×

zbMATH — the first resource for mathematics

Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. I: Theory. (English) Zbl 1116.74412
Summary: This paper is concerned with the development of an improved second-order homogenization method incorporating field fluctuations for nonlinear composite materials. The idea is to combine the desirable features of two different, earlier methods making use of “linear comparison composites”, the properties of which are chosen optimally from suitably designed variational principles. The first method [Ponte Castañeda, J. Mech. Phys. Solids 39, 45-71 (1991; Zbl 0734.73052)] makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain field over the phases, and delivers bounds, but these bounds are only exact to first-order in the heterogeneity contrast. The second method [Ponte Castañeda, ibid. 44, No. 6, 827-862 (1966; Zbl 1054.74708)] makes use of the “tangent” moduli, evaluated at the phase averages (or first moments) of the strain field, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in some special cases. These special cases turn out to correspond to situations, such as percolation phenomena, where field fluctuations, which are captured less accurately by the second-order method than by the bounds, become important. The new method delivers estimates that are exact to second-order in the contrast, making use of generalized secant moduli incorporating both first- and second-moment information, in such a way that the bounds are never violated. Some simple applications of the new theory are given in Part II of this work (Zbl 1116.74411).

MSC:
74Q20 Bounds on effective properties in solid mechanics
74A40 Random materials and composite materials
74E30 Composite and mixture properties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barthélémy, M., Path-integral approach to strongly nonlinear composites, Phys. rev. B, 62, 8576-8579, (2000)
[2] Bobeth, M.; Diener, G., Static elastic and thermoelastic field fluctuations in multiphase composites, J. mech. phys. solids, 35, 37-149, (1987) · Zbl 0601.73001
[3] Bornert, M., Masson, R., Ponte Castañeda, P., Zaoui, A., 2001. Second order estimates for the effective behavior of viscoplastic polycrystalline materials. J. Mech. Phys. Solids, to appear. · Zbl 1047.74529
[4] Budiansky, B., Thermal and thermoelastic properties of composites, J. compos. mater., 4, 286-295, (1970)
[5] Hashin, Z.; Shtrikman, S., On some variational principles in anisotropic and nonhomogeneous elasticity, J. mech. phys. solids, 10, 335-342, (1962) · Zbl 0111.41401
[6] Hill, R., Elastic properties of reinforced solids: some theoretical principles, J. mech. phys. solids, 11, 357-372, (1963) · Zbl 0114.15804
[7] Hill, R., Continuum micro-mechanics of elastoplastic polycrystals, J. mech. phys. solids, 13, 89-101, (1965) · Zbl 0127.15302
[8] Hutchinson, J.W., Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. roy. soc. London A, 348, 101-127, (1976) · Zbl 0319.73059
[9] Kailasam, M.; Aravas, N.; Ponte Castañeda, P., Porous metals with developing anisotropy: constitutive models, computational issues and applications to deformation processing, Comput. modelling eng. sci., 1, 105-118, (2000)
[10] Kreher, W.; Pompe, W., Field fluctuations in a heterogeneous elastic material—an information theory approach, J. mech. phys. solids, 33, 419-445, (1985) · Zbl 0598.73004
[11] Laws, N., On the thermostatics of composite materials, J. mech. phys. solids, 21, 9-17, (1973)
[12] Leroy, Y., Ponte Castañeda, P., 2001. Bounds on the self-consistent approximation for nonlinear media and implications for the second-order method. C.R. Acad. Sci. Paris IIB. 329 (8), 571-577.
[13] Masson, R.; Bornert, M.; Suquet, P.; Zaoui, A., An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals, J. mech. phys. solids, 48, 1203-1227, (2000) · Zbl 0984.74068
[14] Milton, G.W., 2001. The Theory of Composites. Cambridge University Press, Cambridge, to appear.
[15] Moulinec, H.; Suquet, P., A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. methods appl. mech. eng., 157, 69-94, (1998) · Zbl 0954.74079
[16] Nebozhyn, M.V.; Ponte Castañeda, P., Second-order estimates for the effective behavior of nonlinear porous materials., (), 73-88 · Zbl 0914.73080
[17] Pellegrini, Y.-P., Field distributions and effective-medium approximation for weakly nonlinear media, Phys. rev. B, 61, 9365-9372, (2000)
[18] Pellegrini, Y.-P.; Barthélémy, M.; Perrin, G., Functional methods and effective potentials for non-linear composites, J. mech. phys. solids, 48, 429-460, (2000)
[19] Ponte Castañeda, P., The effective mechanical properties of nonlinear isotropic composites, J. mech. phys. solids, 39, 45-71, (1991) · Zbl 0734.73052
[20] 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
[21] 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
[22] Ponte Castañeda, P., 2001. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: II—applications. J. Mech. Phys. Solids, submitted for publication.
[23] Ponte Castañeda, P.; Suquet, P., Nonlinear composites, Adv. appl. mech., 34, 171-302, (1998) · Zbl 0889.73049
[24] Ponte Castañeda, P., Suquet, P., 2001. Nonlinear composites and microstructure evolution. In: Aref, H., Phillips, J.W. (Eds.), Proceedings of the 20th International Congress of Theoretical and Applied Mechanics (ICTAM 2000). Kluwer Academic Publishers, Dordrecht, to appear.
[25] Ponte Castañeda, P., Willis, J.R., 1993. The effective behavior of nonlinear composites: a comparison between two methods. In: Anthony, K.H., Wagner, H.-H. (Eds.), Continuum Models and Discrete Systems (CMDS 7). Trans. Tech., Aedermannsdorf, pp. 351-360.
[26] Ponte Castañeda, P.; Willis, J.R., Variational second-order estimates for nonlinear composites, Proc. roy. soc. London A, 455, 1799-1811, (1999) · Zbl 0984.74071
[27] Sewell, M.J., Maximum and minimum principles., (1987), Cambridge University Press Cambridge · Zbl 0652.49001
[28] 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
[29] Suquet, P., 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, 320, 563-571, (1995) · Zbl 0830.73046
[30] Suquet, P.; Ponte Castañeda, P., Small-contrast perturbation expansions for the effective properties of nonlinear composites, C.R. acad. sci. Paris II, 317, 1515-1522, (1993) · Zbl 0844.73052
[31] Talbot, D.R.S.; Willis, J.R., Variational principles for inhomogeneous nonlinear media, IMA J. appl. math., 35, 39-54, (1985) · Zbl 0588.73025
[32] Talbot, D.R.S.; Willis, J.R., Some explicit bounds for the overall behavior of nonlinear composites, Int. J. solids struct., 29, 1981-1987, (1992) · Zbl 0764.73052
[33] 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
[34] Willis, J.R., Variational and related methods for the overall properties of composites, Adv. appl. mech., 21, 1-78, (1981) · Zbl 0476.73053
[35] Willis, J.R., The overall response of composite materials, ASME J. appl. mech., 50, 1202-1209, (1983) · Zbl 0539.73003
[36] Willis, J.R., On methods for bounding the overall properties of nonlinear composites, J. mech. phys. solids, 39, 73-86, (1991) · Zbl 0734.73053
[37] Willis, J.R., On methods for bounding the overall properties of nonlinear composites: correction and addition, J. mech. phys. solids, 40, 441-445, (1992)
[38] Willis, J.R., The overall response of nonlinear composite media, Eur. J. mech. A/solids, 19, S165-S184, (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.