13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- sij m=min sij * maxt [(sij (t)-sij *)(s ij (t)-sij *)] (10) and the shifted deviator tensor is defined as sijෝ(t)=σij(t)- sij m (11) The principal values of the shifted tensor appear in Eq. (8). The problem in Eq. (10) is solved iteratively using a move limit approach : sij m=min sij * maxt [(sij (t)-sij *)(s ij (t)-sij *)]=min sij * [max t ] (12) with = (t, sij (t), sij *) (13) Choosing an arbitrarily starting value for sij *, for example the average deviatoric stress tensor in the stress history for that material point, then for every iteration we identify the maximum value of . Let tm be the time step at which max happen, then the value of sij * is updated sij *=s ij *+ ds ij * (14) with dsij *= (s ij (tm)-sij *) (15) which can be interpreted as a modified steepest descend method. If at one step increases, is reduced to 0.25 . The iteration is stopped if the norm of the difference between s୧ ∗ ୨ at the current iteration step k and at the previous step falls into a tolerance range: ቛsij *] k-sij *] k-1ቛ≤ εtoll (16) Although a superimposed hydrostatic tension has an effect on the fatigue life in normal cyclic loading [21], several studies [11] have shown that a superimposed mean static torsion has no effect on the fatigue limit of metals subjected to cyclic torsion. The independency of the mean shear stress is correctly predicted through the minimization process in Eq. (10), see also [20]. The Dang Van criterion could also be used with ୫ୟ୶ሺtሻ representing the maximum shear stress at every point of the stress history. Then, one would not account for the experimental observation that in cyclic torsion fatigue failure is independent of the mean shear stress, and this would usually result in lower permitted stress levels. The Dang Van proposal is equivalent to request, in the σH(t)-max(t) plane, that all the representative points of the stress state, fall below the line intersecting the max(t) axis in w with a negative slope of α: if all of the points fulfill this requirement, the criterion predicts a safe life for the component (see Fig. 2). The original Dang Van safe locus predicts a detrimental effect of tensile hydrostatic stress while an over-optimistic positive effect is expected from compressive values. The negative effect of tensile mean stress is well known in literature from classic Haigh diagrams, that also show a flat response for negative stress ratios [22, 23]. For this reason it is not too conservative to choose a different safe locus in the Dang Van plane to be in agreement with this response, for example a bilinear limit curve, as proposed recently in [20]. The safe locus could be therefore identified in two segments, one with a null slope and the other one with a negative slope equal to α (Fig. 2). For σH(t)≥σA the safe region is identical to the original Dang Van region, while for smaller values of σA, the cut-off with the flat curve replaces the Dang Van limit curve by a curve more on the safe side. Values of σA =σw/3 and of A=σw/2 have been proposed in [20], on the basis of experimental results obtained on high-strength steel smooth specimens. However, it is possible to choose a different set of values for (σA, A), though here the same choice has been made. If the ratio of the fatigue limits, σw/w, was equal to 0.5, the value αDV in Eq. (6) would be zero, which is far from reality, as steels usually
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