Zhao, JD, Sloan, SW & Carter, JP 2008, 'Determination of Effective Elastic Properties of Microcracked Rocks Based on Asymptotic Approximation', in Y Potvin, J Carter, A Dyskin & R Jeffrey (eds), Proceedings of the First Southern Hemisphere International Rock Mechanics Symposium
, Australian Centre for Geomechanics, Perth, pp. 601-612, https://doi.org/10.36487/ACG_repo/808_174
In this paper, we present a study on the effective elastic properties of finely fractured rock based on the concept of energy equivalence. The effective elastic properties a rock body weakened by many penny-shaped microcracks, such as the apparent Young’s modulus, shear modulus and Poisson’s ratio, are particularly important in many applications. To determine these macroscopic parameters, we first adopt the dilute solution approach outlined in Kachanov (1992), where a crack compliance tensor is defined and used extensively to express the energy perturbation caused by the presence of the cracks. In parallel, a self-consistent method is employed where an asymptotic form of the Eshelby’s tensor (Eshelby, 1957) is developed to treat a penny-shaped microcrack as an extreme case of a spheroid-shaped inhomogeneity. The asymptotic approximation permits the local Eshelby tensor as well as its global expression to be derived analytically, and to be further used to construct an equivalent eigenstrain problem based on energy equivalence. The effective values for the parameters of interest can then be evaluated. The formulation and results obtained through the asymptotic self-consistent approach are compared to those predicted using a non-interacting scheme.
Benveniste, Y. (1987) A new approach to the application of Mori-Tanaka’s theory in composite materials, Mech. Mater., 6, pp. 147–157.
Berryman, J.G., Pride, S.R. and Wang, H.F. (2002) A differential scheme for elastic properties of rocks with dry or saturated cracks, Geophys., J. Int., 151, pp. 597–611.
Biot, M.A. (1941) General theory of three-dimensional consolidation, J. Appl. Phys., 12, pp. 155–164.
Bristow, J.R. (1960) Microcracks and the static and dynamic elastic constants of annealed and heavily cold-worked metals, British J. Appl. Phys., 11, pp. 81–85.
Budiansky, B. and O’Connell, R.J. (1976) Elastic moduli of a cracked solid, Int. J. Solids Structures, 12, pp. 81–97.
Budiansky, B. (1975) On the elastic moduli of some heterogeneous material. J. Mech. Phys. Solids, 13, pp. 223–227.
Eshelby, J.D. (1957) The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. R. Soc. Lond. Ser., A, 241, pp. 376–396.
Hashin, Z. and Shtrikman, S. (1961) Note on a variational approach to the theory of composite elastic materials, J. Franklin Inst., 271, pp. 336–341.
Hashin, Z. and Shtrikman, S. (1962) A variational approach to the theory of elastic behaviour of polycrystals, J. Mech. Phys. Solids, 10, pp. 343–352.
Hashin, Z. and Shtrikman, S. (1963) A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids, 11, pp. 127–140.
Hashin, Z. (1988) The differential scheme and its application to cracked materials, J. Mech. Phys. Solids, 36, pp. 719–734.
Hill, R. (1963) Elastic properties of reinforced solids: some theoretical principles, J. Mech. Phys. Solids, 11, pp. 357–372.
Hill, R. (1965) A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13, pp. 213–222.
Kachanov, M. (1980) Continuum model of medium with cracks, Journal of Engineering Mechanics Division, 106(EM5), pp. 1039–1051.
Kachanov, M. (1987) Elastic solids with many cracks: a simple method of analysis, Int. J. Solids Structures, 23, pp. 23–43.
Kachanov, M. (1992) Effective elastic properties of cracked solids: critical review of some basic concepts, Applied Mechanics Review, 45(10), pp. 304–335.
Kachanov, M. (1994) Elastic solids with many cracks and related problems, Adv. Appl. Mech., 30, pp. 259–445.
Kachanov, M. (2007) On the effective elastic properties of cracked solids – editor’s comments, Int. J. Fract., 146, pp. 295–299.
Mori, T. and Tanaka, K. (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta. Met., 21, pp. 571–574.
Mura, T. (1982) Micromechanics of defects in solids, 2nd edn., Martinus Nijhoff, Dordrecht.
Norris, A.N. (1985) A differential scheme for the effective moduli of composites, Mech. Mater., 4, pp. 1–16.
Reuss, A. (1929) Berechnung der Fließgrenze von Mischkristallen auf Grund der Plasizitätsbedingung für Einkristalle, Z. Angrw. Math. Mech., 9, pp. 49–58.
Vavakin, A.S. and Salganik, R.L. (1975) Effective characteristics of nonhomogeneous media with isolated inhomogeneities, Mechanics of Solids, 10, pp. 65–75.
Voigt, W. (1928) Lehrbuch der Kristallphysik, Teubner, Leipzig, p. 962.
Walsh, J.B. (1965a) The effect of cracks on the compressibility of rocks, J .Geophys. Res., 70(2), pp. 381–389.
Walsh, J.B. (1965b) The effect of cracks on unaxial compression of rocks, J. Geophys. Res., 70(2), pp. 399–411.
Yang, Q., Zhou, W.Y. and Swoboda, G. (2001) Asymptotic solutions of penny-shaped inhomogeneities in global Eshelby’s tensor, App. Mech. ASME., 68, pp. 740–750.
Zimmerman, R.W. (1991) Elastic moduli of a solid containing spherical inclusions, Mech. Mater., 12, pp. 17–24.