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 and
cross-sectional area
. Suppose we apply a force
to the face of
area
, causing a displacement
along the axis of the rod.
Then Young's modulus
is given by


For wood, Young's modulus is on the order of
N/m
.
For aluminum, it is around
(a bit higher than glass which is near
), and structural steel has
[180].
Young's Modulus as a Spring Constant
Recall (§B.1.3) that Hooke's Law defines a spring
constant as the applied force
divided by the spring
displacement
, or
. 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
), and consider compression by
along the length dimension. The length of each spring in
the bundle is the length of the bar, so that each spring constant
must be inversely proportional to
; in particular, each doubling of
length
doubles the length of each ``spring'' in the bundle, and
therefore halves its stiffness. As a result, it is useful to
normalize displacement
by length
and use relative
displacement
. 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 of the bar. Therefore, the force applied to
each spring is proportional to the total applied force
divided by
the cross-sectional area
. Thus, Hooke's law for each spring in the
bundle can be written


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 used
to stretch it. It is therefore directed along the axis of the string.
A force
must be applied at the endpoint on the right, and a force
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
. (A nonzero
force on a massless point would produce an infinite acceleration.)
If the cross-sectional area of the string is , then the tension is
given by the stress on the string times
.
Next Section:
Wave Equation for the Vibrating String
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Rigid-Body Dynamics