# Making sense of the stress tensor for elastic deformations

I've seen this kind of formula a number of times, in the context of elastic deformations.

$$-\nabla \sigma = f$$

whete $\sigma$ is "the stress tensor" and $f$ is force. I never understood it even though I've coded finite element simulations several times. To me, it all comes down to defining an elastic energy in terms of positions $E(x)$ and then thinking about the body trying to minimize it and that's it, I can use the computer to take care of all the other details. However, I'm trying to understand the terms used in classical mechanics textbooks. First of all, what is $\sigma$ ? It seems to me, from the descriptions I've read that it's just a matrix. If that's true, then why call it a tensor? And also, what's the best way to interpret $\nabla$? Is it a function acting on a matrix? I guess it's some kind of derivative, but it's not clear to me what it computes exactly. Sometimes it appears as $\nabla.\sigma$ (with a dot in between), is it the same thing as $\nabla \sigma$?

The nabla is an operator and it should be with a dot in this case, $\nabla\cdot\sigma$, since it's the divergence operator (which decreases the tensor order by 1), whereas without a dot it denotes a gradient (which increases the tensor order by 1). See here for more about tensor derivatives.
Regarding the nature of $\sigma$ it is both a matrix and a tensor. A matrix is just a rectangular array of numbers; a tensor exhibits certain transformation properties. So if a matrix exhibits those same transformation properties then it is also a tensor (of rank 2). The difference between a matrix and a tensor has been discussed here many times before, e.g. see here.
• For a given mesh, elastic material and nodal positions $x$, I know how to compute the elastic energy $E(x)$. With this information, how do I compute $\sigma$ ? Is it a function of $x$ as well? Commented Jun 18, 2018 at 9:06
• In the context of finite elements, I know how to compute the deformation gradient $F$ for an element. Is there any relation between $F$ and $\sigma$? Commented Jun 18, 2018 at 9:12
• @yewang You can compute the strain tensor from $F$ as follows: $\epsilon=F-I$, and then the stress tensor is related to elastic energy by: $\sigma_{ij}=\partial E/\partial\epsilon_{ij}$. (You should double check that online - I recall a $2$ knocking around somewhere). Commented Jun 18, 2018 at 9:19