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Stress tensor components question

  1. Jul 12, 2010 #1
    1. The problem statement, all variables and given/known data

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

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

    3. 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: Jul 12, 2010
  2. jcsd
  3. Jul 13, 2010 #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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