Elasticity

$$E=\frac{\sigma }{\epsilon }$$

Note

Elasticity is not always isotropic. Because of directional slip planes, the elastic modulus $E$ can vary depending on the alignment with the lattice structure.

Some values for $E$ for metals at different lattice orientations (Values in $GPa$):

Metal	$E_{100}$	$E_{110}$	$E_{111}$
$Al$	36.7	72.6	76.1
$Cu$	66.7	130.3	191.1
$Fe$	125.0	210.5	272.7
$W$	384.6	384.6	384.6

Relation to Condon-Morse Model

The Condon-Morse Model provides us with en explanation as to why certain alloys have a value for $E$.

$$E=\frac{1}{r_m}{U}^{\prime \prime }(r_m)=\frac{1}{r_m}{F}^{\prime }(r_m)$$

Since repulsive forces increase as $T$ increases, the average stable atomic distance $r_m$ also increases, therefore decreasing $E$. (Note that $U^{\prime \prime }(r_m)=\frac{d^{2}U}{d{r}^2}$ at point $r=r_m$).