Green's functions, bi-linear forms, and completeness of the eigenfunctions for the elastostatic strip and wedge

1573-2681

Springer Online Journal Archives 1860-2000

Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics

Physics

Abstract The bi-harmonic Green's functionG(r′,r) for the infinite strip region -1≤y≤1, -∞〈x〈∞, with the boundary conditionsG=∂G/∂y ony=±1, is obtained in integral form. It is shown thatG 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 bi-harmonic in arectangle, and satisfying the same boundary conditions asG, 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 sufficient to ensure that ϕ is exponentially small asx→∞. In particular it is proved that this is so, solely under the condition that ϕ be bounded asx→∞. 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 anannular 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 whichindividually posess the properties (i), (ii). The methods used here extend to all other linear homogeneous boundary conditions for these geometries.

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