Talk:Continuum mechanics

message: the elasticity / continuum mechanics and related articles
1) Strength of materials is definitely NOT the same as the theory of elasticity. It is an engineering approach to solve solid mechanics problems with large contributions from people like Timoshenko in the early 20th century. The 'strength of materials' approach (almost) entirely avoids the elastic field equations and their solutions.
2) The definition of elastic material provided here is incorrect. Briefly, an elastic material is one in which the work done by external forces acting on the body is stored as (and is equal to) the Elastic Potential Energy, which is completely recoverable on unloading. Thus it is perfectly possible to have non-linear elastic materials.
3) You don't 'apply stress', you apply loads. Stress is a DEFINED quantity, as opposed to loads (forces). For example, it is convenient to define the stress as a symmetric second-rank tensor but it is possible to define an alternative, asymmetric stress (Lagrangian stress)
:Thanks for your views. Be bold and make an edit. This whole area would benefit from a few extra informed editors. However, please note that this is an encyclodaedia so we should give a general description that it intelligible to the bright 12-year-old and then proceed to an exact definition that would be meaningful to an expert. Cutler 09:42, 18 February 2006 (UTC)
thread_title:  Not a regular wiki-person, but I saw a few glaring errors in 

message: I've looked around for a while, and couldn't find any article describing the Eulerian and Lagrangian descriptions of a deforming material.  This leads me to suggest that either someone create such an article, or, if it exists, make its presence more obvious, or, if it is plenty obvious, please tell me where to find it.  My homework grade thanks you.  128.83.69.57 02:35, 10 March 2007 (UTC)
thread_title:  link to or section describing lagrangian vs eulerian descriptions? 

message: I started including introductory topics of continuum mechanics. So far, sections on the concept of continuum, configuration of a continuum, and kinematics of a continuum have been included. There is still a lot more to write. Sections on conservation of mass, conservation of momentum, conservation of angular momentum, thermodynamic laws... are needed.
This article should be the portal to solid mechanics and fluid mechanics. There should be a logical flow from this article to these other articles.
Please review in detail what I have written, specially the equations.  I think the writing also needs some improvement.--Sanpaz (talk) 04:10, 14 April 2008 (UTC)
thread_title:  Editions April 13 2008 

message: Why do you (Sanpaz) want convective derivative to be merged into continuum mechanics? Crowsnest (talk) 21:23, 19 April 2008 (UTC)
replies: text: The topic of Convective derivative does not seem to need its own article. This topic can be placed within the Continuum mechanics article and still retain all its content. Additionally, it brings the content of the article into context. However, If the Convective derivative topic needs to be expanded later then the page should be re-open. Sanpaz (talk) 23:16, 19 April 2008 (UTC)
replies: text: As far as I am concerned: go ahead. However, it is important that people can find "convective derivative", if they search for it. So I would suggest that in the case of a merger, convective derivative becomes a redirect to the appropriate article or section. Crowsnest (talk) 07:18, 21 April 2008 (UTC)
replies: text: Please note there are many redirects to here, e.g. advective derivative, substantive derivative, substantial derivative, material derivative, Lagrangian derivative, Stokes derivative and particle derivative. Crowsnest (talk) 11:40, 22 April 2008 (UTC)

text: Ok, I will take the merging label off Sanpaz (talk) 23:09, 10 May 2008 (UTC)
thread_title:  Merger of convective derivative into continuum mechanics 

message: To me the name Continuum mechanics doesn't make any sense and wander why not something like Mechanics of continuous media or even something better. --Gulmammad (talk) 16:04, 20 May 2008 (UTC)
replies: text: Continuum mechanics is a well established term. For instance Google Books gives over 800 results when searching on "continuum mechanics" in the book title, and Google Scholar over 2000 articles with the exact phrase in the title. Further, in my view, the name makes sense: just like particle mechanics describes the mechanics of particles, continuum mechanics describes the mechanics of a continuum. Crowsnest (talk) 21:17, 20 May 2008 (UTC)
replies: text: Actually in above I meant Continuous instead of Continuum. And what I was wandering at was those results of Google, almost no-one uses "my version" of the name, while one of the Landau edition books, vol. 8, has similar name, Electrodynamics of Continuous Media.  --Gulmammad (talk) 21:36, 22 May 2008 (UTC)
replies: text: Continuous media already redirects here. Perhaps Mechanics of continuous media is another good redirect to add, and include this phrase in the head of the article. I would leave the name of the article as it is now. – Crowsnest (talk) 21:56, 22 May 2008 (UTC)
thread_title: The name of the article

