The interfacial shear stresses, \(\tau\), and the fiber normal stresses, \(\sigma_{f}\) acting on a differential element at a distance \(x\) from the end of the fiber are shown in Figure 6.4, where the fiber is assumed to have a circular cross section of diameter \(d\). In some cases, fibers may have cross sections that are more elliptical than circular (see Figure 3.2). For a fiber that has an elliptical cross section of major diameter, \(a\), and minor diameter, \(\mathrm{b}\), find the differential equation governing the variation of the stresses \(\tau\) and \(\sigma_{f}\) along the length of the fiber.

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L O o O o Of x dx FIGURE 6.4 Stresses acting on a differential element of fiber. Of + dof