(* ::Package:: *)

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

ur=Simplify[(D[\[Phi],r]+z*D[\[Omega],r]+2*D[\[Psi],\[Theta]]/r+z*D[\[Chi],r])/(2*\[Mu])]
u\[Theta]=Simplify[(D[\[Phi],\[Theta]]/r+z*D[\[Omega],\[Theta]]/r-2*D[\[Psi],r]+z*D[\[Chi],\[Theta]]/r)/(2*\[Mu])]
uz=Simplify[(D[\[Phi],z]+z*D[\[Omega],z]-(3-4*\[Nu])*\[Omega]+\[Chi]+z*D[\[Chi],z])/(2*\[Mu])]


\[Sigma]rr=Simplify[D[\[Phi],r,r]+z*D[\[Omega],r,r]-2*\[Nu]*D[\[Omega],z]+(2/r)*D[\[Psi],r,\[Theta]]-(2/r^2)*D[\[Psi],\[Theta]]+z*D[\[Chi],r,r]-2*D[\[Chi],z]]
\[Sigma]r\[Theta]=Simplify[(1/r)*D[\[Phi],r,\[Theta]]-(1/r^2)*D[\[Phi],\[Theta]]+(z/r)*D[\[Omega],r,\[Theta]]-(z/r^2)*D[\[Omega],\[Theta]]+(1/r)*D[\[Psi],r]-D[\[Psi],r,r]+(1/r^2)*D[\[Psi],\[Theta],\[Theta]]+z*D[\[Chi],r,\[Theta]]/r-z*D[\[Chi],\[Theta]]/r^2]
\[Sigma]\[Theta]\[Theta]=Simplify[D[\[Phi],r]/r+D[\[Phi],\[Theta],\[Theta]]/r^2+(z/r)*D[\[Omega],r]+(z/r^2)*D[\[Omega],\[Theta],\[Theta]]-2*\[Nu]*D[\[Omega],z]-(2/r)*D[\[Psi],r,\[Theta]]+(2/r^2)*D[\[Psi],\[Theta]]+z*D[\[Chi],r]/r-z*D[\[Chi],\[Theta],\[Theta]]/r^2-2*D[\[Chi],z]]
\[Sigma]rz=Simplify[D[\[Phi],r,z]+z*D[\[Omega],r,z]-(1-2*\[Nu])*D[\[Omega],r]+(1/r)*D[\[Psi],\[Theta],z]+D[\[Chi],r]+z*D[\[Chi],r,z]]
\[Sigma]\[Theta]z=Simplify[(1/r)*D[\[Phi],\[Theta],z]+(z/r)*D[\[Omega],\[Theta],z]-((1-2*\[Nu])/r)*D[\[Omega],\[Theta]]-D[\[Psi],r,z]+D[\[Chi],\[Theta]]/r+z*D[\[Chi],\[Theta],z]/r]
\[Sigma]zz=Simplify[D[\[Phi],z,z]+z*D[\[Omega],z,z]-2*(1-\[Nu])*D[\[Omega],z]+z*D[\[Chi],z,z]]



T=((1-\[Nu])/(\[Mu]*\[Alpha]*(1+\[Nu])))*D[\[Chi],z]
qr=-((1-\[Nu])*K/(\[Mu]*\[Alpha]*(1+\[Nu])))*D[\[Chi],z,r]
qz=-((1-\[Nu])*K/(\[Mu]*\[Alpha]*(1+\[Nu])))*D[\[Chi],z,z]