message: While most believe that matter is a bundle of energy separated by empty space, this is not true. There is no empty space in the universe. All space where matter doesn't exist is permeated by fields: electromagnetic and gravitational to say the least. It is the fundamental essence of space that permits matter to exist. Thought experiment time: close a box and draw a perfect vacuum. Remove all matter and then all energy and fields from it. What are the physical properties of the inside of this box? It has no mass, no fields, no energy. This is the definition of nothing, yet when you open the box the universe fills it right back up. Nothing has the ability to support matter and energy. This is in itself a physical property and therefore the space is not empty. It is potential.  —Preceding unsigned comment added by 75.70.62.142 (talk) 16:09, 30 December 2008 (UTC)
thread_title:  Empty Space 

message: Hi all, I'm wondering if we can remove the coordinate system vector 'b' from this introductory page. It is only used once, which usually leaves me wondering "where did this come from?", and is promptly dispensed with after being mentioned. I am browsing four CM books and only one (Chung) explicitly mentions a warped coordinate system traveling with the particle. I see that such an approach introduces (in addition to 'b') three more axis vectors and lets one write some expressions with a vector-type notation rather than summation or matrix notation, but it is not required to obtain the important CM operators like Jacobian, stress and rate of change of strain tensors, and so on, and I think the article would be less cluttered and better as an intro if we (quietly) don't mention 'b' and its (optional) coordinate system. Thoughts? IDave2 (talk) 20:40, 26 September 2009 (UTC)
thread_title: Question of 'b'

message: I believe it is not reasonable to consider Continuum Mechanics a branch of physics. Only a tiny proportion of research into continuum mechanics over the last 60 years or so has been performed by physicists - (a notable exception being magnetohydrodynamics, because of its importance for the study of plasmas in stars & planets). Most "famous" physics departments (at least in the US) have almost no one working in continuum mechanics. The theoretical work is done by applied mathematicians and practical work by engineers. Very few physicists sit on the editorial boards of mechanics journals. I think it is reasonable to say that physicists have no more than an elementary understanding of the behavior of deformable continua - an average physicist's idea of "mechanics" (at the mesoscale) begins and ends with rigid bodies and "potential" fluid flows. 
The reasons for this are interesting and multi-fold - the explosive, "headline grabbing" developments in atomic and modern physics in the 20th century being one of them. At any rate, I don't think it should be classified as a branch of physics any more.  
Commutator (talk) 07:30, 18 October 2009 (UTC)
replies: text: You just stated an opinion, with no reference. Continuum Mechanics is a branch of physics that uses a mathematical theory (continuum theory). Whether physicists, engineers, or mathematicians use it more or not at all is another issue, which does not take away the fact that it is a theory to interpret the physical world. Please read Continuum Mechanics by Irgens page 2, and also Continuum Mechanics by Ellis Harold Dill pages 1 and 2. sanpaz (talk) 02:16, 19 October 2009 (UTC)
replies: text: Hi Sanpaz, would you object to a sentence in the article stating that most of the research in continuum mechanics is not performed by physicists?  In the sense you describe it, it is physics, but it is certainly an area of physics neglected by physicists. Truesdell says something to this effect in pages 8 & 9  of his book (Nonlinear-field theories of mechanics), I'll try to dig up more modern references about it.Commutator (talk) 13:25, 19 October 2009 (UTC)
replies: text: I think it would be something good to state in the article with Truesdell as the reference. However, it would be good to have a more recent reference (if you could find one) that states that currently research into continuum mechanics is rarely done by physicists. sanpaz (talk) 14:55, 19 October 2009 (UTC)
thread_title:  Physicists have very little to do with Continuum Mechanics 

