Talk:Cauchy stress tensor

Definition of couple stress
The article repeatedly discusses "couple stress", but doesn't give a definition or link to a definition. What is couple stress? --Eihjia (talk) 16:45, 2 October 2013 (UTC)

Article created from "stress" section
This article used to be a section of stress (mechanics); it has been split off because that article was way too long, difficult to read and edit. Another part of that article was split off as Euler-Cauchy stress principle. There are still many rough edges, especially on the partition between these two sister articles. Hopefully we can fix them soon. All the best, --Jorge Stolfi (talk) 22:23, 23 February 2013 (UTC)
 * The Cauchy stress tensor and Euler-Caucy stress priciple articles are better placed together. The principle is needed to explain where the Cauchy tensor comes from. I merged both articles. sanpaz (talk) 19:34, 30 May 2013 (UTC)

Octahedral stress
Shouldn't "octahedral stress" have its own article? (Consider readers who find the term somewhere and want to know what it is. It does not seem useful to redirect them to this article, even if it is to a section.) --Jorge Stolfi (talk) 22:26, 23 February 2013 (UTC)
 * You may be right. The octahedral stress section does not fit in the Cauchy stress tensor article. I do not have any suggestions on where to place it at the moment. I have to think about it. sanpaz (talk) 19:43, 30 May 2013 (UTC)

LaTeX vs HTML
LaTeX for formulas inside text? or better to use HTML? My preference is LaTeX. What is the consensus right now in Wikipedia? sanpaz (talk) 14:35, 9 June 2013 (UTC)

Tensor Type
I've changed the tensor type from (2,0) into (1,1) as it is a linear mapping and not a bilinear form!! Do you agree?
 * No. The section Transformation rule of the stress tensor refers to the stress tensor as contravariant in both indices, so I'm reverting this change for consistency, although the phrasing could still be improved. Since the stress tensor is almost always discussed in the presence of a metric, one may refer to contravariant, covariant, or mixed forms, and this article specifically chose the contravariant form. 98.248.228.159 (talk) 08:14, 17 January 2014 (UTC)


 * I'm confused. If the area normal, n is a vector, and T is a vector, both are of type (1,0) tensors, then the stress tensor must be type(1,1), should it not? Craigde (talk) 20:11, 18 January 2015 (UTC)


 * OK, I see the problem. The index placement on the stress tensor is a matter of convention. Indices of the stress tensor may be raised or lowered with abandon by balanced application of the metric tensor. There are 4 valid choices of index placement. The historical choice is to caste vectors as column vectors, which are implicitly contravariant vectors. If T and n are column vectors, the stress tensor must be type (1,1).


 * What is lacking in this article is a declaration of the convention to be used, as well as the choice of dimensional assignments. This should be stated at the very beginning in a section of its own. The same applies to the mother article "Stress (mechanics)".


 * The section Transformation rule of the stress tensor is peppered with notational errors but I haven't yet put in the hard work to fathom whether the transformation equations given are consistent with the transformation of a type(0,2) tensor. 2001:5B0:2BFF:3EF0:0:0:0:39 (talk) 09:29, 21 January 2015 (UTC)
 * What are the notational errors? The notation used is common in the continuum mechanics literature. sanpaz (talk) 14:10, 21 January 2015 (UTC)


 * In your texts do tensor products use the Einstein summation convention, but where both indices are lower indices or both are upper indices?Craigde (talk)
 * This is one reference Continuum mechanics by Spencer . Please see other references at the bottom of the article. I understand that proper tensor notation has upper and lower indices. But because of the nature of the stress tensor, authors usually only use lower indices. You are right in saying that it is necessary to explain why the convention in the article is the way it is presented.sanpaz (talk) 04:19, 22 January 2015 (UTC)


 * Wow. That text is a Dover reprint of a 1929 publication. It's time to move into the 21 century.Craigde (talk)
 * That is not the only references. Like I said before, check the other more recent reference listed in the article. The one I showed you was the first one it came to mind. sanpaz (talk) 20:49, 22 January 2015 (UTC)


 * Looking over Google Books, it appears that classical physicists and engineers had not yet discovered the power of index placement until T.J.Chung, "Applied Continuum Mechanics", 1996. Craigde (talk) 01:26, 23 January 2015 (UTC)

Notational Blunder
Contrary to convention, throughout the entire article, lower indices are used exclusively in tensor equations. I attempt to mentally edit every tensor equation I see, yet there seems to be insufficient information to do so.

