Stress and Strain Defined:
Stress is the force per unit area; strain is the change in a dimensional quantity divided by the original value of that quantity. Serway does not use any symbols for these; standard engineering symbols are the Greek lower case letters sigma (or tau) and epsilon:
Units of stress are N/m2 (or Pascals Pa); there are none for strain since it is a ratio.
Applications:
When forces are applied to a steel cables, such as those in suspension bridges, the cables stretch under the load. The stress is the tension (force) in the cable divided by the cross sectional area. These are typically very large (order of millions of Pascals or MPa). Stress of this kind, where the force is perpendicular to the area, is called normal (in science & engineering, normal ==> perpendicular) stress. The Greek lower case letter sigma is used to represent normal stress. Serway shows a normal stress being applied to a bar in Figure 12.14 on page 346. The strain in this case is the change in the length divided by the original length. Normal stresses can be tensile (tending to stretch the material) or compressive (tending to make the material shorter). A steel cable cannot be placed in compressive stress. But a rod can support either tensile or compressive stress.
Forces can also be applied to solid materials parallel to a cross sectional area. For example, in Figure 12.14, if the force were being applied vertically downward, the bar might break (shear) across its width. Of course, it might simply bend. But when this test is done on a sample, theforce is very large and is applied suddenly, so that the bar does shear across the cross sectional area A. The stress is still force F divided by area A, but the force is parallel to the cross sectional area. The symbol usually used is the lower case Greek letter tau (don't get it confused with the vector tau, which is used for torque). (Although we shall treat stress as if it were a scalar, it's actually another one of those nine-component things called tensors.) It's a little more difficult to understand strain for this case than when normal stresses are applied. Look at Figure 12.16 on page 347. You can see that the force is parallel to the cross sectional area (Figure 12.16a), causing a "delta-x" to develop. Turn this figure 90o, and you have a model of the bar with a force applied parallel to A, with the bar distorting an amount "delta-x".
"Fluid" is a generic term used to refer to liquids or gases. A fluid cannot support a shearing stress. But normal stresses can be applied if the fluid is confined, as in the cylinder of an automobile engine. Such volumetric stresses are usually called pressures. When a gas is compressed, for example, its volume decreases (and delta-V is negative); the pressure causes a volumetric strain. Pressure can be applied to a solid as well as a fluid. See Figure 12.17 on page 348. A submarine is subjected to immense pressure at 7000 ft depth! It is commonly stated in elementary science courses that liquids (and solids) are incompressible, but this is not true. Sea water at a mile of depth is significantly more dense than the same water at sea level (zero elevation). The densities of solids also change under pressure although not as much as liquids.
Example 1: Stress for various cases
Young's Modulus:
Young's Modulus (engineers call this simply "the elastic modulus") is the normal stress divided by the normal strain:
I do not understand why Serway defines this modulus as "tensile stress/tensile strain". "Normal" is more general; it can mean "tensile" or "compressive". While cables can support only tensile stresses, concrete columns (for example) can support compressive loads. Perhaps Young's work was originally with cables? Anyway, you should use my definition above.
Wires and concrete columns (for example) are similar to springs; when the stress is removed, they return to their original lengths. At least, they do if the stresses are not too large. Just like a spring, we could plot force vs change in length and we'd get a straight line. Here's another way to say this: force and displacement are proportional within the elastic region. But this isn't very useful because there are many cables and columns of many different diameters. For this reason, we divide the force by the cross sectional area to get a quantity (stress) which can be used to characterize the effects on all cables having any diameters. Likewise, the amount that the cable stretches depends not only on the load, but also on the length. Thus, we calculate the stretch per unit length (strain). Rather than plot force vs stretch, we plot stress vs strain. Within the elastic region, this is a straight line. Figure 12.15 on page 353 shows this plot. As the stress is increased beyond the elastic region, the elastic limit is reached. Ferrous materials and non-ferrous materials exhibit somewhat different behaviors in the region of the elastic limit; the curve that Serway shows is more characteristic of aluminum than of steel. (While Serway's plot shows the elastic limit as a point somewhat past the linear section, we shall assume that it is at the end of the linear section.) Brittle materials such as concrete and cast iron rupture suddenly just past the elastic limit. But the linear portion is the same (except for slope) for all of these materials. The table on page 346 gives the slopes (Young's moduli) for a number of materials. Incidentally, engineering tables ordinarily list these in gigapascals (GPa); for example, steel has a Young's modulus of 200GPa.
Example 2: Concrete testing
Example 3: Elastic limit and Offset
Example 4: A Pendulum
The Shear Modulus:
As for Young's modulus, this also is stress dived by strain. However, as explained in the first mini-lecture, the force is parallel to the cross-sectional area. The formula is:
Figure 12.16a on page 347 shows how the symbols are to be interpreted.
Example 5: A rivet under shear
Bulk Modulus:
Of the three elastic moduli equations, this one alone has a negative sign. That's because a positive stress (pressure) results in a decrease in volume. The negative thus cancels with the negative strain and the modulus is positive:
Example 6: Water's compressible!