Talk:Finite strain theory

Finger Tensor

In the process of formulating the Finger tensor, in the equation - F= V R

i believe R is a ROTATION tensor, (as implied by R R^T =I), and NOT an antisymmetric one.

BTW, it's worth mentioning that such a decomposition isn't unique: for F= (-1)I, e.g., the rotation factor R can be extracted at least in 3 different ways. this, of course, does not affect the uniqueness of the Finger tensor, as seen from the definition B = F F^T.

(Ofek Shilon, 7.8.05)

Cleanup

To mention just one issue: the matrix exponential and logarithmic derivative provide the passage between finite and infinitesimal deformations. The discussion of "uni-axial extension" of an "incompressible material" should be clarified: this just means pulling the body from either end along one axis, with the material responding by extending along that axis, while keeping volume constant, so that its cross section shrinks. Then the Finger tensor can be related via logarithmic derivative to the "Cauchy form" of the tidal tensor near a typical compact massive body in Newtonian gravitation (infinitesimal deformation). This will help both physics and engineering students see how two subjects relate to one another, by removing artificial barriers of notation and terminology.

• This is very nice, but I am affraid it is above my command in the clestial mechanics. Can you be bold and made the neccessary changes youself? abakharev 07:22, 13 November 2005 (UTC)

It should also be helpful to write out how the strain deformation tensor defines the linear transformation relating the rest shape to the deformed shape of the body. In one of these articles, the author mentions (but does not explain) that we can regard this as the first term in a Taylor series for a more general transformation. This also should be clarified.

In addition to these, individual articles often use terms without defining them, which makes them much less useful for students, and generally speaking are insufficiently linked to each other and to relevant articles over in the applied math categories.---CH (talk) 00:47, 13 November 2005 (UTC)

Notation

I don't use it myself, but a common and cleaner notation is to use upper-case letters for initial configuration quantities. Primes seem to be more commonly used for a transformation to a new coordinate basis.RDT2 20:21, 19 August 2006 (UTC)

Updating the article

I've been working on this article trying to include what is commonly seen in the the literature for Finite strain theory. I have kept most, if not all, of the content in the previous edits. There are still many things (Figures and text) I will be updating in the following days (please be patient). The derivation of the Finite strain tensor is not 100% fished. I will do that in the next day. I know is a lot in one edit, but I felt this article needed a big boost. Please take your time and read all the additions, as I am sure I may have missed something in the equations, and of course more improvement is needed. Sanpaz (talk) 03:28, 19 June 2008 (UTC)

I forgot to mention that the references are not written yet...working on it. Sanpaz (talk) 04:01, 19 June 2008 (UTC)

Your work is great! Thanks a lot. Hope to see more of this Alex Bakharev (talk) 02:42, 20 June 2008 (UTC)

Added back the sections on deformation tensors. Things were getting hard to find in the long section of polar decompositions. Bbanerje (talk) 04:47, 20 June 2008 (UTC)

Sign wrong in Eulerian-Almansi finite strain tensor?

I could be wrong, but I think there should be a minus sign in front of the nonlinear term in the Eulerian-Almansi finite strain tensor (last equation before the section "Stretch Ratio"). Thanks for a very useful article! 146.64.81.22 (talk) 09:42, 18 August 2008 (UTC)

Good eye. Thanks for spotting the error. Sanpaz (talk) 16:26, 18 August 2008 (UTC)

The second invariant for nearly incompressible material may be incorrectly defined

Hi,

The theory looks beautiful, but I think the second invariant may be incorrectly defined, which has some repercussions on the article on hyperelasticity as well. As far as I am aware, the following definitions hold: $J=\det(\mathbf {F} )$, after which you can define the isochoric deformation gradient ${\bar {\mathbf {F} }}=J^{-1/3}\mathbf {F}$, resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free.

Using this you can subsequently define the isochoric Finger tensor ${\bar {\mathbf {B} }}$, from the regular definition of the Finger tensor, but now using the isochoric deformation gradient: ${\bar {\mathbf {F} }}\cdot {\bar {\mathbf {F} }}^{T}$. And then you can calculate the invariants in the regular way, but now they are isochoric invariants, i.e. volume free. It is easy to see that ${\bar {\mathbf {B} }}=J^{-2/3}\mathbf {B}$.

The invariants are now

${\bar {I}}_{1}={\text{tr}}({\bar {\mathbf {B} }})=J^{-2/3}{\text{tr}}(\mathbf {B} )=J^{-2/3}I_{1}$

${\bar {I}}_{2}={\frac {1}{2}}\left({\text{tr}}({\bar {\mathbf {B} }})^{2}-{\text{tr}}({\bar {\mathbf {B} }}^{2})\right)={\frac {1}{2}}\left(\left(J^{-2/3}{\text{tr}}(\mathbf {B} )\right)^{2}-{\text{tr}}(J^{-4/3}\mathbf {B} ^{2})\right)=J^{-4/3}I_{2}$. This latter result clearly follows from all the squares.

${\bar {I}}_{3}=\det({\bar {\mathbf {B} }})=J^{-6/3}\det(\mathbf {B} )=J^{-2}I_{3}=J^{-2}J^{2}=1$, which was the intent all along.

