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
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.
IMM Technical Report 16/96
Last modified August 8, 1996
Finn Kuno Christensen
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