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:

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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:

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Bulk stress and strain

 

 

Bulk stress and strain

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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
the form

Bulk stress and strain

 

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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),

Bulk stress and strain

 

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Compressibility is the fractional decrease in volume, -?V>V0, per unit increase
?p in pressure. The units of compressibility are those of reciprocal pressure,
Pa-1
 or atm-1
Bulk stress and strain.

 

 

 

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Table 11.2 lists the values of compressibility k for several liquids. For example,
the compressibility of water is 46.4 * 10-6
 atm-1
, 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

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Bulk stress and strain



Frequently Asked Questions

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Ans: The simplest elastic behavior to understand is the stretching of a bar, rod, or wire when its ends are pulled (Fig. 11.12a). Figure 11.14 shows an object that initially has uniform cross-sectional area A and length l0. We then apply forces of equal magnitude F# but opposite directions at the ends (this ensures that the object has no tendency to move left or right). We say that the object is in tension. view more..
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Ans: The rigid body is a useful idealized model, but the stretching, squeezing, and twisting of real bodies when forces are applied are often too important to ignore. view more..
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Ans: Here are some solved examples to help your concepts to be more clear. view more..
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Ans: 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. This is a different situation from the tensile and compressive stresses and strains we have discussed. view more..
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Ans: The third kind of stress-strain situation is called shear. The ribbon in Fig. 11.12c is under shear stress: One part of the ribbon is being pushed up while an adjacent part is being pushed down, producing a deformation of the ribbon. view more..
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Ans: Hooke’s law—the proportionality of stress and strain in elastic deformations— has a limited range of validity. In the preceding section we used phrases such as “if the forces are small enough that Hooke’s law is obeyed.” Just what are the limitations of Hooke’s law? What’s more, if you pull, squeeze, or twist anything hard enough, it will bend or break view more..
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Ans: Fluids play a vital role in many aspects of everyday life. We drink them, breathe them, swim in them. They circulate through our bodies and control our weather. The physics of fluids is therefore crucial to our understanding of both nature and technology view more..
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Ans: A fluid is any substance that can flow and change the shape of the volume that it occupies. (By contrast, a solid tends to maintain its shape.) We use the term “fluid” for both gases and liquids. The key difference between them is that a liquid has cohesion, while a gas does not. The molecules in a liquid are close to one another, so they can exert attractive forces on each other and thus tend to stay together (that is, to cohere). That’s why a quantity of liquid maintains the same volume as it flows: If you pour 500 mL of water into a pan, the water will still occupy a volume of 500 mL. The molecules of a gas, by contrast, are separated on average by distances far larger than the size of a molecule. Hence the forces between molecules are weak, there is little or no cohesion, and a gas can easily change in volume. If you open the valve on a tank of compressed oxygen that has a volume of 500 mL, the oxygen will expand to a far greater volume. view more..
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Ans: A fluid exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid. This is the force that you feel pressing on your legs when you dangle them in a swimming pool. Even when a fluid as a whole is at rest, the molecules that make up the fluid are in motion; the force exerted by the fluid is due to molecules colliding with their surroundings view more..
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Ans: If the weight of the fluid can be ignored, the pressure in a fluid is the same throughout its volume. We used that approximation in our discussion of bulk stress and strain in Section 11.4. But often the fluid’s weight is not negligible, and pressure variations are important. Atmospheric pressure is less at high altitude than at sea level, which is why airliner cabins have to be pressurized. When you dive into deep water, you can feel the increased pressure on your ears. view more..
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Ans: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. view more..
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Ans: If the pressure inside a car tire is equal to atmospheric pressure, the tire is flat. The pressure has to be greater than atmospheric to support the car, so the significant quantity is the difference between the inside and outside pressures. When we say that the pressure in a car tire is “32 pounds” (actually 32 lb>in.2 , equal to 220 kPa or 2.2 * 105 Pa), we mean that it is greater than atmospheric pressure (14.7 lb>in.2 or 1.01 * 105 Pa) by this amount. view more..
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Ans: The simplest pressure gauge is the open-tube manometer . The U-shaped tube contains a liquid of density r, often mercury or water. The left end of the tube is connected to the container where the pressure p is to be measured, and the right end is open to the atmosphere view more..
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Ans: A body immersed in water seems to weigh less than when it is in air. When the body is less dense than the fluid, it floats. The human body usually floats in water, and a helium-filled balloon floats in air. These are examples of buoyancy, a phenomenon described by Archimedes’s principle: view more..
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Ans: We’ve seen that if an object is less dense than water, it will float partially submerged. But a paper clip can rest atop a water surface even though its density is several times that of water. This is an example of surface tension: view more..
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Ans: We are now ready to consider motion of a fluid. Fluid flow can be extremely complex, as shown by the currents in river rapids or the swirling flames of a campfire. But we can represent some situations by relatively simple idealized models. An ideal fluid is a fluid that is incompressible (that is, its density cannot change) and has no internal friction (called viscosity). view more..
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Ans: The mass of a moving fluid doesn’t change as it flows. This leads to an important relationship called the continuity equation view more..




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