Stress tensor force equilibrium in Feynman lecture book

Question

I was reading about tensors and especially stress tensors in Feynman lectures on physics

Which is on the webvsite enter link description here

While reading the “31-6. The tensor of stress”, I was a bit confused.

In the book it states that each tensor component corresponds to a component of the force per unit area that is on a certain plane.

Then it goes onto proving that a strain on any direction of slicing can be completely expressed by the stress tensor that is in terms of x,y, and z. While doing so, it utilizes the fact that all the surface forces on Fig 31-8 should cancel out to write the expression

$\Delta F_{xn}=S_{xx}\Delta y\Delta z+S_{xy}\Delta x\Delta z$

(Where the last two terms correspond to the x-directional force exerted on the two rectangular planes (xy plane and yz plane)).

While reading this, I couldn’t undertand why there shouldn’t be a $-$ sign in front of $\Delta F_{xn}$, since if the forces cancel out we would get $\Sigma F_i =0, ~~\therefore -\Delta F_{xn}=F_{xx}+F_{xy}= S_{xx}\Delta y\Delta z+S_{xy}\Delta x\Delta z $.

Why isn’t there a minus sign?

Also, I noticed that while explaining the stress tensor the book acknowledges that if there is a strain force from region 1 to region 2 (separated by the surface) $\Delta F_i$, there is the reactionary force $-\Delta F_i$ that is exerted from 2 to 1. Which force of the two am I supposed to consider when I, for instance, want to consider the surface forces exerted to a $dx dy dz$ cube?

Answer 1

Signs are often tricky in maths and physics, and this sign is one of the trickiest ones! The stress tensor itself is sometimes defined with an overall sign that not everyone agrees on. The crucial thing is to be clear on what force it is you think your stress tensor is expressing. To get the hang of this, start with something more familiar: pressure in a gas. If the pressure is positive then at the surface where the gas stops and the container begins, there is a force outwards on the container wall. That is unambiguous. The standard definition of the stress tensor then says that the diagonal components for this case are positive. So positive $S_{xx}, S_{yy}, S_{zz}$ corresponds to pressure, and negative $S_{xx}, S_{yy}, S_{zz}$ corresponds to tension.

Consider some small positive $\Delta z$. For a surface parallel to the $xy$ plane, pressure means that the material at $z-\Delta z$ provides a force in the positive $z$ direction on the material at $z + \Delta z$. It equally means that the material at $z + \Delta z$ provides a force in the negative $z$-direction on the material at $z - \Delta z$ (and overall the material between $z-\Delta z$ and $z + \Delta z$ gets squeezed). So you see the sign of the force depends on which material you are considering. More generally, when you pick a cube $dx \, dy \, dz$ you should ask yourself, "do I want to calculate the force exerted by my cube on other things, or do I want to calculate the force exerted by other things on my cube?" It is the same calculation in both cases, but the way you interpret the overall sign will depend on which thing you think you calculated!

The reason why I focused on pressure so far in this answer is because that is the easiest part of this bit of physics to grasp intuitively. What you have to do next is extend the lessons of pressure to the case of sheer stress (the forces along a boundary as opposed to normal to it). All the same issues about signs arise. You wish to make a different sign choice for $\Delta F_{xn}$ than the one made by Feynman. It is not that one is right and the other wrong. Rather, both are right when they are interpreted correctly; both are wrong when they are interpreted incorrectly. You can trust that this bit of Feynman's book is not a typo or a mistake, so he and his co-authors (Leighton and Sands) have made one choice and used it correctly. You just need to figure out which choice they have made! Are they talking about the force exerted by $A$ on $B$ or the force exerted by $B$ on $A$?