Talk:Poisson's ratio

Large Deformation Formula
Can someone tell me what strain measure is used in the graph showing small vs. large deformation lines for the large deformation formula? - EndingPop 18:18, 18 March 2007 (UTC)


 * small deformations assumes that strain is linear (the first derivative is a constant number), large deformation assumes that the strain is not linear (depending on the precision is it not constant up to second or larger "n-th" derivative). If you are talking about "Engineering strain", "true strain", "logarithmic strain" or "lagrange strain" then it is of no importance here: this plot could apply to any of them. The point is that whether you represent strain deformation using a linear or non-liear (more precise) function. Janek Kozicki 20:40, 19 March 2007 (UTC)

Seems like the large deformation formula is wrong - should be $$\left(1+ \frac {\Delta L}{L}\right)^{-\nu} = 1-\frac{\Delta L'}{L}$$. No matter how hard I try I can't get that $$\nu$$ to pop up in the integration. This stackoverflow answer seems to concur. I'll change it for now, feel free to correct and revert back if I'm wrong. -- Adam — Preceding unsigned comment added by 77.103.193.132 (talk) 22:16, 22 December 2016 (UTC)
 * Your version is correct. Bbanerje (talk) 22:07, 25 December 2016 (UTC)

arnt the eqns under orthotropic material true for isotropic?
this should be mentioned. maybe

---

Orthotropic means it is different in different directions. Isotropic means it is the same in all directions. Isotropic equations would be found under the generalize section.--thegreatco (talk) 17:48, 14 May 2008 (UTC)

Area?
The definition given is in terms of strain in the transverse direction. But is that the average strain in the plane orthogonal to the stress, or what? —BenFrantzDale 07:14, 20 October 2005 (UTC)

how about a nice little equation to go along with the description...? --alex

Moved from article
G is actually the Shear Modulus, a materials resistance to a change in shape without a change in volume. G is equal to shear stress, tao, over shear strain, gamma. Tom Harrison Talk 19:50, 24 May 2006 (UTC)

A bit of humor
The Poisson Conspiracy! http://home.messiah.edu/~tvandyke/podcast5.mp3 153.42.211.95 18:44, 5 October 2007 (UTC)

Merge
The result was merge POISSON'S EFFECT into POISSON'S RATIO. The merge was proposed almost two months ago (on 19th March). All opinions (the four expressed below and the one of the user who initially proposed it) are in favor of the merge. -- Federico Grigio, alias Nahraana (talk) 17:54, 25 April 2008 (UTC)

Poisson's effect really should get merged in. That page has some good stuff, so it'll take a little thought, but not much. —Ben FrantzDale (talk) 02:10, 29 April 2008 (UTC)


 * You say the page has good stuff and that you want it to be merged. Now, what is the reason why it should be merged?  Federico Grigio, alias Nahraana (talk) 08:08, 1 May 2008 (UTC)


 * They should be merged because they cover the same topic. Together they contain more details than either one, but the effect and the ratio go hand in hand; the ratio is just a measure of the effect. —Ben FrantzDale (talk) 13:08, 1 May 2008 (UTC)


 * I see. I agree to merge the articles, because of the same reasons as above.  Federico Grigio, alias Nahraana (talk) 08:34, 5 May 2008 (UTC)


 * I agree, they should be merged. They are taught as one in all my engineering courses.  The ratio is the measure of the effect. thegreatco (talk) 17:22, 6 May 2008 (UTC)


 * Merge it. Greg Locock (talk) 01:58, 12 May 2008 (UTC)


 * I'll be honest, I've never merged a page and I don't think Poisson's Ratio is the place to start. --thegreatco (talk) 17:44, 14 May 2008 (UTC)

Hi. I started merging (deleted some repeated information and copied-pasted two sections).

The section "Calculation of Poisson’s effect" is still left in Poisson's effect. Maybe, someone who likes equations can carefully implant its non-repeated bit into this article.

Federico Grigio, alias Nahraana (talk) 11:31, 15 May 2008 (UTC)


 * After we finish the merge (when all content has been removed from Poisson's effect) we can place  on it.  Federico Grigio, alias Nahraana (talk) 11:35, 15 May 2008 (UTC)

Added distinction between pre-and post-yield behavior
I made a slight modification to distinguish between pre-yield and post-yield deformation. This is strongly related to mods I made earlier today on the Necking(engineering) page. Also made it slightly less metals-centric. Crosslink (talk) 22:37, 16 February 2009 (UTC)

Definition of Poissons Ratio
The definition given in the main article refers only to tensional forces (ie: stretching). George Davis (Structural geology) defines it as "the degree to which a core of rock bulges as it shortens" under compression.Iain —Preceding unsigned comment added by 41.0.29.98 (talk) 10:45, 23 October 2009 (UTC)

Greater than 0.5 or not?
I was wondering when I read "ν cannot be less than −1.0 nor greater than 0.5." And later on there is this honeycomb thing with ν up to 3.88. How does that fit together??? (I was wondering even before I read the second thing cause I could easily design a grid which would have ν out of the upper range. Obviously a kind of honeycomb.) So does the first statement have to be relativated somehow? Like "for bulk specimens..." or so? Peterthewall (talk) 17:41, 18 January 2010 (UTC)

Wouldn't be isotropic. Greglocock (talk) 02:52, 19 January 2010 (UTC)


 * Yes, thanks. I came to that conclusion as well now ;-) But it took a while. When I first read it there it seemed to be so general. I added in the intro that there are as well materials exhibiting values > 0.5. Peterthewall (talk) 11:33, 19 January 2010 (UTC)

I read the reference to the paper by Robert (Bob) Park and couldn't where the paper stated Possion's ratio of 0.5 for mild steel in the post-yield range. If anyone can point me towards some information for the value once mild steel has yielded I would be very happy. — Preceding unsigned comment added by 132.181.5.142 (talk) 23:07, 27 November 2014 (UTC)

I too read the paper "The seismic performance of steel-encased reinforced concrete bridge plies" of R.J.T. Park and was unable to find any mention of the post-yield values of poissons ratio. I don't think this paper qualifies as a reference for that statement. - Sebastian Jørgensen, 03 November 2017 "An experimental validation of volume conservation assumption for aluminum alloy sheet metal using digital image correlation method" by X. Xie, J. Li, B. Sia, T. Bai, T. Siebert and L. Yang proves constant volume assumption in the plastic region for an aluminum alloy. Tensile tests of 6082T6 performed by me shows that assuming constant volume over all of the deformation results in a 0,2 proportionality limit and a maximum strength only 0,4 MPa lower than if the correct poissons ratio is used for calculating the area and thus the stresses in the elastic region. - Sebastian Jørgensen, 29th November 2017 — Preceding unsigned comment added by 185.136.116.65 (talk) 20:06, 29 November 2017 (UTC)


 * These guys certainly think so : "For isotropic elastic continuum analysis, a single value of Poisson’s ratio is sufficient to fully characterize the material response since standard plastic flow rules assume incompressibility, and a Poisson’s ratio of 0.5 in the post-yield regime. F"


 * http://www.ecs.umass.edu/~arwade/steelfoam_mat_design.pdf


 * Cheers Greglocock (talk) 00:23, 28 November 2014 (UTC)

Changing the equations for orthotropic and transversely isotropic materials
Some users have been changing the equations for orthotropic and transversely isotropic materials without understanding the significance of the quantities. These equations can be written in many ways. However, the definitions of the quantities in the equations must be consistent throughout the article. They are consistent as of now. For example, changing $$\nu_{xy}/E_x$$ to $$\nu_{xy}/E_y$$ changes the meaning of $$\nu_{xy}$$. The new meaning is non-standard and a explanation has to be given on why such a menaing has been chosen. Also, that change in meaning has to be reflected throughout the article, including in places where various Poisson's ratios of composites have been listed. Please discuss your reasons for changing these equations before going ahead with the changes. Bbanerje (talk) 22:04, 15 February 2010 (UTC)

To be consistent with the matrix for transverse isotropy on the Hooke's Law page, it might be best to change Gyx to Gxy or explicitly state they are equivalent as you have done for other relations. —Preceding unsigned comment added by 18.95.5.74 (talk) 01:55, 1 March 2010 (UTC)

But what does it mean?
transverse shear deformation blah blah blah. I read the whole article and still don't know what it means. This article is obviously written by intelligent, articulate, educated people, but it carries a prerequisite of maybe two semesters of physics or engineering or something.

Could someone please just throw in a statement (preferably in the first paragraph), to the effect of, "a poisson ratio of [?] means it's very strong, or it will break easily when twisted, or whatever it means. Meanwhile, I'm off to find out if a polycarbonate sheet with a poisson ratio of 0.37-0.38 is suitable for my project.

thanks, —Preceding unsigned comment added by Dusty.crockett (talk • contribs) 02:47, 20 June 2010 (UTC)


 * The second paragraph says:


 * When a sample cube of a material is stretched in one direction, it tends to contract (or occasionally, expand) in the other two directions perpendicular to the direction of stretch. Conversely, when a sample of material is compressed in one direction, it tends to expand (or rarely, contract) in the other two directions. This phenomenon is called the Poisson effect. Poisson's ratio ν (nu) is a measure of the Poisson effect.


 * What part of that don't you understand? How can it be improved?PAR (talk) 03:10, 20 June 2010 (UTC)


 * @Dusty.crockett: The Poisson's ratio of a material is a measure of stiffness, not a measure of strength. Someone who knows a bit of undergraduate level mechanics (usually an engineering student) can help in choosing a material with required properties. However, such a task is best left to professionals. Bbanerje (talk) 04:30, 20 June 2010 (UTC)
 * Bbanerje, I don't think it's proper to call it a "stiffness" property; it's a unique intensive material property. Wizard191 (talk) 15:47, 20 June 2010 (UTC)
 * Wizard191, the Poisson's ratio has a precisely defined meaning in linear elasticity as a scaling factor for terms in the stiffness tensor. Therefore it is a measure of stiffness though it is dimensionless.  The Young's modulus, which is the other stiffness measure in linear elasticity, is also an intensive property.  The Poisson effect, which is a qualitative description of the scaling phenomenon, describes a larger range of situations than does the Poisson's ratio which is well-defined only for infinitesimal strains. Bbanerje (talk) 22:56, 20 June 2010 (UTC)
 * Honestly, I'm not familiar with the stiffness tensor at all. However, I can say that this article doesn't mention stiffness once and I've never heard of Poisson's ratio being related to stiffness. However, if you have a reference for your definition, I would like to see it. Wizard191 (talk) 16:50, 21 June 2010 (UTC)
 * The stiffness tensor is also known as the elasticity tensor. It is the "tensor of proportionality" in the tensor expression of Hooke's law which states that $$\sigma_{ij}=C_{ijmn}\varepsilon_{mn}$$, $$C_{ijmn}$$ being the stiffness or elasticity tensor. It corresponds to k in the scalar statement of Hooke's law: F=k x. We should probably make its name consistent across all articles. I am in favor of calling it the stiffness tensor, since as it becomes larger, the force (stress) needed to produce a given displacement (strain) increases. The stiffness tensor will be a function of two of the elastic moduli, one of which may or may not be the Poisson ratio, so you cannot really say the Poisson ratio alone is a measure of stiffness, I think. PAR (talk) 20:02, 21 June 2010 (UTC)


 * "What part of that don't you understand? How can it be improved?PAR"

Thanks for asking. I see that my question might be a bit obtuse; the definition of the ratio itself is pretty clear, but I was hoping to find a more concise "bottom line" explanation. I've worked it out (only took a little bit of effort) -- a material with a ratio close to 0 retains its original proportions when stretched better than one with a larger (positive or negative) ratio. If it expands in the direction perpendicular to the stretch, the ratio is negative; if it contracts, the ratio is positive. I realize that might be too simplistic, there might be other factors that would render such a statement inaccurate. BTW, The table of ratios for various materials -- that's a nice touch. Dusty.crockett (talk) 00:45, 23 June 2010 (UTC)

Now I've gotten around to reading some of this discussion, I see the above quote from George Davis, "the degree to which a core of rock bulges as it shortens" under compression. Although limited in scope, it seems a very clear illustration of exactly what is being measured. For clarification, my only reason for posting here is to provide the perspective of a non-engineer who see the term "Poisson's Ratio" on some product specs, and just wants to fit it into context. I'm basically not qualified to add content to a technical subject like this. Thanks. Dusty.crockett (talk) 13:53, 23 June 2010 (UTC)

bonkers
The way the equation is defined won't give you a poisson's ratio of 0.5 for a perfectly incompressible material. It gives a ratio of 2 as defined in the article. Draw a quick before and after square diagram to see what I mean. I understand the mechanics, but the definition of strain directions in the equation is misleading and gives you the inverse ratio.

