P=1;
EE=1;
nu = 0.25;
Clear[P,EE,nu];
lam = ((EE nu)/((1+nu)(1-2nu))); mu=(EE/(2(1+nu)));

rho[r_,z_]:=Sqrt[r^2+z^2]
u[r_,z_]:=((P r)/(4 Pi mu))((z/(rho[r,z]^3))-((1-2 nu)/(rho[r,z](z+rho[r,z]))))
w[r_,z_]:=(P/(4 Pi mu))(((z^2)/(rho[r,z]^3))+((2(1- nu))/(rho[r,z])))
$Assumptions={r≥0};

sr[r_,z_]:=(P/(2 Pi))(((1-2 nu)/(rho[r,z](z+rho[r,z])))-((3r^2z)/(rho[r,z]^5)));
sz[r_,z_]:=-((3P z^3)/(2 Pi rho[r,z]^5));
st[r_,z_]:=((P(1-2nu))/(2 Pi))((z/(rho[r,z]^3))-(1/(rho[r,z](z+rho[r,z]))))
trz[r_,z_]:=-((3 P r z^2)/(2 Pi rho[r,z]^5))
s[r_,z_]:={{sr[r,z],0,trz[r,z]},{0,st[r,z],0},{trz[r,z],0,sz[r,z]}}
s[r,z]//MatrixForm

Div[s[r,z],{r,t,z},"Cylindrical"]//Simplify

PP=1;EEE=1;nnu=0.25;
repl = {P->PP,EE->EEE,nu->nnu};
{
Plot3D[sr[r,z]/.repl,{r,0,5},{z,-5,0},ImageSize->Medium],
Plot3D[st[r,z]/.repl,{r,0,5},{z,-5,0},ImageSize->Medium],
Plot3D[sz[r,z]/.repl,{r,0,5},{z,-5,0},ImageSize->Medium],
Plot3D[trz[r,z]/.repl,{r,0,5},{z,-5,0},ImageSize->Medium]
}

uu=r/EE(st[r,z]-nu(sr[r,z]+sz[r,z]));
uu-u[r,z]//Simplify

u[r,0]
w[r,0] //Simplify

Ur[r_,z_]:={u[r,z],0,w[r,z]};
U[x_,y_,z_]:={u[Sqrt[x^2+y^2],z]Cos[ArcTan[x,y]],u[Sqrt[x^2+y^2],z]Sin[ArcTan[x,y]],w[Sqrt[x^2+y^2],z]}

gd = Grad[Div[U[x,y,z],{x,y,z}],{x,y,z}] //FullSimplify
lap = Laplacian[U[x,y,z],{x,y,z}] // Simplify

(lam+mu)gd+mu lap//FullSimplify

gd = Grad[Div[Ur[r,z],{r,t,z},"Cylindrical"],{r,t,z},"Cylindrical"] //FullSimplify
lap = Laplacian[Ur[r,z],{r,t,z},"Cylindrical"] // Simplify

(lam+mu)gd+mu lap//FullSimplify