Talk:Von Mises yield criterion

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1 Ratio of the yield and ultimate strengths.
2 Subscript order in third term of local coord equation
3 first equation
4 Re-organization of this article
5 Recent changes (19 Jan 2007)
6 This article lacks layman definition
7 yield strength as critical value
8 maximum octahedral shear stress
9 von Mises stress
10 substitution of the generic critical value by the fracture strength

[edit] Ratio of the yield and ultimate strengths.

I believe your sentence, "It is most applicable to ductile materials where the ratio of the yield and ultimate strengths is near 0.577." should better read " It is most applicable to ductile materials where the ratio of the shear yield to tensile yield strengths is appx. 0.577, or 1 over the square root of 3.

You are right, this was my mistake.

-samba6566

[edit] Subscript order in third term of local coord equation

Since the third term is squared, it does not matter whether the sigma(x) or the sigma(z) come first (mathematically). The mathematical convention, however, is that the terms are presented in order, x then y then z with there complimentary terms following each. —The preceding unsigned comment was added by Samba6566 (talk • contribs) 23:17, 2 March 2007 (UTC).

[edit] first equation

Was this supposed to say sigma_v over square root of 3? Readertonight 01:23, 17 August 2007 (UTC)

You are absolutely right! Thanks! Alex Bakharev 04:09, 17 August 2007 (UTC)

[edit] Re-organization of this article

I started improving the section on the Yield Criterion, and then realized that both sections within this article are redundant. Which made me think that this article needs to be re-organized and include the following: 1- what the von Mises yield criterion states (definition and assumptions) 2- equations and figures for the yield function and yield surface 3- equations for the different stress conditions (triaxial, biaxial, uniaxial, and pure shear) 4- The Flow Rule Associated with the von Mises Yield Function I will try to re-organize this article according to these suggestions. Please comment Sanpaz 00:39, 1 December 2007 (UTC)

I just posted the new version of the article. I know it is a huge change, but I saw necessary to add more content and re-organize the article. There is still more to be added. I apologize for the big change. Please comment Sanpaz 02:39, 4 December 2007 (UTC)

It needs an introductory paragraph and those horizontal lines ought to be removed in favor of sections... I'll get to it later but for now I'll add the page to Wikipedia:WikiProject Engineering. --Explodicle 19:10, 4 December 2007 (UTC)

[edit] Recent changes (19 Jan 2007)

These changes deviate the intent of the article. The article is about the overall yield criterion, not solely about the interpretation of, or the use of it(von vises stress). However, the concept of von mises stress needs to be included. Sanpaz (talk) 18:36, 19 January 2008 (UTC)

I reintroduced the concept of Equivalent stress or von Mises stress into the article, but into its correct spot within the context of the article (i.e. in the uniaxial stress condition section ). Sanpaz (talk) 20:06, 19 January 2008 (UTC)

[edit] This article lacks layman definition

I found a much more succinct and usefully answer on yahoo answers.

"What is Von Mises Stress in layman terms?

Good question. Von Mises Stress is actually a misnomer. It refers to a theory called the "Von Mises - Hencky criterion for ductile failure".

In an elastic body that is subject to a system of loads in 3 dimensions, a complex 3 dimensional system of stresses is developed (as you might imagine). That is, at any point within the body there are stresses acting in different directions, and the direction and magnitude of stresses changes from point to point. The Von Mises criterion is a formula for calculating whether the stress combination at a given point will cause failure.

There are three "Principal Stresses" that can be calculated at any point, acting in the x, y, and z directions. (The x,y, and z directions are the "principal axes" for the point and their orientation changes from point to point, but that is a technical issue.)

Von Mises found that, even though none of the principal stresses exceeds the yield stress of the material, it is possible for yielding to result from the combination of stresses. The Von Mises criteria is a formula for combining these 3 stresses into an equivalent stress, which is then compared to the yield stress of the material. (The yield stress is a known property of the material, and is usually considered to be the failure stress.)

