Everyday physics: the stress tensor.

You might think I have gone bonkers. When have you ever heard of the stress tensor being everyday physics? Don’t let nomenclature fool you. The stress tensor is as a matter of fact something that we have a lot of familiarity with, just not by that name. Consider for example a steel cable made of filaments as in the figure. What is the total tension on the cable? You can say that the tension is built filament by filament, so the total tension (the capacity to pull an object evaluated as a force) is proportional to the number of filaments if all the filaments have the same individual tension. 

 

Each filament contributes to the tension

Since the number of filaments is another way to count areas, we find that the total tension is proportional to the area. If we calculate the tension per unit area, we call that the stress of the material. Now it starts to sound familiar, doesn’t it?

This is very similar to the notion of pressure, as described in the previous post on flotation. Indeed, the units of stress and the units of pressure are the same. The main difference, however, is that the stress is more directional than pressure. So in the picture of the cables above, the force is proportional to the area in the yz plane (the cross sectional area), while the force points in the x direction. However, if we though about it in the vertical direction, the cables are not that good at transmitting force in that direction, so we would expect that the force per unit area going upwards is different. This is on contrast with pressure that goes into all directions at once.

In the picture above, the yz plane has a perpendicular normal vector to it along the x direction. It is natural to associate to this stress both the directionality of the force and the directionality of the area. Each of these is a vector. 

Thus, we find that the natural object is a matrix with two indices: one for directionality in area and another one for directionality of the force. This object is called the stress tensor, and it is usually notated by 

We will use the convention that the first index is the one that indicates the area directionality, while the second index is the one that indicates the directionality of the force. It turns out that a special example of stress is the pressure itself. We can compute the components of the stress tensor associated to pressure, and we find that

The symbol is the Kronecker delta. This is a fancy way to write the identity matrix. It tells us that pressure has the property that the force always points along the direction of the area that you are cutting.

Now that we understand that pressure is a special form of stress it is time to pull the lame joke of the post: if you are feeling under pressure, relax, that is a rather simple form of stress. Imagine what would it be like if you had to put up with all those other components as well!

After that silly break, time for physics again. The same type of arguments that show that a gradient of the pressure gives a force per unit volume can be used to show that a gradient of the stress tensor is a force per unit volume. The difference is that the naive gradient of a stress tensor has three times more components, while a force per unit volume is a vector, so the gradient has to add all forces per unit volume along the same direction. If you remember that the second index is the one associated to Force, this concept is encompassed in the equation

The sign here is a convention (the same minus sign that we had with pressure). This is the force per unit volume on a piece of material subject to a variation of stress.

For example, consider a bar of metal that is suspended from the sides in the presence of gravity (let us say that the piece of metal is supported by the yz plane: this corresponds to a bar elongated along the x direction). Pressure can not result in the bar of metal not falling: there is nothing holding it from below. Instead, the stress on the bar is along the axis (it is supported from the sides along the yz plane), and we find the relation . This can be solved by a stress that is proportional to x (let us also assume that x=0 is the center of the bar of metal). Thus, the material has a lot more stress at the ends than in the middle. This is part of the reason why bridges tend to be fatter on the sides than on the center: to distribute the excess of stress better. If on the other hand, the metal bar was suspended only at one end, and the other end was left dangling, then the stress is more complicated: one needs to compensate the torque of gravity as well. Not to mention that the side on which it is suspended from has to do double duty.

Ok, this is getting long, but bear with me.

If you use Newton’s law, a force is a time derivative of momentum. And a force density is a time derivative of momentum density. Thus, we can write Newton’s equations as

This equation embodies momentum conservation. For a big finite system without external forces acting on it, the stress tenor vanishes away from the object. We can integrate this equation in a volume and use theorems of calculus in several variables (Stokes theorem) to find that the total momentum is conserved.

If you have ever taken a class in relativity, the momentum density is usually grouped with the stress tensor and the energy density into an object called the stress-energy tensor. This is why I made up this component (to make the equations look pretty). A similar equation shows that 

where the components are the energy flux in some direction, and e is the energy density.

 
This collection of equations that represent Newton’s laws: momentum conservation and also energy conservation for materials or spacetime fields are described by saying that the stress-energy tensor is conserved.

Phew! That was long. Boy, what a stressful post.

UPDATE:   By request, here are links:

Stress tensor. Check also here.

Stress energy tensor

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