In the dissertation research, the approach to a homogenization of unidirectional composites with transtropic matrix and hollow transtropic fibers is stated. Using the method of a representative volumetric element, the boundary value problems of longitudinal and transverse elongation, longitudinal and transverse shear for inhomogeneous and homogenized material are solved. Analytical ratios to determine the effective elastic characteristics of these composites were obtained. The kinematic criterion of coordination is chosen as a basis. The first section covers the main approaches to determining the stress-strain state of fibrous composite materials and structures designed using them. Particular attention is paid to taking into account the specific properties of the components, such as viscoelasticity, anisotropy, and others when determining the effective mechanical characteristics of the composite. The issues of homogenization composites, reinforced by hollow fibers, are covered separately. The advantages and disadvantages of numerical and experimental approaches to determining the stress-strain state of such composites, compared to analytical methods, are noted. Main tasks, that need to be solved during the research, are formulated based on analysis. Developed a method of representative volumetric element for homogenization of unidirectional composite material with transtropic matrix and hollow fiber at transverse and longitudinal elongation and shear. It is based on the use of kinematic conditions of coordination of displacements of composite and its components. For the first time, based on the solution of the problem of homogenization on transverse elongation and simple transverse shear, were obtained analytical ratios for determining the transverse modulus of elasticity and Poisson’s coefficient of a unidirectional composite with a transtropic matrix and hollow fiber, depending on the elastic characteristics of its components and volumetric content of fiber and the cavity inside the composite. For the first time, based on the solution of the problem of homogenization of longitudinal elongation and simple longitudinal shear, were obtained analytical expressions for calculating the longitudinal modulus of elasticity, Poisson's ratio, and longitudinal shear modulus of unidirectional composite material with hollow fibers in the case of transtropeness of both components of the composite, depending on the elastic characteristics of its components and volumetric content of fiber and the cavity inside the composite. For the first time calculations according to the obtained formulas were performed, and the dependencies of effective elastic constants composites with hollow fibers on the volumetric content of fiber and cavity inside the composite material were constructed. Calculations of the effective mechanical characteristics of fibrous composite materials according to the obtained formulas demonstrate a high convergence of the results with the calculations based on the ratios of G. A. Vanin and D. M. Karpinos. The practical significance of the work results lies in the possibility of direct use of the obtained analytical relations to determine the effective elastic constants two-phase unidirectional composite materials with hollow fibers, which consisting of both isotropic and transtropic components. The analytical dependences presented in this thesis allow to obtain composite materials with rational characteristics at the design stage, varying the matrix’s material, fiber’s material, volumetric fiber’s content, and volumetric content of the cavity in it.
