Differences between elastic moduli

Question

Is it true that in materials like aluminum and copper, the correct order of elastic moduli, in terms of value are:

Bulk < Young's < shear modulii

If this is correct, why is this the case?

Answer 1

The Poisson ratio gives you the relationship between these different moduli.

For example, bulk modulus $K$ is given by (see reference)

$$K = \frac{E}{3(1-2\sigma)} \tag1$$

and shear modulus

$$ G = \frac{E}{2(1+\sigma)} \tag2$$

If you see a certain ordering of the three, you can deduce the value of the Poisson ratio. So if you find $$K < E < G$$ it follows that

$$ \frac{E}{3(1-2\sigma)} < E < \frac{E}{2(1+\sigma)}\\ 3(1-2\sigma) > 1 > 2(1+\sigma)$$

And thus $\sigma < \frac12$ and $\sigma > -1$.

These are in fact limits on all materials - not just copper and aluminum. They are known as the limits required for "thermodynamic stability". Look at the work that might be done by an object that did not obey these laws - how would it deform when you apply a stress? It would turn out not to be stable; and there's your answer.

Just to clarify that point - $K, G$ and $E$ must all be positive (otherwise, applying a stress would cause strain in the opposite direction!). Taking equations (1) and (2) and combining them, we get

$$K = \frac{2(1+\sigma)}{3(1-2\sigma)}G$$

It is obvious that for $\sigma\geq \frac12$ and $\sigma\leq -1$ the sign of $K$ and $G$ would be opposite - which would mean that the material would not be stable either under shear, or bulk stress. That leaves us with the limits on $\sigma$ stated above, which leads to the ordering of the three moduli. It is a general result for all elastic materials.

Answer 2

No! The two fundamental moduli are the shear modulus $G$, which is the resistance of a material to change of shape under applied forces, and the bulk modulus $K$, which is its resistance to change of volume. Young's modulus $E$ is a hybrid of the two which is only popular because it's the easiest to measure experimentally.

In general: $$G=E/(2+2\nu)$$ For most metals, Poisson's ratio $\nu=0.3$ and so: $$G=E/2.6$$ Also in general: $$K=E/(3-6\nu)$$ and so: $$K=E/1.2$$ So: $$E>K>G$$ For steel, the most common structural metal, in $GPa$: $$E=200, K=160, G=80$$

See Wikipedia 'Elastic modulus' for lots more interrelationships between the various elastic constants.