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# Discussion of fracture paper #26 - Cracks and anisotropic materials

All materials are anisotropic, that's a fact. Like the fact that all materials have a nonlinear response. This we can't deny. Still enormous progress has been made by assuming both isotropy and linear elasticity. The success, as we all know, is due to the fact that many construction materials are very close to being both isotropic and linear. By definition materials may be claimed to be isotropic and linear, provided that the deviations are held within specified limits. Very often or almost always in structural design nearly perfect linearity is expected. In contrast to that quite a few construction materials show considerable anisotropy. It may be natural or artificial, created by humans or evolved by biological selection, to obtain preferred mechanical properties or for other reasons. To be able to choose between an isotropic analysis or a more cumbersome anisotropic dito, we at least once have to make calculations of both models and define a measure of the grade of anisotropy. This is realised in the excellent paper

The study provides a thorough review of materials that might require consideration of the anisotropic material properties. As a great fan of sorted data, I very much appreciate the references the authors give listed in a table with specified goals and utilised analysis methods. There are around 30 different methods listed. Methods are mostly numerical but also a few using Lekhnitskiy and Stroh formalisms. If I should add something the only I could think of would be Thomas C.T. Ting's book "Anisotropic Elasticity". In the book Ting derives a solution for a large plate containing an elliptic hole, which provides cracks as a special case.

The present paper gives an excellent quick start for those who need exact solutions. Exact solutions are of course needed to legitimise numerical solutions and to understand geometric constraints and numerical circumstances that affect the result. The Lekhnitskiy and Stroh formalisms boil down to the "method of characteristics" for solving partial differential equations. The authors focus on the solution for the vicinity of a crack tip that is given as a truncated series in polar coordinates attached to a crack tip.

As far as I can see it is never mentioned in the paper, but I guess the series diverges at distances equal to or larger than the crack length 2*a*. Outside the circle *r*=2*a* the present series for *r*<2*a* should be possible to extend by analytic continuation. My question is: Could it be useful to have the alternative series for the region *r*>2*a* to relate the solution to the remote load?

Does anyone have any thoughts regarding this. Possibly the authors of the paper or anyone wishes to comment, ask a question or provide other thoughts regarding the paper, the method, or anything related.

Per Ståhle

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## Comments

## Region of validity for the series solution

The authors would like to thank you for your interest in our paper. The discussed article is devoted to the numerical calculation of the coefficients of crack tip asymptotic fields in 2-D anisotropic media. The finite element over-deterministic (FEOD) method, which was formerly used in the context of isotropic materials, is successfully extended to anisotropic elasticity in this paper, whereby introducing a simple method for the computation of higher order crack parameters in such medium. The inclusion of higher order crack parameters in the crack tip stress and deformation fields can therefore lead to more precise calculations and more reliable fracture assessments.

The authors would like to thank you for your interest in our paper. The discussed article is devoted to the numerical calculation of the coefficients of crack tip asymptotic fields in 2-D anisotropic media. The finite element over-deterministic (FEOD) method, which was formerly used in the context of isotropic materials, is successfully extended to anisotropic elasticity in this paper, whereby introducing a simple method for the computation of higher order crack parameters in such medium. The inclusion of higher order crack parameters in the crack tip stress and deformation fields can therefore lead to more precise calculations and more reliable fracture assessments.

In line with Section 4.2 of the authors’ article “On higher order parameters in cracked composite plates under far‐field pure shear, Fatigue & Fracture of Engineering Materials & Structures 43, 568–585.”, which introduces exact solutions for crack parameters in a centrally cracked anisotropic plane, it is analytically derived that the crack tip asymptotic stress field holds true only inside the circle with radius r=2a. This can be interpreted as a theoretical limit which prevents the interference of the asymptotic solutions of the two crack tips in a centre crack problem. At distances greater than 2a from each crack tip, the series solution diverges from the exact solution, indicating the inapplicability of the asymptotic series expansion. In such circumstances, the usage of numerical schemes such as the finite element method (FEM) can be beneficial and more efficient compared to the derivation of new analytical solutions. Meanwhile, it is useful to underline that almost in all crack problems in real engineering applications, the stress field within the circle of radius r=a is of particular interest because fatigue and fracture models characterizing the critical conditions corresponding to crack extension are practically based on the stresses within this domain.

Having mentioned the point about centre cracks, a single edge cracked plate has however no interfering secondary stress field, implying that the crack tip asymptotic fields are applicable within the entire domain of the plate. The further the distance from the crack tip, the more the coefficients should be considered in the series solution. The successful usage of the boundary collocation method (that satisfies remote boundary conditions to obtain crack parameters) in single crack and notch problems would have been in fact impossible if the series solution had not been valid within the entire domain. Accuracy of the asymptotic solution in large distances away from the crack tip can be simply benchmarked by the FE results.Having mentioned the point about centre cracks, a single edge cracked plate has however no interfering secondary stress field, implying that the crack tip asymptotic fields are applicable within the entire domain of the plate. The further the distance from the crack tip, the more the coefficients should be considered in the series solution. The successful usage of the boundary collocation method (that satisfies remote boundary conditions to obtain crack parameters) in single crack and notch problems would have been in fact impossible if the series solution had not been valid within the entire domain. Accuracy of the asymptotic solution in large distances away from the crack tip can be simply benchmarked by the FE results.

## Dear Morteza,

Dear Morteza,

Thank you so much for the clarification. I fully agree that the solution near the crack tip is the interesting part for everyone interested in putting the math to work. This is the meaning of life isn't it.

It is interesting what you say about boundary collocation method. For a general application one would chose the Bousinesq-Cerruti solution that has no convergence limitations, I guess. With a 2

along crack I would have tried (a2-z2)-1/2 times a polynomial with free coefficients. Thezwould definitely bez=x±iyfor an isotropic material but for your anisotropic material thez=x±μycould be a good starting point. What are your thoughts? Per## Our Answer

Dear Per,

Thank you for your comment. Yes, exactly, once the general series solution in obtained, the BCM can be used to determine the coefficients of the truncated series. The article below has used the BCM to determine the stress intensity factor for an edge-cracked plate of an anisotropic material.

Zhang Heng, D.M. Cammond, B. Tabarrok, Stress determination in edge-cracked anisotropic plates by an extension of boundary-collocation method, Computer Methods in Applied Mechanics and Engineering 54, 187-195 (1986).