# What do the different elements in strain tensors tell us?

I'm working with strain tensors of all sorts at the moment, and I think I've understood how they're derived. However, I'd like to get more intuition of what they're actually telling us.

More specifically, I want to understand what each element in the stress tensor corresponds to in the sense of deformations. Compare my question to for instance the inertia tensor. In that case, I know that the diagonal elements tell me how easy /hard it's to rotate an element along a certain axis.

So that's my first question. Now I realize that they're tons of different strain tensors out there, for instance the left / right Cauchy-Green strain tensor, the Lagrangian and Eulerian strain tensor. Is there a unified way of describing what each element in such strain tensors represents physically as in the case of the inerta tensor? Which therefore constitutes my second question.

I hope I was clear enough.

Because strains describe normalized deformation, they need to include information about (1) what's being deformed and (2) how it's deformed.

One way to do this is to consider a single unit vector (which can be described in terms of a Cartesian axis or a combination of these axes, for instance) and then to specify how its tip moves relative to its tail (which can also be described in a Cartesian coordinate system).

Thus, we have the strain tensor component $$\varepsilon_{ij}$$: The amount of $$j$$-direction movement of the tip (relative to the tail) of an $$i$$-direction unit vector.

This is pretty straightforward for a small normal strain of a nonmoving object:

(All images from my site.)

Here, one of the vectors we choose (shown in blue) lies in the 1 direction, and its tip (relative to its tail) moves only in the 1 direction (shown in green). The bars underneath show the important distances, and again, the strain represents the ratio of the green distance ($$u$$) to the blue distance ($$x$$).

In this way, we build up the (3×3=)9 components of a tensor describing strain in 3D space.

Things get trickier when we wish to discuss shear strain (e.g., $$\varepsilon_{12}$$), as depending on the coordinate system, we might interpret the strain as a lateral tip movement of one vector or that of an orthogonal vector or both:

Here, shape change is the same for all three animations. Is it meaningful whether vector 1 moves, or vector 2, or both? It shouldn't be, as Nature doesn't care about how we set our coordinate systems.

We address this problem by averaging the two strains. In fact, a definition that works for both normal and shear strains is

$$\varepsilon_{ij}\equiv\frac{1}{2}\left(\frac{\partial u_i}{\partial {x_j}}+\frac{\partial u_j}{\partial {x_i}}\right).$$

For normal strains, the two terms enclosed in parentheses are the same, of course.

In more general cases, we need to avoid ambiguity by specifying whether our coordinate system will deform with the strain or not, and whether it will move with the material or not, for example. (These ambiguities don't arise with infinitesimal strains of solids but do arise with large—sometimes termed finite—strains and with fluids.)