Exact results for weakly nonlinear composites and implications for homogenization methods

Joshua Furer; Martin Idiart; Pedro Ponte Castañeda

doi:10.5802/crmeca.66

Exact results for weakly nonlinear composites and implications for homogenization methods

Joshua Furer ¹ ; Martin Idiart ^2,³ ; Pedro Ponte Castañeda ^4,¹

¹ Graduate Group in Applied Mathematics and Computational Sciences, University of Pennsylvania, Philadelphia, PA 19104-6315, USA
² Centro Tecnológico Aeroespacial / Departamento de Aeronáutica, Facultad de Ingeniería, Universidad Nacional de La Plata, Avda. 1 esq. 47 S/N, La Plata B1900TAG, Argentina
³ Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), CCT La Plata, Calle 8 N ∘ 1467, La Plata B1904CMC, Argentina
⁴ Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA

Comptes Rendus. Mécanique, Volume 348 (2020) no. 10-11, pp. 893-909.

Résumé

Weakly nonlinear composite conductors are characterized by position-dependent dissipation potentials expressible as an additive composition of a quadratic potential and a nonquadratic potential weighted by a small parameter. This additive form carries over to the effective dissipation potential of the composite when expanded to first order in the small parameter. However, the first-order correction of this asymptotic expansion depends only on the zeroth-order values of the local fields, namely, the local fields within the perfectly linear composite conductor. This asymptotic expansion is exploited to derive the exact effective conductivity of a composite cylinder assemblage exhibiting weak nonlinearity of the power-law type (i.e., power law with exponent $m = 1 + δ$ , such that $| δ | ≪ 1$ ), and found to be identical (to first order in $δ$ ) to the corresponding asymptotic result for sequentially laminated composites of infinite rank. These exact results are used to assess the capabilities of more general nonlinear homogenization methods making use of the properties of optimally selected linear comparison composites.

Reçu le : 2020-10-12
Révisé le : 2020-11-25
Accepté le : 2020-11-27
Publié le : 2021-01-13

DOI : 10.5802/crmeca.66

Mots clés : Homogenization, Nonlinear, Heterogeneous materials, Exact solution, Composite materials, Effective conductivity

Affiliations des auteurs :

Joshua Furer ¹ ; Martin Idiart ^{2, 3} ; Pedro Ponte Castañeda ^{4, 1}

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Exact results for weakly nonlinear composites and implications for homogenization methods},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {893--909},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {348},
     number = {10-11},
     year = {2020},
     doi = {10.5802/crmeca.66},
     language = {en},
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Joshua Furer; Martin Idiart; Pedro Ponte Castañeda. Exact results for weakly nonlinear composites and implications for homogenization methods. Comptes Rendus. Mécanique, Volume 348 (2020) no. 10-11, pp. 893-909. doi : 10.5802/crmeca.66. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.66/

Bibliographie
Cité par

[1] J. C. Maxwell A Treatise on Electricity and Magnetism, Clarendon Press, Oxford, UK, 1873 | Zbl

[2] L. Tartar Estimations fines des coefficients homogénéisés, Ennio de Giorgi Colloquium (P. Krée, ed.) (Research Notes in Math.), Volume 125, Pitman Publishing Ltd., London, 1985, pp. 168-187 | Zbl

[3] G. Francfort; F. Murat Homogenization and optimal bounds in linear elasticity, Arch. Ration. Mech. Anal., Volume 94 (1986), pp. 307-334 | DOI | MR | Zbl

[4] A. B. Movchan; S. K. Serkov The Pólya-Szegó matrices in asymptotic models of dilute composites, Eur. J. Appl. Math., Volume 8 (1997), pp. 595-621 | DOI | Zbl

[5] Z. Hashin; S. Shtrikman A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys., Volume 33 (1962), pp. 3125-3131 | DOI | Zbl

[6] K. Schulgasser Sphere assemblage model for polycrystals and symmetric materials, J. Appl. Phys., Volume 54 (1983), pp. 1380-1382 | DOI

