iMechanica - Stress Tensor
http://imechanica.org/taxonomy/term/822
enStress is defined as the quantity equal to ... what?
http://imechanica.org/node/22146
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/822">Stress Tensor</a></div><div class="field-item odd"><a href="/taxonomy/term/11964">definition of stress</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In this post, I am going to note a bit from my <em>personal</em> 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 <em>after</em> my UG days. May be for a decade or even more....</p>
<p>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.''</p>
<p>========================</p>
<p>What we would like to understand is the idea of tensors; the question of what these beasts are really, really like.</p>
<p>As with developing an understanding of any new concept, we first go over some usage examples involving that idea, some instances of that concept.</p>
<p>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!]</p>
<p>Not a bad decision, we think.</p>
<p>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 up and running with the essential idea of the stress tensor right by 1822---i.e., more than half a century <em>earlier</em>. We come to know of this fact, say via James Rice's article on the history of solid mechanics [(.PDF) <a href="http://esag.harvard.edu/rice/163_Ri_Mech_Solids_EB93.pdf" target="_blank">^</a>].</p>
<p>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 could even be used in applications of engineering importance, all for decades before any other tensors were even abstractly conceived of, then, of course, developing a good understanding of the stress tensor ought to provide for a sound pathway to understanding tensors in general.</p>
<p>So, we begin with the stress tensor, and try [very hard] to understand it.</p>
<p>========================</p>
<p>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:</p>
<p><img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?\sigma \equiv\frac{F}{A}" alt="" align="middle" border="0" /> ... Eq. (1)</p>
<p>Admittedly, we had been made aware, that Eq. (1) holds only for the 1D case.</p>
<p>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:</p>
<p>The 3D representation ought to be just a simple generalization of Eq. (1), i.e., it must look something like this:</p>
<p><img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?\overline{\overline{\sigma}}= \frac{\vec{F}}{\vec{A}}" alt="" align="middle" border="0" /> ... Eq. (2)</p>
<p>where the two overlines over <img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?\sigma" alt="" align="middle" border="0" /> represents the idea that it is to be taken as a tensor quantity.</p>
<p>But obviously, there is some trouble with the Eq. (2). This way of putting things can only be wrong, we suspect.</p>
<p>The reason behind our suspicion, well-founded in our knowledge, is this:</p>
<p>The operation of a division <em>by</em> 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 if we were to try an expression of that sort in our exams!]</p>
<p>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).</p>
<p>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.</p>
<p>[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 what we are trying to understand. ... ``Circles in the sand, round and round... .'' In any case, at this stage, the student is just as clueless about divisions by vectors, as he is about tensor products.]</p>
<p>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:</p>
<p><img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?\overline{\overline{\sigma}} \vec{A} = \vec{F}" alt="" align="middle" border="0" /> ... Eq (3)</p>
<p>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.</p>
<p>But a more important consideration is this one: Any quantity must be definable directly, i.e., via an equation that follows the following format:</p>
<p><em>the quantiy being defined, and nothing else but only that quantity, appearing on the left hand-side </em><br /><em>= </em><br /><em>some expression involving some other quantities, appearing on the right hand-side.</em></p>
<p>Let's call this format Eq. (4).</p>
<p>Clearly, Eq. (3) does not follow the format of Eq. (4).</p>
<p>So, despite the rearrangement from Eq. (2) to Eq. (3), the question remains:</p>
<p><em>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)?</em></p>
<p></p>
<p>========================</p>
<p>A few more words:</p>
<p>It would be easy enough to abstractly do just a bit of algebraic manipulation and arrive at the solution. The point isn't that. The point is to understand the physical implications of that manipulation.</p>
<p>And, further, the point is this: If it were that obvious or simple, why is it that not even a <em>single</em> text-book/class-notes ever anticipates the above-mentioned possible line of thought on the part of a beginning student, and therefore, proceeds to provide him with the required definition in direct terms?</p>
<p>And then, as I might note later on, there are a few other conceptual advantages with a direct defintion, too. But more on it, later, if there is enough interest in this topic.</p>
<p>--Ajit</p>
</div></div></div>Mon, 19 Feb 2018 13:15:13 +0000Ajit R. Jadhav22146 at http://imechanica.orghttp://imechanica.org/node/22146#commentshttp://imechanica.org/crss/node/22146Lode Angle History?
http://imechanica.org/node/10199
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
All,
</p>
<p>
In my research, I have been using the Lode angle of the stress tensor in a fracture model. Consequently, I've been reading up on the Lode angle and have found that just about each publication defines their own Lode angle that is slightly different from everyone else. So, I've searched around to find out the history of the Lode angle, but have come up empty-handed. I would imagine that the Lode angle is named after some Mr. Lode, but I cannot even find his first name. Do any of you know where, when, or who came up with the idea of the Lode angle? Particularly, I would like to get a copy of the first publication, if you happen to have one or know where one might be.
</p>
<p>
Thanks,
</p>
<p>
Scot Swan
</p>
</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/109">Ask iMechanica</a></div></div></div><div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/822">Stress Tensor</a></div><div class="field-item even"><a href="/taxonomy/term/966">invariants</a></div></div></div>Mon, 02 May 2011 16:34:32 +0000mswan10199 at http://imechanica.orghttp://imechanica.org/node/10199#commentshttp://imechanica.org/crss/node/10199Stress invariants for anisotropic materials
http://imechanica.org/node/8649
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hello everyone,
</p>
<p>
Does anyone know how to obtain stress invariants for an anisotropic materials, for example a composite material and there are how many of them? Could you pinpoint me to the correct reference?
</p>
<p>
Thanks a lot.
</p>
<p>
Khong
</p>
</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/109">Ask iMechanica</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/822">Stress Tensor</a></div></div></div>Tue, 03 Aug 2010 11:21:11 +0000Khong Wui Gan8649 at http://imechanica.orghttp://imechanica.org/node/8649#commentshttp://imechanica.org/crss/node/8649Plasticity and elasticity questions
http://imechanica.org/node/6979
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/169">Plasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/347">elasticity</a></div><div class="field-item even"><a href="/taxonomy/term/822">Stress Tensor</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span class="Apple-style-span"> Dear All</span><span class="Apple-style-span">I has question about these stresses.</span><span class="Apple-style-span">1.Are deviatoric stress generate plastic deformation?</span><span class="Apple-style-span">2. Are hydrostatic stress generate plastic deformation?</span><span class="Apple-style-span">3.What is the advantage of transferring strain tenser orientation to the principle coordinate axes? </span><span class="Apple-style-span">4. What is the difference between rupture and fracture stress?</span></p>
</div></div></div>Fri, 23 Oct 2009 19:13:17 +0000hiader k. mahbes6979 at http://imechanica.orghttp://imechanica.org/node/6979#commentshttp://imechanica.org/crss/node/6979