* * *
Let's see; for me, if I can map an abstract concept to something readily visual, my understanding is faster. Are there some close visual aids to understanding tensors?
What physical property(ies?) can be mathematically modeled as a tensor?
Imagine a stack of tiles, bottom to top, thin and piled on top of each other. And each tile is connected with its neighbours with springs (not unlike a spring-coiled bed with many layers), like so:
============== ---> Thin tile
\ \ \ \
/ / / / ---> Springs
\ \ \ \
==============
\ \ \ \
/ / / /
\ \ \ \
==============
\ \ \ \
/ / / /
\ \ \ \
==============
Now, we can pull the topmost tile along the stacked direction causing the springs to expand. If we do this and only this, it is pure tensile stress (I am referring to stress in a bit loose way here). We can also sit on that stack and that leads to compressive stress (just a tensile stress with a minus sign). [As a sidenote, bricks can take great compressive forces but can't withstand tensile forces of similar levels. But something like steel has almost symmetric response between tensile and compressive loads].
OK...what else can we do? Can I pull the topmost tile to the right (or left) while holding the bottommost tile still? Sure, and now the stack looks like a rhombus. This is shear stress...in the right-left direction. I could've also pulled the topmost tile towards (or away from) me. That is also shear stress in the front-back direction.
So, for this setup, we can identify three stress components: 1 tensile and 2 shear.
Now, the first tricky bit: imagine a "stack" that is bottom-top, left-right and front-back. There are springs running in all three directions. We have 3 tensile and 6 shear components.
Second tricky bit: shrink that new whole "stack" to a point. We have the stress tensor(!).
s = [s_11 s_12 s_13, s_21 s_22 s_23, s_31 s_32 s_33]
Tensors are used all the time in mechanics. Very very useful stuff.