Rigidity is the required force to produce a unit incrementum of length.
In prismatic beams, the product of EA is known as axial rigidity.
[tex] \delta = \frac{PL}{EA} [/tex]
where [itex] \delta [/itex] is the change in length, P is the force applied at the centroid, L is the original length, E is the modulus of elasticity (assuming the material is at the elastic-linear region) and A is the cross sectional area. Of course this is for Homogenous materials.
In general the rigidity will be a measure of a structural member "opposing the change in length", with rigidity it's often used flexibility, which is inverse to the rigidity.
Maybe you are refering to the modulus of elasticity in shear stress, also know as modulus of rigidity.
According to Hooke's Law in shear (elastic-linear region)
[tex] \tau = G \gamma [/tex]
where [itex] \tau [/itex] is the shear stress, G is the modulus of rigidity or elasticity in shear and [itex] \gamma [/itex] is the angle of distorsion or the unit deformation.
The rigidity here is about measuring the structural element resistance to the "change of its shape".
For the proof you are doing, you'll need to look at the definition of shear strain and how shear strain is related to shear stress in multiple dimensions.
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