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Prologue: I just searched for the word stress on this essay and couldn't find it. So... * * * Let's see; for me, if I can map an abstract concept to something

Prologue: I just searched for the word stress on this essay and couldn't find it. So... * * * 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.

deleted 13y ago
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shiven 13y ago
This is a much clearer explanation than the original blog post (it used terms without explaining them to non-mathematicians i.e. me). Thank you!
- hdevalence 13y ago
  It's a much clearer explanation than the original post, because it explains a different concept, which is easier to understand than the content of the post (what does a tensor product of vector spaces mean?) The distinction is between understanding an instance of a concept (stress is a tensor) and the concept itself (what is a tensor, i.e., what exactly is it that stress shares with all other tensors). This is why the explanation in terms of a universal property is more difficult to understand: because by virtue of stripping away the extraneous details, it shows you the concept as it applies to all instances of the concept.
  - davidw 13y ago
    As someone who had no idea what a tensor was, a rough explanation of what they might sort of be related to is better than something that I don't understand at all, as long as it's clearly labeled that it's not an exact explanation.
pujjad 13y ago
But with Tensors you can not only express compressive/tensile forces on a plane through a point and not only shearing on that plane but also sort of rotation: curls -> non-symmetric across the diagonal elements. I believe Feynman introduced a nice picture for this: imagine you have a perfectly symmetric nutshell with some thin paper blades attached to it, so it looks like a rotor. Put it right into a stream and fix the position. If the stream is flowing faster around on one side, it will start to spin. That's the curl in that point of the stream. Edit: for clarity

deleted 13y ago

[deleted]

shiven 13y ago

This is a much clearer explanation than the original blog post (it used terms without explaining them to non-mathematicians i.e. me). Thank you!

hdevalence 13y ago
It's a much clearer explanation than the original post, because it explains a different concept, which is easier to understand than the content of the post (what does a tensor product of vector spaces mean?) The distinction is between understanding an instance of a concept (stress is a tensor) and the concept itself (what is a tensor, i.e., what exactly is it that stress shares with all other tensors). This is why the explanation in terms of a universal property is more difficult to understand: because by virtue of stripping away the extraneous details, it shows you the concept as it applies to all instances of the concept.
- davidw 13y ago
  As someone who had no idea what a tensor was, a rough explanation of what they might sort of be related to is better than something that I don't understand at all, as long as it's clearly labeled that it's not an exact explanation.

pujjad 13y ago

But with Tensors you can not only express compressive/tensile forces on a plane through a point and not only shearing on that plane but also sort of rotation: curls -> non-symmetric across the diagonal elements. I believe Feynman introduced a nice picture for this: imagine you have a perfectly symmetric nutshell with some thin paper blades attached to it, so it looks like a rotor. Put it right into a stream and fix the position. If the stream is flowing faster around on one side, it will start to spin. That's the curl in that point of the stream. Edit: for clarity