In this review article the three-dimensional linearized theory is presented of the internal and the surface instability of fibrous composite materials. The possible mechanisms of the stability loss in the structure of these materials are investigated. In this investigation the strict model is used of the nonlinearly elastic compressible and incompressible piecewise-homogeneous medium with the arbitrary form of the elastic potential for the theory of finite deformation and for two variants of the theory of small precritical deformations. Problems for a single fiber (fibrous materials with low concentration of the filler, when at stability loss the interaction between fibers is not accounted for), for two fibers (fibrous materials with low concentration of the filler, when as a result of the structure irregularity at the stability loss two neighbouring fibers may interact) for the infinite row and for a doubly periodic system of fibers (fiber materials with nonsmall filler concentration, taking into account fiber interaction), in the infinite and semiinfinite matrix, are considered. Results are obtained for these cases, predominantly when conditions are satisfied of the complete contact on the fiber and matrix polymer or metal interfaces. In the case of the metal matrix at plastic deformations the conception of continuing loading is used, and the change of the unloading zones in the process of stability loss is not accounted for. The influence of the inhomogeneity of the precritical stressed state, resulting from the difference of coefficients of the transverse expansion, of the mechanical properties, and of the volume concentrations of the fibers and the matrix, on the critical parameters is investigated. Application is presented of the obtained results in the fracture mechanics of fibrous composite materials under compression along the reinforcing elements.

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