Problems involving mechanical behavior of materials with microstructure are receiving an increasing amount of attention in the literature. First of all, it can be attributed to the fact that a number of recent experiments shows a significant discrepancy between results of the classical theory of elasticity and the actual behavior of materials for which microstructure is known to be significant (e.g. synthetic polymers, human bones). Second, materials, for which microstructure contributes significantly in the overall deformation of a whole body, are becoming more and more important for applications in different areas of modern day mechanics, physics and engineering.

Since the classical theory is not adequate for modeling the elastic behavior of such materials, a new theory, which allows us to incorporate microstructure into a classical model, should be used.

The foundations of a theory allowing to account for the effect of material microstructure were developed in the middle of the twentieth century and is known now as the theory of Cosserat elasticity. For the last forty years significant results have been accomplished leading to a better understanding of processes occurring in a Cosserat continuum. In particular, progress has been achieved in the area of investigation of three-dimensional, plane-strain problems of Cosserat elasticity and also some problems related to the theory of Cosserat plates and shells.

However, some certain problems of Cosserat elasticity have remained untouched until today. Among them is the anti-plane problem of Cosserat elasticity. Meanwhile, the anti-plane problem is regarded as very important for applications in mechanics, since from the point of view of mechanics, the anti-plane problem with Neumann boundary conditions is the problem of torsion of a beam with significant microstructure.

The objective of this work is to formulate and solve rigorously basic boundary value problems of anti-plane Cosserat elasticity. To achieve this goal we use the boundary integral equation method in order to derive the exact analytical solutions for the corresponding boundary value problems in terms of integral potentials. The exact solutions are then approximated numerically using the method of generalized Fourier series in order to obtain quantitative characteristics of the solutions to the corresponding boundary value problems. In particular, it has been found that in the case of torsion of a circular Cosserat beam, microstructure does have a significant effect on the warping function, provided that the cross-section is elliptic.