A unidirectional composite is to be modeled by the RVE shown in Figure 3.14a, where the fiber and matrix materials are assumed to be isotropic and perfectly bonded together. Using a mechanics of materials approach, derive the micromechanics equations for the effective in-plane shear modulus, \(G_{12}\), for the following cases:

a. The ends of the RVE are perfectly bonded to supports that are rigid against rotation, then subjected to uniform in-plane shear stress, \(\tau_{12}\), by the nonrotating supports, as shown in Figure 3.14b.

b. The top and bottom surfaces of the RVE are subjected to uniform inplane shear stress, \(\tau_{12}\), and the ends of the RVE are free to rotate, as shown in Figure 3.14c.

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(a) 2 Fiber FIGURE 3.14 Matrix 512 Fiber Matrix (c) Fiber Matrix T12 T12 T12