(* Solutions A,B and E in cylindrical polar coordinates *)

ur=Simplify[(D[ϕ,r]+z*D[ω,r]+2*D[ψ,θ]/r)/(2*μ)]
uθ=Simplify[(D[ϕ,θ]/r+z*D[ω,θ]/r-2*D[ψ,r])/(2*μ)]
uz=Simplify[(D[ϕ,z]+z*D[ω,z]-(3-4*ν)*ω)/(2*μ)]

σrr=Simplify[D[ϕ,r,r]+z*D[ω,r,r]-2*ν*D[ω,z]+(2/r)*D[ψ,r,θ]-(2/r^2)*D[ψ,θ]]

σθθ=Simplify[D[ϕ,r]/r+D[ϕ,θ,θ]/r^2+(z/r)*D[ω,r]+(z/r^2)*D[ω,θ,θ]-2*ν*D[ω,z]-(2/r)*D[ψ,r,θ]+(2/r^2)*D[ψ,θ]]

σzz=Simplify[D[ϕ,z,z]+z*D[ω,z,z]-2*(1-ν)*D[ω,z]]

σrθ=Simplify[(1/r)*D[ϕ,r,θ]-(1/r^2)*D[ϕ,θ]+(z/r)*D[ω,r,θ]-(z/r^2)*D[ω,θ]+(1/r)*D[ψ,r]-D[ψ,r,r]+(1/r^2)*D[ψ,θ,θ]]

σθz=Simplify[(1/r)*D[ϕ,θ,z]+(z/r)*D[ω,θ,z]-((1-2*ν)/r)*D[ω,θ]-D[ψ,r,z]]

σrz=Simplify[D[ϕ,r,z]+z*D[ω,r,z]-(1-2*ν)*D[ω,r]+(1/r)*D[ψ,θ,z]]