ClearAll[ux, ν, x, y];

uy[ν_, x_, 
        y_] = (4*ν - 2)*
          y/(x^2 + y^2) + (x^2*y - y^3)/(x^2 + y^2)^2

ux[ν_, x_, y_] = (x^3 - x*y^2)/(x^2 + y^2)^2

Needs["VectorFieldPlots`"]

VectorFieldPlot[{ux[0.3, x, y], uy[0.3, x, y]}, {x, 1, 3}, {y, 0, 
        1}]

ϵxx = D[ux[ν, x, y], {x, 1}]

ϵyy = D[uy[ν, x, y], {y, 1}]

ϵxy = (D[uy[ν, x, y], {x, 1}] + 
            D[ux[ν, x, y], {y, 1}])/2

σyy = ν/(1 - 
                2*ν)*(ϵxx + ϵyy) + ϵyy

y = 0;

Simplify[σyy]

ClearAll[ux, uy, ν, x, y];

phi = -2*π*R^2*Log[Sqrt[x^2 + y^2]]

psi = π*
        R^2*(x^2 + y^2)*(1 - Log[Sqrt[x^2 + y^2]])

Simplify[D[psi, {y, 2}] + D[psi, {x, 2}] - 2*phi]

ux = 
      Simplify[D[psi, x, x, y]/4/π/(1 - ν) - D[phi, y]/
              2/π]

uy = 
      Simplify[D[psi, y, x, y]/4/π/(1 - ν) - D[phi, x]/
              2/π]

ϵxx = Simplify[D[ux, x]]

ϵyy = Simplify[D[uy, y]]

ϵxy = Simplify[(D[uy, x] + D[ux, y])/2]

σyy = 
      Simplify[2*ν/(1 - 2*ν)*(ϵxx + ϵyy) + 
          2*ϵyy]

y = 0;

Simplify[σyy]