On the elastostatic significance of four boundary integrals involving biharmonic functions

For a biharmonic function U, depending upon two space variables, it is known that four curve integrals, which involve U and some derivatives of U evaluated at a closed boundary, must be equal to zero. When U plays the role of an Airy stress function, we investigate the elastostatic significance of the four integrals and we find that it is related to the displacements of the elastic material: Single valued displacements are obtained provided that three of the integrals are zero. (The fourth integral does not provide further information.) It is already known from the classical literature that two of the integrals are related to single valued displacements, but the elastostatical significance of the third integral seems to be a new result.

The method of investigation is unconventional: For "all possible" biharmonic functions, in polar coordinates, we determine stresses, strains, displacements etc. together with the values of the four integrals. The computer algebra system Maple V has been an invaluable tool. By suitable comparisons among the various results obtained we are led to the conclusions about the elastostatic significance of the integrals.