Young's Modulus as a Spring Constant

Recall (§B.1.3) that Hooke's Law defines a spring constant $ k$ as the applied force $ F$ divided by the spring displacement $ x$, or $ F = k x$. 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 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, it is useful to normalize displacement $ \Delta L$ by length $ L$ and use relative displacement $ \Delta L/L$. We need displacement per unit length because we have a constant 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

$\displaystyle \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 since all displacement is assumed to be longitudinal in this model.


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