**Published:** Jan 1967

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**Source: **STP427-EB

Using symmetry arguments, boundary conditions are established which must hold on the lines of symmetry between neighboring filaments in an oriented filamentary composite. These conditions serve to set the problem as a special case of the mixed boundary problem of elasticity. The composite is assumed to deform such that the normal strains in the direction of the filament axes are the same in both media. The equations of elasticity are then solved in polar coordinates so that the solution has the appropriate sixfold symmetry. Because boundary and continuity conditions must be described on both a straight and circular boundary, the solution is not in closed form. The in-plane stresses vanish when the Poisson's ratios of the two media are identical. Three harmonics were used to describe the stress field, which required that fifteen constants be evaluated from the boundary and continuity conditions. Fifteen equations, which depend on the elastic constants of both media and the separation between filaments, were obtained from the continuity conditions at the circular interface between the two media and boundary conditions along the lines of symmetry between neighboring filaments. These equations were solved for an epoxy-fiber glass and a silver-steel composite for a variety of filament spacings. The in-plane stresses are vanishingly small for the epoxy-fiber glass composite. For the silver-steel composite, the in-plane stresses reach about 3 per cent of the average axial stress. When the filaments are close together, the periodic portions of the stress field are most important. As the filaments are more widely spaced, the stress field becomes more circularly symmetric. The phasing of the stress field is such that the radial stress is compressive at the point where the filaments are closest together and tensile at the point where the filaments are farthest apart. In all cases, the composite Young's modulus varies linearly with filament fraction.

**Keywords:**

fiber metallurgy, filaments, composite materials, stress analysis, elasticity

**Author Information:**

Piehler, H. R.*Research assistant, Massachusetts Institute of Technology, Cambridge, Mass*

**Committee/Subcommittee:** D30.07

**DOI:** 10.1520/STP42039S