Building upon previous work, the failure criterion for unidirectional fiber composite materials is examined using a sensitivity analysis as applied to its transverse, matrix controlled failure properties. A new and general relationship is found between these three properties thereby reducing the total number of independent properties needed to calibrate the theory to five. This completes and closes the development of failure criteria for unidirectional fiber composites by the polynomial invariants method. A broad but detailed evaluation of the resulting failure criteria is given. Future applications for these new failure criteria are discussed.