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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. Can we be more precise than that?

 

To address these questions, let’s look at a graph of tensile stress as a function
of tensile strain. Figure 11.19 shows a typical graph of this kind for a metal such
as copper or soft iron. The strain is shown as the percent elongation; the horizontal
scale is not uniform beyond the first portion of the curve, up to a strain of less
than 1%. The first portion is a straight line, indicating Hooke’s law behavior with
stress directly proportional to strain. This straight-line portion ends at point a;
the stress at this point is called the proportional limit.

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From a to b, stress and strain are no longer proportional, and Hooke’s law is
not obeyed. However, from a to b (and O to a), the behavior of the material is
elastic: If the load is gradually removed starting at any point between O and b,
the curve is retraced until the material returns to its original length. This elastic
deformation is reversible.

 

Point b, the end of the elastic region, is called the yield point; the stress at the
yield point is called the elastic limit. When we increase the stress beyond point b,
the strain continues to increase. But if we remove the load at a point like c
beyond the elastic limit, the material does not return to its original length. Instead,
it follows the red line in Fig. 11.19. The material has deformed irreversibly and
acquired a permanent set. This is the plastic behavior mentioned in Section 11.4.

 

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Once the material has become plastic, a small additional stress produces a
relatively large increase in strain, until a point d is reached at which fracture
takes place. That’s what happens if a steel guitar string in Fig. 11.12a is tightened
too much: The string breaks at the fracture point. Steel is brittle because
it breaks soon after reaching its elastic limit; other materials, such as soft iron,
are ductile—they can be given a large permanent stretch without breaking. (The
material depicted in Fig. 11.19 is ductile, since it can stretch by more than 30%
before breaking.)

 

Unlike uniform materials such as metals, stretchable biological materials such
as tendons and ligaments have no true plastic region. That’s because these materials
are made of a collection of microscopic fibers; when stressed beyond the
elastic limit, the fibers tear apart from each other. (A torn ligament or tendon is
one that has fractured in this way.)

 

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If a material is still within its elastic region, something very curious can happen
when it is stretched and then allowed to relax. Figure 11.20 is a stress-strain
curve for vulcanized rubber that has been stretched by more than seven times
its original length. The stress is not proportional to the strain, but the behavior
is elastic because when the load is removed, the material returns to its original
length. However, the material follows different curves for increasing and decreasing
stress. This is called elastic hysteresis. The work done by the material
when it returns to its original shape is less than the work required to deform it;
that’s due to internal friction. Rubber with large elastic hysteresis is very useful
for absorbing vibrations, such as in engine mounts and shock-absorber bushings
for cars. Tendons display similar behavior.

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The stress required to cause actual fracture of a material is called the breaking
stress, the ultimate strength, or (for tensile stress) the tensile strength. Two
materials, such as two types of steel, may have very similar elastic constants but
vastly different breaking stresses. Table 11.3 gives typical values of breaking
stress for several materials in tension. Comparing Tables 11.1 and 11.3 shows that
iron and steel are comparably stiff (they have almost the same value of Young’s
modulus), but steel is stronger (it has a larger breaking stress than does iron).

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The stress required to cause actual fracture of a material is called the breaking
stress, the ultimate strength, or (for tensile stress) the tensile strength. Two
materials, such as two types of steel, may have very similar elastic constants but
vastly different breaking stresses. Table 11.3 gives typical values of breaking
stress for several materials in tension. Comparing Tables 11.1 and 11.3 shows that
iron and steel are comparably stiff (they have almost the same value of Young’s
modulus), but steel is stronger (it has a larger breaking stress than does iron).

 



Frequently Asked Questions

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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: 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 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: 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: summary of equilibrium and elasticity 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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Ans: According to the continuity equation, the speed of fluid flow can vary along the paths of the fluid. The pressure can also vary; it depends on height as in the static situation (see Section 12.2), and it also depends on the speed of flow. We can derive an important relationship called Bernoulli’s equation, view more..
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Ans: To derive Bernoulli’s equation, we apply the work–energy theorem to the fluid in a section of a flow tube. In Fig. 12.23 we consider the element of fluid that at some initial time lies between the two cross sections a and c. The speeds at the lower and upper ends are v1 and v2. In a small time interval dt, the fluid that is initially at a moves to b, a distance ds1 = v1 dt, and the fluid that is initially at c moves to d, a distance ds2 = v2 dt. The cross-sectional areas at the two ends are A1 and A2, as shown. The fluid is incompressible; hence by the continuity equation, Eq. (12.10), the volume of fluid dV passing any cross section during time dt is the same. That is, dV = A1 ds1 = A2 ds2. view more..




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