Issue 33

M.A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 33 (2015) 357-367; DOI: 10.3221/IGF-ESIS.33.39 358 popularity is understandable, since the multi-linear stress-strain curves generated by the Mróz-Garud model provide good results for balanced loadings. However, such multi-linear models cannot predict any uniaxial ratcheting or mean stress relaxation caused by unbalanced loadings, since their idealized hysteresis loops always close because they are unrealistically assumed as perfectly symmetric. In addition, under several non-proportional loading conditions, these models predict multiaxial ratcheting with a constant rate that never decays, severely overestimating the ratcheting effect measured in practice [3]. As a result, multi-surface kinematic hardening models should only be confidently applied to balanced proportional loading histories. To correctly predict the stress-strain history associated with unbalanced loadings, it is necessary to introduce non-linearity in the hardening surface translation equations, the main characteristic of the non-linear kinematic (NLK) models. NLK models are more general than Mróz-Garud because they use non-linear equations to describe the surface translation direction and the value of P , leading thus to a more precise description of the non-linear stress-strain curves. Armstrong and Frederick’s original formulation [4] was improved by Chaboche [5], which indirectly introduced the Mróz nested- surface idea to NLK models, however in a non-linear instead of multi-linear formulation. Therefore, both NLK and Mróz-Garud formulations have several common features, as discussed further in this work. A third class of kinematic hardening models involves the so-called two-surface models, which use a rather simplified formulation that combines elements of both non-linear and Mróz multi-surface kinematic models. In this work, instead of defining the nested hardening surfaces in the 6D stress or 6D deviatoric stress spaces, a 5D reduced order deviatoric stress space E 5s is adopted, using the Mises yield function to describe each surface. This 5D space has two advantages over the usual 6D formulations: it is a non-redundant representation of the deviatoric stresses, which decreases the computational cost of stress-strain calculations; and the radius S of the yield surface is equal to the yield strength without the need to include the scaling factor 2 3 required in 6D formulations. Besides, even though all kinematic hardening equations are presented here in the 5D space, their conversion to 6D versions is trivial. M RÓZ M ULTI -S URFACE M ODELS n Fig. 1, the first (and innermost) circle is the monotonic yield surface, with radius r 1  S Y . In addition, M  1 hardening surfaces with radii r 1 < r 2 < … < r M  1 are defined, along with an outermost failure surface whose radius r M  1 is equal to the true rupture stress  U of the material. Their centers are located at points ci s   with i  2 , 3 , …, M  1 , respectively. These M  1 nested circles cannot cross one another, must have increasing radii, and for a virgin material they all are initially concentric at the origin of the E 5s space, i.e. initially their 0 ci s    . Moreover, the failure surface never translates, i.e. its center always remains at the origin of the E 5s space, 1 0 M c s     . In fact, any stress point that reaches its boundary causes the material to fracture due to ductility exhaustion, which is equivalent to the criterion 1 |s| M U r       . Figure 1 : Yield, hardening and failure surfaces in the  x   xy  3 2D sub-space of E 5s (left) and corresponding radii r i and generalized plastic moduli P i obtained from the piecewise linearization of the uniaxial stress versus plastic strain curve (right). I