Just looking at the figure provided, strain (as elongation) in an x direction is larger than strain in the y direction, which gives a value higher than 1.

It seems more intuitive to just define it as the ratio of contraction to associated longitudinal extension —Preceding unsigned comment added by Georesidue (talk • contribs) 19:30, 25 June 2010 (UTC)


 * This definitely needs to be clarified. Strain is not a dimension (s) of an element, it is a dimensionless quantity ds/s, where ds is the change in a dimension when stressed, so taking a ratio of width to height as the Poisson ratio of a distorted cube is not correct. Also, we have to work in three dimensions, not two.


 * There are three concepts, the position of an infinitesimal element inside the material $$[x,y,z]$$, the displacement of that element $$u=[u_x,u_y,u_z]$$ and the strain tensor $$\varepsilon\,$$ which is a measure of how much the element is distorted. For the simple example of a cube compressed on its x faces, only the diagonal elements of strain are not zero. The diagonal elements of strain are $$\varepsilon=[du_x/dx,du_y/dy,du_z/dz]$$. In the simple example, if you have a cube, one unit of length on a side, and slightly compress it on its +x and -x faces, it will change from having dimensions $$[1,1,1]\,$$ to $$[(1-b), (1+a), (1+a)]\,$$ where a and b are small, (usually) positive values. In other words, an infinitesimal element inside the cube originally at $$[x,y,z]$$ will be moved to $$[(1-b)x,(1+a)y,(1+a)z]\,$$ so that its displacement from its original position will be $$u=[-bx,ay,az]\,$$ and the diagonal elements of the strain will be $$\varepsilon=[-b,a,a]\,$$. Poisson's ratio will then be $$\nu=-\varepsilon_{yy}/\varepsilon_{xx}=-\varepsilon_{zz}/\varepsilon_{xx}=a/b\,$$. If the material maintains constant volume, we must have $$(1-b)(1+a)^2=1\,$$. Solving for Poisson's ratio in terms of a gives $$\nu=(1+a)^2/(2+a)\,$$. In the limit of small values of a, that's $$\nu=1/2\,$$.


 * Now we just have to make that clear in the article.PAR (talk) 12:33, 26 June 2010 (UTC)

Right, that all makes total sense. I think the misleading part in the article (at least for me) comes from the inexact way of defining how the principal stresses are oriented for the example shown. It currently states:
 * "Assuming that the material is stretched or compressed along the axial direction (the y axis in the diagram):"

But for the equation specified, it needs to be the former, not the latter, which also makes the picture backwards. Either the picture is backwards or the equation is inverted. OR just remove any mention of x and y axes and specify directions of compression and tension. I think...

Thanks for the response —Preceding unsigned comment added by Georesidue (talk • contribs) 19:35, 26 June 2010 (UTC)

Figure 1 shows that the red cube is elongated by more than $$\Delta L$$ in x-direction and shortened by more than $$\Delta L'$$. Perhaps by a factor of 2? It could easily be fixed in the figure by deviding all by 2. — Preceding unsigned comment added by 80.254.154.91 (talk) 09:54, 16 June 2015 (UTC)

Incremental Poisson's ratio
I notice that the definition of Poisson's ratio has been replaced with an incremental form

\nu = -\frac{d\varepsilon_\mathrm{trans}}{d\varepsilon_\mathrm{axial}} = -\frac{d\varepsilon_\mathrm{y}}{d\varepsilon_\mathrm{x}}= -\frac{d\varepsilon_\mathrm{z}}{d\varepsilon_\mathrm{x}} $$ Though not incorrect in the strict sense, this definition is does not accurately represent the quantity used by engineers. The reason is that the constant ratio $$\nu$$ is applicable only for linear elastic materials in which case

\frac{d\varepsilon_\mathrm{trans}}{d\varepsilon_\mathrm{axial}} = \frac{\varepsilon_\mathrm{trans}}{\varepsilon_\mathrm{axial}} \,. $$ The definition should agree with recognized standards such as the ASTM standrard "ASTM E132 - 04 Standard Test Method for Poisson's Ratio at Room Temperature" which clearly states that "When uniaxial force is applied to a solid, it deforms in the direction of the applied force, but also expands or contracts laterally depending on whether the force is tensile or compressive. If the solid is homogeneous and isotropic, and the material remains elastic under the action of the applied force, the lateral strain bears a constant relationship to the axial strain. This constant, called Poisson’ratio, is an intrinsic material property just like Young’modulus and Shear modulus."

