A semi-infinite mode I crack partially penetrating a circular elastic inhomogeneity with an imperfect interface

We derive an analytical solution to the plane strain problem of a semi-infinite mode I crack half-way penetrating a circular isotropic elastic inhomogeneity with a spring-type imperfect interface characterized by two non-negative interface parameters. An infinite system of linear algebraic equations is obtained using the expressions for the corresponding two pairs of analytic functions having satisfied the traction-free conditions on the two crack flanks to enforce the interface conditions on the spring-type imperfect interface. A finite system of linear algebraic equations obtained by truncating the infinite system is then solved to arrive at the unknown constants appearing in the two pairs of analytic functions, including the local mode I stress intensity factor at the crack tip. Detailed numerical results for the local mode I stress intensity factor are presented.