Double dot product in Cylindrical Polar coordinates

Hello,



I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows:



2W = σ_ijε_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_xxε_xx + σ_yyε_yy + σ_zzε_zz + 2 σ_xyε_xy + 2 σ_xzε_xz + 2 σ_zyε_zy



My question is how the expression should be for cylindrical polar coordinates (r,θ,z)



Thanks!