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### Re: stress definition

*In reply to Re: Stress Definition*

In the undergraduate linear algebra class, we introduced linear map about two weeks ago. From past experience, I know students need be reminded of what a map (i.e., a function) is. A map requires three sets: domain, codomain, and graph.

For a linear map, the domain must be a vector space, the codomain must be also a vector space, and the graph must be linear.

In saying the stress is a linear map, I stick to this prescription. Thus, the domain is the vector space of areas, the codomain is the vector space of forces, and the graph is the stress. That is your equation (3).

The Cauchy formula traction = (stress)(unit normal) does not follow this prescription. The collection of all unit normal vectors is not a vector space, because the addition of two unit normal vectors do not give another unit normal vector. Also, the collection of all traction vectors is not a vector space, because tractions on different planes are not additive.

Thus, in defining stress, I stick to your Equation (3).

In teaching the undergraduate course on linear algebra, I try to highlight the importance of the scalar set (i.e., the one-directional vector space). Then an n-dimension vector space is the Cartesian product of n scalar sets. A vector-vector linear map is a table of scalar-scalar linear map. Thus all linear map is similar to the fact of nature: a chicken has two feet. This fact is a linear map between two sets: the chicken set and the foot set.

Thus, reducing to one component, stress is just force per unit area. The algebra is similar to that each chicken has two feet.

In the linear algebra class, I then talked about two vector spaces: chicken-rabbit space and head-foot space. Then I talked about the vector-vector linear map: (chicken-rabbit space) → (head-foot space). The linear map is then a table (i.e., matrix) of four facts of life:

- A chicken has 1 head
- A chicken has 2 feet
- A rabbit has 1 head
- A rabbit has 4 feet

This strategy may be parallel to your way of introducing stress.

### Re: Defn of stress: A glaring mistake in my reply to Zhigang

*In reply to Re: Stress Definition*

Please ignore the second derivation, i.e., starting with Cauchy's formula, and proceeding as follows:

{t} = [\sigma]^T {n}

{t}[n] = [\sigma]^T {n}[n] (Post-multiply both sides by {n}-transposed)

{t}[n] = [\sigma]^T ({n}[n] is the dot product, equal to 1)** <------ THIS PART IS WRONG**

[\sigma] = {n}[t] (Take transposes of both sides, and flip LHS and RHS)"

=======

The error is in saying that {n}[n] is the dot product, equal to 1. Very elementary mistake!

=======

I wrote the reply while in office, interrupted by an inaugural function, some 10+ visitors, two meetings, quite a few phone-calls, and a skipped lunch... All simply because I am so enthusiastic about fundamental topics like these, that I can overboard.... But no excuses, so let me say sorry....

I also realize that the characterization of the stress-strain relation as a flux-gradient relation has a very definite merit to it, but the fact is, the strain tensor is just a part of the gradient tensor, not the entire one. I need to think more compreshensively about it.

========

Bye for now, I will go home, think a bit about it, and come back either tomorrow or the day after or so. In the meanwhile, everyone, feel absolutely free to leave your ideas for going from Cauchy's formula to a direct definition of stress, and please see if there is any other error in what I say. Thanks in advance for your feedback...

And sorry, once again, for bothering you all with the mistake....

--Ajit

### Re: Stress Definition

*In reply to Re: Stress is defined as the quantity equal to ... what?*

Dear Zhigang,

Wow, good to know that it's been puzzling for you too...

BTW, yes, I have been aware of your notes, and in fact recently recommended them in a reply at my personal blog, here [^].

Overall, to cut a long story short, here are a few points that I think we should be explaining to students:

====

We should state and explain the concept of stress as the direct end-result of a tensor product.

Here is one way to bring out the required direct definition. Only the UG-level matrix algebra is used.:

Notation: We adopt the convention that a vector is *always* denoted via a *column* vector, *never* using a row vector, and therefore, that a *row* vector *always* represents *only* a transposed vector. Column vectors are denoted using braces {c}, and row vectors and square matrices are denoted with square brackets [r] or [M]. (When using LaTeX, we use the left- and right-floor symbols for the row vectors, and the square brackets for matrices.)

