Stress is defined as the quantity equal to … what?

Update on 01 March 2018, 21:27, IST: I had posted a version of this post also at iMechanica, which led to a bit of a very interesting interaction there [^] too. Check it out, if you want… Also see my today’s post concerning the idea of stress, here [^].


In this post, I am going to note a bit from my personal learning history. I am going to note what had happened when a clueless young engineering student that was me, was trying hard to understand the idea of tensors, during my UG years, and then for quite some time even after my UG days. May be for a decade or even more….

There certainly were, and are likely to be even today, many students like [the past] me. So, in the further description, I will use the term “we.” Obviously, the “we” here is the collegial “we,” perhaps even the pedagogical “we,” but certainly neither the pedestrian nor the royal “we.”


What we would like to understand is the idea of tensors; the question of what these beasts are really, really like.

As with developing an understanding of any new concept, we first go over some usage examples involving that idea, some instances of that concept.

Here, there is not much of a problem; our mind easily picks up the stress as a “simple” and familiar example of a tensor. So, we try to understand the idea of tensors via the example of the stress tensor. [Turns out that it becomes far more difficult this way… But read on, anyway!]

Not a bad decision, we think.

After all, even if the tensor algebra (and tensor calculus) was an achievement wrought only in the closing decade(s) of the 19th century, Cauchy was already been up and running with the essential idea of the stress tensor right by 1822—i.e., more than half a century earlier. We come to know of this fact, say via James Rice’s article on the history of solid mechanics. Given this bit of history, we become confident that we are on the right track. After all, if the stress tensor could not only be conceived of, but even a divergence theorem for it could be spelt out, and the theorem even used in applications of engineering importance, all some half a century before any other tensors were even conceived of, then developing a good understanding of the stress tensor ought to provide a sound pathway to understanding tensors in general.

So, we begin with the stress tensor, and try [very hard] to understand it.


We recall what we have already been taught: stress is defined as force per unit area. In symbolic terms, read for the very first time in our XI standard physics texts, the equation reads:

\sigma \equiv \dfrac{F}{A}               … Eq. (1)

Admittedly, we had been made aware, that Eq. (1) holds only for the 1D case.

But given this way of putting things as the starting point, the only direction which we could at all possibly be pursuing, would be nothing but the following:

The 3D representation ought to be just a simple generalization of Eq. (1), i.e., it must look something like this:

\overline{\overline{\sigma}} = \dfrac{\vec{F}}{\vec{A}}                … Eq. (2)

where the two overlines over \sigma represents the idea that it is to be taken as a tensor quantity.

But obviously, there is some trouble with the Eq. (2). This way of putting things can only be wrong, we suspect.

The reason behind our suspicion, well-founded in our knowledge, is this: The operation of a division by a vector is not well-defined, at least, it is not at all noted in the UG vector-algebra texts. [And, our UG maths teachers would happily fail us in examinations if we tried an expression of that sort in our answer-books.]

For that matter, from what we already know, even the idea of “multiplication” of two vectors is not uniquely defined: We have at least two “product”s: the dot product [or the inner product], and the cross product [a case of the outer or the tensor product]. The absence of divisions and unique multiplications is what distinguishes vectors from complex numbers (including phasors, which are often noted as “vectors” in the EE texts).

Now, even if you attempt to “generalize” the idea of divisions, just the way you have “generalized” the idea of multiplications, it still doesn’t help a lot.

[To speak of a tensor object as representing the result of a division is nothing but to make an indirect reference to the very operation [viz. that of taking a tensor product], and the very mathematical structure [viz. the tensor structure] which itself is the object we are trying to understand. … “Circles in the sand, round and round… .” In any case, the student is just as clueless about divisions by vectors, as he is about tensor products.]

But, still being under the spell of what had been taught to us during our XI-XII physics courses, and later on, also in the UG engineering courses— their line and method of developing these concepts—we then make the following valiant attempt. We courageously rearrange the same equation, obtain the following, and try to base our “thinking” in reference to the rearrangement it represents:

\overline{\overline{\sigma}} \vec{A} = \vec{F}                  … Eq (3)

It takes a bit of time and energy, but then, very soon, we come to suspect that this too could be a wrong way of understanding the stress tensor. How can a mere rearrangement lead from an invalid equation to a valid equation? That’s for the starters.

But a more important consideration is this one: Any quantity must be definable via an equation that follows the following format:

the quantiy being defined, and nothing else but that quantity, as appearing on the left hand-side
=
some expression involving some other quantities, as appearing on the right hand-side.

Let’s call this format Eq. (4).

Clearly, Eq. (3) does not follow the format of Eq. (4).

So, despite the rearrangement from Eq. (2) to Eq. (3), the question remains:

How can we define the stress tensor (or for that matter, any tensors of similar kind, say the second-order tensors of strain, conductivity, etc.) such that its defining expression follows the format given in Eq. (4)?


Can you answer the above question?

If yes, I would love to hear from you… If not, I will post the answer by way of an update/reply/another blog post, after some time. …

Happy thinking…


A Song I Like:
(Hindi) “ye bholaa bhaalaa man meraa kahin re…”
Singers: Kishore Kumar, Asha Bhosale
Music: Kishore Kumar
Lyrics: Majrooh Sultanpuri


[I should also be posting this question at iMechanica, though I don’t expect that they would be interested too much in it… Who knows, someone, say some student somewhere, may be interested in knowing more about it, just may be…

Anyway, take care, and bye for now…]

4 thoughts on “Stress is defined as the quantity equal to … what?

