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| Stretching a block with rectangular cross section. The left end is fixed. |
We create the next displacement field by pulling on the ends of the rectangular shape. This strain is called a tensile strain. (It turns out that other forces are also needed. More of that later.) The origin is fixed at the left end. The block is marked with equally spaced lines to help us track the movement of points within it. A plot of observed displacement of a bit of the block, versus position of that bit, is also shown.
Experimentally, it is found that for small distortions, many useful materials stretch in the following way: The lines remain equally spaced, with the space between each line increasing by the same amount when the block is stretched.
Materials with this property are said to be elastic materials.
(Imagine stretching a good coil spring, and watching the separations between increase as you do it. The distance between each coil is the same.)
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| Stretching a block with rectangular cross section. The center is fixed. |
The figure above is misleading in this sense: It suggests that to stretch the solid, only one force, pulling to the right, is needed. Of course if that were the only horizontal force, the object would accelerate to the right. A second force, on the left end, pulling to the left with the same magnitude as the first, is required to make the net force zero. This force is brought more clearly to mind if we change coordinate systems so that the origin is fixed in the center of the object.
From this second perspective the object stretches in both directions, as in the figure at the right. While the length of the object increases by 0.045cm in both pictures, the distance moved by the right end is 0.045cm for left end fixed, but 0.0275cm for center fixed.
The distortions of the object are superficially different, since in one case the left end moves while in the other it does not. However, the distortion is fundamentally the same. This is shown in the graphs of displacement versus original position of the fiducial marks. Both graphs have the same shape; a straight line, and the same slope. They differ only in the value of the intercept.
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| Displacement field for a stretched block with rectangular |
| cross section. |
To emphasize how different the same strain can appear in different coordinate systems, the displacement field for the stretched rectangle is sketched at the right.
The only difference is in the point chosen to be fixed on the origin. This figure shows unrealistically larger distortions, to make them more visible.
Recall now that if all the displacement vectors were the same, there would be no distortion of the block; no strain. For the block to be strained, different parts of it must be displaced by different amounts. It is the degree to which the displacement is different from point to point that determines the strain.
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| Displacement versus original position for an object with tensile strain. |
The graphs show how the displacement varies. The displacement is plotted versus original position for left-fixed (top graph) and center-fixed (bottom graph). If the displacement were the same for all points, the graphs would be horizontal lines. The obvious way to characterize the variation in displacement is to state the slope of the graph. This ensures an answer of zero for uniform displacement with no strain.
The displacement field for the center-fixed case is easy to write down. It is
where x is positive to the right.
The slope of a graph is
The partial derivative is the one we will want in the future, when more complex motion involving other dimensions is studied.
Because the slopes of the two graphs are the same, we can take
as our
measure of the strain.
As before, there is another sensible way to characterize the strain; as the change in length per unit length. This is easiest to apply when one end is fixed. Then for a sample of length L, the displacement of the right end is aL, and we can write the measure of strain as
often written in terms of the change in length, as
As before the two measures give the same answer and we define tensile strain as
The xx subscript indicates that the strain involves the x component of the displacement and its variation along the x direction.
Note that if
is negative,
it describes a compressive strain; the fiducial marks moving closer together.
Return to index.
Return to survey.
Go forward to shear strain.