E6 Micromechanics of Stress (Shear-Lag)

Topic

The transfer of load from the matrix to the fiber—mediated by the interfacial shear stress described by the model characterizing the matrix-reinforcement interface—does not occur uniformly along the entire fiber; instead, this non-uniform transfer is described by Cox's shear-lag model. Because the load transfer is non-uniform, both the stress within the fiber and the interfacial shear stress at the bond with the matrix vary from point to point along the fiber's length.

According to Cox's shear-lag model, the tensile stress within the fiber is nearly zero at the very ends and increases progressively toward the center as interfacial shear stress transfers load from the matrix into the fiber. As the fiber accumulates tensile stress moving away from its ends, the stress distribution along the fiber takes the form of a curve rising from the ends toward the center.

The rate at which this stress distribution rises from the fiber's ends toward its center is determined by the load transfer parameter: the higher this parameter, the faster the load transfers from the matrix to the fiber, and the shorter the distance required from the fiber end for the tensile stress to reach practically its maximum value. Conversely, a lower load transfer parameter results in slower load transfer, requiring a greater distance from the fiber end to reach that same maximum stress.

This stress distribution—rising from the fiber ends toward the center—is mirrored by the interfacial shear stress, which behaves in the exact opposite manner: it peaks at the fiber ends, where the internal fiber stress is still nearly zero, and decreases toward the center as the fiber absorbs the majority of the load it is capable of taking on. The fact that interfacial shear stress reaches its maximum precisely at the fiber ends means that interface failure is most likely to occur at those very points.

Consequently, this model—based on the load transfer via interfacial shear stress introduced earlier—explains the physical mechanism underlying another model in this set: the one describing critical fiber length. The distance required—starting from the fiber end—for the tensile stress to reach virtually its maximum value constitutes a characteristic distance; in essence, this is the same critical fiber length that the other model calculates based on interfacial shear strength.

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