External crack in torsion in an infinite non-homogeneous medium with a non-homogeneous cylindrical inclusion

This paper deals with the determination of stresses in a non-homogeneous in finite medium containing an external crack surrounding a non-homogeneous cy lindrical inclusion. The crack occupies the region outside the circle surro unding the cylindrical inclusion. The cylindrical inclusion (fiber) and the infinite medium (matrix) have different shear moduli. The external crack i s opened by internal shear stresses acting along the crack. The continuity of stress and displacement is assumed at the common cylindrical surface due to perfect bonding. With reference to the cylindrical coordinates the shea r modulus of the inclusion and matrix are assumed to be of the forms G(1) e (beta \z\) and G(2)r(alpha) c(beta \z\) where G(1), G(2), alpha, and beta a re real constants and alpha > -2. The geometry of the problems is more clea rly shown in fig. 1. The problem is reduced to the solution of a Fredholm i ntegral equation of the second kind, which is solved numerically. A closed form expression is obtained for the stress intensity factor and numerical v alues for stress intensity factor are graphed to demonstrate the effect of non-homogeneity of the inclusion and infinite medium. Finally the order of the singularity is obtained when the crack approaches the inclusion.