Double dot product in Cylindrical Polar coordinates

[jemme]

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!

 

[Chestermiller]

Since cylindrical polar coordinates are likewise orthogonal, it's going to be pretty much analogous.

Chet