Analyzing a shear-lag model, the evolution of the fragment size distribution in the sequential cracking of polymer coatings under uniaxial loading is investigated. This study elucidates the role of a nonlinear elastic stress transfer mechanism at the interface on the fragmentation kinetics. Using a nonlinear expression for the shear stress at the interface, analytical expressions for the stress and the strain in the coating are derived. In the initial stage of cracking, the strain in a fragment equals the substrate's strain everywhere except in the exclusion zone at the fragments' edges. In the later stages of fragmentation, the stress and the strain in a fragment attain a universal scaling form with pronounced maxima in the centers of the fragments. Assuming a three parameter Weibull distribution for the statistical distribution of the coating's strength, analytical expressions for the fragment size distribution in the initial stage and numerical results for the fragment size distribution in the later stages of the cracking process are derived.