Estimation of Fiber Orientation and Length in Fiber Assemblies

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Abstract
General formulas are derived for the estimation of fiber orientation and distribution in fiber assemblies whose constituent fibers are arbitrarily crimped. It is found that the density function of orientation can be estimated by solving an integral equation which relates it to the average numbers of fiber cross sections formed on the unit areas of randomly positioned and variously-oriented secant planes. The formula is: ν(Θ, Φ) = L ∫π0 dθ ∫π 0 dϕ.1(θ, ϕ; Θ, Φ)Ω(θ. ϕ) sin θ , where L is the total length of the fibers in a unit volume, Ω(θ, ϕ) sin θ is the density function of fiber orientation, ν Θ, Φ is the areal density of the number of intersections formed by a secant plane with an orientation of (Θ, Φ), and 1 θ, ϕ; Θ,Φ is defined as | sin θ sin Θ cos(ϕ - Φ) + cosθ cos Φ. It is shown also that the total fiber length can be calculated by two methods.

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