The equivalent stress is often called the "Von Mises Stress" as a shorthand description. It is not really a stress, but a number that is used as an index. If the "Von Mises Stress" exceeds the yield stress, then the material is considered to be at the failure condition.

The formula is actually pretty simple, if you want to know it: (S1-S2)^2 + (S2-S3)^2 + (S3-S1)^2 = 2Se^2 Where S1, S2 and S3 are the principal stresses and Se is the equivalent stress, or "Von Mises Stress". Finding the principal stresses at any point in the body is the tricky part."

I find this a lot with engineering articles, they are all written at graduate student level. But if you think about it, someone of that level, wouldn't need to look this up. Try to write these kinds of articles in such a way that someone can be like "wait, what is von mises stress anyways", and then they can come here and figure it out. Not many people ask "wait, what was the mathematical approximation you used to find this von mises stress" —Preceding unsigned comment added by 75.162.56.142 (talk) 23:43, 9 March 2008 (UTC)

[edit] yield strength as critical value

I have shad more light on the meaning of the yield strength with respect to the yield criterion. IMHO, it makes best sense to describe the yield condition such that it is met when the von Mises stress reaches the yield strength. The previous version, though formally a correct alternative, did not show this. It used a critical stress k that was related but not equal to the yield strength. This is neither instructive nor common practice, at least not in my surrounding.

Unfortunately, my changes induce many consistency changes, namely the factor 3, when the squared alternative of the yield criterion was used. I am afraid, I might have missed some instances, when screening the text. I will check more often the consistency of this article in the coming days and try to eliminate all errors that I might have introduced. However, I would deeply appreciate your assistance with that.

I hope everyone agrees that the yield criterion: influence - resistance = 0 is better stated with the yield strength being the resistance. I am sorry for the amount of work this conceptual change induces. Tomeasy (talk) 09:03, 23 April 2008 (UTC)

I have to review closely what you just changed. My first impression is that the changes you made are not consistent with what most plasticity books present. The subsection you added about Arbitrary load is not really an arbitrary load. This topic was already covered in the "Uniaxial Stress" subsection (Von Mises criterion is just a simplification from Uniaxial to triaxial load). So this subsection you created should be deleted. Any comments?Sanpaz (talk) 15:13, 23 April 2008 (UTC)

You are right, there are probably still inconsistencies with the remainder of the earlier version and I am sorry that. As I said, I am not yet done and I would appreciate your help.

The clue about the arbitrary load section is that, in deed, it holds for all loading cases, so even the triaxial one you mentioned. The only assumption is the symmetry of the stress tensor (which, I think, we should actually mention somewhere. As you probably know, many theories exist where this assumption is not made). The uniaxial stress, however, is much more special in that it has only one principal stress unequal zero. As you can see from my formula, it states (correctly) the dependence on three arbitrary principal stresses, which is a prerequisite if one wants to capture triaxial loading cases. Remains only the question as to whether the formula is correct. I can give you my word that it is and I am working on metal plasticity every day. This statement is of course not sufficient, but there are sources given below. Anyway, since you seem to like what has been in the article before, please observe that my formula is and has been used (underwater) in the section mathematical formulation. I guess you agree that the yield function as it is introduced there is valid for any loading case.

What do you think about my main point, that is to state the yield criterion as a comparisson of the yield strength and the von Mises stress. Actually this is the only thing I changed. Besides this I did not change a single assumption or concept. What was before compared the von Mises stress divided by sqrt(3) to the yield strength divided by sqrt(3). Qualitatively, this is the same, because one is interested in the sign of this function rather than it's value. However, it is uncommon and most important for Wikipedia by far less instructive. As you explained the yield strength is determined in a simple uniaxial loading experiment and we can assume that relatively many people are familiar with that and the result: the yield strength of the material. Therefore, it makes sense to formulate the criterion on basis of this popular quantity.

