I was wondering if anyone knows of an analytic Solution for KI and/or KII in tubes.

I´ve seen an infinite number of equations, shape functions, FEM simulations, standards, etc. But one closed solution for tubes?, an analog to Williams series but for cilindirical surfaces?

I am trying to fit experimental displacement data to Williams series. I´ve done it succesfully for flat specimens (kind of easy actually). But now, we have to do the same for tubes. When cracks are long, you can´t approximate the tube surface to an infinite plate anymore.

Is it dead end?