Double dot product in Cylindrical Polar coordinates - Strain energy I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows:
$$
2W = σ_{ij}ε_{ij}
$$
Where σ and ε are symmetric rank 2 tensors.
For cartesian coordinates it is really easy because the metric is just the identity matrix, hence:
$$
2W = σ_{xx}ε_{xx} + σ_{yy}ε_{yy} + σ_{zz}ε_{zz} + 2 σ_{xy}ε_{xy} + 2 σ_{xz}ε_{xz} + 2 σ_{zy}ε_{zy}
$$
My question is how the expression should be for cylindrical polar coordinates $(r,θ,z)$
Many thanks!
 A: We'll avoid the metric tensor altogether to keep things simple. The demonstration is simpler this way, although not "proper". The polar coordinate system is rather "nice" to deal with, so you don't need the full machinery of tensors in curvilinear coordinates.
First of all, note that W is a scalar. The second fact is that you can think of polar coordinates as a set of rotated Cartesian coordinates at each point, except the angle with respect to the base / natural Cartesian coordinates is a nice function of space (which one?). Once you know that, you know there must be a change of basis matrix $\beta$ which is proper orthogonal such that
$$
\sigma_{ij} =\beta_{pi} \beta_{qj} \sigma'_{pq} \\
\epsilon_{ij} =\beta_{ri} \beta_{sj} \epsilon'_{rs} 
$$ 
Note: No need for upstairs/ downstairs indices; I've used the prime to denote the components in polar. Let's plug in to the expression for 2W.
$$
2W = \sigma_{ij} \epsilon_{ij} \\
=\beta_{pi} \beta_{qj} \beta_{ri} \beta_{sj} \sigma'_{pq} \epsilon'_{rs} \\
=\beta_{pi} \beta^T_{ir} \beta_{qj} \beta^T_{js} \sigma'_{pq} \epsilon'_{rs} \\
=\delta_{pr} \delta_{qs} \sigma'_{pq} \epsilon'_{rs} \\
=\sigma'_{rs} \epsilon'_{rs} 
$$
(Note that $\beta \beta^T = I$).
So you have exactly the same form as in Cartesian. Now just write the components out using the summation and naming conventions. There are 2's because the off-diagonal terms appear twice in the double sum. I'm using $r, \theta, z$ for 1,2,3.
$$
2W = \sigma_{rr} \epsilon_{rr} + \sigma_{\theta\theta} \epsilon_{\theta\theta} + \sigma_{zz} \epsilon_{zz} + 2\sigma_{r\theta} \epsilon_{r\theta} + 2\sigma_{\theta z} \epsilon_{\theta z} + 2\sigma_{rz} \epsilon_{rz}    
$$
