(* ::Package:: *)

   (*  SOLUTIONS A B E AND T IN CYLINDRICAL POLAR COORDINATES  *)

uR=Simplify[(D[\[Phi],R]+R*Cos[\[Beta]]*D[\[Omega],R]-(3-4*\[Nu] \.13)*Cos[\[Beta]]*\[Omega]+(1/R)*D[\[Psi],\[Theta]]+R*Cos[\[Beta]]*D[\[Chi],R]+\[Chi]*Cos[\[Beta]])/(2*\[Mu])]
u\[Theta]=Simplify[((1/R*Sin[\[Beta]])*D[\[Phi],\[Theta]]+Cot[\[Beta]]*D[\[Omega],\[Theta]]-Sin[\[Beta]]*D[\[Psi],R]-(Cos[\[Beta]]/R)*D[\[Psi],\[Beta]]+Cot[\[Beta]]*D[\[Chi],\[Theta]])/(2*\[Mu])]
u\[Beta]=Simplify[((1/R)*D[\[Phi],\[Beta]]+Cos[\[Beta]]*D[\[Omega],\[Beta]]+(3-4*\[Nu])*Sin[\[Beta]]*\[Omega]+(Cot[\[Beta]]/R)*D[\[Psi],\[Theta]]+Cos[\[Beta]]*D[\[Chi],\[Beta]]-\[Chi]*Sin[\[Beta]])/(2*\[Mu])]


