossRef
G.W. Milton, Bounds on the transport and optical properties of a two-component composite material, J. Appl. Phys. 52 (1981) 5294–5304. CrossRef
Milton, G.W., Bounds on the complex permittivity of a two-component composite material. J. Appl. Phys. 52 (1981) 5286–5293. CrossRef
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. CrossRef
Milton, G.W., Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids 30 (1982) 177–191. CrossRef
G.W. Milton, The theory of composites. Cambridge Monographs on Applied and Computational Mathematics 6, Cambridge University Press, Cambridge (2002).
Milton, G.W. and Kohn, R.V., Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36 (1988) 597–629. CrossRef
Milton, G.W. and Serkov, S.K., Bounding the current in nonlinear conducting composites. The J.R. Willis 60th anniversary volume. J. Mech. Phys. Solids 48 (2000) 1295–1324. CrossRef
Morrey, C.B., Multiple integral problems in the calculus of variations and related topics. Ann. Scuola Norm. Sup. Pisa 14 (1960) 1–61.
Murat, F., Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (1982) 69–102.
F. Murat and L. Tartar, Calcul des variations et homogénéisation, in Homogenization methods: theory and applications in physics (Breau-sans-Nappe, 1983), Collect. Dir. Etudes Rech. Elec. France 57, Eyrolles, Paris (1985) 319–369. English translation (see [46]).
F. Murat and L. Tartar H-convergence, Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, mimeographed notes (1978). English translation (see [45]).
F. Murat and L. Tartar, H-convergence, in Topics in the mathematical modelling of composite materials, Birkhauser Boston, Boston, MA, Progr. Nonlinear Differential Equations Appl. 31 (1997) 21–43
F. Murat and L. Tartar, Calculus of variations and homogenization, in Topics in the mathematical modelling of composite materials, Birkhauser Boston, Boston, MA, Progr. Nonlinear Differential Equations Appl. 31 (1997) 139–173.
Nesi, V., Multiphase interchange inequalities. J. Math. Phys 32 (1991) 2263–2275. CrossRef
Nesi, V., Bounds on the effective conductivity of 2-dimensional composites made of $n\geq 3$ isotropic phases in prescribed volume fraction: the weighted translation method. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1219–1239. CrossRef
Prager, S., Improved variational bounds on some bulk properties of a two-phase random media. J. Chem. Phys. 50 (1969) 4305–4312. CrossRef
Procesi, C., The invariant theory of $n\times n$ matrices. Adv. Math. 19 (1976) 306-381. CrossRef
E. Rogora, Invariants of matrices under the action of the special orthogonal group, preprint del Dipartimento di Matematica, Università di Roma “La Sapienza", n. 10/2005, also available at http://www.mat.uniroma1.it/people/rogora/pdf/son.pdf.
Schulgasser, K., Bounds on the conductivity of statistically isotropic polycrystals. J. Phys. C10 (1977) 407–417.
Spagnolo, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa 22 (1968) 577–597.
Šverak, V., New examples of quasiconvex functions. Arch. Rational Mech. Anal. 119 (1992) 293–300. CrossRef
Šverak, V., Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 18–189. CrossRef
Talbot, D.R.S. and Willis, J.R., Bounds for the effective relation of anonlinear composite. Proc. R. Soc. A 460 (2004) 2705–2723. CrossRef
L. Tartar, Estimations de coefficients homogénéisés, in Computing methods in applied science and engeneering (Proc. third Int. Sympos. Versailles, 1977), Lect. Notes Math. 704, Springer Verlag, Berlin (1979) 364–373. English translation in [60].
L. Tartar, Estimations fines des coefficients homogénéisés, in Ennio De Giorgi's Colloquium (Paris 1983), P. Kree Ed., Pitman, Boston (1985) 168–187.
L. Tartar, Compensated compactness and applications to p.d.e. in nonlinear analysis and mechanics, in Heriot-Watt Symposium IV, R.J. Knops Ed., Pitman, Boston (1979) 136–212.
L. Tartar, Estimations of homogenized coefficients, in Topics in the mathematical modelling of composite materials, Birkhäuser, Boston, Proc. Non Linear Diff. Equations Appl. 31 (1997) 9–20.
L. Tartar, An introduction to the homogenization method in optimal design, in Optimal shape design (Tróia, 1998), Springer, Berlin, Lect. Notes Math. 1740 (2002) 47–156. CrossRef
L. Tonelli, Fondamenti di calcolo delle variazioni. Zanichelli, Bologna (1921).
J. Von Neumann, Some matrix inequalities and metrization of metric-space Tomsk Univ. Rev. 1 (1937) 286–300 (also in Collected Works 4, 286–300).
H. Weyl, The classical groups: Their invariants and representations. Fifteenth printing. Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ (1997).
Zhikov, V.V., Estimates for the averaged matrix and the averaged tensor. Russian Math. Surveys 46 (1991) 65–136. CrossRef