Fig. 2 Shear stress distribution in the contact area between ball and raceway. (a) Two-dimensional model of ball and raceway. (b) Nephogram of shear stress distribution in the contact area. (c) Maximum orthogonal shear stress and maximum shear stress on the subsurface under contact load.
According to the classical Hertz contact theory, the rolling contact between ball and raceway can be simplified to ideal line contact under 2D conditions (shown in Fig. 3). p(x) represents the contact load distribution and \(p_{\max}\) represents the maximum contact load which is specified \(p_{\max}\) = 2000 MPa. p(x) is calculated by the following formula:
\begin{equation} {{p\left(d\right)=p}_{\max}\left[1-\left(d/w\right)^{2}\right]}^{1/2}\nonumber \\ \end{equation}
Where w represents the half-width of the contact area and is specified w = 0.25 mm, and d represents distance from the maximum load point. The computational domain is chosen to be 12w length and 5w depth with 0.01w distance between each node horizontally or vertically (shown in Fig. 3a and b). p(x) is set to move uniformly along the x -axis by the DLoad subroutine implemented in ABAQUS, aiming to replace the rolling contact process between ball and raceway. Fig. 3b presents the selected point P1 on the subsurface of the model. Fig. 3c shows the shear stress at point P1 as a function of loading time. The result is that shear stress at a certain point on the subsurface changes alternately with the motion of contact load. Considering the similarity between Fig. 3c and sine curve, the alternating shear stress is replaced by the stress with sinusoidal regularity in the interest of computational simplicity in the following sections.