[7] P. Gilormini Realizable compressibility and conductivity in isotropic two-phase composites, C. R. Acad. Sci. Paris IIb–Mech., Volume 329 (2001), pp. 851-855 | Zbl

[8] G. W. Milton; S. K. Serkov Neutral coated inclusions in conductivity and anti-plane elasticity, Proc. R. Soc. Lond. A, Volume 457 (2001), pp. 1973-1997 | DOI | MR | Zbl

[9] A. M. Dykhne Conductivity of a two-dimensional two-phase system, Zh. Eksp. Teor. Fiz., Volume 59 (1970), pp. 110-115

[10] V. L. Berdichevskii Heat conduction of checkerboard structures, Vestnik Moskov. Univ. Ser. I Mat. Mekh., Volume 40 (1985), pp. 56-63 | MR

[11] D. A. G. Bruggeman Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen, Ann. Phys., Volume 416 (1935), pp. 636-664 | DOI

[12] M. Avellaneda Optimal bounds and microgeometries for elastic two-phase composites, SIAM J. Appl. Math., Volume 47 (1987), pp. 1216-1228 | DOI | MR | Zbl

[13] M. I. Idiart Multiphase conductors realizing Aleksandrov’s mean, SIAM J. Appl. Math., Volume 76 (2016), pp. 1792-1798 | DOI | MR | Zbl

[14] I. Harriton; G. deBotton The nearly isotropic behaviour of high-rank nonlinear sequentially laminated composites, Proc. R. Soc. Lond. A, Volume 459 (2003), pp. 157-174 | DOI | MR

[15] M. I. Idiart Modeling the macroscopic behavior of two-phase nonlinear composites by infinite-rank laminates, J. Mech. Phys. Solids, Volume 56 (2008), pp. 2599-2617 | DOI | MR | Zbl

[16] G. Francfort; P. Suquet Duality relations for nonlinear incompressible two-dimensional elasticity, Proc. R. Soc. Edinb., Volume 131A (2001), pp. 351-369 | DOI | MR | Zbl

[17] J. M. Duva A self-consistent analysis of the stiffening effect of rigid inclusions on a power-law material, J. Eng. Mater. Technol., Volume 106 (1984), pp. 317-321 | DOI

[18] O. Lopez-Pamies An exact result for the macroscopic response of particle-reinforced Neo-Hookean solids, J. Appl. Mech., Volume 77 (2010), 021016 | DOI

[19] X. C. Zeng; D. J. Bergman; P. M. Hui; D. Stroud Effective medium theory for weakly nonlinear composites, Phys. Rev. B, Volume 38 (1988), pp. 10970-10973 | DOI

[20] O. Levy; D. J. Bergman Weakly nonlinear composites of random composites: a series expansion approach, J. Statist. Phys., Volume 82 (1995), pp. 1327-1344 | DOI

[21] O. Levy; R. V. Kohn Duality relations for non-ohmic composites, with applications to behavior near percolation, J. Statist. Phys., Volume 90 (1998), pp. 159-189 | DOI | MR | Zbl

[22] Y. P. Pellegrini Field distributions and effective-medium approximation for weakly nonlinear media, Phys. Rev. B, Volume 61 (2000), pp. 9365-9372 | DOI

[23] J. R. Willis The overall response of composite materials, ASME J. Appl. Mech., Volume 50 (1983), pp. 1202-1209 | DOI | Zbl

[24] Z. Hashin; S. Shtrikman A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids, Volume 11 (1963), pp. 127-140 | DOI | Zbl

[25] D. R. S. Talbot; J. R. Willis Variational principles for inhomogeneous nonlinear media, IMA J. Appl. Math., Volume 35 (1985), pp. 39-54 | DOI | MR | Zbl

[26] P. Ponte Castañeda The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids, Volume 39 (1991), pp. 45-71 | DOI | MR | Zbl