  1. I alway got the feeling, even get it now, that lecturers and authors on the subject themselves do not have a “deep” understanding of the subject matter. My experience has been that when somebody does have a deep understanding of something, they are able to pass it on, but with the subject of tensors, there seems to be something missing, like some kind of eureka, epiphany, whatever, like I experienced when I was teaching myself calculus in my mid teens and had that sudden eureka moment, the fundamental concept of infinitesimals and differentials suddenly made complete sense and a whole new universe abruptly appeared before my eyes. Otherwise, it could simply be that tensors are purely abstract symbolic objects with no way of understanding them in a “geometric” sense, which is enormously frustrating to somebody like me who thinks more in pictures than sums. Sums do not seem to have any reality to them if taken in isolation, but pictures seem a little less other-worldly. One mistake I used to make was thinking of tensors as an object rather than a grouping of mathematically similar objects being represented in a single symbol… and the Einstein summation convention did not do anything to help in that regard!

    • Exactly! Tensors had been enormously frustrating to me, too, for a long, long time, and in some respects they still continue to annoy me (though they have lost the capacity to frustrate me, I guess). And I found them frustrating exactly for the same reason as your mention: I was unwilling to accept them as purely abstract symbols devoid of any geometric (i.e. physical) meaning. However, no text I ran across ever explicitly put across the viewpoint I was looking for.

      A tensor _is_ a single object, not just a grouping of similar objects under a single symbol. The similar objects—i.e. the components of a tensor—may change under coordinate transformations (most notably, under rotations). But the tensor itself stays the same, and so, you have to think of it as a single object. A similar thing happens in the admittedly simpler context of vectors. Think of an arrow shot into the sky. Freeze the frame at any instant. The arrow is where it is, with whatever orientation it has, and with whatever sense of direction it is going into. If n # of pilots flying planes around it take one photograph each, from different angles and positions, etc., then their photographs will depict the same arrow, but the photographs themselves would be concretely different. However, a picture or a depiction (actually, a projection) of an arrow isn’t the same as the arrow itself. The arrow has, physically, always been only one and the same thing. Something similar goes for tensors. It stays the same; only the magnitudes of its components undergo changes with the changes of coordinate frames.

      IMO, a tensor is initially best seen as the end-result produced by the tensor-product of two vectors. Only then is it better to go for the idea that it is nothing but a linear map between vectors. For better understanding the tensor-product, I enthusiastically recommend Prof. Zhigang Suo’s notes on linear algebra posted at iMechanica, here: http://imechanica.org/node/19709. Do go through these, starting with what he notes as the background material quickly.

      Once you develop some understanding of a tensor as the result-object produced by the tensor product, then, the one thing to realize in understanding the _stress_ tensor in particular is that it’s nothing but a flux, essentially speaking! (As a flux, it’s _very_ physical!)

      No, I never bother myself with any algebraic index convention—Einstein’s or any other. I am happy using the column and row vectors and matrices, and the rules for them…. (Guess I gave away a crucial hint for solving the problem in the main post 🙂 )

      … Anyway, thanks, and best,

      –Ajit

  2. We think differently, that is obvious. I do not think of the tensor as being the same thing as what it is modeling, which is why I do not view the tensor representation of a phenomena as unchanged under co-ordinate transformation. For me, the tensor IS the numerical representation, not the phenomena it represents, and I take that view because the tensor IS an abstraction, not an actual object of any physical reality. In any case, I perfectly understand what you are saying, at least that much is no mystery, but the mystery remains in the means by which tensors are explained. I went (partially) through several small books on the subject, each author explaining the ideas with very diverse approaches, because it was my attempt to find somebody that came from a similar initial perspective as myself and, thought I, perhaps foolishly, such would help me, but I had not properly considered the fact that the specific application of the mathematical tool (that is, the direction the author wished to go in) would have such great influence on the particular view he would ultimately take in understanding the idea of a tensor. I did not go right through any particular book because I found that I was not understanding the concepts at a fundamental level, without which I cannot progress. Some authors treat a tensor as purely a mathematical symbol to be manipluated which, addmittedly, would seem the best approach, rather than straining one’s cranial content in an attempt to “visualise” such an abstract object. Others would represent the tensor as an “abstract” physical or geometric object (strange as that may seem), kind of a mechanical thing that interlocks with other similar things to produce different things… it was a weird approach but I could see the merrit, though it did not help me a great deal. What seems often lacking is some proper understanding of what, exactly, the numbers in a tensor are actually telling us. With that it becomes easier to see the meaning of the mathematical manipulations taking place, what they are actually doing to that information, but being treated as a purely mathematical abstraction, the numbers have no meaning — they need application to something practical, first. However, one of the best explanations that helped me the most in only a few pages was a document I found that came from NASA, a document entitled, “An Introduction to Tensors for Students of Physics and Engineering”, by Joseph C. Kolecki (published 2002). Though I finished at uni end of 1986, having lost some of my interest in understanding general relativity over the years and due to a lack of proper instruction at uni regarding tensors, though I had a rudimentary understanding, I went many years without revisiting the subject until about 2007 or so, which is when I came across that NASA document. It helped more than the books.

    Cheers.

    • Thanks for sharing your thoughts.

      For the other readers: Here is the link to the document by Joseph Kolecki [^].

      Yes, Kolecki’s document has helped me a lot, too. But unfortunately, it too did not give me the direct definition that I was looking for. None does…

      I will try to address the other points you mention, at some other time. However, I can note one thing right away.

      The essential difficulty with visualizing a tensor in 3D is that, in the general case, you would have to simultaneously visualize 9 numbers. Not just that, as my next post (already posted) tries to bring it out, the matter gets more complicated because any and all of these 9 numbers may be positive or negative. If the tensor is symmetrical, there is some hope. Out of the 6 degrees of freedom (DOF), we can use 3 different sizes along three different axes for 3 DOF, and 3 different orientations for the other 3 DOF. But the complication due to +ve vs. -ve numbers still remains.

      –Ajit

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