A second-order homogenization method in finite elasticity and applications to black-filled elastomers. (English) Zbl 0984.74070

Summary: We develop an analytical method for estimating the macroscopic behavior of heterogeneous elastic systems subjected to finite deformations. The objective is to generate variational estimates for the effective or homogenized stored-energy function of hyperelastic composites, which will be accomplished by means of a suitable generalization of the “second-order procedure” [the first author, ibid. 44, No. 6, 827-862 (1996)]. The key idea in this method is the introduction of an optimally chosen “linear thermoelastic comparison composite,” which can then be used to convert available homogenization estimates for linear systems directly into new estimates for nonlinear composites. To illustrate the method, an application is given for carbon-black filled elastomers, and estimates analogous to the well-known Hashin-Shtrikman estimates and self-consistent estimates for linear-elastic composites are generated.


74Q20 Bounds on effective properties in solid mechanics
74E30 Composite and mixture properties
74B20 Nonlinear elasticity
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[1] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. rat. mech. anal., 63, 337-403, (1977) · Zbl 0368.73040
[2] Ball, J.M.; James, R.D., Fine phase mixtures as minimizers of energy, Arch. rat. mech. anal., 94, 307-334, (1986)
[3] Bergström, J.S., Boyce, M.C., 1999. Mechanical behavior of particle filled elastomers. Rubber Chem. Technol., accepted
[4] Bhattacharya, K.; Kohn, R.V., Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials, Arch. rat. mech. anal., 139, 99-180, (1997) · Zbl 0894.73225
[5] Bornert, M.; Ponte Castañeda, P., Second-order estimates of the self-consistent type for viscoplastic polycrystals, Proc. R. soc. lond., A 454, 3035-3045, (1998) · Zbl 0916.73025
[6] Braides, A., Homogenization of some almost periodic coercive functionals, Rend. accad. naz. XL, 9, 313-322, (1985) · Zbl 0582.49014
[7] Bruno, O.; Reitich, F.; Leo, P.H., The overall elastic energy of poly-crystalline martensitic solids, J. mech. phys. solids, 44, 1051-1101, (1996)
[8] Budiansky, B., Thermal and thermoelastic properties of composites, J. comp. mater., 4, 286-295, (1970)
[9] Geymonat, G.; Müller, S.; Triantafyllidis, N., Homogenization of nonlinearly elastic materials, macroscopic bifurcation and macroscopic loss of rank-one convexity, Arch. rat. mech. anal., 122, 231-290, (1993) · Zbl 0801.73008
[10] Govindjee, S., An evaluation of strain amplification concepts via Monte Carlo simulations of an ideal composite, Rubber chem. technol., 70, 25-37, (1997)
[11] Govindjee, S.; Simo, J., A micromechanically based continuum damage model for carbon black-filled rubbers incorporating mullins’ effect, J. mech. phys. solids, 39, 87-112, (1991) · Zbl 0734.73066
[12] Guth, E.; Gold, O., On the hydrodynamical theory of the viscosity of suspensions, J. phys. rev., 53, 322, (1938)
[13] Hashin, Z.; Shtrikman, S., On some variational principles in anisotropic and nonhomogeneous elasticity, J. mech. phys. solids, 10, 335-342, (1962) · Zbl 0111.41401
[14] Hashin, Z.; Shtrikman, S., A variational approach to the theory of the elastic behavior of multiphase materials, J. mech. phys. solids, 11, 127-140, (1963) · Zbl 0108.36902
[15] Hill, R., On constitutive macro-variables for heterogeneous solids at finite strain, Proc. R. soc. lond., A 326, 131-147, (1972) · Zbl 0229.73004
[16] Hill, R.; Rice, J.R., Elastic potentials and the structure of inelastic constitutive laws, SIAM J. appl. math., 25, 448-461, (1973) · Zbl 0275.73028
[17] Kailasam, M.; Ponte Castañeda, P., A general constitutive theory for linear and nonlinear particulate media with microstructure evolution, J. mech. phys. solids, 46, 427-465, (1998) · Zbl 0974.74502
[18] Laws, N., On the thermostatics of composite materials, J. mech. phys. solids, 21, 9-17, (1973)
[19] Levin, V.M., Thermal expansion coefficients of heterogeneous materials, Mekh. tverd. tela, 2, 83-94, (1967)