message: Hi, I've just edited the lead in a way in which I hope is helpful, aiming to make it slightly more accessible in the first instance, but keep some of the detail that was there. I'm keen to help out in improving the whole of this branch of Wikipedia, so I'd be interested to see if there is anyone out there watching these articles who I can ask for guidance/advice? Thudso (talk) 23:12, 10 December 2009 (UTC)
replies: text: Hi Thudso, I am also interest in this topic and have worked on this and other pages related to Continuum Mechanics (stress (mechanics), Deformation (mechanics), Infinitesimal strain theory and Finite strain theory. I saw your changes and I think they were good. I'll be around if you want to discuss about these topics. If you feel something needs to change do it and explain why and usually if people agree it will stand. If people disagree then a discussion about the change happens. At least that is my approach. Sometimes is better to say what you want to change and then ask for input (if the change is too big). Apart from that enjoy editing. All these pages on continuum mechanics sure need more working hands like yours. sanpaz (talk) 00:40, 11 December 2009 (UTC)
replies: text: I have edited the first paragraph after the ToC. I looked up the "Hill-Mandel condition" described, and found that Ostaja-Starzewski cites only books from the 1960s which use this term, but sources up to the present day referring to the "Hill condition". I therefore changed "Hill-Mandel" to "Hill", see P.245 in Ostoja-Starzewski for details. I was wondering if Hill condition need a separate article? Probably a little overkill at the moment, since what we already have needs improving first, but at some point in the future, I hope! Having read through the next paragraph, I think the two need to be merged, I might get some time to spend on this a little bit later. Thudso (talk) 16:50, 11 December 2009 (UTC)

text: Just been working on the second paragraph. I noticed that most of the original work there is yours, Sanpaz. I was wondering what the difference between \ \kappa_t and \ \chi that is introduced further down the page is? Thudso (talk) 22:30, 11 December 2009 (UTC)
replies: text: \ \kappa_t and \ \chi are both mapping functions. I just realized I made a mistake (I don't know why I wrote it the way I did). The first function should have a symbol X to identify the particle, which gives the position vector \ \mathbf x =\kappa_t(X), thus mapping the configuration of the body. The second mapping function maps the current configuration to the reference configuration, so it has a position vector \mathbf X as the input. Check the book by  Mase 1999. I need to correct that and write the reference. sanpaz (talk) 23:37, 14 December 2009 (UTC)
thread_title:  An effort to improve the continuum mechanics area 

message: The balance of momentum expressed in the article in the reference configuration, using the transpose of the first-Piola Kirchhoff stress tensor looks WRONG.
It seems to me that the equation should be the same, but using the first-Piola Kirchhoff stress tensor itself and not its transpose.
Reference : Marsden, J., & Hughes, T. (1994). Mathematical foundations of elasticity.
Could anyone confirm or infirm ?
: Quoting Belytschko, Liu, Moran (Nonlinear Finite Elements for Continua and Structures), p. 105 : "The definition of the nominal stress P is similar to that of the Cauchy stress ... The nomenclature used by different authors for nominal stress and first Pioal-Kirchhoff stress is contradictory.  Truesdell and Noll (1965), Ogden (1984), and Marsden and Hughes (1983) use the definition given here, whereas Malvern (1969) calls P the first Piola-Kirchhoff stress."  In the wikipedia article N is the nominal stress and P is the 1st PK stress. The Truesdell-Noll definition is used.  Bbanerje (talk) 22:50, 22 September 2014 (UTC)
thread_title:  Lagrangian balance of momentum 

message: If we use the definitions
{
\begin{align}
P_{ij} &= J \sigma_{ik} F^{-1}_{jk} \; , \\
F_{ij} &= \frac{\partial x_i}{\partial X_j} \; , \\
F^{-1}_{ij} &= \frac{\partial X_i}{\partial x_j} \; , \\
L_{ij} &= \frac{\partial v_i}{\partial x_j} \; , \\
\dot{F}_{ij} &= L_{ik} F_{kj} \; , \; , \\
J &= \text{det} \boldsymbol{F} \; , \; , \\
\end{align}
}
then we get
{
J \boldsymbol{\sigma} : \boldsymbol{L}
= J \sigma_{il} L_{il}
= J \sigma_{il} \delta_{ml} L_{im}
= J \sigma_{il} F_{mj} F^{-1}_{jl} L_{im}
= J \sigma_{il} F^{-1}_{jl} L_{im} F_{mj}
= P_{ij} \dot{F}_{ij}
= \boldsymbol{P} : \dot{\boldsymbol{F}} \; ,
}
which does not fit with the energy conservation law in the Lagrangian frame given by this page, which has the transposed of \boldsymbol{P}.
The same problem could be shown for the momentum equation.
I think that the equations, which are currently,
    {
    \begin{align}
      \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{P}^T -\rho_0~\mathbf{b} & = 0 \; , \\
      \rho_0~\dot{e} - \boldsymbol{P}^T:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0 \; , \\
      \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{N} -\rho_0~\mathbf{b} & = 0 \; , \\
      \rho_0~\dot{e} - \boldsymbol{N}:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0 \; , \\
    \end{align}
    }
  
