Modified Eigenfunctions for Transversely Isotropic Composites with Applications to Stress Analysis of a Broken Fiber

Abstract

When a transversely isotropic elastic body that contains a notch or a crack is under an axisymmetric deformation, the eigenfunction solution near the singular point is in the form of a power series ^δ f(ψ,δ), ^δ+1 f ₁(ψ,δ), ^δ+2 f ₂(ψ,δ) ... in which (,ψ) is the polar coordinate with origin at the singular point and δ is the eigenvalue, or the order of singularity. A difficulty arises when δ as well as δ + k, where k is a positive integer, are also eigenvalues. In this case the higher order terms of the series solution may not exist. A modified solution is required and is presented here. The modified solution has the new terms ^δ+ ^k (ln) F ₁(ψ,δ), ^δ+ ^k ⁺¹(ln)F ₂(ψ,δ).... As an application, we consider the stresses near a broken fiber in a composite which is under an axisymmetric deformation. The interface between the broken fiber and the matrix also suffers a delamination. This creates stress singularities at several points some of which require the modified eigenfunc tions presented here.

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