The standard relations between the Poisson's ratio and the other moduli of elasticity are not generally valid for incrementally linear elasticity. I'd suggest that you revert back to the standard definition and add a caveat saying that incremental Poisson's ratios are sometimes used in nonlinear elasticity. Bbanerje (talk) 02:26, 26 July 2010 (UTC)


 * The standard relations between the Poisson's ratio and the other moduli of elasticity ARE generally valid for incrementally linear elasticity, as long as the other moduli are also defined incrementally. The real question is - what is the (incremental) Poisson's ratio for a stressed material? If it is a constant, independent of the amount of deformation, then the "integrated" Poisson's ratio is a function of the amount of deformation, as presently described in the article. Conversely, if the "integrated" Poisson's ratio is constant, which is what you are talking about, then the incremental Poisson's ratio must vary with the amount of deformation. Only when the deformations are very small with respect to the dimensions of the unstressed test object do the two become very nearly the same. I do not have access to ASTM E132-04, but I cannot imagine that it does not contain caveats about carrying the application of the test stress or strain too far. In other words, the quotation you give cannot be giving the complete picture. If you do have access to this publication, can you investigate what these caveats are? Does it give specific limits on the amount of relative deformation (or strain) under which the measurement is valid? If it does not, it is incomplete, and if it does, are those limits large or small with respect to the unstressed dimensions of the test object? PAR (talk) 13:46, 26 July 2010 (UTC)


 * "Only when the deformations are very small with respect to the dimensions of the unstressed test object do the two become very nearly the same". I don't have the full ASTM standard but it's intended primarily for metals in which the elastic regime is small and relatively linear.  The popular definition of Poisson's ratio is valid only for infinitesimal strains and linear elasticity.  The Poisson effect may be observed beyond that regime. But the Poisson's ratio is neither constant nor does it have any fixed relationship with other moduli beyond the small strain regime.  That can be verified by exploring any of a number of hyperelastic material models.  My point is that using the term "Poisson's ratio" when we mean a more general "Poisson's effect" confuses rather than clarifies.  You can see an example of the problem in P. Gretener, 2003, "AVO and Poisson's ratio", The Leading Edge; January 2003; v. 22; no. 1; p. 70-72, Society of Exploration Geophysicists. Bbanerje (talk) 22:54, 26 July 2010 (UTC)

I'm still having trouble understanding your objection. Maybe it is trouble with definitions. Considering a cube of side L subject to normal stress on the x-faces only, not necessarily small, with its stretched x or axial length being $$L+\Delta L$$ and its contracted transverse length being $$L-\Delta L'$$ we can define the "finite" strains as $$\varepsilon_x=\Delta L/L$$ and $$\varepsilon_y=-\Delta L'/L$$. If we then subject the cube to a further infinitesimal stress, the axial or x faces will move to $$L+\Delta L+\delta L$$ and the transverse faces will move to $$L-\Delta L'-\delta L'$$, where the $$\delta$$'s are infinitesimally small. The infinitesimal strains will be $$d\varepsilon_x=\delta L/(L+\Delta L)$$ and $$d\varepsilon_y=-\delta L'/(L-\Delta L')$$. Now we can define a finite Poisson's ratio as



\nu_{finite}(\varepsilon_x)=\Delta L'/\Delta L $$

where $$\varepsilon_x$$ is chosen as the measure of the pre-existing strain on the cube. An infinitesimal Poisson's ratio can be defined as:



\nu_{inf}(\varepsilon_x)=\delta L'/\delta L $$

And the "usual" Poisson's ratio will then be $$\nu_{inf}(0)$$ which will be equal to the limit of $$\nu_{finite}(\varepsilon_x)$$ as the $$\varepsilon_x$$ approaches zero. This will be a constant of the material, independent of its dimensions. These Poisson's ratios are not necessarily equal to each other except when $$\varepsilon_x$$ approaches zero. The finite Poisson's ratio can be found by integrating the infinitesimal Poisson's ratio from zero out to the final $$\varepsilon_x$$. Linear elasticity assumes infinitesimally small strain variations about $$\varepsilon_x$$. Usually, one considers the case of an initially unstressed material, i.e. infinitesimally small strain variations about $$\varepsilon_x=0$$. The present article, when calculating the large scale deformations, assumes that $$\nu_{inf}(\varepsilon_x)$$ is independent of $$\varepsilon_x$$, allowing the integration to be easily performed.

