The development of microbuckling from a sharp notch under remote compressive loading is analysed for a long fibre composite. The composite is modelled as alternating layers of elastic fibres and a nonlinear matrix. A finite element analysis is used to predict microbuckle initiation. The relationship between the compressive mode-I stress intensity factor K and the microbuckle length l (that is, the compressive R-curve) is calculated. The effect of matrix yield strain on the R-curve is significant, while the influence of the matrix strain-hardening rate and initial fibre waviness is slight. R-curves predicted by the finite element method agree qualitatively with those predicted using a cohesive zone model, although there are quantitative differences. Prediction of the initiation of microbuckling is found to be in good agreement with experimental data for a carbon fibre epoxy composite.