Fourth order tensor
Hi all,
I have a fundamental question on Tensors. The length of a vector (firts order tensor) is independent of the reference co-ordinate system. In case of second order tensor (stress/strain), the invariants (I1, I2, I3) are independent of the co-ordinate system.
If I consider 4th order tensor (of course 3rd order also), say Cijkl, what parameters are constant? (Like length in vector and invariants in second order tensors).
Thanks in advance,
- Ramdas
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Invariants of a fourth order tensor
Hi Ramadas,
One straight forward way of producing invariants is taking contractions until all the indices become dummy. For a second order tensor T_{ij}, for example, you can do one contraction T_{ii} and you get the first invariant. Similary if you take the square T^2_{ik} = T_{ij}T_{jk} and take a contraction T_{ij}T_{ji} you get the trace of the square, which is another invariant. Do the same thing to fourth order tensors. There are more ways to contract a fourth order tensor. C_{ijij}, C_{iijj}, C_{ijji}, C_{iiii} are the ones I can think of. Ofcourse you can take powers of the tensor and do contractions to get more. The contraction is basically multiplication by the kronecker delta. You can also think of multiplication by the Levi-civita symbol. For example, det(T) = 1/6e_{ijk}e_{pqr}T_{ip}T_{jq}T_{kr}.
This is true if you have orthonormal basis. Otherwise use the appropriate definitions, for example, of the Kronecker delta and follow the same procedure.
Good book
Hi Markos,
Thank you. Could you please suggest me some good book on this topic?
Thankk you,
- Ramdas
To Ramdas
Hi Ramdas,
I don't know a book which specifically deals with tensor invariants. As far as tensor analysis is concerned, my favourite is the first chapter from Ogden's book "Nonlinear Elastic Deformations". The later chapters touch on invariance, symmetry etc from the perspective of elasticity in general and Green elasticity in particular.
Re: Fourth order tensor
Dear Ramdas:
What you're looking for is called an integrity basis. Invariance under a change of coordinates is equivalent to invariance under SO(3) (the orientation preserving subgroup of O(3)). Then you want to see what polynomial functions of your fourth-order tensor are invariant under the action of this group. In the case of isotropic tensor functions, one can reduce the dependency of the function of interest to an irreducible basis invariants. As you mentioned, for second-order tensors there are three principal invariants. Equivalently you can rewrite these in terms of basic invariants. Denoting your second-order tensor by A, these are traces of A, A^2=A.A, and A^3=A.A.A. In the case of fourth-order tensors there are six basic (and principal) invariants. Denoting the fourth order tensor by C, these are traces of C^i, i=1,...,6.
Look at the following short paper: J. Betten, Integrity basis for a second-order and a fourth-order tensor, International journal of mathematics and mathematical sciences 5(1), 87-96, 1982.
Regards,
Arash
Dear Arash, It looks I
Dear Arash,
It looks I need to understand more on Tensors. I went through which is required for stress and strain analysis. I downloaded that paper and go through. It looks it is not sufficient.
Could you please suggest some good book exclusively on 'Tensors'?
With regards,
- Ramdas
Re: books on tensors
Dear Ramadas:
Take a look at the following book: Vector and Tensor Analysis with Applications by A.I. Borisenko and I.E. Tarapov. This should be a good book to start with.
Regards,
Arash
Thank you
Hi Arash and Markos,
Thank you very much for your suggestions. I will go through.
With regards,
- Ramdas
Re: tensor book
The best introductory text I've seen so far is
A brief on tensor analysis by James G. Simmonds
-- Biswajit
Soft copy
Hi Biswajit Sir,
Just now I have seen that book in Google books. Nice one.Do you have soft copy of this? If so, could you please share?
With regards,
- Ramdas
Re: Soft copy
Hi Ramdas,
I don't have a soft copy of the book. I don't think a legal soft copy exists *cough*rapidshare*cough*. A good alternative is Prof. Brannon's detailed notes.
I haven't been knighted yet; so the "Sir" upadhi isn't really necessary :)
-- Biswajit
Some links on tensor analysis
Hi all,
Nice discussion going on here. Here's my 5 cents. These are a few links on tensor analysis (all of them introductory) which I found useful when I was learning about tensors.
1) (NASA's brief introductory guide) http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM200221...
2) (Ruslan Sharipov) http://arxiv.org/abs/math.HO/0403252
3) (Joakim Strandberg) http://medlem.spray.se/gorgelo/tensors.pdf
-Arun
One more reference
One more useful reference by R M Brannon:
http://www.mech.utah.edu/~brannon/public/Tensors.pdf
Thank you all!!!
Than you Biswajit, Arun and Sivakumar.
With regards,
- Ramdas
RE: Fourth order tensor
Dear Ramdas
You may be interested in the following paper on 4th order tensors and their manipulations in continuum mechanics:
"Fourth-order tensors - tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis" by O. Kintzel, Y. Ba
ar.
Besides that I would suggest you to have some concepts of symmetric gruops and tensor products also. You may like to read:
"Algebra" by Artin
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your Nasa Link to the Tensor Book http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM2002211716.pdf is very helpful. And than you Biswajit, Arun and Sivakumar for your help to find out more about Tensor. I am happy I have visited this website.
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