Ellipsoidal inhomogeneity in elliptically orthotropic elastic solid

• Published in: International journal of solids and structures Vol. 206; pp. 282 - 291
• Main Authors: Kushch, Volodymyr I., Sevostianov, Igor
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.12.2020; Elsevier BV
• Subjects: Affine transformation; Affine transformations; Boundary value problems; Elastic anisotropy; Ellipsoid; Ellipsoidal harmonics; Elliptical crack; Elliptical orthotropy; Exact solutions; Harmonics; Inhomogeneity; Linear algebra; Mathematical analysis; Maxwell homogenization scheme; Micromechanics; Particulate composites; Stiffness; Stress concentration; Stress intensity factors; Tensors; Elliptical crack; Maxwell homogenization scheme; Elliptical orthotropy; Ellipsoidal harmonics; Affine transformation; Ellipsoid
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2020.09.025

An unbounded elliptically orthotropic elastic solid containing a single anisotropic ellipsoidal inhomogeneity is considered. The problem statement assumes arbitrary orientation of ellipsoid with respect to the orthotropy axes and non-uniform far loading. By applying the appropriate affine transformations, the boundary value problem is reduced to that one with the isotropic constituents. The complete displacement solution to the transformed problem is obtained using the Papkovich-Neuber representation with the scalar potentials written in terms of ellipsoidal solid harmonics. By accurate fulfilling the interface conditions, the boundary value problem is reduced to the system of the linear algebraic equations for the constants entering the potentials. For the Eshelby problem (uniform remotely applied loading and perfect interface), an explicit analytical solution is written in terms of ellipsoidal harmonics. This result yields solution for the elliptical crack problem in the remarkably simple way. The numerical examples illustrate an effect of the matrix anisotropy on the stress concentration and stress intensity factors. An application of the obtained solution to the micromechanics is discussed. The effective elastic stiffness tensor of a particulate composite with elliptically orthotropic matrix is evaluated using Maxwell homogenization scheme.