Bulk stress and strain
When a scuba diver plunges deep into the ocean, the water exerts nearly uniform pressure everywhere on his surface and squeezes him to a slightly smaller volume (see Fig. 11.12b). This is a different situation from the tensile and compressive stresses and strains we have discussed. The uniform pressure on all sides of the diver is a bulk stress (or volume stress), and the resulting deformation—a bulk strain (or volume strain)—is a change in his volume.
If an object is immersed in a fluid (liquid or gas) at rest, the fluid exerts a force on any part of the object’s surface; this force is perpendicular to the surface. (If we tried to make the fluid exert a force parallel to the surface, the fluid would slip sideways to counteract the effort.) The force F per unit area that the fluid exerts on an immersed object is called the pressure p in the fluid:
Pressure has the same units as stress; commonly used units include 1 Pa 1= ( N/m2 ), lb/in.2 (1 psi), and 1 atmosphere 11 atm2. One atmosphere is the approximate average pressure of the earth’s atmosphere at sea level:
1 atmosphere = 1 atm = 1.013 * 105 Pa = 14.7 lb/in.2
Caution pressure vs. force Unlike force, pressure has no intrinsic direction: The pressure on the surface of an immersed object is the same no matter how the surface is oriented. Hence pressure is a scalar quantity, not a vector quantity.
The pressure in a fluid increases with depth. For example, the pressure in the ocean increases by about 1 atm every 10 m. If an immersed object is relatively small, however, we can ignore these pressure differences for purposes of calculating bulk stress. We’ll then treat the pressure as having the same value at all points on an immersed object’s surface.
Pressure plays the role of stress in a volume deformation. The corresponding strain is the fractional change in volume (Fig. 11.17)—that is, the ratio of the volume change ?V to the original volume V0:
Volume strain is the change in volume per unit volume. Like tensile or compressive
strain, it is a pure number, without units.
When Hooke’s law is obeyed, an increase in pressure (bulk stress) produces
a proportional bulk strain (fractional change in volume). The corresponding
elastic modulus (ratio of stress to strain) is called the bulk modulus, denoted
by B. When the pressure on a body changes by a small amount ?p, from
p0 to p0 + ?p, and the resulting bulk strain is ?V>V0, Hooke’s law takes
We include a minus sign in this equation because an increase of pressure always
causes a decrease in volume. In other words, if ?p is positive, ?V is negative.
The bulk modulus B itself is a positive quantity.
For small pressure changes in a solid or a liquid, we consider B to be constant.
The bulk modulus of a gas, however, depends on the initial pressure p0. Table 11.1
includes values of B for several solid materials. Its units, force per unit area, are
the same as those of pressure (and of tensile or compressive stress).
The reciprocal of the bulk modulus is called the compressibility and is denoted
by k. From Eq. (11.13),
Compressibility is the fractional decrease in volume, -?V>V0, per unit increase
?p in pressure. The units of compressibility are those of reciprocal pressure,
Table 11.2 lists the values of compressibility k for several liquids. For example,
the compressibility of water is 46.4 * 10-6
, which means that the volume
of water decreases by 46.4 parts per million for each 1-atmosphere increase
in pressure. Materials with small bulk modulus B and large compressibility k are
easiest to compress
try it with an example: