A theoretical stress analysis has been developed for the fibre fragmentation test in the presence of matrix cracks at sites of fibre breaks. The strain energy release rates for both matrix cracking and interface debonding are calculated for a carbon fibre/epoxy matrix composite. By comparing these strain energy release rates with the corresponding specific fracture resistances, the competition between matrix crack growth and interface debonding has been studied. The distributions of fibre axial stress and interfacial shear stress obtained from the present analysis show that the matrix crack substantially reduces the efficiency of stress transfer from the matrix to the fibre.