# Tensor operation

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.

good tensor calculus book where I can refer for my problem.

Thanks.

### 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}
}

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 