σ[r_,θ_,φ_]= SymmetrizedArray[{{1,1}->σrr[r,θ,φ],{1,2}->σrθ[r,θ,φ],{1,3}->σrφ[r,θ,φ],{2,2}->σθθ[r,θ,φ],{2,3}->σθφ[r,θ,φ],{3,3}->σφφ[r,θ,φ]}, {3,3}, Symmetric[All]];

div=Div[σ[r,θ,φ],{r,θ,φ}, "Spherical"]//Simplify //Normal

u[r_,θ_,φ_] := {ur[r,θ,φ],uθ[r,θ,φ],uφ[r,θ,φ]}

strain=Symmetrize[Grad[u[r,θ,φ],{r,θ,φ}, "Spherical"]]//Simplify;
strain//SymmetrizedArrayRules//Most

stiff = SymmetrizedArray[pos_ :> c@@pos,{3,3,3,3}, Symmetric[All]]
TensorContract[stiff⊗strain, {{3,5},{4,6}}]