Elastic compliances of the “zigzag” and intergranular cracks

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Abstract

Estimates of compliance contributions of 2-D cracks of complex shapes (such as multilink zigzags or curved shapes) observed in brittle materials are suggested. The effects of several geometric factors are clarified. The problem of quantitative characterization of density of such cracks (the identification of the crack density parameter, to be used in the effective media models) is addressed. Applications to intergranular cracks is considered.

Introduction

Cracks in brittle materials often have “irregular”, zigzag-like shapes, due to various factors (see, for example, Begley & Hutchinson, 2017, Zhang, Duan, Li and Zhang (2008), and Pronina, Maksimov & Kachanov, 2020). One of the typical shapes is shown in Fig. 1. For the effective elastic properties of materials with multiple cracks of this kind, the key quantity of interest is the compliance contribution of a single crack of such shape.

Thus, we consider the extra strain Δεij (per reference volume V) due to the presence of the crack, the latter being placed in the applied stress field σ0 (that, in absence of the crack, would have been uniform within its site). It is given by the formula (that follows, for example, from a footnote remark of Hill (1963)):Δεij=12VS(binj+bjni)dSwhere n is a unit normal to the crack and b=u+u is the vector of displacement discontinuity across it (the plus side of the crack face is in the positive direction of n); both vectors are generally variable along the crack surface S. In the 2-D case, reference volume V should be replaced by reference area A and crack surface S – by crack line L.

For a 2-D zigzag (multi-link) crack, the unit normal n is a piece-wise constant function so that the integral (1) reduces to a sum over the links:Δεij=12Ak(binj+bjni)(k)l(k)where the sign ... denotes average over the k-th link. Vectors b(k) are the key quantities of interest; expressing them in terms of the applied stress σ0 (that, in our computational tests, will incorporate two uniaxial loadings and a shear), would results in the relationΔεij=Hijklσkl0 that defines the crack compliance contribution tensor H. In case of multiple cracks, summation of their contributions and expressing the sum kΔεij(k) in terms of σ0 yields the extra compliance due to cracks: ΔSijkl=mHijkl(m).

Remark. The H-tensors entering this summation are usually taken from results for isolated cracks; strictly speaking, this implies the non-interaction approximation (interactions may affect the H-tensors) so that, strictly speaking, the sum mHijkl(m) cannot be applied to interacting cracks. This, however, is commonly done in the effective media theories since the alternative – incorporation of interaction effects into H-tensors – would amount to solving the interaction problem.

The present work focuses on approximate estimates of the compliance contribution tensors Hijkl of cracks of the zigzag (multilink) shapes. The approximate nature of the estimates (errors within 10–12% will be viewed as acceptable), as well as the 2-D treatment, are justified by difficulties of analysis of “irregular” 3-D crack shapes (although certain progress has been made in this direction, see Markov, Trofimov and Sevostianov (2020) and the book of Kachanov and Sevostianov (2018)), and by possibly incomplete information on 3-D crack shapes.

In literature, multilink cracks have been analyzed in several works, mostly by finite elements calculations (see, for example, Liu, Li and Chen (2014) and Meggiolaro, Miranda, Castro and Martha (2005)). They focused primarily on the stress intensity factors (SIFs) – in contrast with the present work that considers crack compliances. The latter are much less sensitive to “details” of the crack geometry than SIFs (as follows, for example, from the work of Rice (1975), as discussed in detail in the mentioned book of Kachanov and Sevostianov (2018)); this low sensitivity allows various approximations considered here.

We first discuss the case when the multilink crack has strongly elongated shape, more precisely, it fits into a narrow ellipse. In this case, a relatively simple estimate for the crack compliance can be given.

In a more general case, our analysis will be based on breaking a multilink into a chain of isolated cracks separated by very small ligaments (Fig. 2). A somewhat different approach was used in earlier works of Martynyuk and Kachanov (2020) where a multilink crack was modeled by a set of two-link zigzags and rectilinear “shortcuts”. Their approach required computations for the ligaments as small as 1080 of the link lengths and hence was limited to small number of links and angles between neighboring links that are not overly large. The present work proposes different approaches that bypass the necessity of dealing with 1080 ligaments and hence are much more economical computationally and can be applied to zigzags with large number of links.