First, define the local field of traction-vector acting on a given surface (whether the surface is internal or external, whether it is used in specifying BCs or not) in the usual way, viz., as the ratio \vec{F} / |\vec{A}| in the limit that |\vec{A}| vanishes. (Or, if you want to avoid limits at least initially, consider a uniform \vec{F} field.)

The denominator here is a scalar (it's only the magnitude of the area vector), and so, the division is not a problem. Basically, before using the area vector further, we isolate apart its two aspects, viz., the scalar magnitude and the direction.

Denote the unit normal to surface as {n} and its transpose as [n].

Start with the transpose of the traction vector, denoted as {t}^{T} and also written as [t].

[t] = [t] (and what else?)

= (1) [t]

= ( [n]{n} ) [t]

= [n] ( {n}[t] )

= [n] [\sigma]

In the last step, we have used the direct definition: [\sigma] \equiv {n}[t].

The above was the round-about way in which I had got to the defintion for the first time in my life. However, another, perhaps simpler, way is this. Start with Cauchy's formula:

{t} = [\sigma]^T {n}

{t}[n] = [\sigma]^T {n}[n] (Post-multiply both sides by {n}-transposed)

{t}[n] = [\sigma]^T ({n}[n] is the dot product, equal to 1)

[\sigma] = {n}[t] (Take transposes of both sides, and flip LHS and RHS)

The matrix-notation expression {n}[t] stands for a *tensor* product taken between the unit surface-normal vector on the left hand-side of the operator, and the traction vector on the right hand-side. Using the vector language, this tensor product is the same as what is expressed by: "\hat{n}\otimes\vec{t}".

This structure of

"{n} \otimes [the surface-intensity of a vector field variable]"

is common to all the *flux*-tensors of *all* the *vector* fields. *The direct definition of stress thus brings out the physical idea that the stress is the flux of the traction vector.*

The idea of the strain is obviously is based on a gradient; it's the symmetrical part of the displacement gradient.

The student can now appreciate that the stress-strain relation is nothing but just a case of a flux-gradient relation. The flux-gradient relations occur in almost *all* other areas of physics; as just one example, consider Fourier's law of heat conduction. So, hopefully, he can see the commonality of analysis with other branches of physics and engineering.

Speaking in general terms, the flux of a *scalar* field is a vector field; the gradient of a *scalar* field is a vector field; and the two vector fields are related via a material (constitutive) law.

Exactly similarly, the flux of a *vector* field is a tensor field; the gradient of a *vector* is a tensor field; and the two tensor fields are related via a constitutive law.

===

The idea that the stress is a linear map, as emphasized by you, is indeed very helpful. It throws light on what kind of purpose it serves. So, it is very valuable.

That's what the indirect equation

{t} = [\sigma]^T {n} (i.e. Cauchy's formula)

shows. It brings out, very clearly, the fact that the stress as a mathematical object is, from the external viewpoint, a linear map (from {n} to {t})

However, IMHO, this idea also should be complemented by pointing out that the stress is a tensor product (which highlights its internal structure) and that it is a flux (which identifies its physical meaning). The direct definition allows one to see both these latter facts.

===

BTW, the idea of a flux does not always refer to something that flows; it can refer to something that simply exists (rather than flows across) a surface. The idea of flux, however, necessarily refers to a reference (planar) surface.

Therefore, the flux nature of stress also helps reinforce the idea that it is purely a surface phenomenon. To define stress, all that you need is a planar element (whose orientation can be made to change) around the point of interest. In particular, you *don't* need a volume element at all. This is important to realize by the text-book writers. To define stress, you *don't* need either Cauchy's tetrahedron or the hexahedral element (as used in deriving the stress-divergence theorem). Both these are volume elements. But stress can be completely defined with just an orientable surface element. This point becomes clear only when you understand the stress as a flux, which itself becomes clear only after you consider the direct definition via tensor product of two vectors.

Finally, just one more point.

Stress analysis is based on the differential equation paradigm. For a problem involving only stresses (i.e., no equations or calculations of strains or displacements being involved), the auxiliary data for such a problem is fully stated in terms of just the unit normals and the traction vectors. The point to note is this: the auxiliary data in fact cannot be stated in any others terms. Specifically, it cannot be stated in terms of forces and areas. When in engineering we say that a force is being specified as a BC for a stress analysis problem, what we are actually doing is to mentally translate the force vectors into the area-intensity vectors, before importing them into the stress analysis problem as traction vectors. The direct definition simply helps makes this part more direct and explicit.

