Derive the "slab model" equation to predict the shear modulus $G_{12}$ considering the figure below and the shear displacements $$ for both the fiber and the matrix and the thickness of the "slabs" of the fiber and matrix, $t_f$ and $t_m.$ The assumption is that there is constant shear stress $$ in both the fiber and matrix.

${}_c={}_f+{}_m$

Shear stress acting on a fiber composite

Shear displacements under stress $($simplified$)$