GREENS-FUNCTIONS, BI-LINEAR FORMS, AND COMPLETENESS OF THE EIGENFUNCTIONS FOR THE ELASTOSTATIC STRIP AND WEDGE

The bi-harmonic Green's function G(r′, r) for the infinite strip region -1≤y≤1, -∞<x<∞, with the boundary conditions G=∂G/∂y on y=±1, is obtained in integral form. It is shown that G has an elegant bi-linear series representation in terms of the (Papkovich-Fadle) eigenfunctions for the strip. This representation is then used to show that any function φ{symbol} bi-harmonic in a rectangle, and satisfying the same boundary conditions as G, has a unique representation in the rectangle as an infinite sum of these eigenfunctions. For the case of the semi-infinite strip, we investigate conditions on φ{symbol} sufficient to ensure that φ{symbol} is exponentially small as x→∞. In particular it is proved that this is so, solely under the condition that φ{symbol} be bounded as x→∞. A corresponding pattern of results is established for the wedge of general angle. The Green's function is obtained in integral form and expressed as a bilinear series of the (Williams) eigenfunctions. These eigenfunctions are proved to be complete for all functions bi-harmonic in an annular sector (and satisfying the same boundary conditions as the Green's function). As an application it is proved that if an elastostatic field exists in a corner region with 'free-free' boundaries, and with either (i) the total strain energy bounded, or (ii) the displacement field bounded, then this field has a unique representation as a sum of those Williams eigenfunctions which individually posess the properties (i), (ii). The methods used here extend to all other linear homogeneous boundary conditions for these geometries. © 1979 Sijthoff & Noordhoff International Publishers.