should be replaced by
    {
    \begin{align}
      \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{P} -\rho_0~\mathbf{b} & = 0 \; , \\
      \rho_0~\dot{e} - \boldsymbol{P}:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0 \; , \\
      \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{N}^T -\rho_0~\mathbf{b} & = 0 \; , \\
      \rho_0~\dot{e} - \boldsymbol{N}^T:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0 \; . \\
    \end{align}
    }
  
Can someone confirm this?
I think different heat flux vectors \boldsymbol{q} should also be used for Lagrangian and Eulerian formulations!  — Preceding unsigned comment added by DeNayGo (talk • contribs) 11:26, 12 January 2017 (UTC)
thread_title:  I think there is a mistake in the balance laws, can someone confirm? 

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thread_title:  External links modified 

message: In the section Concept of a continuum, the following three paragraphs deal with criteria for application of the continuum assumption. These paragraphs are not essential to the article since they do not discuss or explain the continuum concept itself but rather raise possible questions about its validity and its application. For an article which tries to explain a fundamental idea whose basic concept is not excessively difficult to grasp, these paragraphs are too detailed, and I suggest they should be removed. Alternatively – if other editors feel these paragraphs are truly important – they might be moved to the end of the article as a new section.
Here are the three paragraphs:
The validity of the continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure exists. More specifically, the continuum hypothesis/assumption hinges on the concepts of a representative elementary volume and separation of scales based on the Hill–Mandel condition. This condition provides a link between an experimentalist's and a theoretician's viewpoint on constitutive equations (linear and nonlinear elastic/inelastic or coupled fields) as well as a way of spatial and statistical averaging of the microstructure.[1]
When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than that of the representative volume element (RVE) size, one employs a statistical volume element (SVE), which, in turn, leads to random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous.
Specifically for fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made.
I prefer not to remove these paragraphs myself as I am not qualified in this field. I hope someone with more knowledge will consider doing so. Dratman (talk) 18:04, 17 March 2019 (UTC)
replies: text: Moved, also rewrote that section. Still feels like there might be some redundancy between "Explanation" and "Concept of a continuum" sections. Also the validity section could really do with more citations but I haven't looked into that much yet. Adigitoleo (talk) 10:59, 28 December 2022 (UTC)
thread_title:  Concept of a continuum 

message: Hello, for those interested I have just edited the explanation section. Minor rephrasing and more links, see the diff here https://en.wikipedia.org/w/index.php?title=Continuum_mechanics&diff=1130035568&oldid=1126094519 Adigitoleo (talk) 08:54, 28 December 2022 (UTC)
replies: text: I have made further edits to the "concepts of a continuum" section and split off a new section "validity" which ironically lacks citations that I can't provide at the moment. Adigitoleo (talk) 11:01, 28 December 2022 (UTC)
thread_title:  Edit to explanation section 

message: I would suggest removing/replacing the car traffic example. I made some small edits for conciseness, but have left it in at the moment. However, I don't think the example is a great fit for the following reasons:
* This is not a textbook: I'm not convinced that an example belongs in this article, the space could be used for expanding the section on "validity" or creating a section on volume elements (perhaps bringing in the small and lonely Volume element article, unless that is better suited to a finite element article?)
* This is not a PDE article. The example might be better suited in a mathematics article?
* The article focuses on continuous media and approximating the traffic on a highway as a continuum is perhaps not the most intuitive example, because cars are quite obviously discrete even in a macroscopic sense. It is also out of place coming before continuum mechanics#Formulation of models (which is focused on bodies in 3D space).
For now I think I will move the example down further, if there are no objections I will remove it in the next month. Adigitoleo (talk) 11:45, 28 December 2022 (UTC)
replies: text: Car traffic example removed 08/02/2023 Adigitoleo (talk) 07:16, 8 February 2023 (UTC)
thread_title:  Removal of car traffic example