ClearAll[ux, ϕ, x, y];

ϕ = ((x^2 - y^2)/(x^2 + y^2)*(x^2 + y^2 + 
                  R^4/(x^2 + y^2) - 2  R^2) + x^2 + y^2 - 
            R^2*Log[x^2 + y^2])/4

σyy = Simplify[D[ϕ, x, x]]