Manipulation of 2nd, 3rd & 4th order tensor using Index notation

[chowdhury]

If I have an equation, let's say,
$$\mathbf{A} = \mathbf{B} + \mathbf{C}^{Transpose} \cdot \left( \mathbf{D}^{-1} \mathbf{C} \right),$$
1.) How would I write using index notation? Here

• A is a 4th rank tensor
• B is a 4th rank tensor
• C is a 3rd rank tensor
• D is a 2nd rank tensor

I wrote it as $$A_{ijkl} = B_{ijkl} + C_{ijk}^{Transpose} D_{ll}^{-1} C_{ijk} $$
$$A_{ijkl} = B_{ijkl} + C_{kij} D_{ll}^{-1} C_{ijk} $$

2.) How to denote the transpose of a third rank tensor? $$C^{Transpose}$$ or $$C^{t}$$ or is there a succinct way of writing it?

 

[andrewkirk]

Tensor theory does not include a notion of "transpose", which is a concept that applies only to matrices.
Nor does it have a notion of an inverse tensor, so I don't know what to make of your $\mathbf D^{-1}$. Again that is a concept that applies only to matrices.
Lastly, I don't know what you mean by the dot before the parenthesis. Do you mean a dot product (inner product)? If so, that's probably not defined in this context. Alternatively, perhaps you mean the tensor product, but that's usually written with the $\otimes$ symbol.

In summary, you need to make sure first that you've written a valid tensor equation before you start trying to rewrite it in index notation.

 

[fresh_42]

Do you mean a dot product (inner product)?

Yes. He is reading about piezoelectricity (chapter 8) in B. A. Auld, Acoustic fields and waves in solids, vol. 1, 1973.
https://www.physicsforums.com/threads/inverse-of-a-vector.1012428/

@chowdhury You should really read those articles first:
https://en.wikipedia.org/wiki/Einstein_notation
https://en.wikipedia.org/wiki/Raising_and_lowering_indices
https://en.wikipedia.org/wiki/Tensor_contraction
https://en.wikipedia.org/wiki/Covariant_transformation
https://en.wikipedia.org/wiki/Tensor_field

Even
https://www.physicsforums.com/insights/what-is-a-tensor/
should help. And the Wikipedia article on piezoelectricity has some of those formulas, too, and written in a more modern way:
https://en.wikipedia.org/wiki/Piezoelectricity#Mathematical_description

 

[chowdhury]

@andrewkirk and @fresh_42,

I indeed read a book of 300 pages on basic tensor notation applied to mechanics, and what I could grasp as per my understanding, I am writing here.

Just to reiterate my understanding:

• by matrix, we mean (m x n). As a sidenote in Matlab, you can have matrices of ( m x n x L x ...), I think it is called paging.
• Vector (n x 1) or (1 x n)

I learned today, from @andrewkirk, that tensor theory (more than second rank) does not include the concept of transpose and inverse. Thank you.

@fresh_42 : I indeed read few of the wikipedia articles you have provided, but you know, in wiki, many people write many things, not sure whether it is a typo or not. This is a great forum, where I can learn from expert, like you, directly. Thanks for being patient.

 

[chowdhury]

Yes. He is reading about piezoelectricity (chapter 8) in B. A. Auld, Acoustic fields and waves in solids, vol. 1, 1973.
https://www.physicsforums.com/threads/inverse-of-a-vector.1012428/

@chowdhury You should really read those articles first:
https://en.wikipedia.org/wiki/Einstein_notation
https://en.wikipedia.org/wiki/Raising_and_lowering_indices
https://en.wikipedia.org/wiki/Tensor_contraction
https://en.wikipedia.org/wiki/Covariant_transformation
https://en.wikipedia.org/wiki/Tensor_field

Even
https://www.physicsforums.com/insights/what-is-a-tensor/
should help. And the Wikipedia article on piezoelectricity has some of those formulas, too, and written in a more modern way:
https://en.wikipedia.org/wiki/Piezoelectricity#Mathematical_description

@andrewkirk Is the notation in Wikipedia correct? Comparing the dyadic product, it does not seem to be correct.

https://en.wikipedia.org/wiki/Einstein_notation

To me, this $v$ seems not right, as it results a scalar.
$$ v = v^{i} e_{i} =
\begin{bmatrix}
e_{1} & e_{2} & \cdots & e_{n}
\end{bmatrix} \begin{bmatrix}
v^{1}\\
v^{1}\\
\vdots \\
v^{n}
\end{bmatrix}
$$

$$ w = w_{i} e^{i} =
\begin{bmatrix}
w_{1} & w_{2} & \cdots & w_{n}
\end{bmatrix} \begin{bmatrix}
e^{1}\\
e^{1}\\
\vdots \\
e^{n}
\end{bmatrix}
$$

It is mentioned in the above wikipedia article

dyadic product is $A^{i}_{j} = u^{i} v_{j} = (uv)^{i}_{j}$

 

[andrewkirk]

The notation is correct. Under the convention, when the same letter is used for a lower and an upper index in a product, we sum over that index variable, so $u^iv_i$ denotes $\sum_i u^i v_i$, a scalar. Assuming $\mathbf u$ is a row vector and $\mathbf v$ is a column vector, that is the same as the product $\mathbf u \mathbf v$, ie the row vector goes first.

If you don't want to add the terms, you must use different indices, $u^i v_j$ That can be represented by a matrix. Again assuming $\mathbf u$ is a row vector and $\mathbf v$ is a column vector, that is the same as the product $\mathbf v \mathbf u$, ie the column vector goes first.