We also address the problem of proper quantitative characterization of the density of multiple cracks of such shapes, i.e. of identifying the proper crack density parameter in which terms the effective elastic (or conductive) properties are to be expressed. To this end, we explore the possibility of replacing of a zigzag by one or two rectilinear cracks having approximately the same compliance contribution; in cases when this is possible, the conventional crack density parameter can be retained.

In discussing the accuracy of approximations, the difference between tensors Hijkl(1) and Hijkl(2) will be estimated by the normalized Euclidean norm (that, for a fourth-rank tensor, is defined as HHijklHijkl):ξ=H(1)H(2)Hijkl(1)Hijkl(2)·100%=i,j,k,l(Hijkl(1)Hijkl(2))2Hijkl(1)Hijkl(2)·100%

Section snippets

Elongated multilink crack fitting into a narrow ellipse

We consider the case when the crack fits into an ellipse with sufficiently small aspect ratio b/a (a and b are the larger and the smaller ellipse semiaxes), in other words when the ratio of the zigzag amplitude to its length is sufficiently small (Fig. 3). The following observations are relevant to this end:

  • The compliance contribution of the elliptical hole is larger than the one of the crack. This statement (intuitively obvious) follows from Hill's (1965) comparison (or modification) theorem

Estimates based on the ligament scaling factor

In replacing a multilink crack by a chain of isolated cracks, the compliance contribution of the chain approaches the one of the zigzag as the ligaments become smaller. However, this approach is extremely slow, as illustrated on the example of breaking of a rectilinear crack into a chain of collinear cracks (Fig. 4 and Table 1): for the approach to be within several percent, ligaments of the order of 1080 of the link lengths are needed; computationally, this is a challenging problem (Martyniuk

Alternative method. Application to curvilinear cracks

The approach outlined above has the limitation that it has satisfactory accuracy at limited number of links (errors are within 3%, 5%, and 10% at number of links of two, five and ten, respectively; the error increases as the number of links increases). We suggest an alternative approach that does not have this limitation (as has been tested on zigzags having up to eleven links) although it has lower accuracy at small number of links (Fig. 7). This allows application to curvilinear cracks, since

On the possibility of replacing multilink and curvilinear cracks by one or two rectilinear ones. Implications for crack density parameters

In the problem of effective elastic properties of a material with cracks of complex (non-rectilinear, in the 2-D case) shapes, the first challenge is the proper quantitative characterization of cracks, i.e. the identification of the crack density parameter in which terms the effective properties are to be expressed. It must represent individual cracks according to their actual compliance contributions. The conventional crack density parameter ρ is defined for the simplest crack shapes only: in

Examples

We illustrate the developed approaches on two examples of multilink cracks, the one shown in Fig. 1 and the one given by the sketch of Fig. 11.

The crack of Fig. 1 is modeled by a zigzag crack consisting of ten links; we also explored its replacement by a five-link zigzag, by “shortcutting” several links in the middle (Fig. 10). We apply the procedures for finding the crack compliance contributions Hijkl (normalized to Young's modulus) described in Sections 3, 4, and 5. Results for the crack

Intergranular, and similar, cracks: Estimate of the equivalent crack density

We consider a 2-D polycrystal formed by hexagonal grains with partially cracked grain boundaries (Fig 12). The compliance of an intergranular crack of this kind (consisting of N links) can be estimated by replacing it by a pair of orthogonal rectilinear cracks, as shown. We first discuss the results shown in Fig. 9b, at θ=600. The accuracy of the replacement, as measured by the Euclidean norm, is about 10% for the two- and four-link cracks and about 12% for the three-link one. It improves with

Concluding remarks

Several procedures are suggested for approximate estimation of elastic compliances of multilink zigzag cracks (within 10–12% accuracy, as is consistent with the frequently incomplete, or approximate, information on crack shapes).

We first discuss replacing a multilink crack by a chain of closely spaced isolated cracks; as ligaments between them get smaller, the collective compliance contribution of the chain tends to the one of the multilink crack. This limiting transition, however, is very

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by grant 14.Z50.31.0036 awarded to R.E. Alexeev Nizhny Novgorod Technical University by Ministry of Education and Science of the Russian Federation.

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