Anisotropic and Isotropic Elasticity, and its Equivalence for Singularity, Interface and Crack in Bimaterials and Trimaterials
The equivalence between anisotropic and isotropic elasticity is investigated in this study for two-dimensional deformation under certain conditions. That is, the isotropic elasticity can be reconstructed in the same framework of the anisotropic elasticity, when the interface between dissimilar media lies along a straight line. Therefore, many known solutions for an anisotropic bimaterial can be regarded as valid even for a bimaterial, in which one or both of the constituent materials are isotropic. The usefulness of the equivalence is that the solutions for singularities and cracks in an anisotropic/isotropic bimaterial can easily be obtained without solving the boundary value problems directly. Conservation integrals also have the similar analogy between anisotropic and isotropic elasticity so that J integral and J-based mutual integral M are expressed in the same complex forms for anisotropic and isotropic materials, when both end points of the integration paths are on the straight interface. The method of analytic continuation and Schwarz-Neumann's alternating technique are applied to singularity problems in an anisotropic or isotropic 'trimaterial', which denotes an infinite body composed of three dissimilar materials bonded along two parallel interfaces. The method of analytic continuation is alternatively applied across the two parallel interfaces in order to derive the trimaterial solution in a series form from the corresponding homogeneous solution. The trimaterial solution studied here can be applied to a variety of problems, e.g. a bimaterial (including a half-plane problem), a finite thin film on semi-infinite substrate, and a finite strip of thin film, etc. Some examples are presented to verify the usefulness of the obtained solutions.
Kikuo Kishimoto, Masanori Kikuchi, Tetsuo Shoji and Masumi Saka
Y.Y. Earmme et al., "Anisotropic and Isotropic Elasticity, and its Equivalence for Singularity, Interface and Crack in Bimaterials and Trimaterials", Key Engineering Materials, Vols. 261-263, pp. 23-32, 2004