I am in favor of saying that Poisson's ratio is $$\nu_{inf}(0)$$, i.e. the ratio of the transverse to axial changes in distance (or strain) for very small values of those distances (or strains) for an unstressed material. This is given by the "incremental" definition provided. If we say Poisson's ratio is the ratio of the strains for a linear material, we are saying essentially the same thing, since linear implies infinitesimally small variations. If we simply say its the ratio of strains, large or small, then we are giving a definition which is not a constant of the material except under very wierd conditions. PAR (talk) 00:29, 27 July 2010 (UTC)
 * It's a question of definition, i.e., what is the accepted definition of Poisson's ratio and which source should we use to determine that definition?  For instance, the article uses a concept called strain.  What is the definition of strain that is used in the definition of Poisson's ratio?  Is it the infinitesimal strain, the true strain, the Lagrangian Green strain, or some other strain measure?  My take is that we should keep it simple and define the Poisson's ratio as a ratio of infinitesimal strains in linear elasticity and we can call similar ratios in nonlinear or finite strain elasticity something else. Bbanerje (talk) 23:01, 27 July 2010 (UTC)


 * I agree, but isn't the ratio of infinitesimal strains $$d\varepsilon_t/d\varepsilon_a$$ rather than $$\varepsilon_t/\varepsilon_a$$? PAR (talk) 22:10, 28 July 2010 (UTC)
 * We may be talking about two different things. The infinitesimal strain tensor is defined as $$ \varepsilon = \tfrac{1}{2}\left((\nabla\mathbf u)^T + \nabla\mathbf u\right) $$ where $$\|\mathbf u\| \ll 1 \,\!$$ and $$\|\nabla \mathbf u\| \ll 1 $$.  So a ratio of such strains is $$\varepsilon_1/\varepsilon_2$$, for example.  People usually find that ratio easier to understand than ratios of differentials or derivatives.  For the benefit of others here's why I think the simple ratio is good enough.  For a linear material with constant $$\nu$$ (ignoring signs)
 * $$ \varepsilon_1 = \nu\varepsilon_2 \implies d\varepsilon_1/d\varepsilon_2 = \nu$$. Alternatively, we can do
 * $$ d\varepsilon_1/d\varepsilon_2 = \nu \implies \varepsilon_1 = \int \nu d\varepsilon_2 + C = \nu\varepsilon_2 + C \implies \nu = \varepsilon_1/\varepsilon_2 - C/\varepsilon_2$$
 * If $$\varepsilon_1 = 0 \implies \varepsilon_2 = 0$$ we have $$C=0$$ and we get the standard definition of Poisson's ratio.Bbanerje (talk) 01:10, 30 July 2010 (UTC)

Allowable range of ν
Try again from above: Allowable range of ν

It would be useful to have a short section that identifies the allowable range of ν, which is -1 to 1/2, based on the equation 3B(1-2ν) = 2G(1+ν)

132.250.22.8 (talk) 20:26, 6 October 2010 (UTC) goldfish

Cause of the Poisson effect
Is it me, or does the "cause" part simply restate the effect, without explaining the underlying mechanism? To me it reads as "Materials exhibit the Poisson effect because their molecular structure exhibits the Poisson effect." Could someone give a clearer explanation of why this happens? TheCat5001 (talk) 10:16, 3 February 2011 (UTC)

Long lead
Isn't the lead too long? --Mortense (talk) 13:46, 12 January 2013 (UTC)
 * A long lead is not so detrimental as junks in adjacent section, Poisson's ratio. When the bonds elongate in the direction of load, they shorten in the other directions.
 * Obviously, it's not true for liquids and rubber-like materials, and I doubt that a noticeable change in chemical bonds' length in directions perpendicular to the stress actually does exist for most covalent-bound substances. Could you fix both items for the same price? Incnis Mrsi (talk) 15:39, 12 January 2013 (UTC)

Lame's relation
"For isotropic materials we can use Lamé’s relation" However the linked wiki doesn't mention "Lamé’s relation" or the relation of this page. Not sure which to clean up, though...71.29.179.72 (talk) 18:53, 15 October 2020 (UTC)

Incompressible
The statement that rubber, who’s Poisson’s ratio is near 0.5, is “incompressible” is misleading. One way of measuring compressibility is the bulk modulus. Rubber, a polymer, has a bulk modulus that is orders of magnitude lower than metals and ceramics.

