A method to derive new Green’s tensors for polar domains

An efficient unified method to derive Green’s tensors in Cartesian coordinates, called the incompressible influence elements method (IIEM), had been elaborated and published earlier. When extending this method for the domains of polar system of coordinates the additional difficulties have appeared, because the incompressible Green’s tensor does not satisfy, as in Cartesian coordinates, the Poisson’s equation. This paper describes general integral representations for Green’s tensors via special introduced scalar Green’s functions in polar coordinates. Using them the IIEM has been extended to polar domains. To this end, using the IIEM, for deriving the Green’s tensors in polar coordinates, it is necessary either to compute some convolutions of product of scalar Green’s functions or to solve boundary integral equations about elastic dilatation. The paper deals with closed-form Green’s tensors in polar coordinates for an angle and a circle. As particular cases, the respective Green’s tensors for quadrant and half-plane with twice-mixed boundary conditions also are derived.