You mentioned sources. Since the topic is quite basic there are abundant. I have not found any that formulates according to the earlier state of the article. However, I do not doubt it exists, since it is also correct. I give you two text book references from the most famous people who are doing research in my field, that is Computational Mechanics. Both compare the equivalent stress (the von Mises stress) to the yield strength:

* Simo JC and Hughes TJR (1998) Computational Inelasticity, Springer-Verlag, New York.

* Belytschko T, Wing KL, and Moran B (2000) Nonlinear Finite Elements for Continua and Structures, Wiley, Chichester.

Tomeasy (talk) 16:57, 23 April 2008 (UTC)

To clarify: $\sigma^{dev}=s_{ij}(\mathbf e_i \otimes \mathbf e_j)=\boldsymbol{\sigma} - \frac{1}{3}\left( \boldsymbol{\sigma}\cdot\mathbf{I} \right) \mathbf{I}$ . This concept is already in the subsection Uniaxial Stress. The convention on how is written is different, though. I just undid the deletion of the subsection. I will try to understand well your point first.Sanpaz (talk) 17:16, 23 April 2008 (UTC)

Thanks for your efforts. Again you are right, the deviatoric stress is defined twice and we should eliminate one. I allowed myself to correct your formula, since a tensor basis is formed by the dyadic product of two basis vectors and not just one. This way they are in deed identical. Tomeasy (talk) 17:31, 23 April 2008 (UTC)

Thanks for the correction in sij Sanpaz (talk) 17:37, 23 April 2008 (UTC)

I understand what you are suggesting now. Pretty much, we are talking about different ways of showing or expressing the Von Mises Criterion. You are suggesting to express the criterion as a function of $\ \sigma_v$ . However, the original formulation by Von mises,(see [1], says that yielding occurs when $\ J_2$ reaches a critical value $\ k$ . However, the Von mises stress $\ \sigma_v$ is a concept derived or obtained from this formulation. It is an interpretation of the formulation. This interpretation is that to obtain the yield strength of material loaded triaxially (3-D) one needs only the yield strength of the material obtained from uniaxial tests. This uniaxial strength is the Von Mises Stress $\ \sigma_v$ . So, my suggestion is to be general in the formulation, and later talk about the interpretations given to the formulation, which is what the article had before.Sanpaz (talk) 19:40, 23 April 2008 (UTC)

Hill is a perfect reference, but so are mine.

The formulation that I advocate is by no means less general than the one you prefer. In fact, they are equivalent. So please, do not claim otherwise or give a basis to your claim.

As I already explained, the yield strength is the most instructive critical value. Please react on this claim of mine. I find instructiveness is an important point when editing Wikipedia.

to obtain the yield strength of material loaded triaxially (3-D) one needs only the yield strength of the material obtained from uniaxial tests. There is only one yield strength! No triaxial strength, uniaxial strength, etc.

This uniaxial strength is the Von Mises Stress $\ \sigma_v$ . No. Again there is no uniaxial strength. The rest is just wrong. The von Mises stress in not the strength. The strength is a material constant, while the von Mises stress depends on the loading. In the special case of an unloaded material it is simply zero.

OK, with the last two points it may be that I just explained you what you had already known. If so, then my apologies but also please try to formulate more precisely. Tomeasy (talk) 20:09, 23 April 2008 (UTC)

"The formulation that I advocate is by no means less general than the one you prefer. In fact, they are equivalent.":

I have not disagreed with that. That's why I said that we are just talking about different ways of expressing the criterion.

"There is only one yield strength! No triaxial strength, uniaxial strength, etc."

I should have written: to obtain the yield strength of material loaded from any direction (3-D) one needs only the stress at yield of the material obtained from uniaxial tests. And, there are such things as uniaxial strength, shear strength, biaxial strength...

"No. Again there is no uniaxial strength. The rest is just wrong. The von Mises stress in not the strength. The strength is a material constant, while the von Mises stress depends on the loading. In the special case of an unloaded material it is simply zero."'

