A problem of stability of reinforced composite materials under compression is studied well for the case of small under-critical deformations. The investigations were performed both on the basis of a variety of simple models and on the basis of the three-dimensional linearized theory of stability. At that, reinforcing fibers were taken in the form of perfect fibers in all the publications known. We consider the stability of a non-perfect fiber in an elastic matrix, that is, a fiber with a defect of mechanical nature. These defects, weakening, are undercuts, cracks and gaps of technology nature such that a fiber has a discontinuity zone. We consider the case when the first derivative of the displacements function is discontinuous (this case is similar to the weakening of a hinge type). We also suppose that the second derivative related linearly with the discontinuity of the first derivative (this case corresponds to the elastic hinges). To describe the behavior of a matrix, we employ the three-dimensional theory of elasticity. A fiber is considered as an infinite rod with non-deformable circular cross-section. After the loss of stability of the fiber occurs, an axis of the fiber contains a point of discontinuity in which a bending moment is applied. Using the Fourier cosine transform, we reduce the problem to the characteristic equation that should be solved at the point of weakening of the fiber.