Without proper index placement, it is indeterminate how things change under a general coordinate transformation or even a simple rotation.Craigde (talk)
 * The notation used in the article is standard in the continuum mechanics literature and has evolved into this form over the past 60 years to avoid unnecessary complication. The underlying assumption is that the components of the tensor are with respect to an orthonormal basis, i.e., the metric tensor is the rank-2 identity tensor.  That means that raising or lowering the indices does not change the values of the components.  The expressions get much more complicated in general curvilinear bases.  In general, direct tensor notation is preferred so that coordinate systems can be avoided altogether.  But that's not useful for understanding the concepts discussed in this article. Bbanerje (talk) 22:09, 14 April 2016 (UTC)
 * The problem is by not abiding by the einstein summation convention confusion arises. It does not remove unnecessary complication, it adds unnecessary complication. Just because the literature changes slowly it does not mean it is the best way. Rostspik (talk) 23:03, 8 February 2018 (UTC)

Figure 2.3 missing
Where is figure 2.3? Section "Cauchy’s stress theorem—stress tensor" refers.--Das O2 (talk) 20:07, 15 May 2015 (UTC)

Needs introduction for non-experts
Could somebody perhaps rewrite the beginning to explain the basics to somebody who is NOT already an expert in the field? 76.241.142.80 (talk) 03:27, 21 March 2018 (UTC)
 * it contains 9 possible values Zornyank (talk) 02:27, 22 May 2020 (UTC)

About the second invariant of stress and deviator stress
Since this is my first time to observe a (possible) error on my favourite Wikipedia page -- yes, the one about Cauchy stress, I am using since years almost every week -- I thought I rather talk first before I attempt to change it myself. Actually, one of my students pointed me to this inconsistency, and now after hours of checking, I am pretty sure its a typo. I also checked all the definitions, they are ok, its only one tiny typo ...

Invariants in principle don't care about sign, they are invariants anyway. However, if we want to associate J_2 to specific energy, and since we want to take the square root of J_2 many times, it should be positive. If J_2 is positive (all the time) then the sign in the deviator characteristic equation must be positive of what it is now: -J_2 -> +J_2

Can someone pls. confirm/check - maybe I am not seeing the point, but I think its a typo.

best regards Stefan Lui12358 (talk) 13:21, 14 December 2020 (UTC)

I think the problem is not the sign itself, because there exist different conventions in different communities regarding the sign of the second invariant.

The real issue is that the second invariant of the stress tensor (I_2) and the second invariant of the stress deviator tensor (J_2) in this article use opposite conventions. In fact, if you take the formula for I_2 and use it with deviatoric stress components instead of the components of full stress components, you end up with -J_2, which is confusing.

I think consistent conventions should be used throughout the article for all invariants. Mariupolo (talk) 15:14, 1 April 2021 (UTC)


 * Agree. I think the confusion may be related to the fact that a definition of the invariants that gives something that "looks positive" for the second invariant of a matrix will not give something that "looks positive" for the second invariant of the "deviatoric matrix" related to the original matrix.
 * Let me try to explain this. If we define the second invariant as the coefficient in the term proportional to lambda in the determinant of:
 * M - lambda*IdentityMatrix,
 * where M is the diagonal matrix with the numbers a, b, and c on the diagonal, then we get:
 * I2 = -(a*b + b*c + c*a)
 * for the second invariant of the original matrix (i.e., we get something that "does not look positive"). However, if we apply this same definition to the matrix:
 * DM = M - Trace{M)*IdentityMatrix/3,
 * then we get for the second invariant of this matrix:
 * J2 = ( (a - b)^2 + (b - c)^2 + (c - a)^2 )/6,
 * i.e., we get something that certainly is non-negative (for real a, b, and c). Perhaps the definitions used in the article are trying to make sure the invariants are positive (or, at least, non-negative) when the entries of the matrix are positive, but that seems misguided to me. It makes more sense to have a uniform definition of the invariants.
 * Beyond issues of consistency and conventions, the positive sign in the term lambda^3 in the definition of the invariants of the deviatoric stress tensor seems wrong. The sign of J2 also looks wrong (i.e., I do not think the result obtained matches the definition used). Same for the sign of J3.
 * I prefer to define the invariants as:
 * det(M - lambda*IdentityMatrix) = -lambda^3 + I1*lambda^2 + I2*lambda + I3
 * without negative signs, regardless of what matrix I am dealing with. This will give us the result we normally want for the von Mises criterion, sigma_eff^2 = 3*J2 2601:546:C480:750:5062:4183:12DF:1DF0 (talk) 19:53, 31 May 2022 (UTC)

Why include so much detail on eigenvalue calculation?
The section "Principal stresses and stress invariants" includes an exceedingly thorough calculation of the eigenvalues of a 3x3 matrix, up to and including a thorough derivation of the characteristic polynomial and detailed step by step computations of each of its terms. Why is this here? Can't the article just link to Eigenvalues_and_eigenvectors, add a sentence or two of intuition for what they mean for the benefit of people who haven't studied linear algebra, and then just say that the eigenvalues are the principal stresses? (After introducing the three stress invariants, it could even mention in passing that those combinations are the coefficients in the characteristic polynomial.) It just seems very unnecessary to include so much detail about an absolutely routine bit of math in this article. What am I missing? Steuard (talk) 01:35, 9 January 2023 (UTC)