Thin Inclusions and Cracks in Anisotropic Media

Abstract

By using the solution of eigenstrain problems in anisotropic media, flat ellipsoidal inclusions and crack problems are investigated. When a dilatational misfit strain is defined in a flat ellipsoidal inclusion (i.e., disk-shaped precipitate), the elastic strain energy associated with the misfit becomes minimum when the plane of the inclusion coincides with one of the crystalline planes, where the matrix and the inclusion are assumed as two different cubic crystals having the same crystalline directions. The minimum value, however surprisingly, depends only on the elastic moduli of the inclusion. When crystalline directions of the inclusion are parallel to the principal axes of the ellipsoid, the elastic strain energy is independent of the orientation of the inclusion with respect to the crystalline directions of the matrix and the constant value is equal to the minimum value in the first case. If an inclusion is simply defined by giving a uniform dilatational misfit strain in an ellipsoidal domain in a homogeneous material (i.e., matrix and inclusion are same material), the condition for the minimum elastic strain energy is the same as that of the first case, namely the minimum occurs when the plane of the inclusion is parallel to one of the crystalline directions. The method employed here can be applied equally to generally anisotropic materials. The method is also applicable to fracture problems by taking the elastic moduli of the inclusion as zero. As examples, an elliptical crack is considered for a simple tension and a pure shear. The interaction energy between the applied stress and a crack is calculated. By using this result, the Griffith criterion for fracture is derived for the penny-shaped crack. Some numerical results for cubic crystals are shown.