I won't argue that. What I was trying to state was that the uniaxial stress at yield is the same Von Mises Stress $\ \sigma_v$ .

I still think the criterion should be stated based on J2 and not $\ \sigma_v$ . I prefer the idea of starting the formulation with J2 and then later coming to the definition of the von mises stress $\ \sigma_v$ . I still see $\ \sigma_v$ as a conclusion or interpretation of the criterion, and not the original thinking of the criterion. See [2] Perhaps there in a better way to emphasizes your point about $\ \sigma_v$ is some other way without changing the general formulation.
I know what is the intent of the Arbitrary Loading Condition section. Before, this condition was addressed in the general explanation for the criterion, it was implied in the Mathematical Formulation, as it was presented as a general state of stresses, and it was not particularly included in the section about Different Stress Conditions (this section was for particular stress cases such as uniaxial, pure shear, etc). I think that is fine that you have included the general condition.

Sanpaz (talk) 01:30, 24 April 2008 (UTC)

It's not only the uniaxial stress at yield [that] is the same [as the] von Mises stress. Even at any other point of the uniaxial loading curve the uniaxial stress is equal to the von Mises stress. Reason is, as the article shows, that under uniaxial loading the uniaxial stress is identical to the von Mises stress. Further, at yield (the event that you pointed at) it additionally holds that this unixaial stress cum von Mises stress equals in magnitude also the yield strength and the yield function is zero.

Sorry for being so picky on words, but I think we should try to make correct statements only.

So, my suggestion is to be general in the formulation, that's why reacted a little bit affected. Thanks for your clarification.

OK, one more source. Again a very good one. I do not doubt there are hundreds. Shall I show you more sources that employ my formulation?

Proposition: Let's simply discuss, why giving your or my proposition preference is better.

My main point is (1) instructiveness. But also (2) the physical unit, which is Pa for the von Mises stress, whereas J2 first has to be raised to 0.5 to match it with some critical strength, which is not even the yield strength. Another point is (3) that von Mises plasticity namely applies to metals and their strength is not measured by a shear experiment, which is probably the most simple to measure for soils, but in a uniaxial tensile test. I have not seen similar arguments to your position apart from pointing at (very good) references, which, however, I am equally doing.

Tomeasy (talk) 11:43, 24 April 2008 (UTC)

I have to be honest with you, I really do not like much the new formulation. Not that it is not correct, but it is not what most books that I am familiar with would explain (a good example is Hill's book which is a very well known book). And if I were new to Wikipedia and I knew about the topic and I saw this formulation as it is now, I would be hesitant about the whole formulation. I already explained a few comments above the reason for my position on the subject, but I would like to state it again: The original formulation by Von Mises stated that yielding occurred when the value of the second deviatoric stress invariant $J 2$ reaches a certain value. The formulation you are proposing, to me, is just an interpretation which substitutes $J 2$ as a function of a new stress value called Von Mises stress (I do not know who came up with that term, it would be interesting to know though). Perhaps the fields we worked on treat the criterion differently. I am a civil engineer so I am use to the other formulation.

So, what I would prefer is to state the criterion the classic way (the way it was before), and expand the formulation to account for the Von Mises stress

σ v

as you are proposing now. This way, I think it will give more clarity to the article. Think about what I just wrote, you do not need to reply now nor change anything in the article. But consider what I am suggesting. - Sanpaz (talk) 00:25, 28 April 2008 (UTC)

Ok, you are really challenging my references, claiming their formulation was less classical. Note, that I had appreciated your sources and that my argumentation was so far was not based on whose references are the better. Since you choose not to argue about the points that I find important with respect to wikipedia editing, I turn to the only point that seems to interest you: What is more classical, common, or accepted.

The authors of the text books have according to ISI - web of sience the foolowing h-indices: Belytschko (59), Hughes (57), Simo(50). For sure, Hill (I wasn't able to get his exact number for there are to many R. Hill), as a great scientist, has also a high h-index, but probably lower than the one's I mentioned. At least one can say, according to this most objective numerical measure, it appears inappropriate to deem their way of writing just an interpretation of the original form. Actually we are not talking here about an interpretation as the forms are mathematically identical. I find the term interpretation derogative in this discussion.

