- #1

blue24

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- Homework Statement
- Show that 2nd order tensor, a*d_ij, where a is an arbitrary constant, retains its form under any transformation, Q_ij. This form is then an isotropic 2nd order tensor.

- Relevant Equations
- a'_ij = Q_ip * Q_jq * a_pq (General transformation relation for 2nd order tensor)

**Backstory**- I have not been in school for 5ish years, and am returning to take some grad classes in the field of Solid Mechanics. I am freaking out a bit about the math (am rusty). I have not started class yet, but figured I would get my books and start working through problems. This problem is from my Theory of Elasticity Class.

The book provides the general transformation relation for a 2nd order tensor. Applying a rotation to the given tensor, a*d_ij:

a*d'_ij = Q_ip * Q_jq * a*d_pq

I am not sure where to go from here. My understanding is that the repeated indices "p" and "q" imply summation, so then the first term in the matrix a*d'_ij would be:

a*d'_11 = Q_1p*Q_1q*a*d_pq = a*[Q_11*Q_11*d_11 + Q_12*Q_12*d_22 + Q_13*Q_13*d_33]

Since d_11 = d_22 = d_33 = 1, then a*d'_11 = a*[Q_11*Q_11 + Q_12*Q_12 + Q_13*Q_13]

But my guess is I've gone wrong somewhere, because I don't know what I do with this. My apologies if this is a dumb question. Still trying to get my head wrapped around this. Thank you.