Question

In a Fibre-reinforced composite, the "Critical Fibre Length" ($l_c$) is defined as the length required for

• A. the fibre to break before the matrix fails
• B. the maximum tensile stress to be transferred from the matrix to the centre of the fibre
• C. the composite to become transparent
• D. the matrix to undergo recrystallization

Hint:

For fibers to reinforce a matrix effectively, their length ($l$) must be much greater than the critical length ($l_c$).
Fibers with $l > 15 \ l_c$ are generally considered continuous, while $l < l_c$ fibers act primarily as particulate fillers.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the definition of "Critical Fibre Length" ($l_c$), which is a fundamental concept in the mechanics of fiber-reinforced polymer (FRP) composites governing stress transfer and reinforcement efficiency.

Step 2: Key Formula or Approach:

• In a short-fiber composite, stress is transferred from the matrix to the fiber through shear stresses acting along the fiber-matrix interface.

• The critical fiber length ($l_c$) is given by the formula: \[ l_c = \frac{\sigma_f^* d}{2\tau_c} \] where:
$\sigma_f^*$ is the ultimate tensile strength of the fiber,
$d$ is the fiber diameter,
$\tau_c$ is the fiber-matrix interfacial shear strength (or matrix yield strength in shear).

Step 3: Detailed Explanation:

• When a composite is subjected to an external tensile load, the deformation of the matrix shears the fiber surface, transferring stress into the fiber.

• The tensile stress in the fiber is zero at its ends and reaches a maximum value at its longitudinal center.

• If the fiber is too short ($l < l_c$), the maximum stress built up at the center of the fiber is less than its ultimate tensile strength ($\sigma_f^*$), and the fiber will pull out of the matrix rather than fracture.

• If the fiber length is exactly equal to the critical length ($l = l_c$), the maximum tensile stress transferred from the matrix reaches the fiber's ultimate tensile strength at its exact midpoint, allowing the fiber to achieve its maximum reinforcement potential.

• This explains why option (B) is the correct definition.

Step 4: Final Answer:
The critical fiber length is the length required for the maximum tensile stress to be transferred from the matrix to the center of the fiber.