Looking what the internet, here google, offers to von Mises yield criterion is also revealing. I just researched the first three non-wikipedia hits. Those were the hits # 2, 3, and 4. Wikipedia was on top :-) Hit #2: Engineering Fundamentals makes the statement as I propose. #3: NASA ditto. #4 University of Cambridge formulates the criterion in a way that is neutral to your and my proposition. When becoming specific, however, they first turn to the uniaxial case and consequently write From above, if, s1 = Y, s2 = s3 = 0, then the constant is given by 2Y2. This is the von Mises Yield Criterion. Your proposition is later attended by the words The von Mises criterion can therefore be expressed as:. I really did not look into more items on google. Therefore, I expect the large majority will follow the formulation that I advocate.

So much to the point that appears of sole significance to you. Nevertheless, I find instructiveness, applicability important points to consider as well. It would be nice, if you attended those concerns as well or give reasons for ignoring this line of thought.

Tomeasy (talk) 00:28, 29 April 2008 (UTC)

It is not a discussion of who has the best reference. It is a discussion of what is the best way to present the criterion. Von Mises did not formulate his criterion thinking about a Von Mises Stress. Some body else came up with that term later. He simply stated that yielding occurred when $\ J_2$ reached a certain numerical value. He just formulated his criterion on mathematical grounds. Knowing that the formulation is $\ f(J_2) = \sqrt{J_2} - Constant = 0$ , one can deduce from the case of pure shear that the Constant is the yield stress in pure shear, $\ k$ . Later, in the case of uniaxial stress conditions, one can deduce that $\ k = \frac{\sigma_y}{\sqrt{3}}$ , where $\ \sigma_y$ is the yield stress in uniaxial tension, which you called $\ \bar \sigma$ in your formulation. And that is the logical order in which the article should present the information. Your formulation simply jumps straight away into formulating the criterion as a function of the yield stress in uniaxial tension, $\ \bar \sigma$ , and modifies the first term of the criterion into the von Mises stress by multiplying the original formulation by $\ \sqrt 3$

$\ \begin{align} f(J_2) &= \sqrt 3\left (\sqrt{J_2} - k\right ) = 0 \\ f(J_2) &= \sqrt{3J_2} - \bar{\sigma} = \sigma_v - \bar{\sigma} = 0 \end{align}$

and that is why you obtain your formulation, which is correct, but is not how the criterion was formulated.

Maybe that is the conclusion to our discussion: Just to formulate the criterion in that order.

And that is it. That is my reasoning which I think is correct. I don't have anymore to say about the subject. I cannot be more clear than that. This discussion was good to make things more clear about the subject, but I don't see the point of going on and on about it. Cheers - Sanpaz (talk) 00:22, 1 May 2008 (UTC)

[edit] maximum octahedral shear stress

I am unable to comprehend what is said about the maximum octahedral shear stress. Especially I have a problem with the second part of this equation

$\tau_{oct} = \sqrt{\frac{2}{3}J_2} = \sqrt{\frac{2}{3}}k$ .

The first part simply states the shear stress acting on the octahedron, but where does the second part come from. It is important to mention that the equation stated above is still based on a formulation of the yield criterion that compares the square root of $J 2$ to a critical value, which is equal to the shear stress at the onset of yielding in a pure shear experiment. Please note, that at present the remainder of the article is based on a yield formulation that compares the von Mises stress to the yield strength, which is equal to the tensile stress at at the onset of yielding in a uniaxial tensile experiment.

While trying to adjust this section to the latter formulation, I am in trouble, because I do not understand the second part of the equation in the first place. According to my naive thinking the shear stress on the octahedron at at the onset of yielding should (with the first formulation of the yield criterion in mind) simply equal the critical value k, and not $\sqrt{\frac{2}{3}}k$ . I probably miss a point here, perhaps some geometric reason. Can anyone help me out so that a consistent version can finally be established?

