Effects of Terms of High Order in Synthesized Polynomial Series Approximation for Fields Associated with the Crack Tip in Anisotropic Media. Part 1. Comparison of Exact and Asymptotic Solutions for Determining Stresses at the Crack Tip

The article presents the analysis of asymptotic series expansions generalized to the case of anisotropic linearly elastic media representing fields of displacements, strains and stresses around the tip of an acute crack in anisotropic media. We study anisotropic materials with the simplest cubic symmetry and fields in the proximate neighbourhood of the crack tip. The asymptotic series are constructed on the essential principles of the classical theory of elasticity of an anisotropic body. By analyzing the series associated with the fields at the crack tip, it is shown that higher approximations with coefficients called generalized stress intensity factors have an essential impact on the accurate and relevant representation of the stress field to expand the zone of the asymptotic solution. Using the example of a plane problem for an infinite anisotropic plane with the cubic symmetry of properties (with different tensors of elastic modules having three independent elements) with different crack orientations relative to the axes of the symmetry of elastic properties, it is shown that in the generalized series, in addition to the first two terms (containing stress concentration factors and T-stresses), terms of higher orders of smallness should be preserved. The circumferential -apportionments of the stress components at various spans from the crack tip are assembled with the retention of a different quantity of series terms to obtain sufficiently accurate approximations. A comparison of the dependencies of the tensor components on the polar angle obtained taking into account the different quantity of series terms clearly indicates the need to retain the higher approximations of the series. In order to expand the area in which the solution in the series is valid, it is necessary to preserve a larger number of terms. All computations were performed for real materials which elastic constants were determined using the molecular dynamics method for single-crystal substances with a face-centered cubic (FCC) lattice.