Finite–element analysis of the effect of material properties and fibre shape on stresses in an elastic fibre embedded in an elastic matrix in a fibre–composite material

The finite–element method was used to calculate the axial stress in an elastic fibre embedded in an elastic matrix to model a fibre–composite material. Axisymmetric models were created for cylindrical, ellipsoidal, paraboloidal and conical fibres embedded in a matrix and characterized by a fibre axial ratio, q. The effects of varying q, from 200 to 1000, and the ratio of the Young moduli of the fibre and the matrix, Ef/Em, from 50 to 104, were investigated. For a cylindrical fibre, the axial stress distribution along the fibre axis was similar for all values of q and Ef/Em; it was greatest at the centre, decreased steadily over most of the fibre length and fell rapidly to zero near the fibre end. For fixed q, the magnitude of this stress increased with increasing Ef/Em, whereas for fixed Ef/Em the variation with q was small. There was good qualitative agreement between these data and previous analytical models. The axial stress in the conical fibre was a minimum at the fibre centre and rose gradually to a maximum close to the fibre end. This was most pronounced for small values of q and at large values of Ef/Em. Stress distributions for the paraboloid and ellipsoid lay between those for the cylinder and the cone. For small values of Ef/Em, both the magnitudes and the axial distributions of axial stress were almost indistinguishable for all shapes of fibre and all values of q studied.