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446 CHAPTER 14 Statics and Elasticity

parallelepiped to a rhomboidal parallelepiped (see Fig. 14.25a). During this deforma-
tion, the parallel layers of the body slide with respect to one another just as the pages of
a book slide with respect to one another when we push along its cover (see Fig. 14.25b).
If the force is applied from all sides simultaneously, by subjecting the body to the pres-
sure of a fluid in which the body is immersed, then the deformation is a compression of
the volume of the body, without any change of the geometrical shape (see Fig. 14.26).
In all of these cases, the fractional deformation, or the percent deforma-
Force pushes tion, is directly proportional to the applied force and inversely proportional to
tangentially along the area over which the force is distributed. For instance, if a given force pro-
one side.
duces an elongation of 1% when pulling on the end of a block, then the
(a)
A x same force pulling on the end of a block of, say, twice the cross-sectional
F 1
area will produce an elongation of %. This can be readily understood
2
h if we think of the block as consisting of parallel rows of atoms linked by
springs, which represent the interatomic forces that hold the atoms in
their places (see Fig. 14.27). When we pull on the end of the block with
a given force, we stretch the interatomic springs by some amount; and
when we pull on a block of twice the cross-sectional area, we have to
Other side stretch twice as many springs, and therefore the force acting on each
(b) of body is
held fixed. spring is only half as large and produces only half the elongation in each
Deformation F spring. Furthermore, since the force applied to the end of a row of atoms
is a shear.
is communicated to all the interatomic springs in that row, a given force
produces a given elongation in each spring in a row. The net elongation
of the block is the sum of the elongations of all the interatomic springs
in the row, and hence the fractional elongation of the block is the same
as the fractional elongation of each spring, regardless of the overall length
of the block. For instance, if a block elongates by 0.1 mm when subjected
to a given force, then a block of, say, twice the length will elongate by
0.2 mm when subjected to the same force.
FIGURE 14.25 (a) Tangential force To express the relationships among elongation, force, and area mathematically,
applied to the side of a block of material consider a block of initial length L and cross-sectional area A. If a force F pulls on the
causes shear. (b) When such a tangential end of this block, the elongation is L, and the fractional elongation is L L. This
force is applied to the cover of a book, the
fractional elongation is directly proportional to the force and inversely proportional
pages slide past one another.
to the area A:
¢L 1 F
elongation and Young’s modulus (14.18)
L Y A

Here the quantity Y is the constant of proportionality. In Eq. (14.18) this con-
stant written as 1/Y, so it divides the right side, instead of multiplying it (this is anal-
ogous to writing Hooke’s Law for a spring as x (l/k) F, where x is the elongation
Deformation is
a compression.

A

F F

An equal force per
unit area is applied FIGURE 14.27 Microscopically, If force pulling on end is
to each side. a block of solid material may be distributed over larger area,
thought of as rows of atoms linked more springs need to be
stretched, and smaller
FIGURE 14.26 Pressure applied to all sides by springs. The springs stretch when deformation will result.
of a block of material causes compression. a tension is applied to the block.