New numerical resolution of the elastic quarter-space, eighth-space and finite-length-space contact problems

In this paper, a new algorithm to solve the elastic quarter-space, the eighth-space and the finite-length-space contact problems is proposed. This corresponds to an extension of the Hertz theory. The theoretical foundations of such a problem are limited, due to the presence of displacements at the free edges- or stresses at the virtual edges — resulting to complex boundary conditions. The new approach presented here is 3D and based on Guilbault’s ingenious fast correction method. In this approach, the edge effects are taken into account by introducing two corrective factors ${\psi }_1$, ${\psi }_2$ respectively on the $\left(Ox\right)$ and $\left(Oz\right)$ axes to replace the mirror pressure iterative process of Hetenyi. The exact numerical values of these two correction factors are derived analytically. The results show that the free edge can substantially increase locally the contact pressure and therefore the stresses and displacement fields if close to the contact area. So the pressure field and the contact zone present an asymmetry which is more pronounced as the free edge is getting closer. This study is carried out on spaces with one, two and four free edges which will be respectively called: quarter-space, eighth-space and finite-length-space. A validation is performed using a Finite Element Method (FEM) analysis. A parametric study is also performed to exhibit the differences with the Hertz solution, including in the situation where one expects the truncation of the contact area when the free edge is virtually located within the Hertz contact area.