Smolyankova T. Mathematical modeling of the mechanical characteristics of a fibrous composites with multi-modular components

In this thesis, I determined the effective mechanical characteristics of the multi-modular elastic fibrous composite material at stretching and compression. I designated the effective technical characteristics of these composites – longitudinal and transverse modulus of elasticity and Poisson coefficients. So I used the kinematic conditions of displacements for points of the homogenized composite and its components such as matrix and fiber at their common deformation. The main part of the thesis consists of an introduction, four sections, and conclusions. In the introduction, I proved my choice of thesis theme, clearly defined the main aim and tasks, determined the methods of researches, scientific novelty, and practical value of outcome measures. I described the personal role in the researches made in the co-authorship includes. Also I submitted the data of approbation the outcome measures. In the first section of the scientific research paper, I explored the current situation and main approaches in solving the problem of the homogenization of the multi-modular composites. To define the entity of the following problem I considered the basic mechanical characteristics of composites. It requires a definition in the process of homogenization. I expounded the essence of the experimental approach and the mathematical modeling for finding the effective mechanical characteristics of the fibrous composites. I also reviewed the best-known researches in these areas. I noted that there were several methodological approaches in exploring multi-modular materials. So I provided the short characteristics of the main publications within each of the approaches. In this section, I also defined the theoretical grounds of the thesis as the main formula between tensions and deformations. These formulas are formulated for the isotropic material according to the classic theory of elastic materials. It was suggested by the Ambartsumyan and distributed for the case of the multi-modular transversally-isotropic material. While, it was assumed the hypothesis about the stability of the shear module at stretching and compression. In the second section, I determined the effective longitudinal modulus of elasticity and the Poisson coefficient of the multi-modular transversally-isotropic composite at stretching and compression. An elementary cell of a composite represents by a cylinder. It consists of a hollow cylinder as a matrix and a solid cylinder as a fiber. Firstly, I solved the problem of joint deformation of the components of a representative composite cell as a matrix and fiber to obtain the effective elastic constants. The radial displacements and tensions on a surface of contact of matrix and fiber thus are continuous. A similar problem of determining the components of the stress-strain state at longitudinal stretching and compression is solved in the case of homogeneous composite material. Effective elastic constants such as a longitudinal modulus of elasticity and the Poisson coefficient are determined from the equality of the axial deformation in the point of the elementary cell of a homogeneous composite, matrix and fiber. And it is also based on the equality of the axial displacements at the outer surface of matrix and elementary cell of homogeneous composite. In the third section, I explored the homogenization of the multi-modular transtropic composite at transverse stretching-compression. Firstly I obtained the boundary problem of determining the components of the stress-strain state of matrix and fiber at given radial stress on an outer surface of the matrix. Nevertheless, radial displacements and tensions on an outer surface of matrix and fiber are continuous; the axial deformations of matrix and fiber are equal. At the next stage, I solved the problem of finding the displacements, tensions and deformations for a homogeneous composite at the same boundary conditions. The conditions for the alignment of displacements at the points in a homogeneous composite cell are the equality of axial displacements for the arbitrary point and the radial displacements at the outer surface of the matrix and homogeneous composite cell. Using these conditions for the alignment allowed us to obtain a formula between effective elastic constants in the plane of isotropy of a composite. In the fourth section, I found the effective transverse shift module of elasticity and the Poisson coefficient in the plane of isotropy of a transversally-isotropic multi-modular composite. So I used the assumptions of equality of the shift module at stretching and compression of the multi-modular material; the ratio between effective characteristics of a composite in the plane of isotropy from the third section. The conclusions were formulated based on the completed thesis.