Within the scope of the piecewise homogeneous body model through the use of the three-dimensional geometrically non-linear exact equations of the theory of elasticity, an approach for the investigation of problems with the micromechanics of a periodically curved fiber near the free convex cylindrical surface is proposed and employed. The main difficulties in finding the solution to these problems are caused by the impossibility of employing the summation theorem for cylindrical functions to satisfy the boundary conditions on the cylindrical surface. For this purpose the cosine and sine Fourier series presentation of the sought values is proposed to satisfy the boundary conditions. The coefficients of these series are calculated numerically through the integrals of the cylindrical functions whose argument depends on the integrating variable in the complicated form. This approach is employed successfully both for the solution to the corresponding boundary value problems in the determination of the self-balanced stress, which is caused primarily by the periodical curving of the fiber, and for the solution to the problems of micro-buckling near the surface through the use of the initial imperfection criterion.