The set of invariants which are now used to define the behaviour deviatorically are the first two invariants of the isochoric Finger tensor, (which are identical to the ones for the right Cauchy Green stretch tensor), and add $J$ into the fray to describe the volumetric behaviour.

The first invariant thus contains a first power of $J^{-2/3}$, the second invariant a second power, and the third invariant the third power. These definitions are also used by e.g. Abaqus a finite strain finite element solver, which you can find there in their theory manual.

--Dzjurdie (talk) 10:10, 17 October 2008 (UTC)

Thanks for pointing that out. I've updated the equations in Hyperelastic material. Could someone check them for correctness.Bbanerje (talk) 03:44, 3 November 2008 (UTC)

Figure 3 is drawn incorrectly

Notice that if you follow the top path, then the original top surface is stretched to a location farther away from what was the bottom surface, i.e. extension occurs in what was the X_3 direction. But if you follow the bottom path, the original right side is stretched away from the original left side, i.e. extension occurs in what was the X_2 direction. If you number the faces, the error will be more apparent.

Anyway, it looks to me like you rotated the cube around the wrong edge in the top pathway...

Otherwise it's a very useful page. Thanks! Therealbrewer (talk) 15:15, 2 December 2008 (UTC)

I think that you are referring to the additional translation of the center of the cube. There should be only rotation of the cube with respect to its centre. I will correct that. Thanks for spotting the mistake. - Sanpaz (talk) 23:13, 5 December 2008 (UTC)

The Right Cauchy-Green deformation tensor

Is the index notation of the Right Cauchy-Green tensor really correct? Shouldn't it be something like: $\qquad C_{IJ}={\frac {\partial x_{k}}{\partial X_{I}}}{\frac {\partial x_{k}}{\partial X_{J}}}$ with uppercase letters? Arichelo (talk) 20:00, 21 December 2008 (UTC)

Fixed. Bbanerje (talk) 01:29, 6 January 2009 (UTC)

Rename

I think that this article would be more accessible if it were simply named "Finite strain", which starts with a simple explanation of finite strain and then goes on to spend most of its time on finite strain theory. Opinions? Awickert (talk) 19:36, 19 January 2009 (UTC)

I disagree (need to think about it more). Please see Talk:Deformation_(mechanics)#Merging_articles Sanpaz (talk) 00:00, 20 January 2009 (UTC)

OK - I'll be replying there. Awickert (talk) 00:07, 20 January 2009 (UTC)

Notational confusion

I'm a bit confused about the notation being used in the article. At the beginning the displacements are defined as

$\mathbf {u} \equiv \mathbf {u} (\mathbf {X} ,t)~;~~\mathbf {U} \equiv \mathbf {U} (\mathbf {x} ,t)$

Later on, during the definition of strain, derivatives are taken with respect to what seems to be the dependent variable, i.e.,

${\cfrac {\partial U_{i}}{\partial X_{j}}}~;~~{\cfrac {\partial u_{i}}{\partial x_{j}}}$

I can't understand why we need such convoluted derivatives when the simpler

${\cfrac {\partial u_{i}}{\partial X_{j}}}~;~~{\cfrac {\partial U_{i}}{\partial x_{j}}}$

is what we usually use. Bbanerje (talk) 03:27, 2 October 2009 (UTC)

Link to two-point tensor

You might want to add a link to the two-point_tensor article when claiming F is a two point tensor.

http://en.wikipedia.org/wiki/Two-point_tensor

It may be useful.

Etaoin Shdrlu (talk) 06:59, 30 October 2010 (UTC)

Done, thanks for pointing out the link. Mark viking (talk) 05:50, 14 January 2013 (UTC)

Confusion on Eulerian and Lagrangian coordinate description

It is written under Material coordinates (Lagrangian description) that a description of the lagrangian coordinates is :

"Using ${\displaystyle \mathbf {X} \,\!}$ in place of ${\displaystyle P_{i}\,\!}$ and ${\mathbf {x} }\,\!$ in place of ${\displaystyle p_{i}\,\!}$, both of which are vectors from the origin of the coordinate system to each respective point, we have the Lagrangian description of the displacement vector: ${\mathbf {u} }({\mathbf {X} },t)=u_{i}{\mathbf {e} }_{i}\,\!$

Where ${\displaystyle \mathbf {e} _{i}\,\!}$ is the unit vector that defines the basis of the spatial (lab-frame) coordinate system."

And it is written under Spatial coordinates (Eulerian description) :

"In the Eulerian description, the vector joining the positions of a particle ${\displaystyle P\,\!}$in the undeformed configuration and deformed configuration is called the displacement vector:

${\displaystyle \mathbf {U} (\mathbf {x} ,t)=U_{J}\mathbf {E} _{J}\,\!}$

Where ${\displaystyle \mathbf {E} _{i}\,\!}$ is the unit vector that defines the basis of the material (body-frame) coordinate system. "

I think there is a confusion here, the Eulerian (spatial coordinate) are the one a "spatial (lab-frame) coordinate system" and the Lagrangian (material coordinates) are the one related to the "material (body-frame) coordinate system"

Akadalow (talk) 07:56, 27 March 2017 (UTC)Akadalow

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