On the elastostatic significance of four boundary integrals involving biharmonic functions
Four boundary integrals are formulated for a biharmonic function \(U\) depending on two space variables. Expressions for boundary integrals contain the next four functions: \(U\), \({\partial U\over \partial \nu}\), \(\Delta U\), \({\partial\over \partial\nu} \Delta U\). Here \(\partial/ \partial\nu\) denotes the differentiation in the direction of outward normal to the boundary, and \(\Delta\) is the Laplace operator. The purpose is to give an elastostatic interpretation of the above-mentioned boundary integrals if \(U\) is an Airy stress function for a plane strain problem. At the first step, the author collects ``all possible'' biharmonic functions in polar coordinates. The second step is to compute for these functions strains, stresses, displacements and resultant forces couples. The third step is to evaluate the boundary integrals by means of Maple V. Then the author is able to explain the mechanical meaning of the results obtained. It turns out that the first two integrals are related to single-valued displacements, and the interpretation of the third integral seems to be a new result. The vanishing of the fourth integral does not give more information as do the first three integrals. The results presented can be used in the solution of plaine strain problems by means of boundary integral equations.
scientific article; zbMATH DE number 1143345