[27] P. Ponte Castañeda Bounds and estimates for the properties of nonlinear heterogeneous systems, Phil. Trans. R. Soc. Lond. A, Volume 340 (1992), pp. 531-567 | MR | Zbl

[28] P. Suquet Overall potentials and extremal surfaces of power law or ideally plastic materials, J. Mech. Phys. Solids, Volume 41 (1993), pp. 981-1002 | DOI | Zbl

[29] P. Gilormini A shortcoming of the classical nonlinear extension of the self-consistent model, C. R. Acad. Sci. Paris IIb–Mech., Volume 320 (1995), pp. 115-122 | Zbl

[30] R. Hill Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, Volume 13 (1965), pp. 89-101 | DOI | Zbl

[31] J. W. Hutchinson Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. R. Soc. Lond. A, Volume 348 (1976), pp. 101-127 | Zbl

[32] E. Kröner Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls, Z. Phys., Volume 151 (1958), pp. 504-518 | DOI

[33] P. Suquet Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte Castañeda’s nonlinear variational procedure, C. R. Acad. Sci. Paris II, Volume 320 (1995), pp. 563-571 | Zbl

[34] P. Suquet; P. Ponte Castañeda Small-contrast perturbation expansions for the effective properties of nonlinear composites, C. R. Acad. Sci. Paris II, Volume 317 (1993), pp. 1515-1522 | Zbl

[35] P. Ponte Castañeda Exact second-order estimates for the effective mechanical properties of nonlinear composites, J. Mech. Phys. Solids, Volume 44 (1996), pp. 827-862 | DOI | MR | Zbl

[36] P. Ponte Castañeda Stationary variational estimates for the effective response and field fluctuations in nonlinear composites, J. Mech. Phys. Solids, Volume 96 (2016), pp. 660-682 | DOI | MR

[37] M. I. Idiart; P. Ponte Castañeda Field statistics in nonlinear composites. I. Theory, Proc. R. Soc. A, Volume 463 (2007), pp. 183-202 | DOI | MR | Zbl

[38] J. Furer; P. Ponte Castañeda A symmetric fully optimized second-order method for nonlinear homogenization, Z. Angew. Math. Mech., Volume 98 (2018), pp. 222-254 | DOI | MR

[39] J. R. Willis Variational estimates for the overall response of an inhomogeneous nonlinear dielectric, Homogenization and Effective Moduli of Materials and Media (J. L. Ericksen, ed.), Springer-Verlag, New York, 1986, pp. 247-263 | DOI | Zbl

[40] P. Ponte Castañeda; P. Suquet Nonlinear composites, Adv. Appl. Mech., Volume 34 (1998), pp. 171-302 | DOI | Zbl

[41] G. W. Milton The Theory of Composites, Cambridge University Press, Cambridge, UK, 2002 | Zbl

[42] M. I. Idiart; P. Ponte Castañeda Estimates for two-phase nonlinear conductors via iterated homogenization, Proc. R. Soc. A, Volume 469 (2013), 20120626 | DOI | MR | Zbl

[43] P. Ponte Castañeda; G. deBotton; G. Li Effective properties of nonlinear inhomogeneous dielectrics, Phys. Rev. B, Volume 46 (1992), pp. 4387-4394 | DOI

[44] P. Ponte Castañeda; M. Kailasam Nonlinear electrical conductivity in heterogeneous media, Proc. R. Soc. Lond. A, Volume 453 (1997), pp. 793-816 | DOI | MR | Zbl

[45] J. Furer; P. Ponte Castañeda On the optimality of the variational linear comparison bounds for porous viscoplastic materials, J. Mech. Phys. Solids, Volume 138 (2020), 103898 | DOI | MR

[46] J. C. Michel; P. Suquet Effective potentials in nonlinear polycrystals and quadrature formulae, Proc. R. Soc. Lond. A, Volume 473 (2017), 20170213 | MR | Zbl

[47] I.S. Gradshteyn; I. M. Ryzhik Table of Integrals, Series, and Products, Academic Press, London, UK, 1994 | Zbl

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