===

I have grown so convinced about the necessity to have a direct definition that I am seriously thinking of writing a journal paper on the topic, just to let the argument have a permanent place in the archives.

===

But, thanks, once again, for your interest,

Best,

--Ajit

### paper is accepted in Fatigue & Fracture of Engineering Materials

*In reply to On the connection between Palmgren-Miner’s rule and crack propagation laws*

Soon we can send you the final form.

MC

### Re: Stress is defined as the quantity equal to ... what?

*In reply to Stress is defined as the quantity equal to ... what?*

Ajit: Your question has also puzzeled me. I now favor your equation (3). Here is a post called "a state of stress is a linear map".

Incidentally, I am teaching undergarduate linear algebra now. Here are class notes that try to explain where we get vector spaces and linear maps.

### of course, it is in our to-do-list, but the idea came recently!

*In reply to BAM model and "Contact Challenge" surface*

You are not disrespectful at all. I agree with you.

The BAM model was validated, in the original paper, with pull-off data, which are much MORE difficult to predict with any other theory ---- including all Persson's theory.

The case of the Contact Challenge is relatively TRIVIAL, and with BAM model, given it is a very first approximation single closed form equation, I think we should be able to do it without any effort!

Let me talk to my collaborators if we find a couple hours free time.

### Re: VEMLab

*In reply to VEMLab: a MATLAB library for the virtual element method*

This one used to be visible at the front page of iMechanica until just this afternoon (IST).

Why was it taken off it? I wonder.

Quite unlike what this forum *used* to be like. ...

Best,

--Ajit

### BAM model and "Contact Challenge" surface

*In reply to by the way, I forgot to check if my BAM model works fine *

Dear Prof. Ciavarella,

I truly do not wish to be "cheeky" or disrespectful, but I find it remarkable that any researcher could refrain from the tantalising task of checking their own model against published data: is it not one of the most exciting aspect of modelling?

I hope you will find the time to check your BAM model with the "Contact Challenge" surface: regardless of the results it would certainly add value to this interesting discussion.

Best Regards

### BAM model and "Contact Challenge" surface

*In reply to by the way, I forgot to check if my BAM model works fine *

Dear Prof. Ciavarella,

I truly do not wish to be "cheeky" or disrespectful, but I find it remarkable that any researcher could refrain from the tantalising task of checking their own model against published data: is it not one of the most exciting aspect of modelling?

I hope you will find the time to check your BAM model with the "Contact Challenge" surface: regardless of the results it would certainly add value to this interesting discussion.

Best Regards

### Review with good friends top tribologists, including Muser

*In reply to a "contact sport" between academics*

Modeling and simulation in tribology across scales: An overview

http://imechanica.org/node/22143

### by the way, I forgot to check if my BAM model works fine

*In reply to a "contact sport" between academics*

By the way, this imechanica discussion makes me wonder why I did not check if my BAM model

http://imechanica.org/node/22131

works fine with the "Contact Challenge" surface. One reason may be that I developed BAM after the contact challenge results were collected.

It should be easy to do it, except to find the time!

Anyone willing to do it?

### Interested in the position

### the big review paper with my good friend Muser has appeared!

*In reply to a "contact sport" between academics*

Modeling and simulation in tribology across scales: An overviewTribology International

