$A \approx \frac{W_0*L_0}{L}*\frac{T_0*L_0}{L}=\frac{W_0*T_0*L_0^2}{L^2}$
If my interpretation is correct, you are assuming that $W*L=W_0*L_0$ and $T*L=T_0*L_0$
That would make the volume: $W_0*T_0*L_0^2/L$, which decreases when you stretch the material. For small strain, the Poisson ratio would approach 1. Poisson ratio should be between -1 and 0.5 for a stable, isotropic, linear elastic material. I don't know what the material is you're testing, but..
$A \approx \frac{A_0*L_0}{L}$ seems more suitable imo.
The modulus of elasticity (Young's Modulus), $E$, and the shear modulus, $G$, are related by the equation:
$$G=\frac{E}{2(1+ν)}$$
Where $ν$ is Poisson's ration = -(lateral strain)/longitudinal strain).
As you can see, the two are proportional to one another. I personally never heard of the shear modulus being called modulus of rigidity and I agree with you it doesn't seem to make sense to call one "rigidity" and the other "elasticity" when the are linearly related. You'd think they would both be called rigidity or elasticity, but not the opposite.
Young's modulus involves longitudinal stress/strain (tension/compression). The shear modulus involves transverse or lateral stress/strain (shear), so it is logical they are related to each other by Poisson's ratio (ratio of lateral to longitudinal strain). You can also see this because when you longitudinally compress or stretch something it laterally expands and contracts, respectively, as well.
My question was, 'Why is it called modulus of rigidity?'. The above
answer does not answer that question
The only reason I can think of is to avoid confusion in the use of terms. If both the shear modulus and Young's modulus were referred to as "modulus of elasticity", or, for that matter, "modulus of rigidity" how would we know which modulus was being referred to? What I was trying to say is there should be no technical reason for the difference in terms for $G$ and $E$, since they both refer to resistance to deformation (lateral and longitudinal).
Hope this helps.
Best Answer
Yes, this relation is true for isotropic materials and the relation is independent of temperature, so it is valid at any constant temperature. This is why the experiments find values of $E$ and $G$ at various constant temperatures.
But, it's a relevant question, as argued here since the error induced in experiments that assume Poisson's ratio is constant while the temperature varies is non-negligible in most cases. You can see that in figure 8, for these composites, varying $\nu$ has virtually no effect on the Stress but does greatly effect the width strain and thus effecting the complex modulus.
There have been numerous experiments for various materials. See figures 8 and 9 here, figures 9 and 10 here and the figures here.
Depending on the material in question, the Poisson ratio can either increase or decrease with increasing temperature, however either way it is monotonic.
For more interesting materials, some work has been done to find generalized equations for $E$ and $G$ and then you can use your relation to find the Poisson ratio, i.e. composite materials.
And here's a nice nature article about modern materials and Poisson's ratio.
If you come across a paywall, email me and I can send you a pdf.