Some reverse mappings in 3-D crack and dislocation problems

Abstract

A new division of geometric regions for internal and external planar cracks is introduced. This allows some reverse mappings between internal and external problems to be revealed in connection with 3-D planar crack and dislocation problems in an infinite elastic medium. Transform factors are given for a mode I crack with arbitrary shape in an infinite inhomogeneous isotropic medium where the elastic modulus is an exponential function of depth. Arbitrary loading for the inhomogeneous material and coplanar dislocation-crack interaction for a homogeneous isotropic medium are analyzed through easy derivations. As one of the applications, the known solution of an external elliptical crack is used to obtain the exact solution in closed form for the crack with the boundary contour % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdiNaai% ikaiabeI7aXjaacMcacqGH9aqpcaWGIbWaaWbaaSqabeaacaaIYaaa% aOWaaOaaaeaadaWcaaqaaiGacogacaGGVbGaai4CamaaCaaaleqaba% GaaGOmaaaakiabeI7aXbqaaiaadggadaahaaWcbeqaaiaaikdaaaaa% aOGaey4kaSYaaSaaaeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaai% aaikdacqaH4oqCaaaakeaacaWGIbWaaWbaaSqabeaacaaIYaaaaaaa% aeqaaaaa!4F66!\[\rho (\theta ) = b^2 \sqrt {\frac{{\cos ^2 \theta }}{{a^2 }} + \frac{{\sin ^{2\theta } }}{{b^2 }}} \]subjected to a pair of concentrated forces at the crack center.