Available online 12 February 2018

In Press, Accepted Manuscript — Note to users

Modeling and simulation in tribology across scales: An overview ☆

- a Advanced Production Engineering, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Nijenborg 4, 9747 AG Groningen, The Netherlands
- b MINES ParisTech, PSL Research University, Centre des Matériaux, CNRS UMR 7633, BP 87, F 91003 Evry, France
- c Univ Lyon, Ecole Centrale de Lyon, ENISE, ENTPE, CNRS, Laboratoire de Tribologie et Dynamique des Systèmes LTDS, UMR 5513, F-69134, Ecully, France
- d Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands
- e Department of Industrial Engineering, University of Padova, Via Venezia 1, 35015 Padua, Italy
- f Tribology Group, Department of Mechanical Engineering, Imperial College London, South Kensington Campus, Exhibition Road, London SW7 2AZ, UK
- g Division of Machine Elements, Luleå University of Technology, Luleå, Sweden
- h IMT School for Advanced Studies Lucca, Multi-scale Analysis of Materials Research Unit, Piazza San Francesco 19, 55100 Lucca, Italy
- i Department of Mechanical Engineering, Technical University of Denmark, DK-2800, Kgs. Lyngby, Denmark
- j National Centre for Advanced Tribology at Southampton (nCATS), Bioengineering Science Research Group, Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, UK
- k Biomechanics and Mechanobiology Laboratory, Biomedical Engineering Division, Department of Human Biology, Faculty of Health Sciences, University of Cape Town, Anzio Road, Observatory, 7925, South Africa
- l LSMS, ENAC, Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland
- m Department of Engineering, Aarhus University, Inge Lehmanns Gade 10, 8000 Aarhus C, Denmark
- n SKF Engineering & Research Centre (ERC), SKF B.V., Nieuwegein, The Netherlands
- o Department of Physics, King's College London, Strand, London WC2R 2LS, England, UK
- p Hamburg University of Technology, Department of Mechanical Engineering, Am Schwarzenberg-Campus 1, 21073 Hamburg, Germany
- q Department of Control Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, Karlovo Namesti 13, 12135, Prague 2, Czech Republic
- r Politecnico di Bari, V. le Gentile 182, 70125 Bari, Italy
- s Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland
- t Department of Physics and Nanostructured Interfaces and Surfaces Centre, University of Torino, Via Pietro Giuria 1, 10125 Torino, Italy
- u Laboratory of Bio-Inspired & Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123 Trento, Italy
- v Ket Lab, Edoardo Amaldi Foundation, Italian Space Agency, Via del Politecnico snc, 00133 Rome, Italy
- w School of Engineering and Materials Science, Queen Mary University of London, Mile End Road, E1-4NS London, United Kingdom
- x Department of Materials Science and Engineering, Saarland University, 66123 Saarbrücken, Germany

- Received 18 November 2017, Revised 1 February 2018, Accepted 4 February 2018, Available online 12 February 2018

### some more clarifications

*In reply to a "contact sport" between academics*

I would specify that my statements are general and do not mean to say that Muser was in any way dishonest or falsifying any of the process of the "contact mechanics" challenge.

But the situation is simply that I raise points, also in my "comment" to the "contact challenge", which are difficult questions and do not necessarily pertain to the contact-mechanics challenge.

In a sense, it may look misleading and too much an attack to Muser that I raise these questions.

Muser is a very strong scientist, with very strong honest behaviour, and I have no intention to claim otherwise.

These questions are purely scientific debates. Unfortunately the way the "contact challenge" is organized, being Persson's theory, for that specific case, very well organized, makes all other theories less strong.

Muser was not in a good position to answer all my questions. His reply is a very interesting scientific paper.

The fact that we collaborated on a review paper shows that we are very good collegues and friends!

Future directions in tribology require more discussion.

You should make clear these things clear in an extra blog. I am seriously concerned about the damage your blog did and I want damage control to the largest possible degree. You are the only person who can fix this right now.

### I apologize for one of the questions to Muser -- 9

*In reply to a "contact sport" between academics*

I must apologize for one point 9 in my questions to Muser. It is not true that he did not permit me to partecipate. I apologize and corrected.

