On stress concentrations caused by spheroidal inhomogeneities

The problem of spheroidal inclusion in an infinite homogeneous isotropic elastic media is considered. On the base of Eshelby’s method of equivalent inclusions the stress concentrations inside and at the inclusion boundary are written down for the inclusions, which elastic constants (shear moduli and Poisson’s ratios) are different from the elastic constants of the media (matrix). Expressions for stress concentrations both inside and at the boundary were obtained by asymptotic expansions and limit transitions of the general solution for the number of particular important cases. Parameters determining the type of asymptotical behavior are the ratio of the inclusion semi-axes and ratios of the shear moduli of the matrix and inclusion (or the inversed values). The Poisson’s ratios of matrix and inclusion have less effect. For simultaneous high deviation of these parameters from unity seven non-overlapping regions are distinguished corresponding to various successive limit transitions of these parameters to zero either to infinity. These seven regions correspond to various physical situations corresponding to penny-shaped and needle-like inclusions of high and low rigidity. Although the obtained solutions were either known before, or might be obtained from the known particular solutions by algebraic manipulations, the advantage of the used method is in simultaneous estimation of the applicability ranges of the obtained solutions rather than obtaining the solution themselves. The applicability ranges of the obtained asymptotics were also obtained by numerical comparison with the exact solutions, which confirm the theoretical estimations.