A composite consists of carbon fibers with a diameter of 7.0 µm and a tensile strength of 3.5 GPa in a resin whose fiber–matrix interfacial shear strength is 50 MPa. Which value is closest to the critical fiber length lc = σf·d/(2τc) needed to make full use of the fiber strength?

Answer: C. About 0.25 mm

The critical fiber length is the length needed for stress transferred by interfacial shear from the fiber ends to reach the tensile strength at the middle of the fiber. lc = σf·d/(2τc) = 3.5×10⁹ × 7.0×10⁻⁶ / (2 × 50×10⁶) = 2.45×10⁻⁴ m, about 0.25 mm. About 0.49 mm forgets the 2 in the denominator, and about 0.12 mm uses the radius instead of the diameter. Fibers much longer than lc give reinforcement close to that of continuous fibers, while fibers shorter than lc pull out of the resin before they break. This is why fiber shortening from breakage during molding is a concern in injection-molded short-fiber-reinforced plastics.