Micromechanical modeling of a cracked elliptically orthotropic medium
Résumé
In this paper, we derive the second-order crack opening displacement tensor for an arbi- trarily oriented elliptical crack in an elliptically orthotropic (EO) matrix. This result is ob- tained 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 nei- ther the same in-plane aspect ratio nor the same vanishing aspect ratio. It requires a cor- rection 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 de- tail 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.
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