Maybe:
solDispRR = DSolve[strainDispRR == 0, uR, {r, \[Theta]}] // Flatten;
solDisp\[Theta]\[Theta] =
DSolve[strainDisp\[Theta]\[Theta] == 0, u\[Theta], {r, \[Theta]}] //
Flatten;
uRFunctionTemp = uR[r, \[Theta]] /. solDispRR[[1]]
u\[Theta]FunctionTemp = (u\[Theta][r, \[Theta]] /.
solDisp\[Theta]\[Theta][[1]] /. solDispRR[[1]]) // Activate //
ExpandAll
Looks like MMA can't integrate, a workaround:
u\[Theta]FunctionTemp = (Integrate[#, {K[1],
1, \[Theta]}] & /@ (u\[Theta]FunctionTemp[[1, 1]])) +
u\[Theta]FunctionTemp[[2]]
(*Integrate[-C[1][K[1]], {K[1], 1, \[Theta]}] + (2*P*\[Nu]^2*Log[r]*(Sin[1] - Sin[\[Theta]]))/(Pi*\[DoubleStruckCapitalE]) +
(2*P*\[Nu]*(-Sin[1] + Sin[\[Theta]]))/(Pi*\[DoubleStruckCapitalE]) + (2*P*\[Nu]^2*(-Sin[1] + Sin[\[Theta]]))/(Pi*\[DoubleStruckCapitalE]) +
(2*P*Log[r]*(-Sin[1] + Sin[\[Theta]]))/(Pi*\[DoubleStruckCapitalE]) + C[1][r]*)
In this line:
Integrate[-C[1][K[1]], {K[1], 1, \[Theta]}]
what answer do you expect?