Applicability of half-space-based methods to non-conforming elastic normal contact problems

The half-space assumption has been employed in many solution methods for non-conforming contact problems in elasticity such as the Hertz theory and the Kalker's variational theory. It is generally believed that to guarantee acceptable accuracy in these half-space-based methods, the characteristic size (twice length of one semi-axis) of the contact patch should be much smaller than the significant dimensions (i.e. the height, width, length and the principal radii of curvature) of each body in contact. In engineering practice, the 3x rule is often employed, which requires that the significant dimensions be at least three times as large as the characteristic size. However, this requirement has not been justified. In this paper, the applicability of half-space-based methods is investigated by comparing the solutions obtained using the Hertz theory and the Kalker's theory with those of the Finite Element (FE) method which is not limited to the half-space assumption. Different combinations of significant dimensions in terms of height, width and length are studied. Various contact patch eccentricities and contact body shapes are considered. It is found that the half-space-based methods yield high-accuracy calculation for non-conforming contact problems. Even when the significant dimensions are as small as 1.1x the characteristic size, the differences between the solutions of the half-space-based methods and the FE method are within 9%. The findings of this paper indicate that the typically assumed 3x restriction can be greatly relaxed. Since a clear estimation of the deviation of the results of half-space-based methods from those of the FE method is provided, the applicability of half-space-based methods in mechanical engineering can be much better understood.