Crack–inclusion interaction for mode II crack analyzed by Eshelby equivalent inclusion method

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Abstract

The interaction between an inclusion of arbitrary shape and a crack subjected to a remote mode II stress is investigated. The approximate solutions to predict the variations in crack-tip stress intensity factors of mode II crack induced by the inclusion have been developed both for the cases where the crack tip is near or within the inclusion. The solution technique is based on the transformation toughening theory. The transformation strain induced by the inhomogeneity between matrix and inclusion is estimated based on the Eshelby equivalent inclusion method. As validated by numerical examples, the approximate solutions have satisfactory accuracy for a variety of inclusion shapes and a wide range of the modulus ratios between inclusion and matrix.

Introduction

The interaction between a crack and an inclusion has received considerable research over years because it is evidently important for understanding the mechanisms of strengthening and toughening, as well as material damage and fracture for composite materials discontinuously reinforced by particles or fibers. The study is traditionally based on solution of appropriate boundary value problems in the linear theory of elasticity. Only a few and highly idealized cases, such as circular inclusion [1], [2], [3], [4], [5], elliptical inclusion [6], [7], have been obtained analytical solution due to the complexity of this kind of problem. Most of the studies have been performed by numerical approaches, such as finite element method [8], [9], [10], [11], boundary element method [9], [12], [13], [14], and singular integral equation method [15], [16], the weight function method [17]. In these numerical analyses, the effects of inclusion shape, size, location and stiffness on crack-tip field have been widely investigated, which provide some insight into understanding interactions between crack and inclusion. However, these analyses are not easy to use in practice, and the numerical results are limited to the fixed calculation parameters. No generalizations could be drawn from these numerical results because of the intricacy of the results in the individual situations. Thus, the knowledge of the interaction between a crack and an inclusion comes by slow accumulation of results for special cases, rather than by establishment of general propositions.

A general solution of the interaction between a crack and an inclusion of arbitrary configuration is very difficult to obtain from linear theory of elasticity. In our previous studies [18], [19], approximate solutions to the interactions for a mode I crack near and partially penetrating an inclusion of arbitrary shape have been obtained based on Eshelby equivalent inclusion theory. As validated by corresponding numerical analyses, the approximate solutions give good prediction for the interaction between mode I crack and an inclusion of arbitrary shape.

In the present study, the interaction between mode II crack and an inclusion of arbitrary shape will be solved based on transformation toughening theory and Eshelby equivalent inclusion method. The general solutions to determine the near-tip stress intensity factor (SIF) of mode II crack are obtained both for the cases where the crack tip is near or within the inclusion. From the general solutions, a set of simple formula for several special inclusion shapes is also suggested. It is shown that, in comparison with corresponding numerical results, the present solutions have satisfactory accuracy.

Section snippets

Mode and formulation

This paper will deal with two cases: a mode II crack near an inclusion and partially penetrating an inclusion. Throughout this paper it will be assumed that the size of the inclusion is small compared with the length of the crack. The moduli of the inclusion and matrix are Ei and Em, respectively. For simplicity, it is assumed that their Poisson’s ratios are the same, denoted by ν, and that the bonding between matrix and inclusion is perfect. The crack is subjected to a remote stress field

Some special cases

Some simple forms of the approximate solutions can be obtained for several special inclusion shapes:

  • (a)

    A lamellar inclusion of length 2l and width W(lW) perpendicular to crack plane and its center is located at (r0,0) (Fig. 4(a)). By using of dA≈Wrdθ/cosθ in (16), it follows thatΔKIItip=KIIπ−θ0θ0Wrcosθ[C1cosθ+C2cos2θ+C3cos3θ]dθ=WKIIπr02C1sinθ0+C2sin0+2C33sin0whereθ0=arctglr0When l/r0≫1, we haveΔKIItip=WKIIπr02C123C3

  • (b)

    A lamellar inclusion of length l and width W(lW) parallels to the x-axis

Numerical examples

In order to verify the accuracy of the fundamental formulas , five simple problems are analyzed by detailed finite element method: a lamellar inclusion symmetrically perpendicular to (Fig. 5) and an equilateral triangle inclusion lying on the extension of the crack-tip line (Fig. 6); a lamellar inclusion (Fig. 7) and a circular inclusion (Fig. 8) centered at crack tip; Fig. 9 shows the case where the crack passes through a rectangular inclusion. The crack is subjected to a remotely applied SIF

Conclusions and discussion

In this paper, a closed-form solution for predicting the interaction between mode II crack and an inclusion of arbitrary shape is obtained based on the transformation toughening theory. The transformation strains induced by the inhomogeneity between matrix and inclusion are estimated by the Eshelby equivalent inclusion method. Since the transformation toughening theory for homogeneous inclusion [24] is based on Eshelby inclusion technique, it is natural to treat the problem for inhomogeneous

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      Many of which either treated the material matrix as homogeneous or ignored the interactions of the reinforcing phases (Castañeda, 1991; Jasiuk and Kouider, 1993; Pal, 2005; Yun and Park, 2017). The Galerkin vectors (Liu and Wang 2005, Liu et al., 2012; Mindlin and Cheng, 1950; Yu and Sanday, 1991) and Green’s function method (Chen et al., 2014; Lee et al., 2015; Mura, 2013) have been regarded as efficient approaches for finding the closed-form solutions to the inclusion problems with eigenstrains, while Eshelby’s equivalent inclusion method (EIM) (Eshelby, 1957) provides an effective means to convert inhomogeneities to eigenstrains, as demonstrated by the works on ellipsoidal inhomogeneities (Moschovidis and Mura, 1975; Shodja and Sarvestani, 2001), inclusion of arbitrary shape (Li and Shi, 2002), and cracks (Li and Chen, 2002; Shodja et al., 2003; Yang et al., 2004). The EIM has also been employed to the anisotropic structures.

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