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I'm confused what the relationship between each of the shear moduli are in an orthotropic material. From what I understand, there are 3 independant major shear moduli, G12, G13, and G23. They each have their corresponding minor shear moduli G21, G31, and G32. Are each major/minor pair equal to one another? Are they inversely proprtional in the same way Poisson's ratios are? Or, are they related to one another through other engineering constants?
What you have shown is the general case of pure shear stresses on a three-dimensional element. The relationship between the stresses are:
$\sigma_{12} = \sigma_{21}$, $\sigma_{13} = \sigma_{13}$, $\sigma_{32} = \sigma_{23}$
The proof is shown on the two-dimensional plane element below - when a shear stress is applied on the top edge/face, then there must be an equal but opposite shear stress on the bottom edge/face to maintain translational equilibrium. Now, the stresses pair forms a couple, which in turn must be countered by a pair of shear stresses on the two vertical edges/faces to maintain rotational equilibrium. Note that this proof holds true for the other two perpendicular planes too, which, in combination, validates the relationship of the stresses stated above.
