In the present paper, we apply the real boundary integral equation method to obtain the solution of the inclusion boundary-value problem with an imperfect interface arising in the theory of antiplane elasticity with significant microstructure. We find the solution in the form of the integral potential and employ the boundary element method to derive an approximate representation for the corresponding integral density. Finally, we consider an example of an elliptic inclusion with a homogeneously imperfect interface and find the distribution of shear stresses along the inclusion interface to demonstrate the effectiveness of the method. The method is very general in nature so it can be applied for the treatment of problems relating to inclusions of arbitrary shape, different types of interface and general forms of applied loading.
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