  We analyse a problem of anti-plane shear in a bi-material plane containing a
semi-infinite crack situated on a soft imperfect interface. The plane also
contains a small thin inclusion (for instance an ellipse with high
eccentricity) whose influence on the propagation of the main crack we
investigate. An important element of our approach is the derivation of a new
weight function (a special solution to a homogeneous boundary value problem) in
the imperfect interface setting. The weight function is derived using Fourier
transform and Wiener-Hopf techniques and allows us to obtain an expression for
an important constant (which may be used in a fracture criterion) that
describes the leading order of tractions near the crack tip for the unperturbed
problem. We present computations that demonstrate how this constant varies
depending on the extent of interface imperfection and contrast in material
stiffness. We then perform perturbation analysis to derive an expression for
the change in the leading order of tractions near the tip of the main crack
induced by the presence of the small defect, whose sign can be interpreted as
the inclusion's presence having an amplifying or shielding effect on the
propagation of the main crack.
