The Fundamental Solution for a Shallow Shell With an Arbitrary Quadratic Midsurface

The complex-valued differential operator associated with the linear Marguerre equations for a constant thickness, elastically isotropic shallow shell with an arbitrary quadratic midsurface consists of the biharmonic operator minus the imaginary unit i times a constant coefficient second-order differential operator. The fundamental solution of this differential operator is denoted by g(r, θ; κ), where r and θ are polar coordinates and κ is the (dimensionless) Gaussian curvature of the midsurface at the origin. Via a double Fourier transform, a plane wave or Whittaker representation is obtained for g(r, θ; κ). From this is extracted a series representation for g(r, θ; κ) in powers of r², with coefficients depending on ln r, θ, and κ. The results are shown to agree with the known special cases for κ = 1, 0, and −1.