
   (*  SOLUTIONS A B AND E IN CARTESIAN COORDINATES  *)

ux = Simplify[(D[\[Phi], x] + z*D[\[Omega], x] + 2*D[\[Psi], y])/(2*\[Mu])]
uy = Simplify[(D[\[Phi], y] + z*D[\[Omega], y] - 2*D[\[Psi], x])/(2*\[Mu])]
uz = Simplify[(D[\[Phi], z] + 
          z*D[\[Omega], z] - (3 - 4*\[Nu])*\[Omega])/(2*\[Mu])]
\[Sigma]xx = 
  Simplify[D[\[Phi], x, x] + z*D[\[Omega], x, x] - 2*\[Nu]*D[\[Omega], z] + 
      2*D[\[Psi], x, y]]
\[Sigma]xy = 
  Simplify[D[\[Phi], x, y] + z*D[\[Omega], x, y] + D[\[Psi], y, y] - 
      D[\[Psi], x, x]]
\[Sigma]yy = 
  Simplify[D[\[Phi], y, y] + z*D[\[Omega], y, y] - 2*\[Nu]*D[\[Omega], z] - 
      2*D[\[Psi], x, y]]
\[Sigma]xz = 
  Simplify[D[\[Phi], x, z] + 
      z*D[\[Omega], x, z] - (1 - 2*\[Nu])*D[\[Omega], x] + D[\[Psi], y, z]]
\[Sigma]yz = 
  Simplify[D[\[Phi], y, z] + 
      z*D[\[Omega], y, z] - (1 - 2*\[Nu])*D[\[Omega], y] - D[\[Psi], x, z]]
\[Sigma]zz = 
  Simplify[D[\[Phi], z, z] + z*D[\[Omega], z, z] - 
      2*(1 - \[Nu])*D[\[Omega], z]]

