Compliance and Hill polarization tensor of a crack in an anisotropic matrix

Abstract : This work aims at developing an efficient method to compute the compliance due to a crack modeled as a flat ellipsoid of any shape in an infinite elastic matrix of arbitrary anisotropy (Eshelby problem) when no closed-form solution seems currently available. Whereas the solution of this problem usually requires the calculation of the so-called fourth-order Hill polarization tensor if the ellipsoid is not singular, it is shown that the crack compliance can be derived from the first-order term in the Taylor expansion of the Hill tensor with respect to the smallest aspect ratio of the ellipsoidal inclusion. For a 3D ellipsoidal crack model, this first-order term is expressed as a simple integral thanks to the Cauchy residue theorem. A similar method allows to express the same term in the case of a cylindrical crack model without any integral. A numerical example is finally treated.
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Jean-François Barthélémy. Compliance and Hill polarization tensor of a crack in an anisotropic matrix. International Journal of Solids and Structures, Elsevier, 2009, 46 (22-23), pp.4064-4072. ⟨10.1016/j.ijsolstr.2009.08.003⟩. ⟨hal-01977051⟩

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