13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- of pre-existing residual stresses in the material, resulting from hardening process. Using the Dang Van criterion, different residual stresses and hardening distributions are studied, and results are compared. 2. Problem formulation Part of the initial geometry of the inner ring of the roller bearing is illustrated in Fig. 1. The inner ring and the shaft have been considered as one body of external radius R=Rs+tk, where Rs is the shaft radius and tk is the thickness of the inner ring. This assumption is equivalent to neglecting contact stresses related to the mounting and any local stress concentrations at the interface between ring and the shaft. In order to reduce the computational time, only an angular sector of the solid, with angular width α=10°, has been modeled. Far away from the surface, the region analyzed is terminated by a circular arc boundary with radius r. Along the edges, the solid is free to slide in the radial direction, being constrained in the direction perpendicular to the edges. A cartesian coordinate system Oxyz is used, with the origin O in the center of curvature of R, the axis z pointing out of the paper, and the axes x and y, respectively, horizontally and vertically aligned. As a 2-D model is studied, no edge effects in the direction perpendicular to the plane of the model are accounted for. The pressure acting on the raceway and resulting from the contact with the roller, is evaluated according to classical Hertzian theory, and is considered identical in any plane parallel to xy: pሺx,yሻ=p0 [1-( x-xp a )2-( y-yp a )2]0.5 (1) In Eq.(1), p0 is the maximum value of the pressure, xp and yp the coordinates of the center of the contact area, a the semi-width of the contact area under the roller and x and y the coordinates of a generic point on the surface in the contact area. The value of p0 is related to the force acting on the roller by the relation p0= ට q (2) where is function of the Young moduli Ei and Poisson ratios i of the roller and the inner race, here assumed of the same material. The constant is a pure function of the curvature radii and q=F/L is the force per unit length acting on the roller. A bearing with the inner ring thickness tk=19 mm, mounted on a shaft of Rs=200 mm, has been used in the simulations. Furthermore values of 70 mm and 20 mm, respectively, are assumed for the length and the radius of the roller. A load of 37 KN is considered pushing the roller against the inner race, resulting in a static Hertzian maximum pressure p0~1 GPa. The contact is assumed continuous without any vibrations effects. No friction or sliding are here accounted for. The pressure distribution, that simulates the contact, is assumed to move along the surface, in a region where the mesh is uniform. Far away from the zone affected by the contact stresses, instead, the elements are stretched, both in the radial and in the tangential direction, close to the edges. The material is considered isotropic, with Young modulus E=210 GPa and Poisson ratio =0.3. In terms of the displacement components ui on the reference base vectors the strain tensor is given by εi,j= 1 2 (ui,j +uj,i) (3) where (),j denotes partial differentiation. The equilibrium equations, written in terms of the stress tensor σij and the strain tensor εij, are obtained by the use of the principle of virtual work: σijδεij V dV= Tiui S dS (4)
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