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

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 1, page 1-34
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topNesi, Vincenzo, and Rogora, Enrico. "A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials." ESAIM: Control, Optimisation and Calculus of Variations 13.1 (2007): 1-34. <http://eudml.org/doc/249995>.

@article{Nesi2007,

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.
},

author = {Nesi, Vincenzo, Rogora, Enrico},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Compensated compactness; rank-r convexity; effective
conductivity; quadratic forms.; effective conductivity; extremal 2-forms},

language = {eng},

month = {2},

number = {1},

pages = {1-34},

publisher = {EDP Sciences},

title = {A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials},

url = {http://eudml.org/doc/249995},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Nesi, Vincenzo

AU - Rogora, Enrico

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/2//

PB - EDP Sciences

VL - 13

IS - 1

SP - 1

EP - 34

AB -
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.

LA - eng

KW - Compensated compactness; rank-r convexity; effective
conductivity; quadratic forms.; effective conductivity; extremal 2-forms

UR - http://eudml.org/doc/249995

ER -

## References

top- G. Allaire and G. Francfort, Existence of minimizers for non-quasiconvex functionals arising in optimal design. Ann. Inst. H. Poincaré Anal. non Linéaire15 (1998) 301–339. Zbl0913.49008
- 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. Zbl0806.73038
- G. Allaire and V. Lods, Minimizer for a double-well problem with affine boundary conditions. Proc. Roy. Soc. Edinburgh Sec. A129 (1999) 439–466. Zbl0958.49008
- G. Allaire and H. Maillot, H-measures and bounds on the effective properties of composite materials. Port. Math. (N.S.) 60 (2003) 161–192. Zbl1075.74068
- M. Avellaneda, A.V. Cherkaev, K.A. Lurie and G.W. Milton, On the effective conductivity of polycrystals and a three dimensional phase interchange inequality. J. Appl. Phys.63 (1988) 4989–5003.
- M.J. Beran, Nuovo Cimento38 (1965) 771–782.
- D.J. Bergman, The dielectric constant of a composite material: a problem in classical physics. Phys. Rep.43 (1978) 377-407.
- D.J. Bergman, Rigorous bounds for the complex dielectric constant of a two-component composite. Ann. Physics138 (1982) 78–114.
- R. Bhatia, Matrix Analysis. Graduate texts in Mathematics, Springer-Verlag, New York (1997).
- J.G. Berryman and G.W. Milton, Microgeometry of random composites and porous media. J. Phys. D: Appl. Phys.21 (1988) 87–94.
- A. Cherkaev, Variational methods for structural optimization. Applied Mathematical Sciences 140, Springer-Verlag, Berlin (2000). Zbl0956.74001
- A.V. Cherkaev and L.V. Gibiansky, The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites. Proc. Roy. Soc. Edinburgh Sect. A122 (1992) 93–125. Zbl0767.73061
- 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. Zbl0823.73043
- G. Dal Maso, An introduction to $\Gamma $-convergence. Progress in Nonlinear Differential Equations and their Applications 8, Birkhauser Boston, Inc., Boston, MA (1993). Zbl0816.49001
- E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine. Bull. Un. Mat. Ital (4)8 (1973) 391–411. Zbl0274.35002
- G. Dell'Antonio and V. Nesi, A scalar inequality which bounds the effective conductivity of composites. Proc. Royal Soc. London A431 (1990) 519–530.
- A.M. Dykhne, Conductivity of a two-dimensional two-phase system. Soviet Physiscs JETP32 (1971) 63–65.
- I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal.30 (1999) 1355–1390. Zbl0940.49014
- 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. Zbl1055.74552
- 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). Zbl0928.74077
- L.V. Gibiansky and A.V. Cherkaev, Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite. J. Mech. Phys. Solids41 (1993) 937–980. Zbl0776.73044
- L.V. Gibiansky and S. Torquato, Link between the conductivity and elastic moduli of composite materials. Phys. Rev. Lett.71 (1993) 2927–2930.
- L.V. Gibiansky and S. Torquato, Connection between the conductivity and bulk modulus of Isotropic composite materials. Proc. Roy. Soc. London A452 (1996) 253–283. Zbl0872.73032
