10.6 Modulus of Elasticity, Modulus of Rigidity, and Bulk Modulus of Compressibility 283
The modulus of elasticity, or Young’s
modulus, is computed by calculating the
slope of a stress-strain diagram over the elas-
tic region. Themodulus of elasticityis a
measure of how easily a material will stretch
when pulled (subject to a tensile force) or
how well the material will shorten when
pushed (subject to a compressive force). The
larger the value of the modulus of elasticity,
the larger the force required to stretch or
shorten the material by a certain amount. In
order to better understand what the value of
the modulus of elasticity represents, consider
the following example: Given a piece of rub-
ber, a piece of aluminum, and a piece of steel, all having the same rectangular shape, cross-
sectional area, and original length as shown in Figure 10.29, which piece will stretch more
when subjected to the same forceF?
As you already know from your experience, it will require much less effort to elongate the
piece of rubber than it does to elongate the aluminum piece or the steel bar. In fact, the steel
bar will require the greatest force to be elongated by the same amount when compared to the
other samples. This is because steel has the highest modulus of elasticity value of the three
samples. The results of our observation — in terms of the example cited — are expressed, in a
general form that applies to all solid materials, by Hooke’s law. We discussed Hooke’s law ear-
lier when we were explaining springs. Hooke’s law also applies to stretching or compressing a
solid piece of material. However, for solid pieces of material, Hooke’s law is expressed in terms
of stress and strain according to(10.24)
where
snormal stress (N/m
2
or lb/in
2
)Emodulus of elasticity (or sometimes called Young’s modulus), a property of material
(N/m
2
or lb/in
2
)enormal strain, the ratio of change in length to original length (dimensionless)
As we explained earlier, the slope of a stress – strain diagram is used to compute the value of the
modulus of elasticity for a solid specimen. Equation (10.24) defines the equation of the line that
relates the stress and strain on the stress – strain diagram.
As an alternative approach, we will next explain a simple procedure that can be used to mea-
sureE. Although the following procedure is not the formal way to obtain the modulus of elas-
ticity value for a material, it is a simple procedure that provides additional insight into the
modulus of elasticity. Consider a given piece of rectangular-shaped material with an original
lengthL, and cross-sectional areaA, as shown in Figure 10.30. When subjected to a known
forceF, the bar will stretch; by measuring the amount the bar stretched (final length of the bar
minus the original length), we can determine the modulus of elasticity of the material in thesE e
F
Rubber
F
Aluminum
F
Steel
■Figure 10.29 When subjected to the same force,
which piece of material will stretch more?L
A
F■Figure 10.30
A rectangular bar subjected to a
tensile load.62080_10_ch10_p251-302.qxd 5/22/10 12:32 AM Page 283
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