Issue 33

M. Vormwald, Frattura ed Integrità Strutturale, 33 (2015) 253-261; DOI: 10.3221/IGF-ESIS.33.31 256 where the x -coordinate is the crack area normal direction. The stress tensor components have to be provided in this coordinate system. Next, the crack opening strain is determined in the ascending  x -  x path. The effective crack driving force is determined on the descending  x -  x path,   ,cl ,max ,eff ,max d x x x x x x W          . (5) The effective ranges associated with shear modes are estimated by defining an efficiency factor with                 1 1 eff fric F fric F ln cos ( / 2 ) / 2( ) / ln cos ( / 2) / 2 U                , (6) where fric act n,max      (7) is a model parameter requiring two fitting constants,  act and  , and  n,max is the maximum normal stress during a shear cycle. These remarks on the crack closure phenomenon and the corresponding effective ranges should have made clear that the problems is away from being solved generally and provides another challenge for future research. Challenge 3: Definition of a damaging event – or a cycle Most procedures apply rainflow counting of stress or strain components acting on the critical plane. There is hardly an alternative for defining a cycle although the physical basis of the rainflow algorithm is lost in non-proportional loading. A desirable approach would yield an incremental crack growth for an infinitesimal variation of the crack driving force. Such an approach has been proposed for example by Lu and Liu [21] for uniaxial loading. For multiaxial loading some trials have been published, e.g. [22], to use plastic work as the responsible quantity for incrementally increasing the fatigue damage. The models have not yet achieved a wide acceptance. Challenge 4: Stress-strain paths The input for the cycle identification and counting must be provided by a calculation of components of the stress and strain tensor as a function of time. Solving this task requires application of a cyclic plasticity model. Contrary to a uniaxial case, the differential equations of incremental plasticity have to be solved. These models should address two phenomena which emerge in non-proportional loading: multiaxial ratcheting and non-proportional hardening. Here, a model suggested by Döring et al. [23] was used. Fig. 4 shows a comparison of measured and calculated stresses for a strain controlled butterfly path. An animation showing the calculated stress path together with the kinematics of the yield surface is provided by link. The accuracy is better than acceptable; the numerical expense is enormous. -500 -250 0 250 500 -300 -200 -100 0 100 200 300  in MPa  in MPa measured calculated elastic   Figure 4 : Measured and calculated stress response, strain controlled butterfly loading of a thin-walled tube, S460N, Döring et al. [23]. Challenge 5: Mixed mode hypothesis The combined effect of mode-related crack driving forces on the crack increment has to be evaluated. Little is known concerning mixed mode hypotheses for non-proportional fatigue [24]. Generally, the hypothesis has to specify the crack