# Young's modulus and shear modulus relation - temperature effect

Young's modulus and shear modulus are related by $$E=2G(1+\nu)$$ (for isotropic and homogeneous materials), $$E$$ is Young's modulus, $$G$$ is shear modulus and $$\nu$$ is Poisson's ratio. I can do experiment to measure Young's modulus and shear modulus as a function of temperature (for structural steels). If I use the above relation, can I get Poisson's ratio at that temperature? In other words, does the above relation hold true at elevated temperature too?

• I dont see why it should change, but I'm not sure about it. Here's a link (bssa.org.uk/topics.php?article=139) with measurements of the three variables at different temperatures, maybe you can test the formula by yourself. – user190081 Oct 17 '18 at 23:36
• The equation is independent of temperature and applies to all linearly elastic isotropic materials. – Chet Miller Oct 18 '18 at 2:24

If I use the above relation, can I get Poisson's ratio at that temperature? In other words, does the above relation hold true at elevated temperature too?

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.