UDC 539.3 in the compression of composite materials with a clearly expressed structural orientation, one possible failure mechanism is stability loss in the structure of the material [7]. Investigations of the mechanisms of failure and stability loss in the structure of reinforced materials are usually based on various simplified calculation schemes.Within the framework of the continuum approach, using three-dimensional linearized stability theory, it has been possible [6,8] to elucidate the character of failure and determine the theoretical strength of a reinforced material under compression along the reinforcing elements. At first, a rigorous approach (within the framework of piecewise-homogeneous media) to the investigation of the stability of fibrous composite materials, using three-dimensional linearized theory, was proposed [5]. In [8,9], methods of solving stability problems for reinforced materials were formulated and their mathematical basis was completely established. Also in [8,9], in the case of homogeneous subcritical states, general solutions of three-dimensional linearized equations were constructed, and the corresponding qualitative and quantitative analysis of the results was undertaken. The problem of the stability of a fiber in an elastic matrix was considered in [i] within the framework of threedimensional linearized theory, with small homogeneous subcritical deformations. Within the framework of three-dimensional linearized theory, the results of [2, I0] regarding the stability and mechanics of brittle failure of composite materials in compression were reviewed and analyzed. In most works, the stability loss of composite materials was investigated with homogeneous subcritical states. Stability loss of the fiber in the matrix was studied in [12,20] in the case of small inhomogeneous subcritical deformations, but the matrix material was assumed to be incompressible.In the present work, the problem of the stability of a rectilinear elastic fiber of circular cross section (radius R) in an infinite elastic medium with inhomogeneous subcritical deformations is considered.

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