just out of interest, third order tensors arise in nonlocal models of
gradient plasticity such as that proposed by Gurtin and Anand (J.
Mech. Phys. Solids 53 (2005) 1624–1649). In the paper you will find
their definition for symmetry and various other operations (pg 1630).
Regards
Andrew
On 16 Nov 2009, at 4:43 PM, Toby D. Young wrote:
Is this a widely used notion in physics? Where is it used?
Widely? Definately third-order tensors are not widely used in
Physics to
start with, but when they are, this seems to be the preferred
notation.
The piezoelectric modulii are one concrete example, often used to
compute
piezoelectric effects in nanostructures, for example, in semiconductor
materials or devices.
A slightly more "far-out" example would be the 2d+1d space-time
manifold I
guess. There are probably examples in hadron Physics, (eg: strong-
force)
but I would have to check this before claiming it to be ipso facto.
I read that this is sometimes called "the alternating tensor of third
order", or, "psuedotensor". Which is one thing that made me suspicious
to start with... As far as I can tell the difference between
choosing the
symmetry properties:
A_{ijk} = A_{ikj}, and A_{ijk} = A_{kij}
ie, last two, or first two indices are symmetric, is wholly a matter
of
convention.
Note that I am not opposed if you wanted to implement this, I just
want to
make sure we have the semantics right.
I know Wolfgang, and me too; that's one good reason we are having this
conversation. Anyways, I enjoy the debate :-)
The real problem, I guess, is that I can not multiple a
SymmetricTensor
with a Tensor (apparently). Otherwise I may be happy enough to do:
SymmetricTensor<1,dim> * Tensor<3,dim>.
Did you mean SymmetricTensor<**2**,dim> * Tensor<3,dim>? If so,
would the
product be a contraction over indices, or an outer product?
Yes, first-order = second-order * third-order. By that I meant a
contraction over outer indices (or a double contraction).
Example ->
X_{ij} = \sum_k A_{ijk} B_{jk}
X_i = \sum_{jk} A_{ijk} B_{jk}
Best,
Toby
-----
Toby D. Young
Assistant Professor
Philosophy-Physics
Polish Academy of Sciences
Warszawa, Polska
www: http://www.ippt.gov.pl/~tyoung
skype: stenografia
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