Issue 37

C. Madrigal et alii, Frattura ed Integrità Strutturale, 37 (2016) 8-14 DOI: 10.3221/IGF-ESIS.37.02 9 number of proposals in this respect [11, 12]. The third step is probably the most difficult and it is the area where more work has been invested so far: the multiaxial cycle counting and fatigue life criteria. There are too many of them to single any one out. A comparison of several criteria is provided in [13]. They need the stresses and strains as inputs and therefore they also depend on the two previous steps. We are concerned here with the first step. Our theory does not make use of yield or loading surfaces that move about in stress space, a common ingredient of existing cyclic plasticity theories. It uses the concept of distance in a stress space endowed with a certain metric measurable from the yield criterion. The full mathematical details of the method have been given elsewhere [1-6] and we would just like here to provide a first insight of this idea of distance in the stress space and show some comparison with experimental results. To keep the discussion at the simplest possible level we will restrict the treatment given here to the case of combined tension and torsion loading. P LASTIC STRAINS CALCULATIONS he local strain method revolves around a simplified description of the stress-strain behaviour. A very characteristic feature of the calculations of plastic strains in low cycle fatigue problems is the clear distinction between loading and unloading . In the uniaxial case, one speaks of loading when the stress goes up in the cycle of applied stress and of unloading when it goes down. During the first quarter of the very first cycle, we “move” along the cyclic curve (dashed line in Fig. 1) until unloading starts, marking the first point of load reversal (point A). We then “depart” from the cyclic curve and switch to the hysteresis loop. After a while moving along the descending branch of the hysteresis loop another point of load reversal (point B) will be reached and we will leave the current branch of the loop being traversed and start a new branch going up, and so on. One of the key elements in the simulation of the    behaviour at a notch for variable amplitude loading is the correct application of the memory effect (see [9, chapters 12-14] and [10, chapter 5]), both for closing hysteresis loops and for switching the axes where Neuber’s hyperbolas are drawn for each load excursion. This is shown to occur in Fig. 1 as one moves, for example, from point D to point E. After reaching point E the strain is then decreased to point F, following the path determined by the hysteresis loop shape. Upon re-loading, after reaching point E- E', the material continues to point A along the hysteresis path starting from point D, proceeding just as if the small loop E-F-E had never occurred. The same thing happens in the loop B-C-B. As we will point out later on, this memory rule is a simplified representation of the so-called kinematic hardening. Figure 1 : Uniaxial memory effect. T