Biharmonic potentials and plane isotropic displacement fields (Q1114473)

In the case of isotropic elastic bodies at small deformations the displacement functions u(x,y) and v(x,y) satisfy the system of differential equations \(\Delta u+\nu \partial \theta /\partial x=0\), \(\Delta v+\nu \partial \theta /\partial y=0\), where \(\nu\) is a material constant, and \(\theta =div \vec s\). The functions u(x,y) and v(x,y) are biharmonic and the function \(\theta\) (x,y) harmonic. Using the theory of complex functions as applied to plane elasticity, the authors derived the dependence between the plane displacement field at small deformations and the biharmonic potential as defined in an earlier paper of the same authors [Dokl. Akad. Nauk Ukr. SSR, Ser. A 1981, No.8, 26-28 (1981; Zbl 0472.31001)].