[20] Meinecke, E.A.; Taftaf, M.I., Effect of carbon-black on the mechanical properties of elastomers, Rubber chem. technol., 61, 534-547, (1988)
[21] Mullins, L.; Tobin, N.R., Stress softening in rubber vulcanizates. part I: use of strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber, J. appl. polymer sci, 99, 189-212, (1965)
[22] Müller, S., Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. rat. mech. anal., 99, 189-212, (1987) · Zbl 0629.73009
[23] Ogden, R., Large deformation isotropic elasticity: on the correlation of experiment and theory for compressible rubberlike solids, Proc. R. soc. lond., A 328, 567-583, (1972) · Zbl 0245.73032
[24] Ogden, R., Extremun principles in non-linear elasticity and their application to composites: part I, Int. J. solids structures, 14, 265-282, (1978) · Zbl 0384.73022
[25] Ponte Castañeda, P., The overall constitutive behaviour of nonlinearly elastic composites, Proc. R. soc. lond., A 422, 147-171, (1989) · Zbl 0673.73005
[26] Ponte Castañeda, P., The effective mechanical properties of nonlinear isotropic composites, J. mech. phys. solids, 39, 45-71, (1991) · Zbl 0734.73052
[27] Ponte Castañeda, P., New variational principles in plasticity and their application to composite materials, J. mech. phys. solids, 40, 1757-1788, (1992) · Zbl 0764.73103
[28] Ponte Castañeda, P., Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J. mech. phys. solids, 44, 827-862, (1996) · Zbl 1054.74708
[29] Ponte Castañeda, P.; Suquet, P., Nonlinear composites, Adv. appl. mech., 34, 171-302, (1998) · Zbl 0889.73049
[30] Ponte Castañeda, P.; Willis, J.R., On the overall properties of nonlinearly viscous composites, Proc. R. soc. lond., A 416, 217-244, (1988) · Zbl 0635.73006
[31] Ponte Castañeda, P.; Willis, J.R., The effect of spatial distribution on the effective behavior of composite materials and cracked media, J. mech. phys. solids, 43, 1919-1951, (1995) · Zbl 0919.73061
[32] Ponte Castañeda, P.; Willis, J.R., Variational second-order estimates for nonlinear composites, Proc. R. soc. lond., A 455, 1799-1811, (1999) · Zbl 0984.74071
[33] Sewell, M.J., Maximum and minimum principles, (1987), Cambridge University Press Cambridge · Zbl 0652.49001
[34] Smallwood, H., Limiting law of the reinforcement of rubber, J. appl. phys., 15, 758-766, (1944)
[35] Smyshlyaev, V.; Willis, J.R., A non-local variational approach to the elastic energy minimization of martensitic polycrystals, Proc. R. soc. lond., A 454, 1573-1613, (1998) · Zbl 0932.74055
[36] Suquet, P., Overall potentials and extremal surfaces of power law or ideally plastic materials, J. mech. phys. solids, 41, 981-1002, (1993) · Zbl 0773.73063
[37] Suquet, P.; Ponte Castañeda, P., Small-contrast perturbation expansions for the effective properties of nonlinear composites, C.R. acad. sc. Paris II, 317, 1515-1522, (1993) · Zbl 0844.73052
[38] Talbot, D.R.S.; Willis, J.R., Variational principles for inhomogeneous nonlinear media, IMA J. appl. math., 35, 39-54, (1985) · Zbl 0588.73025
[39] Talbot, D.R.S.; Willis, J.R., Some simple explicit bounds for the overall behavior of nonlinear composites, Int. J. solids structures, 29, 1981-1987, (1992) · Zbl 0764.73052
[40] Talbot, D.R.S.; Willis, J.R., Bounds of third order for the overall response of nonlinear composites, J. mech. phys. solids, 45, 87-111, (1997) · Zbl 0969.74570
[41] Treolar, L.R., The physics of rubber elasticity, (1975), Oxford University Press Oxford
[42] Willis, J.R., Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J. mech. phys. solids, 25, 185-202, (1977) · Zbl 0363.73014
[43] Willis, J.R., Variational and related methods for the overall properties of composites, Adv. appl. mech., 21, 1-78, (1981) · Zbl 0476.73053
[44] Willis, J.R., The overall elastic response of composite materials, J. appl. mech., 50, 1202-1209, (1983) · Zbl 0539.73003
[45] Willis, J.R., Variational estimates for the overall response of an inhomogeneous nonlinear dielectric, (), 247-263 · Zbl 0649.73013
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