Additional Separated-Variable Solutions of the Biharmonic Equation in Polar Coordinates

From the biharmonic equation of the plane problem in the polar coordinate system and taking into account the variable-separable form of the partial solutions, a homogeneous ordinary differential equation (ODE) of the fourth order is deduced. Our study is based on the investigation of the behavior of the coefficients of the above fourth order ODE, which are functions of the radial coordinate $r$⁠. According to the proposed investigation additional terms, $φ¯−m(r,θ)$$(1≤m≤n)$ other than the usually tabulated in the Michell solution (1899, “On the Direct Determination of Stress in an Elastic Solid, With Application to the Theory of Plates,” Proc. Lond. Math. Soc., 31, pp. 100–124) are found. Finally the stress and the displacement fields due to each one additional term of $φ¯−m(r,θ)$ are determined.