The non-homogeneous biharmonic plate equation: fundamental solutions

Real materials and structural components are often non-homogeneous, either by design or because of the physical composition and imperfections in the underlying material. Thus, analytical solutions for non-homogeneous materials under mechanical loads are of considerable interest to engineers and have widespread applications, given the prevalence of these materials in fields as diverse as aerospace, construction, electronics, etc. More precisely, those are essentially composites with carefully manufactured properties that yield desirable mechanical characteristics and properties, such as optimal arrangement of the material, minimum weight, etc. To this end, the displacement fundamental solution (or Green’s function) corresponding to a point force for the non-homogenous biharmonic equation in two dimensions are derived in this work by employing a conformal mapping technique in conjunction with the Radon transformation. These functions, besides being useful in their own right, can also be used within the context of integral equation formulations for the solution of boundary-value problems. Finally, a series of numerical examples that deal with the non-homogeneous plate on elastic foundation problem serve to illustrate the present method.