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Einstein Notation for Stored Energy Function

  1. Jun 22, 2013 #1
    Dear all,

    I encounter the following formula (for stain energy function) a lot in physics literature:

    [itex]
    W(\epsilon_{kl}) = \int_0^{\epsilon_{kl}} \sigma_{ij} \textrm{d}\epsilon_{ij}
    [/itex]

    where all indices ranges from 1 to 3, both [itex]\epsilon[/itex] and [itex]\sigma[/itex] are 3x3 matrices.

    My question is what exactly does it mean in non-Einstein notation? Is it an abuse of notation? Because when I see [itex]W(\epsilon_{kl})=\cdots[/itex], I feel it should interpreted as 9 separate equations (with [itex]kl[/itex] replaced by 11, 12, 13, 21, 22, 23, 31, 32, 33), which does not make sense in this case.

    Thanks a lot!
    James
     
  2. jcsd
  3. Jun 22, 2013 #2

    tiny-tim

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    Science Advisor
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    Hello James! Welcome to PF! :smile:
    No, it does make sense!

    W(εkl) = … is nine separate equations in Einstein notation.

    The i and j on the RHS are "dummy" indices, but the k and l on both the RHS and LHS are not "dummy" indices. :wink:
     
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