Convergence of solutions for noncanonical bodies found by the boundary-shape perturbation method

Abstract

The exact solution of the problem of the torsion of a solid cylindrical shaft with an ellipsoidal cavity is expanded into a series in powers of an eccentricity parameter. At the points of the cavity where the stress concentration is maximum (minimum), the expansion coefficients for the exact and approximate solutions coincide. The approximate solution is obtained by the first version of the boundary-shape perturbation method, which is used to solve three-dimensional problems of elasticity for bodies of revolution with nearly canonical shape. Successive approximations as an iteration process are shown to converge to the exact solutions Keywords: theory of elasticity, boundary-shape perturbation method, solid cylindrical shaft, ellipsoidal cavity, series expansion of exact solution, approximate solutionIntroduction. The perturbation method is used to solve some problems of continuum mechanics and outlined in [7]. The method of perturbation of elastic properties was used in [11] to solve problems of stress concentration around holes in physically nonlinear plates [4]. In [10], this method was applied to orthotropic materials using the elastic properties of a transversely isotropic material as a zero approximation. The boundary-shape perturbation method was used in [2, 3, 6] to study the stress concentration in elastic plates and shells with curvilinear holes.Three-dimensional problems of elasticity were solved in [2, 3] using two versions of the boundary-shape perturbation method for noncanonical bodies under static forces: (i) bodies with nearly canonical inclusions and cavities bounded by orthogonal surfaces and (ii) bodies bounded by nonorthogonal surfaces [8,9].The former method [3] shows practical convergence of approximate solutions of stress-concentration problems for elastic bodies with ellipsoidal cavities (a shaft under torsion or a medium under uniform tension-compression), which permit exact solutions [5,12]. Other approaches to solving problems for bodies with stress concentrators such as thin inclusions and cavities by exact and approximate methods are proposed in [13][14][15][16].In the present paper, we will expand the exact analytic solutions of stress-concentration problems for a shaft with an ellipsoidal cavity under torsion in powers of the eccentricity parameter e. We will compare the coefficients of the expressions of the exact [5] and approximate [3] solutions.1. Exact solution for a shaft with an ellipsoidal cavity under torsion is as follows [5]: