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# A new theory of stress?

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I was browsing the discussion page for Stress in Wikipedia when I came upon this interesting comment:

"

**Refutation of Cauchy stress**

*The theory of stress based on Euler & Cauchy is now refuted. The profound incompatibility of this theory with the rest of physics, especially the theory of potentials and the theory of thermodynamics, has been documented in*

Koenemann FH (2001) Cauchy stress in mass distributions. Zeitschrift für angewandte Mathematik & Mechanik (ZAMM) **81**, suppl.2, pp.S309-S310

Koenemann FH (2001) Unorthodox thoughts about deformation, elasticity, and stress. Zeitschrift für Naturforschung **56a**, 794-808

Furthermore, three articles are due to appear in print in the **International Journal of Modern Physics B** (accepted for publication May 2008, expected publication date August 2008).

In the first paper "On the systematics of energetic terms in continuum mechanics, and a note on Gibbs (1877)" I show that the First Law of thermodynamics has been routinely turned upside-down in continuum mechanics.

In the second paper "Linear elasticity and potential theory: a comment on Gurtin (1972)" I show that a well-known continuum mechanicist must have discovered the fatal flaw in the Euler-Cauchy theory in 1972, but he did his best to mislead his readers.

In the third paper "An approach to deformation theory based on thermodynamic principles" I give an outline of the new approach, which is basically a transformation of the theory of thermodynamics from the scalar form (implying that it is isotropic) into vector field form, in order to consider anisotropic boundary conditions and/or materials. Fully satisfactory predictions for a number of phenomena are presented which were considered unsolved so far, such as kinematics of plastic simple shear, cracks in solids, turbulence in viscous flow,

elastic-reversible dilatancy and others.

The new theory has no precursors, except for two papers by Rudolf Clausius (1870) and Eduard Grueneisen (1908) which were completely ignored by the continuum mechanics professional group. The Clausius paper is essentially a modern counter-proposition to the Navier-Stokes equations.

All the papers mentioned above, including the Clausius and Grueneisen papers (in English), can be downloaded from my homepage, see [1]

Falk H. Koenemann

Aachen, Germany, 1 July 2008 —Preceding unsigned comment added by 217.250.179.59 (talk) 08:29, 1 July 2008 (UTC)

"

The page that Koenemann links to is http://www.elastic-plastic.de/ . Cranks are not uncommon in physics but very few point their attention towards continuum mechanics. I wonder what iMechanicians have to say about Koenemann's ideas.

-- Biswajit

- Biswajit Banerjee's blog
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## Comments

## Seriously?

Biswajit,

Is there anything specific on this guy's website that you think is of interest? I have glanced at some of the papers and I am not sure where I would start in pointing out the flaws. One example is that he claims that the continuity equation "has the side effect that energy is conserved". Have you looked at this? Do you understand where these comments are coming from? At first glance I cannot imagine how he has gotten these papers past review.

Chad

## Re: Seriously?

Chad,

I had a brief glance at the page last night. But the more interesting point, in my view, was that he had managed to get his papers published! Now that we have had a few comments on iMechanica, I'll post a link to this discussion on the discussion page on Wikipedia for the benefit of other confused readers there.

-- Biswajit

## Biswajit, these wiki people seem confused about other issues...

We really need some good people to deal with this wiki page, if you read my recent post Some news from WIKIMANIA / change in the system this is one case were we really need some "administrator" with better grasp on the field, before Wikipedia gets it published. Maybe you should contact the wikipedia top people to ask if "imechanica" can be in turn this "administrator" !

Read the other point and how poorly it starts :(

Brittle and ductile

I am skeptical about "By definition, brittle materials fail under

normal stress, and plastic or ductile materials fail under shear

stress." Don't ductile materials undergo plastic yield before failing

whereas brittle materials simply break? (I am in the midst of a stress

class; if nobody touches this, I'll come back to it later.)

## Re: Problems with WIkipedia

You're right Mike.

A problem that I've run into is that some people tend to be very possessive of a particular wiki page. Sometimes a correction that I make is reverted back by the "owner" of the page. What's amazing is that Wikipedia still contains some valuable and accurate information - though I wouldn't bet anything on the accuracy bit.

-- Biswajit

## I am the author

Hi Biswajit,

Monday 16 Feb 09 a friend informed me of this blog, and that my theory has been discussed there. I have to say I am disappointed that no one sent me an email. I have not known about this site and the discussion which took place 7 months ago.

Summing up the comments I saw so far, you guys weren't impressed, but so far no one bothered to look at the actual papers. I can say that all my papers are now published, and out on the shelf, all of them in the Int. J. Modern Physics B: my theory came in July 08, the Gurtin script in October, and Systematics & Gibbs in November. And certainly I did not win the reviewer approval in the lottery.

To give some background: I am not confused by the basics of continuum mechanics (henceforth CM), but I can make decisions. In my intro class 1980 at UCD I asked a number of rather simple questions, and they weren't answered. After six years of learning I summed up and concluded:

1. there were far too many items in the deformation theory that I had to swallow due to grade pressure, but nobody would explain them;

2. I had an observation in my mind which I wanted to understand, and which made me learn CM in the first place, and after so much learning I had not gotten a millimeter closer to a better understanding;

3. I had taken extensive math classes at UCD and UNH, and these applied math methods never left anything to faith or force, they worked and convinced; mathemetics was very reliable.

I started to read the fundamental texts, but with new eyes: Truesdell, Gurtin, Eringen, many others; then I decided that CM is for the birds: it is simply not possible to get from CM to thermodynamics, but without a question an elastic deformation is a change of state in the sense of the First Law.

However, these applied mathematics methods I had learned seemed to be connected by a scheme on a grander level, and I deduced that as good as I could. Only in 1994 I discovered that this way, I had reinvented potential theory, without knowing that this theory is known for 150 years. I found the book by Kellogg, Springer 1929, and all I had to change in my scripts was to throw out my self-developed terminology and replace it by that of Kellogg, it fit like a glove.

Historically, this misfit is understandable: the foundations of CM are far older than potential theory. Today I maintain that at the latest by 1850 they should simply have started CM all over again, based on the First Law. This was not done, so I did it. You will not find an Eulerian concept in my work. To sum up my theory in a few words:

The theory of thermodynamics that you know is commonly taught in scalar form (P, V). I rewrote that theory in vector field form (f, r). For isotropic boundary conditions my theory gives the same result as standard thermostatics. But I could now consider anisotropic boundary conditions, and could now treat elastic deformation as a change of state, I can calculate all its properties. (Please do not be confused by "Thermo"dynamics; I do not have the power to change the terminology, but I use the term in lieu of "Physics of changes of state", be it the dw part or the dq part. In my work I simply set dq = 0, making the theory adiabatic.)

To demonstrate just one success: from experiments it is known that in the elastic field, a simple shear deformation costs ca.12-15% more Joule per unit strain than a pure shear; in the plastic field a simple shear costs 30% less Joule per unit strain than a pure shear. Thus there is an energetic inversion across the reversible-irreversible boundary, and the differences are pretty odd. I had known about this energetic pattern, but I was hunting for something else; but when my approach was finished, the correct predictions for the numbers above just fell out.

That was before 1994, and I only realized then that I must have found something of an entirely new quality. Since I found that book by Kellogg, I know why, and I am absolutely certain now: CM based on Euler and Cauchy is for the birds.

So, let's see if anyone answers.

Falk

## Responses to Mr. Koenemann

Although I doubt Mr. Koenemann will read this post, I thought I might try to address some of the questions he posts on his site. These are very short answers, and in some cases there is a deeper discussion that may be required. The numbered questions are his, and the answers are mine.

For his comments on these questions go to:

http://www.elastic-plastic.de/Hp-5point.htm

It actually might be a good exercise for students to think about these questions and how to answer them.

1. Why do you use an equation of motion and not an equation of state?

Ans: In continuum mechanics one uses both an equation of motion and an equation of state.

2. Why do you use Newton's equilibrium condition and not the thermodynamic equilibrium condition which distinguishes system and surrounding?

Ans: Newton's equilibrium condition and what I believe you mean by "thermodynamic equilibrium" are independent concepts. Again, continuum mechanics uses both.

3. There are bonds in solids, but there's no mention of bonds in this theory. Aren't bonds important for the understanding of a solid?

Ans: Certainly bonds are important for understanding the behavior of solids. With the appropriate assumptions the stored energy in the bonds can be related to the strain energy of the solid (i.e. the equation of state of the solid).

4. Newton defined a rotating force as being perpendicular to the radius of a body. Here you define a shear force as being perpendicular to the normal of a planar element, which is an unit vector. These definitions are incompatible with one another because the magnitude of the radius vector can vary with direction whereas the unit vector cannot. Why do you believe that Newton's definition is wrong?

Ans: Although I am not familiar with Newton's definition of a rotating force (did he really define this?), these definitions appear to be independent and so are not incompatible. I do not see what the issue is here.

5. If you deform a body, say, a circle, by stretching it in X, work is done upon the body in X. Let's say this is negative work. But the body will contract in Z, so positive work is done in Z. If the volume remains constant, the work in X and the work in Z must balance, so no net work is done. Isn't this impossible?

Ans: First, if you stretch a body, the work done to stretch it will not be negative unless the body is unstable. The contraction in "Z" has no work associated with is unless there is a force in the "Z" direction during the "X" stretching process.

## Reply to Chad Landis

Blue: my five arguments from my homepage; maroon: Chad's comments; black: my replies

Chad,

1. Why do you use an equation of motion and not an equation of state?

Ans: In continuum mechanics one uses both an equation of motion and an equation of state.

A perfect mix of apples and oranges.

Newtonian work is the transformation of E_pot into E_kin and vice versa, such that their sum is invariant. The system is isolated, the process takes place

withinthe system. Thermodynamic work is PdV-work, requiring that the system interacts with its surrounding. PdV-work is work doneuponthe system. Newtonian work and PdV-work have the same units, but they are entirely different in nature. Either a process is conservative, then E_kin + E_pot = const applies; or a process is nonconservative (i.e. a change of state), then you are in the realm of dU = dw + dq. Mixing up the two work terms is arguably the worst error one can commit in physics this side of Planck. But it is very characteristic of current continuum mechanics to blur this most fundamental distinction. I had a reason to write in my own blog that the First Law of thermodynamics is emasculated in continuum mechanics (details in "Systematics & Gibbs ").2. Why do you use Newton's equilibrium condition and not the thermodynamic equilibrium condition which distinguishes system and surrounding?

Ans: Newton's equilibrium condition and what I believe you mean by "thermodynamic equilibrium" are independent concepts. Again, continuum mechanics uses both.

No, it does not. Thermodynamics distinguishes system and surrounding, continuum mechanics does not. You and I had a discussion of precisely that point, and you conked.

3. There are bonds in solids, but ther's no mention of bonds in this theory. Aren't bonds important for the understanding of a solid?

Ans: Certainly bonds are important for understanding the behavior of solids. With the appropriate assumptions the stored energy in the bonds can be related to the strain energy of the solid (i.e. the equation of state of the solid).

Assumptions are not a valid replacement for a physical term, especially not in the equilibrium conditions. If bonding forces are not mentioned, half of the acting forces are left out of consideration.

4. Newton defined a rotating force as being perpendicular to the radius of a body. Here you define a shear force as being perpendicular to the normal of a planar element, which is an unit vector. These definitions are incompatible with one another because the magnitude of the radius vector may vary with direction whereas the unit vector cannot. Why do you believe that Newton's definition is wrong?

Ans: Although I am not familiar wiht Newton's definition of a rotating force (did he really define this?), these definitions appear to be independent and so are not incompatible. I do not see what the issue is here.

The issue is

f.randfxrversusf.nandfxnbecausenis an unit vector,ris not. If you user, you need more information about the shape of the body to complete the equilibrium condition. If you taken, you implicitly assume that the body is a sphere. That's too simple. Newton usesr. Euler usesn. I cannot see that Newton was wrong. Euler is wrong.5. If you deform a body, say, a circle, by stretching it in X, work is done upon the body in X. Let's say this is negative work. But the body will contract in Z, so positive work is done in Z. If the volume remains constant, the work in X and the work in Z must balance, so no net work is done. Isn't this impossible?

Ans: First, if you stretch a body, the work done to stretch it will not be negative unless the body is unstable. The contraction in Z has no work associated with it unless there is a force in Z during the stretching in X.

Not so. If you stretch a bar in X and you prevent it from attenuating in Z, the work done per chosen change of length in X will be much larger than if you let it attenuate. Hence the surrounding does extensional work (negative) on the bar in X, whereas the bar does contracting work (positive) on the surrounding in Z. The reason for the attenuation is not a mysterious material property called Poisson's ratio, but simply the Law of Least Work (which so far has never been considered in this context before). I explained that to you, and you did not counter.

Chad: you and I had an exchange of >30 emails in late Feb 09, and you kept running out of arguments every time. Like so many before you, you always changed the subject when your arguments fell short. When I asked you to directly respond to my refutation of the Cauchy stress tensor (we had discussed this at some length, and I had sent you the page from Kellogg with Lemma 1 which Cauchy violated) you broke off the exchange, calling me intellectually dishonest; instead you wrote to my editor to wail, calling my papers shoddy, but without submitting a formal 'Comment' to which I can'Reply'. That speaks for itself: you have no arguments. You scored once: I quoted a wrong experimental paper. But that is easily corrected. I stand by my predictions.

I make the very general claim that continuum mechanics in its current form is massively and profoundly at variance with classical physics. That's the issue here.

Falk H. Koenemann

## Absurd

Falk,

It is absurd for you to claim that I ran out of arguments. The fact is that you never gave a rigorous response to any of my simple and direct questions. The reason I contacted the editors of IJMPB, was to find out if someone had "fallen asleep at the wheel" so to speak. I found my answer.

I will wait unit you correct your quotation error before I try to show you the errors in your other arguments.

Chad

## Chad,

come over to my blog .

Falk

## Trying to understand.

It's certainly interesting to challenge our understanding of these underlying parts of our discipline. However, I really have trouble understanding what is being talked about. I thought the poster and ultra-short five-point summary would help me get what Koenemann was saying, but they proved unhelpful to me.

The math on the poster page will not display right, and I just have to guess too much. I was glad to see a link to a .doc file at the bottom that I might be able to view and was glad when I saw it was missing but there was a similar PDF file posted. However, this was only for a small part of the page containing only images and plain text, which are easily displayed on the web.

The five point summary leaves me without any real answers. I am very sad if no one tried to express Konemann's concerns, but I have a hard time believing that is really the case. Perhaps no one addressed them to his satisfaction. (In a more cynical mood, I might just say that no one told him he was right.) Forgive me for being harsh, but Konemann was very harsh as well.

His five points are

This doesn't really help me.

Number 3 confuses me. It's hopelessly vague. It has nothing to do with whether continuum mechanics presents a reasonable way to understand the behaviour of materials.

Number 4 is equally confusing. There are several differences between Newtonian mechanics and continuum mechanics, why isn't this one of them? If I understand the criticism correctly, this is indeed a simplification of physics we take at the infinitesimal level to be able to solve problems.

Number 5 is not helpful for me. Firstly, the description is poor. I suppose we are talking about a disc in the X-Z plane? Secondly and more importantly, I don't understand how the problem exists. What work is done in Z? There's no force spoken of in Z? The work done in X balances with the internal work of the body, if I understand. The only external work is done by the place there is external force.

Numbers 1 and 2 are more the key parts of what he is saying, I think. I don't really understand where continuum mechanics is supposed to have erred, though. Certainly, some equilibrium conditions are enforced, but perhaps not the right ones?

Maybe I need to read the papers.

## I doubt you need to read the papers.

Michael,

Reading his papers is not likely to be worth your time. From your post it is clear that you have a better understanding of mechanics than Mr. Koenemann. There are certainly fundamental questions to be addressed in continuum mechanics and thermodynamics, but I do not think these are the ones.

Chad

## Thanks

Thanks for the advice. I am not taking Koenemann extremely seriously, but I figure there might be something there. His website is not as cooky as I'd usually expect from a troll. It looks like he tried to plainly show what he was talking about on the poster page, but the math was unreadable due to display issues. At the least, it might sharpen my idea of why we do continuum mechanics like we do.

## Koenemann's papers

First I am relieved to see Chad's reponses to the five questions. I was thinking that would be a good place to start in order to address Koenemann's issues. Then I thought it may be better to read a couple of his papers to know more about his thinking. I printed out his "comment on Gurtin (1972)" and "note on Gibbs (1877)". After reading the first, my desire to read on of his papers drops significantly. I can't understand how the paper is accepted for publication in a physics journal. Does anyone know about International Journal of Modern Physics B? The entire paper is based on his own guess or assumption of what Gurtin knew. It reads like a presonal attack in politics.

I am not sure if I am going to read the second paper.

RH

## Koenemann's papers

I came across this website a few years ago and thought it interesting

but got no response on the Usenet groups to discuss these ideas so I am glad to see something on imechanica. I found some of it hard to follow.

The website used to carry some tales of his efforts to get papers published which were entertaining, but these appear to have disappeared and I can't find them archived (the URL was different from my memory).

I guess I would be keen to make sure there is nothing of value in these ideas, and/or they have serious flaws, which appears to be the impression from the posts so far.

Anyone care to drop the chap an email?!!!

Regards

Charles

## read Kellogg

Rui,

instead of ranting last July, it would have been better to contact me then. Your distaste for reasons fo style is granted, but only facts count, and they aren't pretty. The absolutely only way to deal with this matter is this one: put Kellogg's and Gurtin's book on one desk, use my paper as a guide, and make up your own mind.

If you read Kellogg, you will get a decent introduction into a mathematical theory and how to use it, be it Laplace problems or Poisson problems; and when you are done, you see physics with new eyes. If you are familiar with various methods of engineering mathematics, but not yet with potential theory (which was my situation until 1994) you will have numerous aha-moments while reading because Kellogg delivers the strings that tie all these methods together.

If you read Gurtin's presentation of Kellogg, you will be told that everything is in order, so you do not have to look it up yourself. But if you know potential theory already, and then read Gurtin, you will find that his collection of items is incoherent and absolutely useless. He tells you nothing about the framework or the purpose of the theory, he does not tell you how it works, he does not even tell you how a vector field is derived. But the gaps from a clear pattern: he spared out everything that could possibly hint to the reader that Poisson problems even exist.

Kellogg's book leads you properly just if you read the Table of Contents. If Gurtin did not follow that lead, there is only one explanation.

- What do you think of an experimenter who invents his data?

- Can a theoretician cheat?

Case closed. I look forward to receiving your informed judgment.

Falk

## I am not sure any of his papers are accepted, plus zero cites !

This guy is a joke. Don't waste time. You find only preprints in Archiv http://arxiv.org/ftp/physics/papers/0103/0103010.pdf

and there the few citations are self-citations.

Maybe the guy is real and is just playing with you. Now he had his 5 minutes celebrity, that's it !

Regards Mike

Query: "Koenemann FH": all

Summary: <<

Papers: 9 Cites/paper: 1.00 h-index: 2 AWCR: 1.00

Citations: 9 Cites/author: 9.00 g-index: 3 AW-index: 1.00

Years: 16 Papers/author: 5.83 hc-index: 2 AWCRpA: 1.00

Cites/year: 0.56 Authors/paper: 1.44 hI-index: 2.00

hI,norm: 2

Hirsch a=2.25, m=0.13

Contemporary ac=1.00

Cites/paper 1.00/0.0/0 (mean/median/mode)

Authors/paper 1.44/1.0/1 (mean/median/mode)

1 paper(s) with 0 author(s)

4 paper(s) with 1 author(s)

3 paper(s) with 2 author(s)

1 paper(s) with 3 author(s)

>>

Cites,Authors,Title,Year,Source,Publisher,ArticleURL,CitesURL

4,"FH Koenemann","Cauchy stress in mass distributions",2001,"Arxiv preprint physics/0103010","arxiv.org",

"http://arxiv.org/abs/physics/0103010",

"http://scholar.google.com/scholar?num=100&hl=en&lr=&cites=25719674733111...

3,"FH Koenemann","Unorthodox Thoughts about Deformation, Elasticity, and Stress",2001,"Zeitschrift für Naturforschung. A, A Journal of physical …","znaturforsch.com","

http://znaturforsch.com/aa/v56a/56a0794.pdf","http://scholar.google.com/...

2,"FH Koenemann","Tectonics of the Scandian orogeny and the Western Gneiss Region in southern Norway",1993,"International Journal of Earth Sciences","Springer","

http://www.springerlink.com/index/Q8520JU4TK523267.pdf","http://scholar....

0,"FH Koenemann, I Johannistal","On the systematics of energetic terms in continuum mechanics, and a note on Gibbs (1877)",0,"elastic-plastic.de","","

http://www.elastic-plastic.de/Gibbs.pdf","http://scholar.google.com/scho...

0,"H Forster, FH Koenemann, U Knittel","Regional framework for gold deposits of the Odzi-Mutare-Manica greenstone belt, Zimbabwe-Mozambique",1996,"Transactions of the Institution of Mining and Metallurgy. …","","","

http://www.google.com/search?hl=en&lr=&q=Forster+Regional+framework+*+gold"

0,"","Linear Elasticity and Potential Theory: a Comment on Gurtin (1972)",0,"","","","

http://scholar.google.com/scholar?num=100&hl=en&lr=&q=related:xa3JQfIzLW...

0,"FH Koenemann, I Johannistal","A Reevaluation of the Cauchy Stress Hypothesis",0,"elastic-plastic.de","","

http://www.elastic-plastic.de/re-eval.pdf","http://scholar.google.com/sc...

0,"FH Koenemann, I Johannistal","An approach to deformation theory based on Boyles law. IV. Application to a discrete body problem",0,"elastic-plastic.de","","

http://www.elastic-plastic.de/Theo4.pdf","http://scholar.google.com/scho...

0,"FH Koenemann","Origin of Oblique Microfabric Orientation in Simple Shear Zones",2000,"","agu.org","

http://www.agu.org/cgi-bin/wais?q=T11C-04","http://66.102.1.104/scholar?...

PDF] Unorthodox Thoughts about Deformation, Elasticity, and Stress - all 5 versions »

FH Koenemann - Zeitschrift für Naturforschung. A, A Journal of physical …, 2001 - znaturforsch.com

The nature of elastic deformation is examined in the light of the potential

theory. The concepts and mathematical treatment of elasticity and the choice of

equilibrium conditions are adopted from the mechanics of discrete bodies, ...

Cited by 3 - Related Articles - View as HTML - Web Search - BL Direct

[CITATION] Linear Elasticity and Potential Theory: a Comment on Gurtin (1972)

FH Koenemann

Related Articles - Web Search

[PDF] On the systematics of energetic terms in continuum mechanics, and a note on Gibbs (1877)

FH Koenemann, I Johannistal - elastic-plastic.de

The systematics of energetic terms as they are taught in continuum mechanics

deviate seriously from the standard doctrine in physics, resulting in a profound

misconception. It is demonstrated that the First Law of Thermodynamics has ...

Related Articles - View as HTML - Web Search

[PDF] A Reevaluation of the Cauchy Stress Hypothesis

FH Koenemann, I Johannistal - elastic-plastic.de

The theory of stress is solidly based on the cut model of Euler which was used

by Cauchy to derive the stress tensor. The cut model considers a group of planes

passing through a given point Q in space, and the system of forces acting ...

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PDF] An approach to deformation theory based on Boyles law. IV. Application to a discrete body problem

FH Koenemann, I Johannistal - elastic-plastic.de

The approach to deformation theory is used to model the distribution of the

failure potential in a discrete body subjected to a specific loading

configuration in 2 dimensions. The Fourier series method is applied, and it ...

Related Articles - View as HTML - Web Search

[CITATION] Linear Elasticity and Potential Theory: a Comment on Gurtin (1972)

FH Koenemann

Related Articles - Web Search

[PDF] On the systematics of energetic terms in continuum mechanics, and a note on Gibbs (1877)

FH Koenemann, I Johannistal - elastic-plastic.de

The systematics of energetic terms as they are taught in continuum mechanics

deviate seriously from the standard doctrine in physics, resulting in a profound

misconception. It is demonstrated that the First Law of Thermodynamics has ...

Related Articles - View as HTML - Web Search

## OK, someone should cite it for the first time!

someone should cite it for the first time! at least an erratum.

## But if the paper is NOT published, I cannot cite it !!! ;(

michele ciavarella

www.micheleciavarella.it

## ZAMM

I had seen questionable papers in ZAMM but not very radical! who is reviewer?

## This discussion is curious: you hope to find the "holy graal"..

I fully understand that it would be good to discover something great of the past that has been completely misunderstood or neglected.

So the entire Chauchy stress framework is flawed and only this guy knows it.

I also understand that there is some kind of annoyment to be aligned like number 12500 who cites the "fractal geometry of nature" from Benoit Mandelbrot

Similar pattern is found for the great Benoit Mandelbrot (H=62)

Yet, I suspect it is much more profitable to read Benoit Mandelbrot despite 12500 papers have already found and cited him (so I suspect at least 100 000 have read it), rather than following the chimera that Chauchy was wrong and that this guy who maybe does not even exist, and who has no previous record, nor present record, claims.

So why not finding the 10 000+ books and papers that really matter, and start discussing, or perhaps mixing them up?

I have provided 2 examples. The discovery of Carbon Nanotubes, and the "discovery" of fractals. Please provide other examples, and we shall make progress. I do not beleive in Holy Graal!

Actually, if you really want to hope to discover something, there is more chance to do it in Leonardo da Vinci. His latest books were found in Madrid in 1970, and contain a lot of mechanics. Roberto Ballarini recently wrote that in this codes, Leonardo clearly had already solved Beam Theory, well before Euler and Bernoulli, let alone the wrong theory of Galileo, and before not only Chauchy concept of stress, but even Hooke's concept of Elasticity!

"The

Da Vinci-Euler-BernoulliBeamTheory," ME Online Web...I challenge you then to go to Madrid and read Leonardo -- you may discover, like Ballarini, that he had something there.

I have for example just finished reading Capra's book on Leonardo

Fritjof Capra- The Science ofLeonardoIt is a good book, and opens the mind. There is a lot of Leonardo that is being rediscovered and is still not yet appreciated, particularly his ideas about ecology.

New Book:

The Science of LeonardoFrom the Preface:

Leonardo da Vinci, perhaps the greatest master painter and genius of

the Renaissance, has been the subject of hundreds of scholarly and

popular books. His enormous

oeuvre,said to include over 100,000 drawings and over 6,000 pages of notes,

and the extreme diversity of his interests have attracted countless

scholars from a wide range of academic and artistic disciplines.

However,

there are surprisingly few books about Leonardo's science, even though

he left voluminous notebooks full of detailed descriptions of his

experiments, magnificent drawings, and long analyses of his findings.

Moreover, most authors who have discussed Leonardo's scientific work

have looked at it through Newtonian lenses, and I believe this has

often prevented them from understanding its essential nature.

Leonardo

intended to eventually present the results of his scientific research

as a coherent, integrated body of knowledge. He never managed to do so,

because throughout his life he always felt more compelled to expand,

refine, and document his investigations than to organize them in a

systematic way. Hence, in the centuries since his death, scholars

studying his celebrated Notebooks have tended to see them as

disorganized and chaotic. In Leonardo's mind, however, his science was

not disorganized at all. It gave him a coherent, unifying picture of

natural phenomena — but a picture that is radically different from that

of Galileo, Descartes, and Newton.

Only

now, five centuries later, as the limits of Newtonian science are

becoming all too apparent and the mechanistic Cartesian worldview is

giving way to a holistic and ecological view not unlike Leonardo's, can

we begin to appreciate the full power of his science and its great

relevance for our modern era.

My

intent is to present a coherent account of the scientific method and

achievements of the great genius of the Renaissance and evaluate them

from the perspective of today’s scientific thought. Studying Leonardo

from this perspective will not only allow us to recognize his science

as a solid body of knowledge. It will also show why it cannot be

understood without his art, nor his art without the science.

As

a scientist and author, I depart in this book from my usual work. At

the same time, however, it has been a deeply satisfying book to write,

as I have been fascinated by Leonardo da Vinci's scientific work for

over three decades. When I began my career as a writer in the early

1970s, my plan was to write a popular book about particle physics. I

completed the first three chapters of the manuscript, then abandoned

the project to write

The Tao of Physics, into which Iincorporated most of the material from the early manuscript. My

original manuscript began with a brief history of modern Western

science, and opened with the beautiful statement by Leonardo da Vinci

on the empirical basis of science that now serves as the epigraph for

this book.

Since

paying tribute to Leonardo as the first modern scientist (long before

Galileo, Bacon, and Newton) in my early manuscript, I have retained my

fascination with his scientific work, and over the years have referred

to it several times in my writings, without, however, studying his

extensive Notebooks in any detail. The impetus to do so came in the

mid-1990s, when I saw a large exhibition of Leonardo's drawings at The

Queens Gallery at Buckingham Palace in London.

As

I gazed at those magnificent drawings juxtaposing, often on the same

page, architecture and human anatomy, turbulent water and turbulent

air, water vortices, the flow of human hair and the growth patterns of

grasses, I realized that Leonardo's systematic studies of living and

nonliving forms amounted to a science of quality and wholeness that was

fundamentally different from the mechanistic science of Galileo and

Newton. At the core of his investigations, it seemed to me, was a

persistent exploration of patterns, interconnecting phenomena from a

vast range of fields.

Having

explored the modern counterparts to Leonardo's approach, known today as

complexity theory and systems theory, in several of my previous books,

I felt that it was time for me to study Leonardo's Notebooks in earnest

and to evaluate his scientific thought from the perspective of the most

recent advances in modern science.

Although

Leonardo left us, in the words of the eminent Renaissance scholar

Kenneth Clark, "one of the most voluminous and complete records of a

mind at work that has come down to us," his Notebooks give us hardly

any clues to the author's character and personality. Leonardo, in his

paintings as well as in his life, seemed to cultivate a certain sense

of mystery. Because of this aura of mystery and because of his

extraordinary talents, Leonardo da Vinci became a legendary figure even

during his lifetime, and his legend has been amplified in different

variations in the centuries after his death.

Throughout history, he personified the age of the Renaissance, yet each era "reinvented" Leonardo according to the

zeitgeistof the time. To quote Kenneth Clark again, "Leonardo is the Hamlet of

art history whom each of us must recreate for himself." It is

therefore inevitable that in the following pages I have also had to

reinvent Leonardo. The image that emerges from my account is, in

contemporary scientific terms, one of Leonardo as a systemic thinker,

ecologist, and complexity theorist; a scientist and artist with a deep

reverence for all life, and as a man with a strong desire to work for

the benefit of humanity.

Click here for the book tour schedule.

## Re: A new theory of stress?

(0.0) I went through some of Mr. Koenemann's papers / documents / Web pages today. Here are my initial impressions (which are unlikely to change much).

(1.0) Initial impression gathered from his writing style: Even in his serious papers, he seems to jump from topic to topic far too easily---even carelessly. For example, see the "Conclusion" part of his paper "Gibbs.pdf," the one which is supposedly accepted for pub. in IJMPB. (BTW, I checked the site of this journal, but they do not list any of their forthcoming papers.) It is next to impossible to even guess what he might be thinking in going from one step to another step---if the statements can be called "steps".

(2.0) His "Logic": In his easiest to read (for me) paper, i.e. "Systematics.pdf" (published in 2004), the "position" from which his arguments flow began to become somewhat clearer. The way I understand his position, his essential logic seems to be the following:

(2.1) He says in this paper (and I quote) that "div \vec{f} is a measure of the work done by/upon a system."

Hello? My understanding is that for certain vector fields like the static electric field, \vec{f} can be given as the gradient of a scalar potential function, say, \phi. (This, of course, is not universally true though that's what he seems to assume.) In the cases where this condition holds, it is \phi which represents work/energy---not div \vec{f}. What does divergence of \vec{f} have to do with __work__? I fail to see this. In fact, in electrostatics, div \grad\phi would give you the stored __electric charge density__, not the stored __work__. But still, in this paper, work, it becomes. Why? Apparently, simply because he thinks so.

I wonder why none noticed such a prominent and so simple a mistake in the review process... (Has anyone checked whether these papers were even actually published in those journals?)

(2.2) He then tries to apply this (wrong) premise of his to the div of the stress vector. Now, in stress analysis, div \vec{T} = 0 (using momentum and torque balance). On this basis, he concludes (using his above-mentioned wrong idea) that no __work__ is done during a volume-constant deformation....

Now that is some conclusion to draw!!

(3.0) In another paper: "thoughts.pdf" (published in 2001), he says: "General solutions for the Poisson equation exist only for reversible processes, e.g. the Helmholtz equation." [sic]

Phew! At this point, I stopped reading all the __published journal papers__ of his.

(4.0) The interesting part is not whether what he has written is right or wrong (i.e. true or false). That issue is relatively simple to settle: obviously, what he writes is so absolutely false.

But a more interesting part is: What does it say about the review process of today's journal articles? ... You see, it was only yesterday that I was talking to some gentleman who mentioned to me: "But all that you have published during your PhD studies are conference papers, no journal papers... Your conference papers will not be counted..." And I found myself wondering aloud, once again: How does it matter? I even remembered the Bogdanov brothers incidence (which I had come to know from David Harriman's article). I mean, it does not seem likely, but still, it is possible that some journal might publish such articles...

If the journal review system also is not going to provide that gentleman with any guaruntees or assurances of the kind that he was seeking and presuming, why does he insist on that? Now, that is one issue which is much more interesting to me.

(5.0) However, the most interesting issue, for me, is that when a piece of writing like this does come along, what does it do to you.... Doesn't it force you to examine the clarity of your own fundamentals, even if only for an hour or two---before you find the author out? Even if only because the speaker has been using the terms so casually and indiscriminately? Not as a matter of some naive or honest mistake, but out of a deliberate kind of gliding over of the relevant facts? One is not accostomed to that kind of writing in science, and so, simply because the author throws a lot of incommensurate concepts together in a rapid succession, it begins to challenge your mind. So, there is a value to it in a weird, even humorous, sort of way...

I mean, in a way, a "paper" of this kind does serve to expose the weak spots in your own understanding, too...

For instance, have a look at these concepts/ideas which come up repeatedly while going through some of Mr. Koenemann's published journal papers.... Some aspects of these basic concepts have stumbled me quite a lot in the past... Why, some aspects have stumped me even in a very recent past---as late as a year or two back:

-- What is the essential difference between a field theory and a particle theory? (This question is relevant because what he describes as Newton's theory actually seems to refer to the "particle" kind of description. This, he seems to want to differentiate from the field abstraction.)

-- Why is pressure generally regarded as a scalar quantity when it is well known that the air actually is forcing the rubber walls of a balloon out?

-- What conceptual steps are involved before the Newtonian idea of force (something which acts on a particle to change the course of its motion) can be brought into the analysis of a continuum phenomenon, i.e. a field? For example, a vector field? A scalar field? How about tensor fields? Is particle really a point-phenomenon? Or does it, too, represent a differential-element-based abstraction?

-- What precisely are the physical dimensions of the potential functions used in the analytical stress theory? More important: Why do respectable authors never mention this in their books? (Here, feel free to pick any book/paper you like.)

-- What, precisely, is the difference between the meaning of the term "potential" when this term is used in electrostatics as against in stress analysis?

-- Can you reduce (3D) stress, a tensor, to a "collection" of three vectors? Why? Why not?

(6.0) All in all, an interesting Web site!!

## Quite a rich comment. You are right, we need to ask ourselves...

... about the clarify of our "fundamentals" and indeed this guy has been asking himself so much that he is now all confused! ;)

## Comment on Cauchy's stress theorem

I have been bugged by this question ever since I studied cauchy's stress theory 10 years ago.

In cauchy's analysis of solids, there two popular geometries.

First one is a unit cube used for deriving equations of equilibrium.

And the second one is a tetraheadon used for deriving Cauchy's stress theorem.

In both the cases limit of force equilibrium as solid geometry shrinks to a point is derived.

My question has been, how come limit of force equilibrium at a point inside a solid depenent on the path taken by limiting process.

If this process were to be rigorous enough, any arbitrary geometry shrunk to a point should yeild the same limit of force equilibrium.

