Estimation of crack density due to fragmentation of brittle ellipsoidal in- homogeneities embedded into a ductile matrix M. Korobeinik *1 , K.C. Le, and K. Hackl 1 Lehrstuhl f ¨ ur Allgemeine Mechanik Ruhr-Universit¨ at Bochum 44879 Bochum, Germany An estimation is found for the energy release due to fragmentation of a brittle inhomogeneity of ellipsoidal shape embedded in a ductile matrix under remote static loading.The energy release calculated for prolate spheroidal inhomogeneities is used in the balance of energy to determine the crack density. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The intense fragmentation of brittle inhomogeneities embedded in a ductile matrix under high remote stresses has been re- ported in a number of geological studies [1, 2]. Cracks formed during fragmentation are generally restricted to the inhomo- geneity and do not propagate into the surrounding matrix. As a rule, fragmentation does not appear in inclusions with low aspect ratios and there is no correlation between the inclusion size and the crack density. The aim of this paper is to apply the well-known Eshelby’s equivalent inclusion method [5,6] to determine the homogeneous stress field inside the inhomogeneity before fragmentation and to calculate the energy release due to fragmentation under the condition of static loading. 2 Energy release due to fragmentation Consider an ellipsoidal elastic isotropic inhomogeneity occupying the finite domain Ω which is embedded into an infinitely extended elastic isotropic matrix occupying the remaining domain R 3 \ Ω. The elastic moduli of the inhomogeneity and matrix are λ * ,µ * and λ, µ respectively, the bond between matrix and inhomogeneity is assumed to be perfect. Let the stresses applied at infinity be σ ∞ ij . The presence of an inhomogeneity in the region Ω is simulated by a fictitious eigenstrain ǫ * ij in the same domain in an infinite homogeneous body with elastic moduli λ, µ. The energy of the composite body before fragmentation reads [6] E = - 1 2  Ω σ ∞ ij ǫ * ij d 3 x. In the state of completed fragmentation the inhomogeneity does not resist to further loading, and, therefore, can be approximately treated as a void. The energy of the matrix with a void is given by E 0 = - 1 2  Ω σ ∞ ij ǫ *0 ij d 3 x. The difference between these expressions gives us the estimation for the energy release available for fragmentation ΔE = 1 2 σ ∞ ij (ǫ *0 ij - ǫ * ij )|Ω|. (1) 3 Prolate spheroidal inhomogeneities Let us consider a prolate spheroidal inhomogeneity of the form x 2 1 a 2 + x 2 2 b 2 + x 2 3 b 2 ≤ 1, where a > b,κ = a b > 1. The remote stress tensor at infinity consists of the normalized hydrostatic pressure ¯ p = p µ and the normalized shear stress ¯ τ = τ µ = α ¯ p acting in the (y,z)-plane. The strain at infinity can be found using Hooke’s law. In order to calculate the eigenstrain tensor inside the inhomogeneity we use the following formulae [5] ǫ * α = - ¯ µ * - 1 2(¯ µ * - 1)S αα +1 ǫ ∞ α (no sum!), α =4, 5, 6, (2) ¯ A αβ ǫ * β = ¯ f α , α, β =1, 2, 3, (3) where ¯ µ * = µ * µ . Note that ¯ A αβ and ¯ f α depend on normalized elastic constants and the aspect ratio κ. After the eigenstrain is found the normalized stress inside the inhomogeneity can be determined according to Hooke’s law and the magnitude of maximum principal stress can be found for specific values of parameters. * Corresponding author: e-mail: korobeinik@am.bi.ruhr-uni-bochum.de, Phone: +49 234 322 6035, Fax: +49 234 321 4154 PAMM · Proc. Appl. Math. Mech. 5, 339–340 (2005) / DOI 10.1002/pamm.200510146 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim