Only a portion of the elongation in wire $\left(1\right)$ is due to the force in the wire. Part of the elongation results from the temperature increase.

Since

${\mathrm{\sigma }}_1=\frac{{\mathrm{F}}_1}{{\mathrm{A}}_1}$

this relationship can be rewritten as:

${\mathrm{\delta }}_1={\mathrm{\sigma }}_1\frac{{\mathrm{L}}_1}{{\mathrm{E}}_1}+{\mathrm{\alpha }}_1∆{\mathrm{TL}}_1$