This is the third article in a series dealing with the mechanical behavior of a system of fibrils, capable of a phase change, embedded in a matrix. The fibrils are characterized by a “critical” stress fcwhich is required to initiate the formation of the secondary phase B from the primary phase A, and a lower “equilibrium” stress fenecessary to maintain the thermodynamic equilibrium between the two phases. The effects of possible slip at the interface between fibril and matrix are calculated. The results are shown to be also applicable to a system of continuous or discontinuous fracturing fibrils in a matrix.