The Fundamental (Normal Point Load) Solution for a Shallow Hyperbolic Paraboloidal Shell

Source abstract

A complete characterization is given for the fundamental solution of the complex-valued differential operator governing the linear behavior of a shallow, elastically isotropic hyperbolic paraboloidal shell. The only previously known fundamental solutions in shell theory have been those for the sphere and the shallow cylinder. Physically, the real and imaginary parts of these fundamental solutions represent the normal deflection and Airy’s stress function for an unbounded shell under a normal point load. The differential operator for the hyperbolic paraboloid consists of the biharmonic operator minus the imaginary unit i times the second order wave operator. Unlike the case with the sphere and the cylinder, this operator cannot be factored, and this accounts for the relative complexity of the analysis. The fundamental solution is expressed as a Fourier series in the polar angle, and the radially dependent Fourier coefficients are represented by a contour integral. From this, explicit series and asymptotic expansions are obtained. An alternate asymptotic expansion, useful near the characteristics of the wave operator, is developed in terms of an Airy function and its indefinite integral. Properties of this latter function are derived in an Appendix.