Effects of Terms of High Order in Synthesized Polynomial Series Approximation for Fields Associated with the Crack Tip in Anisotropic Media. Part 2. Accuracy Estimates of Asymptotic Solutions

The second part of the article discusses the accuracy of generalized asymptotic series representing the stress and displacement fields associated with the tip of an acute crack in orthotropic materials in the formulation of the plane problem of the theory of anisotropic elasticity. We compare the exact analytical solution to the problem of stretching an infinite anisotropic plane with an inclined central crack obtained using methods of the complex variable theory and an approximate solution found using the power series expansion method. For the first time, the fields of absolute error allowed for the truncation of the asymptotic series on a different number of terms for materials with a cubic symmetry of elastic properties are obtained. The analysis of the fields of the decadic logarithm of absolute errors showed that near the crack tip of all types of combined (mixed) deformation there are geometric points – loci of accuracy, in which the approximate solution almost coincides with the exact solution, which can be used in the interpretation of experimental and computational data obtained in order to determine the coefficients of asymptotic series for stresses and displacements near the tip of a crack or notch. To quantify the errors allowed when truncating the asymptotic series on the th term, a relatively accurate analytical solution obtained on the basis of the theory of the function of a complex variable the -norm is introduced into scrutiny, which gives an opportunity to choose and specify the number of terms of the series necessary to achieve the required accuracy when presenting the asymptotic ansatz with a multi-coefficient asymptotic series, for a wide range of slope angle values of the crack to the vertical axis (the axis of action of the applied tensile load) and the angle, setting the location of the anisotropy axes of the elastic properties of the material.