Influence of high-order terms in the solution generalizing the approach of M. Williams, taking into account the anisotropy of the material

K. A. Mushankova ; L. V. Stepanova

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 29 (2023) no. 2, pp. 30-39

Résumé

The research is devoted to the study of the stress field at the crack tip in an anisotropic material with three mutually orthogonal axes of symmetry of the fourth order (with cubic symmetry). A plane case is considered when one of the axes of symmetry is orthogonal to the cracked plate, and the remaining two axes lie in the plane of the plate. The paper presents an asymptotic analysis of the contribution of higher approximations in the generalized asymptotic decomposition of mechanical fields near the crack tip in a linearly elastic anisotropic material with cubic symmetry of its elastic properties. In the article, based on the obtained solution of M. Nejati and co-authors for an infinite anisotropic plate with a central crack, circumferential apportionments of the stress tensor components at the crack tip at various distances from the crack tip are constructed, which makes it possible to estimate the contribution of non-singular (regular) terms to the general asymptotic representation of mechanical fields generated by an acute crack. In the work of M. Nejati, the contribution of exclusively T-stresses is analyzed, then, as shown in this work, the terms following the T-stress play a significant role in describing the fields induced by the crack. A comparison of the angular distributions of the stress tensor components constructed at different distances from the crack tip indicates that with the increase of distances from the crack tip, it is required to preserve in asymptotic series representing stresses, displacements and strains near the tip of the crack, the terms of high order of smallness. The preservation of the terms of high order of smallness can be used to expand the domain in which the asymptotic solution in the series is valid.

Détail

Keywords: asymptotic expansion, stress fields, cubic syngony, high-order terms.

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