-Somesh

## Re: Comment on Cauchy's stress theorem

Hi Somesh,

0. In terms of a proper hierarchical order, I think, the second one you mention comes first.

1. Cauchy's stress theorem refers to the state of stress "at" a point. What it essentically states is the tensorial nature of stress, in particular, its invariance under rotation of the refernce frame. To bring out the mathematics in its generality, it's necessary to have the reference plane (i.e. the plane of the imaginary cut through the solid) oriented at nonzero angles to the three Cartesian planes (the ones normal to the x-, y- and z-axes). Hence the tetrahedron.

I deliberately put scare-quotes ("") around the word "at." The reason is: if you shrink an area in a limiting process, what you get, *IMO*, is: *not* a zero area but an infinitesimal area. (A great deal of epistemological confusion exists on this point, but then, it's a different story.) Thus, when we say "stress at a point," it's just a loose way of speaking. What we really mean then, IMO, is this: the state of stress in an infinitesimal region around the point. Or, to put it in terms of a Taylor series expansion, what we mean is: keeping only the zeroth order term and neglecting all the higher order terms.

Thus, the tetraheron is (usually) meant (only) for expressing to the zeroth-order measure of stress---and its invariance under the coordinate transformation of rotation. The invariance refers to that point around which the infinitesimal tetrahedral plane area is constructed.

2. In contrast, the unit cube geometry is used when it comes to determining the higher-order variations in the local stress values at various points within a solid. (Usually, we are interested in only the first-order variations.) Thus, the unit cube geometry serves to express the *changes* in the local stress states "at" all the points lying in between the opposite faces of the cube. For instance, see the diagram here [^]. (Nothing special about it; it just happens to be the first link showing a variation in stress values, after Googling on "stress equilibrium equation 3D").

Realize that here, as with any application of the Taylor series, the \delta x etc. are sufficiently small distances that the idea of the Taylor series is applicable. (Do we have to assume \delta x etc. as finite? Or do they have to be infinitesimal? I *think* that the former is correct, but comments on this matter are most welcome!)

3. I don't think putting this point is necessary, but just for the sake of completeness:

To relate the two geometries: Take the first sub-eq. of eq. (3.2) on the linked page. Here, you have to imagine that the tetrahedral (or the inclined) plane is aligned with the plane normal to the x-axis at the left-hand side of the cube, at the right-hand side of the cube, and at each point in between the two. That's the relation between the two geometries.

4. Which raises an interesting question. Can there be an infinity of such planes even if the \delta x etc. distances are infinitesimally small? In other words: Are there an infinity of points lying on an infinitesimal line segment? I would love it if a mathematician can clarify on this point. My position is: there indeed should be an infinity of them in there, for the same reason that an infinitesimal is not a zero.

5. Finally, let me pose one open question, along the same lines, as food for thought: Determine if the definition of stress requires a reference *plane*, or is it rather a reference *plate* of infinitesimally small thickness.

Answers, criticism, and opinions are most welcome, esp. from the analytical mathematicians, or from those mechanicians who are so inclined. :)

--Ajit

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## Difference between derivation of Stress theorem and Eq'um Eq'ns

Hello Somesh,

A really good question (at least because I think I know the answer!). Many a times the teachers do not make this point very clear, probably because they feel the students will not appreciate the matter (or they themselves don't know!).

The tetrahedron used in derivation of Cauchy stress theorem represents four planes passing through exactly the same point. It is not really a limiting process (even though, often it is made to appear so). It is used as a convenient way of representation (Can tou think of any better way of representing more than one plane passing through a point, still allowing the use of the concept of equilibrium?).

The cube used in the derivation of equilibrium equation is actually a cube, though quite small so as to make a first order approximation of stress variation acceptable. Please remember that if you have only one point, the spatial derivatives do not make sense. Why we use a cube (or really a brick) is that the normal and shear stresses are perfectly aligned in the coordinate directions, which makes writing the equilibrium equations easier.

At the risk of contradicting myself, I would like to mention the following point also. Thinking from a Taylor series point-of-view, in derivation of Cauchy stress theorem we use only Zeroth order terms, while in the derivation of equilibrium equations we use the First order terms also.

Regards,

Jayadeep

## Re: Difference between derivation of Stress theorem and Eq'um Eq

Hi Jayadeep,

Wow! We were almost simultaneous in replying. Indeed, when I began writing, your reply had not yet come in. (I think of the patents system and certain of Stalin et al's mind-related experiments.)

Anyway, coming back to the topic, yes, you are right. That indeed is a Taylor-series expansion.

However, I am not too sure if the \delta x etc. are to be taken as finite or infinitesimal. I *think* that for Taylor series in general, these were meant to be taken as finite. However, I don't know. As I said, I would really like to have an answer from a specialist mathematician.

In our case (the cube) there is no harm in taking them to be either finite or infinitesimal because, if the product terms giving area (like \delta y X delta z) can be taken to cancel out without giving any problem even when the terms are finite, then there certainly won't be any issue even if they were to be made infinitesimal. (The converse won't be true in general.) (The area-giving product terms can always be cancelled because spatial relations are taken to be homogeneous and linear in this context.)

BTW, your point about the tetrahedron as composed of four planes through the *same* point, but with the three orthogonal ones being merely translated for convenience of representation, was great!

--Ajit

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## force equilibrium at a point

Hello Jayadeep,

Thanks for your response.

I believe concept of force equilibrium is with reference to following cases

1. Forces acting on a point

2. Forces acting on a body

Could you please clarify me (or site some other references) on how force equilibrium

at a point can taken as sum of forces acting on different planes passing though

the same point.

Even if it is so, why just four planes(three mutually perpendicular and one oblique)?

While there are infinite number of planes passing through a point,

why these four planes are to be selected for describing force equilibrium.

with regards

Somesh

## Re: force equilibrium at a point

Hi Somesh,

A disclaimer to start with: A perfect answer on this matter, as Ajit was mentioning above, can come only from an expert in mathematical theory of elasticity. I am obviously not one, hence my answer can not be perfect. However, I will try my best.

I believe your points about force equilibrium (either at a point or that of a body) is perfectly valid. So you can think of Cauchy stress theorem to be corresponding to force equilibrium at point, and the derivation of equilibrium equations to be corresponing to equibrium of a small, but finite-sized, cube/brick.