\[Sigma]RR=Simplify[D[\[Phi],R,R]+R*Cos[\[Beta]]*D[\[Omega],R,R]-2*(1-\[Nu])*Cos[\[Beta]]*D[\[Omega],R]+(2*\[Nu]/R)*Sin[\[Beta]]*D[\[Omega],\[Beta]]+(1/R)*D[\[Psi],R,\[Theta]]-(1/R^2)*D[\[Psi],\[Theta]]+R*Cos[\[Beta]]*D[\[Chi],R,R]+2*Sin[\[Beta]]*D[\[Chi],\[Beta]]/R]
\[Sigma]R\[Theta]=Simplify[(1/(R*Sin[\[Beta]]))*D[\[Phi],R,\[Theta]]-(1/(R^2*Sin[\[Beta]]))*D[\[Phi],\[Theta]]+Cot[\[Beta]]*D[\[Omega],R,\[Theta]]-(2*(1-\[Nu])*Cot[\[Beta]]/R)*D[\[Omega],\[Theta]]+(1/(R^2*Sin[\[Beta]]))*D[\[Psi],\[Theta],\[Theta]]-Sin[\[Beta]]*D[\[Psi],R,R]-(Cos[\[Beta]]/R)*D[\[Psi],R,\[Beta]]+(2*Cos[\[Beta]]/R^2)*D[\[Psi],\[Beta]]+(Sin[\[Beta]]/R)*D[\[Psi],R]+Cot[\[Beta]]*D[\[Chi],R,\[Theta]]]
\[Sigma]\[Theta]\[Theta]=Simplify[(1/R)*D[\[Phi],R]+(Cot[\[Beta]]/R^2)*D[\[Phi],\[Beta]]+(1/(R*Sin[\[Beta]])^2)*D[\[Phi],\[Theta],\[Theta]]+(1-2*\[Nu])*Cos[\[Beta]]*D[\[Omega],R]
+(2*\[Nu]/R)*Sin[\[Beta]]*D[\[Omega],\[Beta]]+(Cos[\[Beta]]^2/(R*Sin[\[Beta]]))*D[\[Omega],\[Beta]]+(Cot[\[Beta]]/(R*Sin[\[Beta]]))*D[\[Omega],\[Theta],\[Theta]]
+(1/(R*Sin[\[Beta]])^2)*D[\[Psi],\[Theta]]-(1/R)*D[\[Psi],R,\[Theta]]-(Cot[\[Beta]]/R^2)*D[\[Psi],\[Beta],\[Theta]]+(Cot[\[Beta]]/(R*Sin[\[Beta]]))*D[\[Chi],\[Theta],\[Theta]]-Cos[\[Beta]]*D[\[Chi],R]+((1+Sin[\[Beta]]^2)/(R*Sin[\[Beta]]))*D[\[Chi],\[Beta]]]
\[Sigma]\[Beta]R=Simplify[(1/R)*D[\[Phi],\[Beta],R]-(1/R^2)*D[\[Phi],\[Beta]]+(1-2*\[Nu])*Sin[\[Beta]]*D[\[Omega],R]+Cos[\[Beta]]*D[\[Omega],\[Beta],R]-(2*(1-\[Nu])*Cos[\[Beta]]/R)*D[\[Omega],\[Beta]]+(1/R^2)*D[\[Psi],\[Theta],\[Beta]]+(Cot[\[Beta]]/R)*D[\[Psi],\[Theta],R]-(2*Cot[\[Beta]]/R^2)*D[\[Psi],\[Theta]]+Cos[\[Beta]]*D[\[Chi],R,\[Beta]]-Sin[\[Beta]]*D[\[Chi],R]]
\[Sigma]\[Theta]\[Beta]=Simplify[(1/(R^2*Sin[\[Beta]]))*D[\[Phi],\[Theta],\[Beta]]-(Cot[\[Beta]]/(R^2*Sin[\[Beta]]))*D[\[Phi],\[Theta]]+(Cot[\[Beta]]/R)*D[\[Omega],\[Beta],\[Theta]]+(2*(1-\[Nu])/R)*D[\[Omega],\[Theta]]-(1/(R^2*Sin[\[Beta]]))*D[\[Omega],\[Theta]]+(Cot[\[Beta]]/(R^2*Sin[\[Beta]]))*D[\[Psi],\[Theta],\[Beta]]-(Sin[\[Beta]]/R)*D[\[Psi],R,\[Beta]]-(Cos[\[Beta]]/R^2)*D[\[Psi],\[Beta],\[Beta]]+(1/(R^2*Sin[\[Beta]]))*D[\[Psi],\[Beta]]+(Cot[\[Beta]]/R)*D[\[Chi],\[Beta],\[Theta]]-D[\[Chi],\[Theta]]/(R*Sin[\[Beta]]^2)]
\[Sigma]\[Beta]\[Beta]=Simplify[(1/R)*D[\[Phi],R]+(1/R^2)*D[\[Phi],\[Beta],\[Beta]]+(Cos[\[Beta]]/R)*D[\[Omega],\[Beta],\[Beta]]+(1-2*\[Nu])*Cos[\[Beta]]*D[\[Omega],R]+(2*(1-\[Nu])*Sin[\[Beta]]/R)*D[\[Omega],\[Beta]]+(Cot[\[Beta]]/R^2)*D[\[Psi],\[Theta],\[Beta]]-(Cot[\[Beta]]/R)^2*D[\[Psi],\[Theta]]+Cos[\[Beta]]*D[\[Chi],\[Beta],\[Beta]]/R-Cos[\[Beta]]*D[\[Chi],R]]



T=((1-\[Nu])/(\[Mu]*\[Alpha]*(1+\[Nu])))*(Cos[\[Beta]]*D[\[Chi],R]-Sin[\[Beta]]*D[\[Chi],\[Beta]]/R)
qR=((1-\[Nu])*K/(\[Mu]*\[Alpha]*(1+\[Nu])))*(Sin[\[Beta]]*D[\[Chi],R,\[Beta]]/R-Cos[\[Beta]]*D[\[Chi],R,R]-Sin[\[Beta]]*D[\[Chi],\[Beta]]/R^2)
q\[Theta]=((1-\[Nu])*K/(\[Mu]*\[Alpha]*(1+\[Nu])))*(D[\[Chi],\[Beta],\[Theta]]/R^2-(Cot[\[Beta]]/R)*D[\[Chi],R,\[Theta]])
q\[Beta]=((1-\[Nu])*K/(\[Mu]*\[Alpha]*(1+\[Nu])))*(Sin[\[Beta]]*D[\[Chi],R]/R-Cos[\[Beta]]*D[\[Chi],R,\[Beta]]/R+Sin[\[Beta]]*D[\[Chi],\[Beta],\[Beta]]/R^2+Cos[\[Beta]]*D[\[Chi],\[Beta]]/R^2)
