eLastiCitY and pLastiCitY
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).