For simplicity, the following analysis considers a homogeneous cylinder—such as a thin copper wire—with length $\ell$ and cross-sectional area $A$. The results also apply to bodies of arbitrary shape, although the calculations are typically more complex.
When a tensile force acts on the side surfaces of the cylinder, it stretches by a length $\Delta \ell$. The volume generally remains nearly constant, meaning the cross-sectional area usually decreases. This effect, known as lateral contraction, can be clearly observed in muscles, for example.
Experimentally, it is observed that $\Delta \ell$ is proportional to the original length $\ell$—i.e., the longer the wire, the more it stretches. Furthermore, $\Delta \ell$ is proportional to the applied force and inversely proportional to the cross-sectional area of the wire. This relationship can be expressed as:
$$
\Delta \ell = \frac{1}{E} \frac{F}{A} \ell
$$
The quantity $E$ is called the Young's modulus (or elastic modulus). It is a material constant that can be looked up in reference tables.
The reason for expressing the formula with the reciprocal $1/E$ becomes clearer when rewriting the equation as:
$$
\frac{\Delta \ell}{\ell} E = \frac{F}{A}
$$
Replacing $\Delta \ell / \ell$ with the strain $\varepsilon$, and $F/A$ with the stress $\sigma$, we obtain:
$$
\sigma = E \varepsilon
$$
This form resembles Hooke’s law for spring extension and is therefore also referred to as the generalized Hooke’s law.
The connection between both laws becomes clear when the factor $EA/\ell$ is replaced by the spring constant $k$, and the length change $\Delta \ell$ is replaced by the spring extension $s$.