Calculation of high-modulus cross-reinforced composite shells

Abstract

For the regular structure of a composite which is typical of the great majority of the reinforced materials, these difficulties can be overcome using the asymptotic method of averaging out the periodic structures [3-5]. The subsequent application of this method makes it possible, on the basis of solving auxiliary local problems in the periodicity cell, not only to determine by means of a justified approach the effective (averaged-out) characteristics of the composite but also determine with high accuracy the local distribution of the stresses and strains and, on the basis of this, calculate the strength of composites. In [6, 7] on the basis of modification of the averaging method without making any simplifying hypotheses of the Kirchhoff-Love type of other hypotheses, the authors transferred from spatial problems of the theory of elasticity, heat conductivity, thermal elasticity for a thin distorted composite layer with a regular structure to a model of the averaged-out shell. Its effective characteristics were determined from the solution of the local problems in the periodicity cell [6, 7] which also make it possible to reproduce with high accuracy the three-dimensional local structure of the examined fields. In this work, the method developed in [6, 7] is used to calculate the high-modulus cross-reinforced shells. We derive analytical equations for the effective stiffness moduli in tension-shear, skew symmetry, and bending moduli, and analyze deformation of the fibers and the binder on the microlevel. Specifically, we describe the effect of failure of the binder with close packing of the fibers. The results obtained in [8] in determining the effective stiffnesses of skeleton shells are used. We examine a composite shell with a regular structure formed by plies of parallel fibers. The plies are parallel to the 0=I~ 2 surface (~l, ~2 is the Gaussian coordinate system introduced on the median surface of the shell, at ~ = 0; 7 is the rectilinear transverse coordinate). The angle formed by the fibers of the i-th ply with the coordinate line O~ I is denoted by ~i, i = i, 2 .... , M (M is the number of reinforcing plies); the thickness of the reinforcing plies is denoted by e, and it is assumed that the dimensions of the periodicity cell (in the plane of the shell) are also determined by the low value of c and are equal to ~h I, eh 2 (h I , h 2 are the constants of the order of unity).