Micromechanical modeling of a cracked elliptically orthotropic medium

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Abstract

In this paper, we derive the second-order crack opening displacement tensor for an arbitrarily oriented elliptical crack in an elliptically orthotropic (EO) matrix. This result is obtained in explicit closed form. The approach is based on the Saint-Venant’s idea of linear transformation between boundary value problems for elliptically orthotropic and isotropic bodies. The solution utilizes the classical representation of an ellipsoid crack where the smallest aspect ratio approaches zero and the transformation of the Taylor expansion of the corresponding Hill tensor. It is shown, in particular, that transformed cracks have neither the same in-plane aspect ratio nor the same vanishing aspect ratio. It requires a correction factor in the crack opening displacement tensor. Some specific relative orientations of the crack with respect to the symmetry planes of the EO matrix are considered in detail and effective properties are calculated in the case of randomly distributed cracks. The result is also extended to the case of a cylindrical (plane strain) crack.

Introduction

In this paper, we focus on the effect of an elliptical crack on the overall compliance of an elliptically orthotropic elastic material. First evaluation of a crack contribution to the elastic moduli of isotropic material has been done by Sack Sack (1946) for a penny-shaped crack. His result was used by Bristow Bristow (1960) for calculation of the overall elastic properties of an isotropic material containing multiple randomly oriented circular cracks. O’Connell and Budiansky O’Connell and Budiansky (1974) and Budiansky and O’Connell Budiansky and O’Connell (1976) proposed a methodology to evaluate effects of planar cracks of the elliptical shape in isotropic materials based on the knowledge of the stress intensity factors. Similar method has been used by Rice Rice (1975) for a crack of any shape (in a material of any symmetry) for which stress intensity factor is known as a function of a position vector and crack size. The results obtained by this method are summarized by Kachanov and Sevostianov Kachanov and Sevostianov (2012). Detailed review of the results on crack contribution to overall elastic properties is given by Kachanov Kachanov (1993).

The problem of an elliptical crack in an anisotropic medium was first addressed by Willis Willis (1968) who relied on the Fourier transform to express the stress field around the crack as well as the crack opening displacement in a general compact form by means of contour integrals. Hoenig (Hoenig (1977), Hoenig (1978)) used the properties of the J integral (Rice (1968), Budiansky and Rice (1973)) to derive formulas (in the integral form) for the stress intensity factors and crack opening displacements for an elliptical crack in a generally anisotropic media. He showed that the integrals can be evaluated in closed form only for a transversely isotropic material, if the crack is parallel to the plane of isotropy. He also calculated the change in moduli for transversely isotropic elastic media with cracks parallel to the plane of isotropy. Later, results of Hoenig (Hoenig (1977), Hoenig (1978)) on elliptical crack in a 3D anisotropic material have been recovered by different methods (or specified for some particular cases, very often without proper citation). They have also recently been exploited by Laubie and Ulm Laubie and Ulm (2014) to examine the problem of propagation of an elliptical crack. Fabrikant Fabrikant (1989) calculated crack opening displacement tensor for a single circular crack embedded in a transversely isotropic material parallel to the plane of isotropy using the method of potential functions. The closed form analytical expressions for a spheroidal inhomogeneity embedded in a transversely isotropic matrix have been obtained by various methods in papers of Laws Laws (1985), Withers Withers (1989), Yu et al. Yu, Sanday, and Chang (1994), Sevostianov et al. Sevostianov, Yilmaz, Kushch, and Levin (2005) and Barthélémy Barthélémy (2020) for the case when rotation axis of the spheroid is aligned with the symmetry axis of transverse isotropy (a circular crack has been considered as a limiting case). These results have been generalized to the case of piezoelectric materials by Dunn and Wienecke Dunn and Wienecke (1997), Levin et al. Levin, Michelitsch, and Sevostianov (2000) and Mikata Mikata (2000). Kanaun and Levin Kanaun and Levin (2009) presented an integral equation for anisotropic medium with elliptical cracks and its solution with constant and linear polynomial external fields. They also considered the problem about overall properties of an anisotropic media with multiple cracks.

Guerrero et al. Guerrero, Sevostianov, and Giraud (2008) showed that effect of an arbitrarily oriented crack in a transversely isotropic material on the overall elastic behavior can be evaluated from the compliance contribution tensor of a crack in an isotropic material if the extent of anisotropy is mild. Inspired by the observation of Tsukrov and Kachanov Tsukrov and Kachanov (2000) who showed that, in 2D, the second-order crack opening displacement tensor is independent of the crack orientation if the coordinate system coincides with the principal directions of anisotropy, Guerrero et al. Guerrero, Sevostianov, and Giraud (2007) showed that similar approximately holds in 3D if tensor of elastic stiffness of the material can be expressed in terms of a second-order tensor Sevostianov and Kachanov (2008). This result was used by Seyedkavoosi et al. Seyedkavoosi, Vilchevskaya, and Sevostianov (2018) to calculate overall properties of a transversely isotropic material with any orientation distribution of penny-shaped cracks.