Independent from this, I acknowledge that it is not yet decided which formulation should stand in the end. However, my question to the above formula exist even if the yield criterion is stated as in the earlier version. Moreover, I am trying to establish a consistent version of the yield strength based formulation, so that in the end we can simply choose between the two, both being formally OK. This section is the last piece that lacks this consistency. Tomeasy (talk) 08:34, 25 April 2008 (UTC)

With the previous formulation, the value of the octahedral shear stress is

$\tau_{oct} = \sqrt{\frac{2}{3}J_2} = \sqrt{\frac{2}{3}}k$ , which is obtained by replacing the von mises criterion equation $\ J_2=k^2$ with k being the yield stress in pure shear. But with the new formulation that you are proposing you have to change the value of k for $\ \bar{\sigma}$ - Sanpaz (talk) 00:25, 28 April 2008 (UTC)

I was afraid it is. Well then we should delete this whole paragraph, since it does not state anything else then what the octahedral shear stress is. One might argue to leave that, even though one can find it also on the respective page. Definitely misleading is the conclusionthat anything reduces here to the yield criterion, as if the yield criterion was derived her. That's how I erroneously understood it. What is actually done here, as you explain, is that the yield criterion is inserted in the formula for the octahedral shear stress, nothing more. Any further claim is a short circuit. Of course, the formula will then contain this information. Note, this thread is independent from the discussion on how to formulate the criterion. Even with the criterion being based on the yield strength, the reduced to formulation is senseless. Tomeasy (talk) 17:24, 28 April 2008 (UTC)

Another thing: The precedent text states: yielding begins when the octahedral shear stress reaches a critical value k. And then it is stated \tau_{oct} =...= \sqrt{2/3} k. To say the least this is very confusing. You have explained that therein the yield criterion is used. This is admitted if the stated formula applies to the state at the onset of yielding, which is also implied by the text. So the text and the formula should in deed harmonize, which they don't. Hence, it's not only confusing, but also wromg somewhere. Btw, nothing to do with k being the yield strength or the root of J2. Tomeasy (talk) 17:49, 28 April 2008 (UTC)

For reference, Elastic and Inelastic Stress Analysis by Shames and Cozzarelli first describe the Tresca yield criterion, then the Mises yield criterion in the classic

J 2

way. Then they describe the maximum distortion strain-energy theory and the octahedral stress theory and then shows those two to be the same as the Mises theory. It gives a footnote:

"The physical interpretation of the Mises yield criterion via the maximum distortion energy theory was suggested by H. Hencky in 1924. The interpretation based on the octahedral stress concept was proposed by A. Nadai in 1937. Other interpretations have also been proposed. See R. Hill, The Mathematical Theory of Plasticity (New York: Oxford University Press, 1950), Chap. 2."

—Ben FrantzDale (talk) 01:30, 29 April 2008 (UTC)

[edit] von Mises stress

I have set up a new article for the von Mises stress. I think can take a little bit the pressure away as to how formulate things on this page. Actually, one could delete now the equations stated in the subsection arbitrary loading conditions as they are double now on the new page. I will leave this deletion to other users, since I am not disturbed by this redundancy. Tomeasy (talk) 06:01, 27 April 2008 (UTC)

I don't think there is a need to create another article about this, if it is already in this article. I would delete that article.Sanpaz (talk) 23:34, 27 April 2008 (UTC)

[edit] substitution of the generic critical value by the fracture strength

I have substituted the value k by the fracture strength, as can be done if the yield criterion is stated the way it currently is. This simplifies the whole article enormously. That's why I chose to do so. However, should we decide in the future to step back to the generic critical value (the one that equals the square root of J2) than we will also revert these edits. Please, comment on what you prefer. Tomeasy (talk) 06:05, 27 April 2008 (UTC)