Comparative micromechanical analysis of nonelastic behavior of unidirectional plastics with tetragonal and hexagonal structures

Abstract

.4 mathematical model for determining the effective elastic properties and describing the processes of inelastic deformation and damage accumulation of unidirectional fiber-reinforced composites with tetragonal and hexagonalModels with a periodic structure serve as good approximations in modeling the mechanical behavior of composite materials. It is assumed that, in an elementary macrovolume of such media, the strain fields are periodic, i.e., during calculation of structural stresses and strains and evaluation of the effective properties, periodic problems can be considered, taking into account that the stresses averaged over the periodicity cells must be equal to the given macroscopic stresses. Efficient methods have recently been developed for the solution of periodic problems [1][2][3] in the mechanics of composites.This study deals with investigation of nonlinear stress and strain fields, modeling of failure processes in the matrix and fibers, and calculation of the effective characteristics of unidirectional composites with tetragonal and hexagonal layout of fibers in the matrix (Fig. 1). The tetragonal model is employed because of the minimum distances between the fibers at a given volume fraction [1]. Thus, we can expect that the strength properties of a composite predicted according to this model provide the lower estimates of the actual characteristics. However, the structure of an actual composite can differ considerably from the model representations. In creating struCaLreS from fibrous composites, there exists a tendency to increase the volume fraction of fibers in these materials. In this case, the geometry of the structure approaches the hexagonal model, which ensures the highest theoretically possible degree of packing for fibers of identical radius in a unidirectional fibrous composite. Therefore, along with examination of a material with a tetragonal packing of fibers, investigation of the behavior of a composite with a hexagonal structure is also of importance.To calculate the fields of structural stresses and strains in a composite, the method of local approximation, described in detail in [4][5][6], is applied. According to this method, the problem ofmicromechanics is solved using the structural fragments shown in Fig. 2. For a composite material with a tetragonal structure, such a fragment consists of nine cells, while for that with a hexagonal structure--of seven cells. The boundary conditions on the contour of the fragment, 8#, are chosen so that the stresses averaged over the central periodicity cell be equal to the given macroscopic stresses ,.~.The mathematical statement of the problem on the generalized plane deformation is described by the following system of equations: