Abstract
This work is concerned with the application of a new general procedure for estimating the overall constitutive behavior of nonlinear composites to porous materials with statistically isotropic microstructure. For two-phase systems, the procedure involves a linear-elastic comparison composite with the tangent moduli of the constituent phases evaluated at appropriately chosen estimates for the average strains in the phases. The procedure can thus be used to generate estimates for the effective behavior of a nonlinear composite, directly from corresponding estimates for a linear comparison composite. One significant advantage of the procedure, over other procedures that are currently available in the literature, is that it leads to estimates that are exact to second order in the contrast. In addition, the predictions for the effective behavior of isotropic composites with isotropic nonlinear phases are found to depend on the third invariant of strain. The procedure will be used here to obtain estimates of the Hashin-Shtrikman and self-consistent types for isotropic, power-law, porous materials.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Budiansky, B. (1965) On the elastic moduli of some heterogeneous materials, J. Mech. Phys. Solids, 13, pp. 223–227.
Budiansky, B. (1970) Thermal and thermoelastic properties of composites, J. Comp. Mater., 4, pp. 286–295.
Duva, J. M. and Hutchinson, J. W. (1984) Constitutive potentials for dilutely voided non-linear materials, Mech. Mater., 3, pp. 41–54.
Duva, J.M. (1986) A constitutive description of nonlinear materials containing voids, Mech. Mat., 5, pp. 137–144.
Eshelby, J. D. (1957) The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. R. Soc. Land. A, 241, pp. 376–396.
Gurson, A.L. (1977) Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile materials Int. J. Eng. Mail. Tech., 99, pp. 2–15.
Hashin, Z., and Shtrikman, S. (1963) A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids, 11, pp. 127–140.
Hill, R. (1965) Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, 13, pp. 89–101.
Kachanov, L. M. (1971) Foundations of the Theory of Plasticity. North-Holland, Amsterdam, p 20.
Laws, N. (1973) On the thermostatics of composite materials, J. Mech. Phys. Solids, 21, pp. 9–17.
Levin, V. M. (1967) Thermal expansion coefficients of heterogeneous materials, Mekh. Tverd. Tela, 2, pp. 83–94.
Olson, T. (1994) Improvements on Taylor’s upper bound for rigid-plastic bodies, Mater. Sc. Engng. A, 175, pp. 15–20.
Ponte Castañeda P. (1991) The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids, 39, pp. 45–71.
Ponte Castañeda P. (1992) New variational principles in plasticity and their application to composite materials, J. Mech. Phys. Solids, 40, pp. 1757–1788.
Ponte Castañeda P. (1996) Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J. Mech. Phys. Solids, 44, pp. 827–862.
Ponte Castañeda P., and Nebozhyn, M. (1997) Exact second-order estimates of the self-consistent type for nonlinear composite materials, Mech. Mater., submitted.
Ponte Castañeda P., and Willis, J.R. (1988) On the overall properties of nonlinearly viscous composites, Proc. R. Soc. Land. A, 416, pp. 217–244.
Suquet, P. (1993) Overall potentials and extremal surfaces of power law or ideally plastic materials, J. Mech. Phys. Solids, 41, pp. 981–1002.
Suquet, P. (1995) Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte nonlinear variational procedure, C.R. Acad. Sc. Paris II, 320, pp. 563–571.
Suquet, P., and Ponte P. (1993) Small-contrast perturbation expansions for the effective properties of nonlinear composites, C.R. Acad. Sc. Paris II, 317, pp. 1515–1522.
Talbot, D.R.S., and Willis, J.R., (1985) Variational principles for inhomogeneous nonlinear media, IMA J. Appl. Math., 35, pp. 39–54.
Talbot, D.R.S., and Willis, J.R., (1992) Some explicit bounds for the overall behavior of nonlinear composites, Int. J. Solids Struct., 29, pp. 1981–1987.
Willis, J. R. (1977) Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J. Mech. Phys. Solids, 25, pp. 185–202.
Willis, J. R. (1981) Variational and related methods for the overall properties of composites, In: Advances in Applied Mechanics, Vol. no. 21, (C. Yih, ed.), Academic Press, New-York, pp. 1–78.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Kluwer Academic Publishers
About this paper
Cite this paper
Nebozhyn, M.V., Ponte Castañeda, P. (1998). Second-Order Estimates for the Effective Behavior of Nonlinear Porous Materials. In: Bahei-El-Din, Y.A., Dvorak, G.J. (eds) IUTAM Symposium on Transformation Problems in Composite and Active Materials. Solid Mechanics and its Applications, vol 60. Springer, Dordrecht. https://doi.org/10.1007/0-306-46935-9_6
Download citation
DOI: https://doi.org/10.1007/0-306-46935-9_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-5122-1
Online ISBN: 978-0-306-46935-0
eBook Packages: Springer Book Archive