13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Figure 3. The Dang Van criterion: in order that the failure does not occur, all the representative points should be inside the safe region delimited by the limit curves. In this figure, and for the problem considered, only the representative points corresponding to max value of the damage factor are plotted for all the integration points in the region analyzed. Figure 4. Damage factor versus distance from surface (a) and versus distance from surface non dimensionalized by the half contact width (b). 3.1 Hardness variation The relationships between fatigue strength, the hardness and the ultimate tensile strength are used, in this section, to study the influence of the hardness variation in the inner ring. Since fatigue crack initiation is mainly caused by slip within grains, the yield stress, in the past, has been thought to have the strongest correlation with the fatigue limit. However Murakami [25] has found better correlations between tensile strength, hardness and fatigue limits. In order to correlate the hardness to the fatigue limit, w, this limit has first been related to σUTS through an approximate expression proposed in [26] for low-alloy steels: w=σw/√3≈ 0.274 σUTS (20) Denoting the Brinell hardness by HB and using an approximate relationship found in [27] σUTS=0.0012 HB 2+3.3 HB (21) an approximate final relation between w and HB can be written as w=0.274 (0.0012 HB 2+3.3 HB) (22) In the following we assume that the fatigue limit w is given by the Eq. (22). If another expression w(HB) applies for a material, this will not in principle change the procedure. In fact, all we need is
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