Help applying summation convention to tensors(generalised Hooke's law)

[K29]

I understand the simplest application of the summation convention.

$x_{i}y_{i}$

I create a sum of terms such that in each term the subscripts are the same i.e.
$x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+...$

But now when I look at understanding summation convention applied to the generalised Hooke's law:

$\tau _{ik}=C_{ikrs}E_{rs}$

I'm a bit unsure what this means. I understand that C is a tensor with 81 components and that by applying the summation convention I should get linear combinations for every $\tau$ component (I think). But as for applying summation convention to see what exactly those sums look like I'm unsure.

The statement below doesn't seem to help my understanding either

"$\tau _{ik}$ is a linear combination of all strain components $E_{ik}$"

Any assistance would be appreciated.

The Attempt at a Solution

I could hazard I guess and say I only sum when r or s equals i or k, and when r=s and i=k. Still unsure if that is correct though.

 

[cepheid]

From your statement about C_ikrs having 81 components, I am assuming that each index ranges over 3 possible values i.e. i = {1,2,3}, and likewise for k, r and s. since only repeated indices are summed over, this means that i and k are not summed over, and you actually have nine different equations represented here, one for each component of tau. So just cycle through all the components of tau (ie all possible values of i and k ):

$$\tau_{11} = C_{1111}E_{11} + C_{1112}E_{12} + C_{1113}E_{13} + C_{1121}E_{21} + C_{1122}E_{22} + C_{1123}E_{23} + C_{1131}E_{31} + C_{1132}E_{32} + C_{1133}E_{33}$$

And that's just ONE of the nine equations. You still have to do the same for tau_12, tau_13, tau_21 et cetera.

 

[cepheid]

If it helps, the NON shorthand way of writing your tensor expression would have been: $$ \tau_{ik} = \sum_{r=1}^3\sum_{s=1}^3 C_{ikrs}E_{rs}$$

 

[K29]

Thank you for the help. Both replies helped me understand.

 

[paranormal]

You are on the right track with your understanding of the summation convention. In general, the summation convention allows us to simplify the notation when working with tensors by replacing repeated indices with summation signs. So in the case of the generalised Hooke's law, we can rewrite it using the summation convention as:

\tau_{ik} = C_{ikrs}E_{rs} = \sum_{r=1}^{3}\sum_{s=1}^{3}C_{ikrs}E_{rs}

This means that for each component of \tau_{ik}, we sum over all possible combinations of r and s, which gives us a total of 9 terms (since there are 3 possible values for r and 3 possible values for s). This is where the statement "\tau_{ik} is a linear combination of all strain components E_{ik}" comes from, as each component of \tau_{ik} is a sum of products of the components of the C tensor and the E tensor.

To better understand this, let's look at a specific example. Let's say we want to calculate \tau_{11}. Using the summation convention, we have:

\tau_{11} = \sum_{r=1}^{3}\sum_{s=1}^{3}C_{11rs}E_{rs}

Now, since we are looking at \tau_{11}, we know that i=k=1. This means that for each term in the sum, r and s can take on the values of 1, 2, or 3. So for the first term, we have:

C_{1111}E_{11} = C_{11rs}E_{rs} (since r=s=1)

Similarly, for the second term, we have:

C_{1121}E_{21} = C_{11rs}E_{rs} (since r=1 and s=2)

And so on for the remaining terms. So in the end, we have:

\tau_{11} = C_{1111}E_{11} + C_{1121}E_{21} + C_{1131}E_{31} + C_{1112}E_{12} + C_{1122}E_{22} + C_{1132}E_{32} + C_{1113}E_{13} + C_{1123}E