In the present work we consider an elliptical crack embedded in elliptically orthotropic material introduced by Saint-Venant (Barré de Saint-Venant (1863b), Barré de Saint-Venant (1863a)). As shown by Pouya and Zaoui Pouya and Zaoui (2006) and Pouya Pouya (2011), boundary value problems for bodies having elastic symmetry of this type can be reduced to the boundary value problems for isotropic bodies (of, generally, different shape) by simple affine transformation of coordinates. Sevostianov and Kushch Sevostianov and Kushch (2020) and Kushch and Sevostianov Kushch and Sevostianov (2020) showed that many real orthotropic materials can be approximated with good accuracy as elliptically orthotropic (EO) materials and showed how compliance contribution tensors can be calculated for ellipsoidal inhomogeneities embedded in them. In the text to follow, we explore the fact that affine transformation of coordinates reduces the problem about an elliptical crack in EO material to elliptical crack of different aspect ratio in an isotropic matrix. In Section 2, after recalling the definition of the EO behavior resulting from the transformation of an isotropic one, the relationships between contribution tensors of transformed problems, at the origin of further developments about cracks, are put in evidence. The link between the fourth-order compliance contribution tensor and the second-order crack opening displacement tensor is then detailed in Section 3 for arbitrary anisotropic matrix with a peculiar emphasis on the equivalence between the ellipsoidal representation of a crack and the interface model obtained as a limit when the smallest aspect ratio tends towards 0. The second-order crack opening displacement tensor of an elliptical crack in an EO matrix is fully derived in Section 4 with a deep analysis of its properties and of the approximation of the contribution tensor by its counterpart of an ellipsoid with finite aspect ratio. The result is then applied on some particular cases of matrix anisotropy and relative orientation of the crack with respect to the matrix axes leading to condensed analytical formulas and illustrative graphs in Section 5. In addition effective stiffness tensors are calculated in Section 6 with a randomly oriented distribution of cracks in order to evaluate the effect of cracking on the anisotropy level. Section 7 is finally dedicated to the careful adaptation of the previous developments to the case of a crack seen as a long cylinder of flat elliptical section and 2D crack.

Section snippets

Problem of a single ellipsoidal inclusion in an EO matrix and relationships between polarization tensors

Eshelby’s work Eshelby (1957) is one of the fundamental bases of micromechanics. The first problem (inclusion problem) consists in a linear elasticity problem posed on an infinite medium of homogeneous stiffness tensor C while considering a uniform eigenstrain ɛ* within an ellipsoidal domain and zero displacement at infinity. The important result is that the strain field solution to this problem is uniform within the ellipsoid where it writes SEsh:ɛ*. The fourth-order tensor SEsh is called

General results on the compliance of a crack in a matrix of arbitrary anisotropy

This section recalls some important background about the crack compliance contribution and opening displacement tensors in an anisotropic matrix before application to the particular case of an EO matrix in Section 4.

A crack is defined here as a void flat pore of elliptical shape (see Fig. 1b), in other words the set of points IA such thatx̲IAx̲·n̲=0and(x̲·̲)2+(x̲·m̲η)2a2As shown in numerous works (Laws (1977), Laws (1985), Nemat-Nasser and Hori (1999), Barthélémy (2009), Dormieux and

Derivation from the transformation method

This section develops the methodology to build the crack contribution tensor in an EO matrix from that of a transformed crack in an associated isotropic matrix.

As shown in the literature (Laws (1977), Laws (1985), Kachanov (1992)), Nemat-Nasser and Hori (1999), Barthélémy (2009), Kachanov and Sevostianov (2018)), the effect of an open crack in the compliance of the infinite medium in which it is embedded relies on the singularity of Q when ω is set to 0 and, as recalled in Section 3, the

Particular cases of crack opening displacement tensor in an EO matrix

In this section some illustrations of the B tensor of a crack in an EO matrix are proposed in the particular cases of a circular crack firstly parallel and secondly rotated with respect to a symmetry plane of the matrix. Then a more general case of crack shape and orientation is finally proposed.

Effective elasticity of a cracked EO medium

Once in possession of a practical way to calculate the crack opening displacement tensor for an arbitrarily oriented crack embedded in an EO matrix as derived in Section 4.1 (see (52)) and subsequently the corresponding compliance tensor from (30), it is straightforward to apply the classical homogenization schemes allowing to estimate the effective elasticity of a representative volume element of cracked medium. The derivation of schemes is not detailed here (one can refer to Dormieux and

Cylindrical crack

This section revisits the main results of the paper in the case of a cylindrical crack, i.e. such that the domain defined by (24) is transformed inx̲IAx̲·n̲=0and|x̲·m̲|b

This domain is obviously invariant by translation along ̲ and asymptotically corresponds to the definition of the ellipsoid (17) with A given by (18) such that c tends towards 0 and a tends towards infinity. In addition to the two aspect ratios η and ω, a third one ρ is conveniently introduced hereη=ba;ω=ca;ρ=cb=ωηWhereas

Conclusion

We derived explicit expression for crack opening displacement tensor of an arbitrarily oriented elliptical crack in an elliptically orthotropic matrix. The approach is based on Saint-Venant’s classical idea of linear transformation between boundary value problems for elliptically orthotropic and isotropic bodies. This new result opens the way to analytical evaluation of the overall properties of 3D anisotropic materials containing multiple microcracks using traditional homogenization techniques

Declaration of Competing Interest

My co-authors and I declare that there is no conflict of interest related to this submission.

Acknowledgements

The authors are grateful to Prof. Mark Kachanov for his very helpful comments on this work.

The author JFB thanks Cerema and Université Gustave Eiffel for allowing the creation of the joint research team entitled ”Équipe de Recherche Commune sur les Matériaux pour une Construction Durable (ERC MCD)”, within which these research works were carried out.

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