### in the commercial site of Persson surfaces are non Gaussian!!!

*In reply to a "contact sport" between academics*

http://multiscaleconsulting.com/index.php/software/power-spectrum

The influence of roughness on the adhesion and frictional properties is mainly determined by the surface roughness power spectrum C(q) (or power spectral density) which is the most important quantity characterizing roughness. The surface roughness power spectrum fully characterizes all statistical properties of a measured surface. This means that all available information on the roughness is uniquely preserved in this quantity. It can be calculated directly from nearly any measured topography using our power spectrum software. This means that the power spectrum software is the perfect tool to further analyze your topography files and to calculate the input files necessary for our contact mechanics and rubber friction software. Note that while the root mean square roughness is usually dominated by the longest wavelength surface roughness components, higher order moments of the power spectrum such as the average slope or the average surface curvature are dominated by the shorter wavelength components. All these roughness parameters have therefore one thing in common, they do not describe the surface properties on different length scales. However it has been found that this is necessary to understand the true contact area between a tire with the road surface. It is hence not enough to gather information about a measured topography by only calculating the standard roughness parameters which exist for a long time already. The surface roughness power spectrum is the only statistical quantity which covers roughness properties over all length scales without loosing information over the surface measured. This is actually very important because practically all macroscopic bodies have surfaces with roughness on many different length scales. When two bodies with nominally flat surfaces are brought into contact, real (atomic) contact will only occur in small randomly distributed areas, and the area of real contact is usually an extremely small fraction of the nominal contact area. The contact regions can be visualized as small areas where asperities from one solid are squeezed against asperities of the other solid; depending on the conditions the asperities may deform elastically or plastically. How large is the area of real contact between a solid block and a substrate? This fundamental question has extremely important practical implications. For example, it determines the contact resistivity and the heat transfer between the solids. It is also of direct importance for wear and sliding friction, e.g., the rubber friction between a tyre and a road surface, and has a major influence on the adhesive force between two solid blocks in direct contact. The power spectrum calculator offers a quick, intuitive and hence easy to use software for calculating the surface roughness power spectrum of all kinds of different input formats.The figure below shows the latest Windows version of the program. After specifying the format type and some other important information about the topography file, like how many points in x and y-direction or the lattice constant between two points, one can choose between calculating the full, top or bottom power spectrum. Windows Version of the power spectrum software

When the calculation is finished successfully it is possible to directly check the power spectrum as shown below. The power spectrum software will give you in addition to the power spectrum many other important parameters. This includes different roughness parameters as for example the rms roughness value or the rms slope.

The surface roughness power spectrum

Here we show two other quantities which can be very to check after the power spectrum has been calculated as they contain very useful information. On the left is the height probability distribution while on the right we show the slope probability distribution. It is recommended to check these two curves after the calculation is finished to make sure that the results are reasonable and consistent with the power spectrum.

### Thanks for your feedback on the HASEL technology!

*In reply to Dear Christoph,*

Dear Tongqing,

thank you very much for your positive comment!

Yes, Zhigang mentioned some still unpublished efforts on GEO in an earlier comment; I am curious to see this when the papers go online. GEO is without doubt a great combination of materials that is exceptionally rich in mechanics and materials innovation. As for GEO, I would suggest that you maybe consider making your analysis even broader. We have had some very nice results with Peano-HASELs, which do not rely on elastomers and can also be made with thin metal films as electrodes -- and still, Peano-HASELs can serve as very good electrohydraulic artificial muscles. Maybe some of the mechanics concepts you are working on related to GEO could be generalized and made broader to also apply to systems, that consist of a liquid dielectric, thin polymer films (including but not limited to elastomers) and conductors (hydrogels, but also thin metal films or other types of stretchable conductors). Of course, you would have to adapt the beautiful name of GEO which you might not want to do. In some sense GEO can be viewed as a specific case of HASEL, using elastomers and hydrogels. I look forward to seeing your results and a discussion in iMechanica!

Here are answers to your questions:

1) Self-healing is only one advantage of using liquid dielectrics. I think what might be even more or equally important is the ability to have direct electrical control over soft hydraulic actuators, which are incredibly versatile and can achieve a lot of different actuation modes. We are still exploring what we can do with different types of oil with high values of permittivity or dielectric strength, both of which influence Maxwell stress. It is also a nice feature that HASEL actuators can be optimized for a specific application by making use of hydraulic amplification, as we have shown with donut HASEL actuators.

2) We have shown that donut HASEL actuators undergo a pull-in transition, which can be clearly seen from our experimental data. Whether this is a desired feature or not depends on the specific application. Some applications might benefit from bistable behavior (maybe soft mechanical switches or triggers, maybe also valves), other applications do not benefit from pull-in. The design of HASEL actuators is very versatile and we already have versions of these actuators that do not show this type of highly nonlinear response, but react with monotonic, almost linear behavior (which can be a simplifying feature for controls in robotics applications).

Happy new year of the dog!

Best,

Christoph

### A new article in national press in Corriere della Sera!

*In reply to Corriere della Sera: recruitment in Italian Academia*

See here. How petition has collected almost 13 000 signatures, including prestigeous names. See the link here.

### Unable to find the job link

*In reply to Postdoctoral Position in FEM Flow Modeling*

Hi,

I am trying to apply for this position but unable to find this job in all job section. Could you please share the application link.

Thanks

Abhishek

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