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Perturbations of Cracks | SpringerLink

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Perturbations of Cracks

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Mechanics and Physics of Fracture

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Abstract

This chapter, of theoretical character, is devoted to the description of various methods of analysis of geometric perturbations of cracks in linear elastic media, in both 2D and 3D. Important applications to the prediction of crack paths are presented.

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Notes

  1. 1.

    When the vectors \({\textbf {e}}_i\) and \({\textbf {e}}_j\) are distinct, such a unit dipole is just a unit point torque.

  2. 2.

    The symbol \(^*\) in the parameter \(a^*\) is intended to underline its relevance to the geometry after the kink.

  3. 3.

    This does not mean that the kink has no effect; the geometric parameters \(\alpha \) and \(a^*\) of the crack extension indeed appear in the arguments of the functional \({\textbf {L}}^*\left[ \alpha ,R,c,a^*;. \right] \).

  4. 4.

    This equation was solved incorrectly, analytically, in the first two works, but correctly, numerically, in the third one.

  5. 5.

    The numerical evaluation of the coefficients was still based on the induction formulae, not on numerical calculation of integrals. Thus the errors made (minimized by performing the calculations in quadruple precision) arose only from the replacement of polynomials in \(\pi \) by numbers with a limited number of digits.

  6. 6.

    For through-the thickness cracks propagating in thin plates for instance, the kink angles observed on the two surfaces of the plate are often notably different, for no apparent reason.

  7. 7.

    Note that this justification of the PLS, based on essentially logical arguments, has nothing to do whatsoever with Goldstein and Salganik (1974)s obscure physical justification.

  8. 8.

    Note that the crack is initiated numerically close to, but not exactly on, the symmetry plane of the plate coinciding with the left boundary of the region modelled—the aim being to avoid simulating a crack forking into two symmetric branches.

  9. 9.

    Specified in Sect. 3.3 below.

  10. 10.

    In other subsubsections the definition (69) leads to the simplest possible expression of the second-order variation of the SIF, devoid of cumbersome factors of \(2\pi \); but conversely in this subsubsection use of this definition would generate such factors in nearly all equations.

  11. 11.

    In Leblond et al. (1996) our function \(\overline{\mathcal F}\) is denoted \(\overline{f}\), and our function \(\overline{f}\) is denoted \(\widehat{f}\).

  12. 12.

    The function f here has of course nothing to do with that in Sect. 3.5.

  13. 13.

    For a given local SIF, the local energy-release-rate for the symmetric problem is obviously twice that for the non-symmetric one; but since this true for both \(\delta G\) and \(G^0\), their ratio remains the same.

  14. 14.

    Local and semi-local terms were also found in the work of Leblond (1999); but he could not get Movchan et al. (1998)s fully non-local terms since the perturbations he considered were confined to the immediate vicinity of the crack front.

  15. 15.

    In the work of Leblond (1999), the notations \(T_{II}^0(s) \equiv T_{xz}^0(s)\), \(T_{III}^0(s) \equiv T_{zz}^0(s)\) were used instead of \(T_{II}^0(s) \equiv T_{zz}^0(s)\), \(T_{III}^0(s) \equiv T_{xz}^0(s)\). The present notations lead to a more natural-looking \({\textbf {G}}\)-matrix, see Eq. (144)\(_1\) below.

  16. 16.

    The sole, purely formal difference lies in the absence of a second index in the 2D functions; this index was unnecessary in the 2D case since there was only one non-singular stress then.

  17. 17.

    This assumption was already implicit in the disregard of the variation of the unperturbed SIFs with the position of the unperturbed front within the crack plane.

  18. 18.

    There were other non-local terms in Movchan et al. (1998)s original formulae, connected to the unperturbed non-singular stresses and higher-order constants characterizing the initial stress field, which are disregarded here.

  19. 19.

    The unperturbed SIF \(K_{II}^0\) may harmlessly be assumed to be nonzero; this has no impact upon the results of this subsubsection.

  20. 20.

    This now widely adopted terminology was first suggested by Hourlier and Pineau (1979).

  21. 21.

    The facet wavelength does not, however, remain small when the crack propagates, due to some “coarsening” of facets resulting from several coalescence events. But this phenomenon is ignored in the present analysis limited to incipient facets.

  22. 22.

    The precise minimum value of \(\nu \) is given in Leblond et al. (2018).

  23. 23.

    Note that the comparison between equations (178) (without mode II) and (186) (with mode II) is not as simple as it may seem at first sight: it does not suffice to set \(\varphi ^0=0\) or \(R^0=0\) in (186) to get (178), because the former equation was obtained for a small \(\rho ^0\) in contrast to the latter.

  24. 24.

    A more refined analysis presented in Vasudevan et al. (2019) shows that in fact, a drifting motion of instability modes exits even when \(G_c\) is independent of \(\rho \); this is due to terms of order 2 or more in the pair \((\varphi ^0,\rho ^0)\), disregarded in the present analysis. But the corresponding drift velocity is much smaller than that calculated here for a \(\rho \)-dependent \(G_c\).

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Leblond, JB. (2023). Perturbations of Cracks. In: Ponson, L. (eds) Mechanics and Physics of Fracture. CISM International Centre for Mechanical Sciences, vol 608. Springer, Cham. https://doi.org/10.1007/978-3-031-18340-9_2

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