(* Definitions of the displacement functions. *)
uR=Simplify[(D[ϕ,R]+R*Cos[β]*D[ω,R]-(3-4*ν)*Cos[β]*ω+(2/R)*D[ψ,θ])/(2*μ)]

uθ=Simplify[((1/R*Sin[β])*D[ϕ,θ]+Cot[β]*D[ω,θ]-2*Sin[β]*D[ψ,R]-(2*Cos[β]/R)*D[ψ,β])/(2*μ)]

uβ=Simplify[((1/R)*D[ϕ,β]+Cos[β]*D[ω,β]+(3-4*ν)*Sin[β]*ω+(2*Cot[β]/R)*D[ψ,θ])/(2*μ)]

(* The stress functions. *)

σRR=Simplify[D[ϕ,R,R]+R*Cos[β]*D[ω,R,R]-2*(1-ν)*Cos[β]*D[ω,R]+(2*ν/R)*Sin[β]*D[ω,β]+(2/R)*D[ψ,R,θ]-(2/R^2)*D[ψ,θ]]

σθθ=Simplify[(1/R)*D[ϕ,R]+(Cot[β]/R^2)*D[ϕ,β]+(1/(R*Sin[β])^2)*D[ϕ,θ,θ]+(1-2*ν)*Cos[β]*D[ω,R]+(2*ν/R)*Sin[β]*D[ω,β]+(Cos[β]^2/(R*Sin[β]))*D[ω,β]+(Cot[β]/(R*Sin[β]))*D[ω,θ,θ]+(2/(R*Sin[β])^2)*D[ψ,θ]-(2/R)*D[ψ,R,θ]-(2*Cot[β]/R^2)*D[ψ,β,θ]]

σββ=Simplify[(1/R)*D[ϕ,R]+(1/R^2)*D[ϕ,β,β]+(Cos[β]/R)*D[ω,β,β]+(1-2*ν)*Cos[β]*D[ω,R]+(2*(1-ν)*Sin[β]/R)*D[ω,β]+(2*Cot[β]/R^2)*D[ψ,θ,β]-2*(Cot[β]/R)^2*D[ψ,θ]]

σθβ=Simplify[(1/(R^2*Sin[β]))*D[ϕ,θ,β]-(Cot[β]/(R^2*Sin[β]))*D[ϕ,θ]+(Cot[β]/R)*D[ω,β,θ]+(2*(1-ν)/R)*D[ω,θ]-(1/(R*Sin[β]^2))*D[ω,θ]+(Cot[β]/(R^2*Sin[β]))*D[ψ,θ,θ]-(Sin[β]/R)*D[ψ,R,β]-(Cos[β]/R^2)*D[ψ,β,β]+(1/(R^2*Sin[β]))*D[ψ,β]]

σβR=Simplify[(1/R)*D[ϕ,β,R]-(1/R^2)*D[ϕ,β]+(1-2*ν)*Sin[β]*D[ω,R]+Cos[β]*D[ω,β,R]-(2*(1-ν)*Cos[β]/R)*D[ω,β]+(1/R^2)*D[ψ,θ,β]+(Cot[β]/R)*D[ψ,θ,R]-(2*Cot[β]/R^2)*D[ψ,θ]]

σRθ=Simplify[(1/(R*Sin[β]))*D[ϕ,R,θ]-(1/(R^2*Sin[β]))*D[ϕ,θ]+Cot[β]*D[ω,R,θ]-(2*(1-ν)*Cot[β]/R)*D[ω,θ]+(1/(R^2*Sin[β]))*D[ψ,θ,θ]-Sin[β]*D[ψ,R,R]-(Cos[β]/R)*D[ψ,R,β]+(2*Cos[β]/R^2)*D[ψ,β]+(Sin[β]/R)*D[ψ,R]]