Analytical solution of elastic deformations inside and outside circular contact area between tilted rigid punch and elastic half space

This paper proposes an analytical model for the Boussinesq problem between a tilted rigid punch and an elastic half space to enable the analysis of elastic deformations inside and outside a contact area. Inside the contact area, two types of pressure distributions are applied: one generates a flat elastic deformation, and the other produces a tilted elastic deformation. The projection of this elastic deformation varies depending on the observed horizontal direction because the elastic deformation is non-axisymmetric. To calculate integrals for the non-axisymmetric elastic deformation, we use the polar coordinate system with two angular coordinates, which can enable the calculation of an integral at any arbitrary point. The proposed model can obtain the relationship between the pressure distribution and the elastic deformations inside and outside a contact area from any arbitrary direction. In addition, the normal load and torque applied inside the contact area are obtained, and these parameters are normalized using the contact radius and the elastic modulus. At the zero-pressure point around the contact edge, the elastic deformation is smooth.