Weak solutions of the transmission problem in anti-plane Cosserat elasticity

The theory of Cosserat elasticity (also known as micropolar or asymmetric elasticity) was introduced by Cosserat brothers [1] and further developed by Eringen [2] to model the mechanical behavior of materials for which the microstructure is significant (see [3] for a review of works in this area and an extensive bibliography). The theory was intended to eliminate discrepancies between the classical elasticity and experiments since the classical elasticity failed to produce the acceptable results when the effects of material microstructure were known to significantly affect the body’s overall deformation, for example, in the case of granular bodies with large molecules (e.g. polymers), human bones, porous and cellular solids or foams (see, for example, [4–6]). These cases are becoming increasingly important in the design and manufacture of modern day advanced materials as small-scale effects become paramount in the prediction of the overall mechanical behavior of these materials. Recent experiments [6] show that Cosserat theory gives more accurate results in comparison with the classical theory in modeling of many modern materials, such as fiber-reinforced composites, synthetic polymers or metallic foams. In [7], the classical (Dirichlet, Neumann, mixed) boundary value problems of the three dimensional theory of elasticity were shown to be well-posed and subsequently solved in a rigorous setting using the boundary integral equation method. In a series of recent papers [8–12], the boundary integral equation method proposed in [7] has been extended for the rigorous treatment of the corresponding plane and antiplane problems of Cosserat elasticity and the boundary value problems arising in the theory of plates. The inclusion problems have been receiving an increasing amount of attention in the literature [13–16]. This interest is caused by the rapid development and growing use of composite materials, since the interaction of fibers with the surrounding matrix can be modeled as an inclusion problem. The inclusion problem in micropolar elasticity is especially challenging due to the presence of independent micro-rotations. The solutions available in the literature are restricted by assumptions of a simplified shape of the inclusion or a form of the applied loading. For example, Cheng and He in [17,18], Hansong and Gengkai in [19] considered a spherical, cylindrical and ellipsoidal inclusions, respectively, and derived the analytical expressions of the corresponding micropolar Eshelby tensors [20]. In [21, 22] the rigorous mathematical analysis of an inclusion problem in plane and anti-plane Cosserat elasticity, respectively, was presented. The boundary value problem for an inclusion of arbitrary shape with a homogeneously imperfect interface was formulated and showed to be well-posed, and subsequently reduced to the system of boundary integral equations with unique solutions. Despite the fact, that the inclusion problem was considered in a very general formulation, i.e. without restrictions for the shape of the inclusion and type of the boundary conditions, the analysis was done under assumption that the boundary of the inclusion is a C2-curve. Therefore, a wide class of boundary shapes was excluded from consideration, such as for example, polygons and piecewise-defined curves. These cases are very important for applications, such as finite element method and boundary element method, but also very difficult for analysis due to the