A new analysis for stress transfer from the fiber to the matrix through an imperfect interface was derived using a Bessel-Fourier series. The new analysis is specific to the fragmentation test. It satisfies equilibrium and compatibility everywhere and satisfies most boundary conditions exactly. The only approximation is the axial stress boundary condition in the fiber which is satisfied in an averaged sense rather than exactly. Two important advantages of the stress transfer analysis are that it can analyze anisotropic fibers and it can include imperfect interfaces or interphases. Theoretical predictions of stress transfer were compared to experimental Raman spectroscopy results. The predictions agree well with the experiments and they can be used to measure interface properties. The stress transfer analysis was coupled with an interfacial failure criterion to model the fragmentation test. The results of modeling fragmentation data as a function of applied strain suggested new ways of interpreting fragmentation experiments and cast doubt on the relevance of the commonly measured interfacial shear strength.