A linear, periodic, three-dimensional shear-lag model of unidirectionally-reinforced composites that allows for fibre breakage, and matrix failure is proposed. Matrix failure can take the form of matrix splitting or interfacial debonding. A computationally efficient scheme for its solution is developed. This scheme exploits the translation invariance of the elastostatic fields due to failed elements in the periodic cell, and is asymptotically faster than the classical eigensolution-based approach. The new computational scheme is used to illustrate the influence of matrix failure on the elastostatic fields induced by small clusters of fibre breaks in several test problems. Monte Carlo simulations of fracture in model three-dimensional composite specimen with Weibull-distributed fibre segment strengths are also performed. Matrix failure is found to considerably alter fracture development, to weaken the median specimen, and to reduce the variability in composite strength. © 2020, Springer Nature B.V.