Static and dynamic analysis of the fracture tests of fiber composites in hydraulically servo-controlled testing machines currently in use shows that their grips are much too soft and light for observing the postpeak softening. Based on static analysis based on the second law of thermodynamics, confirmed by dynamic analysis of the test setup as an open system, far stiffer and heavier grips are proposed. Tests of compact-tension fracture specimens of woven carbon-epoxy laminates prove this theoretical conclusion. Sufficiently, stiff grips allow observation of a stable postpeak softening, even under load-point displacement control. Dynamic analysis of the test setup as a closed system with proportional-integrative-differential (PID)-controlled input further indicates that the controllability of postpeak softening under crack-mouth opening displacement (CMOD) control is improved not only by increasing the grip stiffness but also by increasing the grip mass. The fracture energy deduced from the area under the measured complete load-deflection curve with stable postpeak is shown to agree with the fracture energy deduced from the size effect tests of the same composite, but the size effect tests also provide the material characteristic length of quasibrittle (or cohesive) fracture mechanics. Previous suspicions of dynamic snapback in the testing of stiff specimens of composites are dispelled. Finally, the results show the stress- or strain-based failure criteria for fiber composites to be incorrect, and fracture mechanics, of the quasibrittle type, to be perfectly applicable.