The Physics Philes, lesson 68: Cracking Under the Shear Pressure
Last week we started talking about stress and strain, and I attempted to explain tensile stress and strain. But did you know there are a whole two other types of stress and strain? There are! These stresses and strains are called bulk and shear.
Let’s first tackle bulk stress and strain. The stress when you have an object submerged in a fluid (either gas or liquid). When that happens, you get a uniform pressure on all sides, which results in a change in volume. Kind of like this:
See? The perforated circle represents the volume before the uniform pressure is applied, and the inner circle represents the volume after.
The stress in this situation is bulk stress, and resulting deformation – the volume change – is called bulk strain. Because this type of stress and strain results in a change in volume, it’s often called volume stress and strain.
Bulk stress is called pressure and it’s calculated by dividing the force exerted on the submerged by by the area. The pressure increases with depth. But if the immersed object is small enough, we can pretend that pressure differences between the top and bottom don’t exist. All that means is that, if you’re sitting at the bottom of a pool, we can assume the pressure at your head and the pressure at your feet are the same. This pressure is the same regardless of how the object is oriented because pressure has no intrinsic direction.
We learned last week that strain is basically the way we describe how an object is deformed. In the case of bulk (or volume) strain, the deformation is the change in volume. So we can define bulk strain as the change in volume divided by the original volume.
Remember last week when we talked about Hooke’s (not really a) law? In the context of tensile stress and strain, if the stress is sufficiently small, then the stress and strain are proportional. This holds true with bulk stress and strain, as well. If the bulk stress is sufficiently small, the pressure produces a proportional bulk strain. This proportional relationship is called the bulk modulus, which we can write as:
Notice the negative sign. This just indicates that an increase in pressure always causes a decrease in volume. The bulk modulus (B) is positive.
So far, these are all concepts we’ve seen before, just applied in a slightly different context. Now we have a different concept, but don’t worry. It’s not too difficult. With bulk stress and strain we can calculate something called compressibility, which is the reciprocal of the bulk modulus. Denoted by k, it’s the fractional decrease in volume per unit increase in pressure.
A small bulk modulus and a large compressibility means an object is easier to compress.
We have two types of stress and strain under our belt, but we’re not done yet. We have one more type to discuss, that that is shear stress and strain.
Shear stress is a bit different from tensile and bulk stress. Tension and pressure involve forces that are perpendicular to the surface of the object. But shear stress involves forces that are parallel to the surface of the object. Maybe a picture would make it more clear.
Here is a rectangle with height h:
A force acts parallel to this rectangle, we get something like this:
See what happens? The parallel force distorts the rectangle. Shear stress is defined in basically the same way the same way the other stresses are defined: the force divided by the area.
Now let’s think about shear strain. On the above diagram I labeled the height of the rectangle with an h. The force causing the shear stress on the top layer is acting on the layers all the way down the distance h. The displacement we can define as x. So we define shear strain as the ratio of the displacement x over the height h.
Hooke’s law applies to shear stress and strain, as well. If the shear stress is small enough, then there is a proportional relationship between the shear stress and shear strain:
This relationship is called the shear modulus. (You may have spotted a pattern.)
Shear stress, strain, and modulus are different from tension and pressure in one more important aspect. It only applies to solid objects. Once a shear force is removed, objects snap back into place. This doesn’t happen with liquids and gases because those don’t have definite shape.
That’s all the physics nourishment I have for you today. But be sure to check back next week. I’ll be looking at Hooke’s law in more detail. We’ll finally find out what it means for a force to be “sufficiently small.” I can hardly wait!
Featured image credit: Harry Pherson