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Tensor operation
Mon, 2010-07-12 23:36 - sidd04mech
I am trying to make a material model for my FEM code and
while deriving Elasticity tensor I found a term where I have to do some
tensor operation. It is as below
P:C:PT where all the terms are 4th
order tensors. Now I konw the result of this should be a 4th order
tensor only as this term is in addition with other 4th order terms. Here
any (A:B) indicates Double dot product between A and B.
Please help me out with this if you know or suggest some
good tensor calculus book where I can refer for my problem.
Thanks.
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Re: Tensor operation (fourth-order tensors)
I've not found any textbooks that deal in any detail with fourth-order tensors. However, i've found the following papers useful.
1) @article{itskov2000theory,
title={{On the theory of fourth-order tensors and their applications in computational mechanics}},
author={Itskov, M.},
journal={Computer Methods in Applied Mechanics and Engineering},
volume={189},
number={2},
pages={419--438},
year={2000},
publisher={Elsevier}
}
2) @article{betten1982integrity,
title={{Integrity basis for a second-order and a fourth-order tensor}},
author={Betten, J.},
journal={International Journal of Mathematics and Mathematical Sciences},
volume={5},
number={1},
pages={87--96},
year={1982}
}
3) @article{jog2006concise,
title={{A concise proof of the representation theorem for fourth-order isotropic tensors}},
author={Jog, CS},
journal={Journal of Elasticity},
volume={85},
number={2},
pages={119--124},
year={2006},
publisher={Springer}
}
4) @article{moakher2008fourth,
title={{Fourth-order cartesian tensors: old and new facts, notions and applications}},
author={Moakher, M.},
journal={The Quarterly Journal of Mechanics and Applied Mathematics},
volume={61},
number={2},
pages={181},
year={2008},
publisher={Oxford Univ Press}
}
Quick answer
If these are tensors whose components are expressed in an orthonormal basis (orthogonal base vectors of unit length), and PT means the transpose of P, also assumed to be a 4th order tensor, the ijkl components of the product are as follows:
P(ijmn)C(mnop)P(klop)
Matt Lewis
Los Alamos, New Mexico
Transpose of a 4th-order tensor
Let me add a little bit of confusion to the proceedings by pointing out that there can be more than one type of transpose defined for fourth-order tensors. The reference to Itskov's paper defines some of those.
The transpose that's most commonly used is defined by inner products as
U : (A^T : V) = V : (A : U)
where U,V are 2nd order tensors and A, A^T are 4th order. This definition leads to Matt's expression
A_ijkl = A^T_klij
-- Biswajit
I had the same problem, a
I had the same problem, a couple of students put the equations in their Dissertations, but people kept asking this same question, so I put everything we used (to formulate elements, micromechanics, etc.) as anAppendix in http://www.amazon.com/exec/obidos/ASIN/1420054333/booksoncomposite
Dr. Ever Barbero