Matrix and fibre stress and strain magnification using strain energy equivalence and the Halpin–Tsai equations

Abstract

Derivations for the transverse matrix strain magnification and fibre strain reduction, based on the principle of strain energy applied to a representative volume element and using the semi-empirical model of Halpin–Tsai are presented here. The arising values of fibre/matrix strain reductions/magnifications decrease/increase with increasing volume fractions of the reinforcing fibre. Significant differences are seen between the transverse, matrix/fibre strain magnification/reduction for the present case where no assumptions are made about the existence of the iso-stress state and others where the assumption applies. The transverse stresses in the composite, matrix and fibre are shown here to be different, which implies that the assumption of transverse iso-stress that is normally made in the study of composite materials is wrong. The fibre stresses are found to be greater in magnitude than the matrix stresses for all volume fractions and up till 71% reinforcing volume fraction for the hexagonal and square arrays, respectively. The fibre/matrix strain reduction/magnification, fibre and matrix/central sub-region stress ratios are all shown here to be dependent on the fibre geometry for all volume fractions of reinforcing fibre, with the exception of the fibre/central sub-region stress ratio where this characteristic only prevails below 76% reinforcing volume fraction.