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Tensor operation

Submitted by sidd04mech on

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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Double dot product of 4th order tensor
Biswajit Banerjee

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}
}

Wed, 07/14/2010 - 03:33 Permalink
Matt Lewis

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

Wed, 07/14/2010 - 14:12 Permalink
Biswajit Banerjee

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

Thu, 07/15/2010 - 22:52 Permalink