Young's Modulus as a Spring Constant

Recall that Hooke's Law defines a spring constant $k$ as the applied force $F$ divided by the spring displacement $x$, or $F = k x$ (see §J.1.3). An elastic solid can be viewed as a bundle of ideal springs. Consider, for example, an ideal bar (a rectangular solid in which one dimension, usually its longest, is designated its length $L$), and consider compression by $\Delta L$ along the length dimension. The length of each spring in the bundle is proportional to the length of the bar, so that each spring constant $k$ must be inversely proportional to $L$; in particular, each doubling of length $L$ doubles the length of each ``spring'' in the bundle, and therefore halves its stiffness. As a result, to obtain a counterpart to Hooke's law for solids, we must normalize displacement $\Delta L$ by length $L$ and use relative displacement $\Delta L/L$. We need displacement per unit length because we are working with a spring compliance per unit length.

The number of springs in parallel is proportional to the cross-sectional area $S$ of the bar. Therefore, the force applied to each spring is proportional to the total applied force $F$ divided by the cross-sectional area $S$. Thus, Hooke's law for each spring in the bundle can be written

$$\frac{F}{S} = Y \frac{\Delta L}{L}$$

where $Y$ is Young's modulus.

We may say that Young's modulus is the Hooke's-law spring constant for the spring made from a specifically cut section of the solid material, cut to length 1 and cross-sectional area 1. The shape of the cross-sectional area does not matter.