- L.V. Gibiansky and S. Torquato, Phase-interchange relations for the elastic moduli of two-phase composites. Internat. J. Engrg. Sci.34 (1996) 739–760. Zbl0899.73313
- G.H. Goldsztein, Rigid-pefectly-plastic two-dimensional polycrystals. Proc. Roy. Soc. Lond. A457 (2003) 1949–1968. Zbl1066.74528
- Z. Hashin and S. Shtrikman, A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys.33 (1962) 3125–3131. Zbl0111.41401
- 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).
- J.B. Keller, A theorem on the conductivity of a composite medium. J. Math. Phys.5 (1964) 548–549. Zbl0129.44001
- R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems I. Comm. Pure Appl. Math.39 (1986) 113–137. Zbl0609.49008
- R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems II. Comm. Pure Appl. Math.39 (1986) 139–182. Zbl0621.49008
- R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems III. Comm. Pure Appl. Math.39 (1986) 353–377. Zbl0694.49004
- K.A. Lurie and A.V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportions. Proc. Roy. Soc. Edinburgh Sect. A99 (1984) 71–87. Zbl0564.73079
- 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.
- G.W. Milton, Bounds on the transport and optical properties of a two-component composite material, J. Appl. Phys.52 (1981) 5294–5304.
- G.W. Milton, Bounds on the complex permittivity of a two-component composite material. J. Appl. Phys.52 (1981) 5286–5293.
- G.W. Milton, 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. Zbl0751.73041
- G.W. Milton, Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids30 (1982) 177–191. Zbl0486.73063
- G.W. Milton, The theory of composites. Cambridge Monographs on Applied and Computational Mathematics 6, Cambridge University Press, Cambridge (2002). Zbl0993.74002
- G.W. Milton and R.V. Kohn, Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids36 (1988) 597–629. Zbl0672.73012
- G.W. Milton and S.K. Serkov, Bounding the current in nonlinear conducting composites. The J.R. Willis 60th anniversary volume. J. Mech. Phys. Solids48 (2000) 1295–1324. Zbl0991.78019
- C.B. Morrey, Multiple integral problems in the calculus of variations and related topics. Ann. Scuola Norm. Sup. Pisa14 (1960) 1–61. Zbl0094.08104
- F. Murat, 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. Zbl0464.46034
- 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 Zbl0920.35019
- 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. Zbl0939.35019
- V. Nesi, Multiphase interchange inequalities. J. Math. Phys32 (1991) 2263–2275. Zbl0807.73040
- V. Nesi, Bounds on the effective conductivity of 2-dimensional composites made of $n\ge 3$ isotropic phases in prescribed volume fraction: the weighted translation method. Proc. Roy. Soc. Edinburgh Sect. A125 (1995) 1219–1239. Zbl0852.35016
- S. Prager, Improved variational bounds on some bulk properties of a two-phase random media. J. Chem. Phys.50 (1969) 4305–4312.
- C. Procesi, The invariant theory of $n\times n$ matrices. Adv. Math.19 (1976) 306-381. Zbl0331.15021
- 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 . Zbl1136.16020URIhttp://www.mat.uniroma1.it/people/rogora/pdf/son.pdf
- K. Schulgasser, Bounds on the conductivity of statistically isotropic polycrystals. J. Phys.C10 (1977) 407–417.
- S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa22 (1968) 577–597.
- V. Šverak, New examples of quasiconvex functions. Arch. Rational Mech. Anal.119 (1992) 293–300. Zbl0823.26009
- V. Šverak, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A120 (1992) 18–189. Zbl0777.49015
- D.R.S. Talbot and J.R. Willis, Bounds for the effective relation of anonlinear composite. Proc. R. Soc. A460 (2004) 2705–2723. Zbl1072.74058
- 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 SymposiumIV, 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. Zbl0920.35018
- 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.
- 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 Works4, 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).
- V.V. Zhikov, Estimates for the averaged matrix and the averaged tensor. Russian Math. Surveys46 (1991) 65–136. Zbl0751.15014

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.