Stress tensor components question

  • #1

Homework Statement



stress tensor in cartesian components.
[tex]\sigma[/tex] is the stress tensor.
[tex]e_i[/tex] are the basis vectors

Homework Equations



[tex]\sigma \cdot \sigma[/tex]

The Attempt at a Solution


I tried to write out the components with a cartesian basis:
[tex]\sigma=\sigma_{ij} (e_i \otimes e_j)[/tex]
But then I'm stuck on
[tex]\sigma \cdot \sigma = \sigma_{ij} (e_i \otimes e_j) \cdot \sigma_{ji} (e_j \otimes e_i) [/tex]

How can that be a scalar, since it is the scalar product...

I have no idea if this is the right approach, should I explicit use the unit vectors e_i to emphasize the cartesian components?

Thanks in advance.
 
Last edited:

Answers and Replies

  • #2
Okay, I made my way through this.

[tex]
\sigma_{ij} (e_i \otimes e_j) \cdot \sigma_{kl} (e_k \otimes e_l) = \sigma_{ij} \delta_{ik} \delta_{jl} \sigma_{kl}
= \sigma_{ij}^2
[/tex]
This is indeed a scalar, since there is no tensor space to span.
The key was to create the kronecker deltas.

Thanks anyway.
 

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