The off-axis Young's modulus for a particular unidirectional fiber-reinforced orthotropic composite lamina is given by

\[ E_{x}=E_{1}-\left(E_{1}-E_{2}\right) \sin \theta \]

where \(\theta\) is the lamina orientation in radians, \(E_{1}\) is the longitudinal Young's modulus of the lamina, and \(E_{2}\) is the transverse Young's modulus of the lamina.

(a) For the material described above, find the equation for the Young's modulus of the composite if the fibers are randomly oriented with respect to \(\theta\). Express your answer in terms of \(E_{1}\) and \(E_{2}\). Derive your own equation and do not use the Tsai-Pagano equation (Equation 6.44).

(b) For the equation derived in part (a), what would be the appropriate micromechanics equations to use in the derived equation if the fibers are continuous?

(c) for the equation derived in part (a), what would be the appropriate micromechanics equations to use in the derived equation if the fibers are discontinuous?

\(\tilde{E}=\frac{3}{8} E_1+\frac{5}{8} E_2, \quad \tilde{G}=\frac{1}{8} E_1+\frac{1}{4} E_2 \tag{6.44}\)