The length needed for the stress at the middle of a short fiber to reach the fiber's tensile strength through interfacial shear. It is given by lc = σf·d/(2τc); fibers shorter than this pull out.
Japanese name: 臨界繊維長
Critical fiber length appears in 2 practice questions across 2 topics, including Polymers and composites, Composite materials.
Field: Composite materials
Load is transferred to a short fiber in a matrix mainly through shear at the interface along the fiber's sides. At the fiber ends the stress is nearly zero; moving away from an end, load enters gradually through the interfacial shear stress τc and the fiber stress increases. The length at which load entering from both ends brings the stress at the fiber's middle exactly to its tensile strength σf is the critical fiber length lc, and force balance gives lc = σf·d/(2τc) (d is the fiber diameter). For carbon fiber in epoxy, lc is a few tenths of a millimeter to about 1 mm.
If a fiber is shorter than lc, its stress never reaches the tensile strength, and the fiber pulls out of the matrix before it breaks, so its strength cannot be fully exploited. If the fiber is much longer than lc, the low-stress end portions make up a small fraction, and reinforcement close to that of continuous fibers is obtained. As a rough guide, fibers more than about fifteen times lc are regarded as close to continuous. The stronger the interfacial adhesion, the larger τc and the shorter lc, and the thinner the fiber, the shorter lc.
In injection-molded short-fiber-reinforced plastics, fibers break and shorten during compounding and molding, and when the average fiber length approaches lc, the reinforcing effect drops sharply. Long-fiber pellets that keep fibers long and adjustments to molding conditions are therefore used. Pull-out also absorbs energy during fracture, so a certain amount of pull-out can help toughness. In ceramic matrix composites, the interface is made weak to deliberately induce pull-out, preventing sudden fracture and raising toughness.
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.
Yes. lc is inversely proportional to the interfacial shear strength τc, so strengthening adhesion with a coupling agent or similar shortens it, letting even short fibers use their strength.
lc is proportional to the fiber diameter d, so thinner fibers have shorter critical lengths. For fibers of the same length, thinner ones have a larger ratio of length to lc, improving reinforcing efficiency.
Even with continuous fibers, when a single fiber breaks somewhere along its length, a length of about lc/2 on each side of the break, about lc in total, is needed for the stress to recover. This length also bears on how a fiber break affects neighboring fibers and on how strength scatter is treated.
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.
A composite consists of carbon fibers with a diameter of 7.…The length needed for this maximum to reach the tensile strength of the fiber is the critical fiber length lc, and the strength is reached at a distance lc/2 from the end.
When a short fiber embedded in a matrix is loaded in tensio…Ceramic matrix composite CFRP (carbon-fiber-reinforced plastic) Delamination GFRP (glass-fiber-reinforced plastic) Metal matrix composite Quasi-isotropic laminate Silane coupling agent