Issue 33

M.A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 33 (2015) 357-367; DOI: 10.3221/IGF-ESIS.33.39 361 But Prager-Ziegler’s translation rule is too simplistic to model kinematic hardening. An improved rule was adopted by Mróz [1], who assumed that the translation i d    of the active surface occurs in a direction i v   defined by the segment that joins the current stress state c i i s s n r          with the corresponding “image stress point” 1 1 c i M i s s n r            that has the same normal unit vector n   at the next hardening surface i  1 , see Fig. 3. The Mróz translation direction is then 1 1 1 1 nt point ( ) ( ) ( ) M i c c c c i i i i i i i i i i i s image poi current r v s n r s n r n r r s s n r                                                    (Mróz) (4) Note that the Mróz translation direction is the combination of Prager-Ziegler’s hardening term with a “dynamic recovery” direction i    . This last term induces the center of the active surface to translate back towards the center of the next hardening surface, attempting to dynamically recover from the hardening state described by the surface backstress i    that separates their centers. This dynamic recovery term is able to consider a fading memory of the plastic strain path, which is necessary to model mean stress relaxation and ratcheting effects. However, the Mróz rule can induce a few numerical problems, which can result in surfaces intersecting in more than one point under finite load increments. Garud proposed an empirical correction that avoids intersection problems even for coarse integration increments, as detailed in [2]. A major concern of the Mróz multi-surface formulation is that the directions of the calculated stress or strain paths may significantly vary depending on the number of surfaces used. Even worse, better predictions are not necessarily obtained from using a larger number of surfaces. As a result, the number of hardening surfaces that results in the best calculation accuracy is a finite number that would also need to be calibrated, an undesirable feature. Moreover, the multi-linear formulation adopted by Mróz and Garud models is not able to correctly predict ratcheting and mean stress relaxation effects. A better approach is to replace such multi-linear models with a non-linear kinematic hardening formulation, described next. N ON -L INEAR K INEMATIC (NLK) H ARDENING M ODELS he multi-linear stress-strain curves generated by the Mróz and Garud multi-surface models provide good results for balanced proportional loadings, which by definition do not induce ratcheting or mean stress relaxation. However, such piecewise linear models cannot predict any uniaxial ratcheting or mean stress relaxation effects caused by unbalanced proportional loadings. This shortcoming is due to the linearity of the Mróz and Garud surface translation rules and the resulting multi-linearity of the approximated stress-strain representation, which describes all elastoplastic hysteresis loops using multiple straight segments, instead of predicting the experimentally observed curved paths caused by those non-linear effects. Such straight segments generate unrealistic perfectly symmetric hysteresis loops that always close under constant amplitude proportional loadings. In addition, for NP loadings, the Mróz and Garud multi-surface models may predict multiaxial ratcheting with a constant rate that never decays, severely overestimating the ratcheting effect measured in practice. As a result, Mróz or Garud kinematic strain-hardening models should only be applied to balanced loading histories, severely limiting their applicability. Another serious flaw of the Mróz and Garud models becomes evident for a NP path example where the stress state s   follows the contour of the active yield surface. In this example, the stress increment ds   would always be tangent to such active surface, therefore it would induce 0 T ds n       and no plastic strain would be predicted from the normality rule. This conclusion is physically inadmissible, since it would assume zero plastic straining along a load path with radius r A larger than the yield surface radius r 1 . The Mróz and Garud models would predict that the yield surface can translate tangentially to the active surface without generating plastic strains. These major drawbacks are a consequence of Mróz and Garud multi-surface kinematic hardening models being of an “uncoupled formulation” type, as qualified by Bari and Hassan in [8]. Such uncoupling means that the generalized plastic modulus P  P i in this formulation is not a function of the load translation direction i v   , since it is a constant for each surface. Such “uncoupled procedure” (where P and i v   are independent) provides undesirable additional degrees of freedom to the Mróz and Garud models that allow, for instance, 90 o out-of-phase tension-torsion predictions of resulting plastic strain amplitudes that are not a monotonic function of the applied stress amplitudes, as they should be [9]. These T