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Properties of Elastic Solids

Young's Modulus

Young's modulus can be thought of as the spring constant for solids. Consider an ideal rod (or bar) of length $ L$ and cross-sectional area $ S$. Suppose we apply a force $ F$ to the face of area $ S$, causing a displacement $ \Delta L$ along the axis of the rod. Then Young's modulus $ Y$ is given by

$\displaystyle Y \isdefs \frac{\mbox{Stress}}{\mbox{Strain}} \isdefs \frac{F/S}{\Delta L/L}

F &=& \mbox{total applied force}\\
S &=& \mbox{area over whic...
...a L/L &=& \mbox{\emph{strain} = displacement per unit length}\\
For wood, Young's modulus $ Y$ is on the order of $ 10$ N/m$ \null^2$. For aluminum, it is around $ 70$ (a bit higher than glass which is near $ 65$), and structural steel has $ Y\approx 200$ [180].

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.

String Tension

The tension of a vibrating string is the force $ F$ used to stretch it. It is therefore directed along the axis of the string. A force $ F$ must be applied at the endpoint on the right, and a force $ -F$ is applied at the endpoint on the left. Each point interior to the string is pulled equally to the left and right, i.e., the net force on an interior point is $ F + (-F) = 0$. (A nonzero force on a massless point would produce an infinite acceleration.) If the cross-sectional area of the string is $ S$, then the tension is given by the stress on the string times $ S$.
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Wave Equation for the Vibrating String
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Rigid-Body Dynamics