It’s not that rubber is incompressible. It’s that the ratio of bulk modulus to shear modulus is very high. Or, more helpfully, its shear modulus is very low compared to its bulk modulus. Calling rubber "incompressible" without any other context makes it sound like materials who's Poisson's ratio is lower, like steel or diamond, are more compressible, which is of course silly. Hermanoere (talk) 15:46, 3 January 2021 (UTC)
 * I think you are confused about what "incompressible" means. In engineering, it means that the volume does not change when then material is compressed or extended.  If you apply a uniform pressure to a solid ball of nearly-incompressible rubber, the volume will not change measurably.  A consequence is that an ideal incompressible ball can support an infinite amount of pressure. Bbanerje (talk) 04:43, 4 January 2021 (UTC)
 * The change in volume when the material is compressed or extended is measured using the bulk modulus. Steel and diamond, and other non-polymeric solids, have huge bulk moduli compared to rubber and other polymers. (Check out typical values here.) In other words, they are orders of magnitude more incompressible than rubber is. But as a result of this, we can't say their Poisson's ratios are 0.5. They are more like 0.3.


 * The reasoning presented in the article is backwards. Rubber's Poisson's ratio is not 0.5 because it is incompressible. Instead, it can be shown that if the bulk modulus is much larger than the shear modulus (check out reference 1 in this very article), then poisson's ratio is close to 0.5. A result of this (not the cause of this) is that the material can be considered essentially incompressible in some calculations (like in a uniaxial tension or compression test). Rubber is not infinitely stiff in isostatic compression. Hermanoere (talk) 16:23, 5 January 2021 (UTC)
 * Perhaps it's time to look at some equations for isotropic solids. In particular, the relationship between $$K$$, $$G$$, and $$\nu$$, i.e, $$\nu = \tfrac{3K-2G}{2(3K+G)}$$.  At $$\nu = 0.5$$, we get the requirement that $$ G = 0$$ while $$ K $$ can have any value as long as $$ K \ne 0 $$. Since one such value is $$K \rightarrow \infty$$ (excluding $$\infty$$ where division is not defined), by your own reasoning, the material is incompressible. But we know that rubber has a non-zero shear modulus.  Let's see what happens when we choose $$\nu$$ = 0.499.  In that case, $$ K/G $$ = 499.7 (if my algebra is correct).  You can play that game with values closer and closer to 0.5 and, of course, the ratio will increase until it's infinity.  Since all solids have a non-zero shear modulus (by definition), to get an infinite ratio, the bulk modulus must be infinity. You will notice that all material data sheets give low values for rubber shear modulus, e.g., 0.0002-0.03 GPa.  But you will not find many experiments that directly measure this value.  Instead, they measure the tangent Young's modulus and back out the shear modulus assuming (or measuring) a Poisson's ratio close to 0.5. Bbanerje (talk) 02:51, 6 January 2021 (UTC)


 * Well, like you say, $$K$$ cannot be $$\infty$$, since it would lead to the ratio $$K$$/$$K$$ being mathematically indeterminate. And your assertion that $$K$$ must somehow approach $$\infty$$ is unnecessary and inaccurate (as seen in typical values for rubber and other materials). Upshot is, if $$G$$ is zero, and $$\nu$$ is 0.5, then $$K$$ can be anything except zero or $$\infty$$. So saying rubber is incompressible is flawed. Like I said before, it is essentially "incompressible" if you're just doing uniaxial loading, but $$K$$ is not infinite. Therefore, saying that "rubber is incompressible, so $$\nu$$ is 0.5" is flawed, and you should stick to the more correct wording of reference 1 in the article. Hermanoere (talk) 23:53, 7 January 2021 (UTC)