Einstein Notation for Stored Energy Function

[singularme]

Dear all,

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

$W(\epsilon_{kl}) = \int_0^{\epsilon_{kl}} \sigma_{ij} \textrm{d}\epsilon_{ij}$

where all indices ranges from 1 to 3, both $\epsilon$ and $\sigma$ 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 $W(\epsilon_{kl})=\cdots$, I feel it should interpreted as 9 separate equations (with $kl$ replaced by 11, 12, 13, 21, 22, 23, 31, 32, 33), which does not make sense in this case.

Thanks a lot!
James

 

[tiny-tim]

Hello James! Welcome to PF!

… when I see $W(\epsilon_{kl})=\cdots$, I feel it should interpreted as 9 separate equations (with $kl$ replaced by 11, 12, 13, 21, 22, 23, 31, 32, 33), which does not make sense in this case.

No, it does make sense!

W(ε_kl) = … 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.