Since a point is acted upon by the effect of a continuous distribution of matter in its neighbourhood (continuum hypothesis), the force acting on a point can only be represented by a stress tensor (stress at a point). The reason for using four planes is to show that we can represent the force on any arbitrary plane using traction vectors on three orthogonal planes. From a Taylor series perspective, this is effectively an analysis using only the zeroth order terms in stress "variation"(!!?).

When we think of equilibrium of a finite-sized object, we need to consider the variation of stresses within it. So by assuming the body to be sufficiently small, we approximate the spatial variation of stresses by using only zeroth and first order terms in Taylor series, while deriving the equlibrium equations.

I hope I could throw some light into the matter.

Regards,

Jayadeep

## Understanding from Taylor series perspective

Hello Jayadeep and Ajit,

I am glad both of you have responded to my comments.

Jayadeep: We have two connection as I am an alumnus of IISc (M.E. Civil(Structures), 2000-02) and an ex-EACoE employee(2002-2003) and I am enthralled to see your 10/10 score. I only have 7.4/8.0

I am trying to understand the Taylor series perspective you mentioned in your messages.

A first order approximation Taylor series approximation of any function shall also contain the zeroth order terms along with first order terms. By excluding the first order terms in the approximation we shall obtain the zeroth order approximation of the function. I do not see this happening when we right the force equilibrium equations in both cases and compare them.

with regards

Somesh

## Reply to Somesh

Hi Somesh,

1. First of all, a few corrections/clarifications to what I said above.

(1.1) I spoke of "invariance" of stress in my above reply [^]. For example, here is a quote from my above reply: "Thus, the tetraheron is (usually) meant (only) for expressing to the zeroth-order measure of stress---and its invariance under the coordinate transformation of rotation. The invariance refers to that point around which the infinitesimal tetrahedral plane area is constructed."

It was a loose way of using the term "invariance." By "invariance," I meant something like saying that: Stress (at a given point) remains the same, regardless of the orientation of the coordinate system. Its components, when resolved in two differently oriented coordinate systems, may change; however, the mathematical object that is stress, remains the same.

For an advanced---and far more educative---discussion than what I know, see the iMechanica thread started by Prof. Nemat-Nasser, with many wonderful comments by other iMechanicians, here [^].

(1.2) Similarly, I used the term "variation" rather loosely, as being synonymous with "change." Not necessarily as the variation of the variational calculus.

(1.3) Finally, I was in the Eucledian geometry throughout, for instance, when I spoke of homogenity and uniformity of space (in my above reply to Jayadeep, here [^ ]).

2. Now coming to your concern here. Jayadeep's recent comment [^] clarifies it all. However, let me think it all aloud. Doing so may be exasperating to the reader, perhaps, but it will also help pinpoint any mistakes in my thinking. So, I request your patience.

The Taylor-series expansion is about expressing the unknown value of a variable y at a point Q, in terms of its (known) value at a reference point P that lies in the close neighbourhood of Q. In short: given y_P, find y_Q as a function of: y_P and the relative position of Q with respect to P. That, in essence, is the concern of the Taylor series.

If there is no change in y(x) values at x = P and x = Q, then y_Q = y_P and so, no further manipulation is necessary. Thus, y_P itself appears as the zeroth term in the Taylor series expansion. It denotes the case of, say, stress being constant at all x values within a solid (e.g., approximately speaking, at all points along the length of a thin guitar wire suspended from ceiling with a weight attached to it).

However, if there indeed is a change in y(x) value in going from P to Q, then we have to add (or subtract) something to the value y_P, so as to get the estimated value y_Q. Taylor said, in effect, that this can be accomplished by following his method---the form of his series.

For a linear change betweeen y_P and y_Q, you would have y_Q = y_P + (slope of y(x) vs x "at" P)X(distance of Q from P). This exactly is the expression appearing as the first-order term in the Taylor series.

If there also is a *further* parabolic change sitting on top of that linear change, then you have to consider the second-order term... And so on. The terms after the zeroth-order denote changes at a particular order of differentiation (i.e. a particular wiggliness to be built into the shape of the curve representing the change of y from P to Q).

Now, you cannot have a Taylor series expansion of a quantity y unless you first have defined that quantity. Defining stress at a point is what the tetrahedron situation is all about. Determining the first-order change is what the cube situation is all about. Unless stress waves exist, we aren't bothered about the higher-order terms.

Usually, teachers of mathematics simply give you the formula for the Taylor series, and expect you to be willing to immediately jump to expanding, say, the sine function around x = 0, and thereby win great marks, graduate with honors, go to a top-ranked US university, be surrounded by ample flows of milk and honey, make (or grab) money (it doesn't matter how, any more), win friends, influence people (esp. those in India), etc. But the mathematicians don't explain the fact that there are two different points, P and Q involved in it. (No professor at COEP did that, anyway. Neither does, say, the Wiki page.) Mathematicians, as a rule (barring exceptions) simply are not physical enough a people. (And these days, physicists themselves compete mathematicians out as far as "thinking" in that mode is concerned---"shut up and ...".)

That's why, today, it's very easy to think that the Taylor series is all about having progressively more curvy curves at the same point P. That is a misconception. The Taylor series isn't primarily about expanding a complicated function (e.g. a trigonometric or a transcendental one) in the form of an (algebraic) polynomial. Primarily, it's about approximating the value at Q, given P---two distinct points.

If I have made any essential mistakes in this position of mine (the expression being akin to a loud thought), then I would like to have those corrected. Accordingly I request the reader to please do so. Thanks in advance.

3. And, if the above is correct, then, in connection with Prof. Nemat-Nasser's above-mentioned thread [^], I would like to know whether the discussion there was about stress being taken as a field quantity (as a variable that tries to analytically characterize the entirety of the continuum in one expression), or whether it was about the stress at a point. I think the former. If not that, why should concerns like non-Eucledian geometry and structure of manifolds etc. enter into that discussion? I see no reason! But then, as I said, I don't know that part---not well enough, anyway. [Never had a course-work on that part, and so, no chance for me to carry even misconceptions about it :)] I would like to know a bit about this part, too. Thanks in advance, again.

--Ajit

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