In[]:=
Clear["Global`*"]​​lineLoad=P;​​poissonR=ν;​​modulusE=;​​Ω=0;​​ϕ=-(lineLoad/Pi)*(r*θ*Sin[θ]);​​σrr=(((D[ϕ,{r,1}])/r)+((D[ϕ,{θ,2}])/(r^2))-Ω)//FullSimplify;(*
-Ω*)​​σθθ=D[ϕ,{r,2}]-Ω;(*
-Ω*)​​σrθ=((D[ϕ,{θ,1}])/(r^2))-((D[ϕ,r,θ])/r)//FullSimplify;(*
*)​​σθr=((D[ϕ,{θ,1}])/(r^2))-((D[ϕ,r,θ])/r)//FullSimplify;(*
*)​​​​stress={{σrr},{σθθ},{σrθ}};​​elasticCompliance=((1+poissonR)/modulusE)*{{1-poissonR,-poissonR,0},{-poissonR,1-poissonR,0},{0,0,2}};​​strain=elasticCompliance.stress//Flatten;​​​​ϵrr=strain[[1]];​​ϵθθ=strain[[2]];​​ϵrθ=strain[[3]];​​​​(*TheStrain-DisplacementPDEquationsForLinearElasticity*)​​strainDispRR=D[uR[r,θ],r]-ϵrr;​​strainDispθθ=uR[r,θ]/r+((1/r)*D[uθ[r,θ],θ])-ϵθθ;​​strainDispRθ=1/2*(((1/r)*D[uR[r,θ],θ])+D[uθ[r,θ],r]-uθ[r,θ]/r)-ϵrθ;​​​​(*SolvingthePDEsforDisplacements*)​​solDispRR=DSolve[strainDispRR0,uR,{r,θ}]//Flatten;​​solDispθθ=DSolve[strainDispθθ0,uθ,{r,θ}]//Flatten;​​uRFunctionTemp=uR[r,θ]/.solDispRR[[1]]​​uθFunctionTemp=uθ[r,θ]/.solDispθθ[[1]]​​​​​​​​​​
Out[]=
2P(-1+
2
ν
)Cos[θ]Log[r]
π
+

1
[θ]
Out[]=

1
[r]+
θ
∫
1
2PνCos[K[1]]+2P
2
ν
Cos[K[1]]-πuR[r,K[1]]
π
K[1]