Issue 33

M.A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 33 (2015) 357-367; DOI: 10.3221/IGF-ESIS.33.39 364 Figure 4 : Geometric interpretation of the three components of the translation direction i v   of hardening surface i : Prager-Ziegler’s, dynamic recovery, and radial return terms, where the equivalent parameter  i *   i *  m i *   i . In this model, every time an elastic stress state s   reaches the yield surface, the length of the current Mróz translation vector between the yield and bounding surfaces is stored as the initial reference length 1 | | |s | in M v v s          , where s M   is the Mróz (stress) image point, see Fig. 5. The generalized plastic modulus P is then calculated at each stress increment from the current length 1 | | v   of the Mróz translation vector and its initial value v in through: 2 1 1 ( ) | | ( | |) in in P P f v v v v         (9) where P 2 is the value of P calibrated for the bounding surface, and f(v in ) is a continuous material function that needs to be calibrated. The generalized plastic modulus P continuously varies in a non-linear way between the elastic value P  ∞ and the saturated P  P 2 , instead of assuming a piecewise-constant value P  P i (from the active surface i  i A ) as in the original Mróz multi-surface formulation. Some implementations [22] of this model adopt a slightly different metric to define generalized plastic modulus P and the initial reference length v in , replacing 1 | | v   with the projection 1 T v n      onto the yield surface normal vector. The translation direction of the surface backstress 1    is defined by 1 1 1 M v s s n r                  , the Mróz surface translation rule, while the translation direction of the yield surface can adopt any linear or non-linear hardening rule such as the ones listed in Tab. 1. A non-linear rule is suggested, to avoid the same drawbacks from the Mróz and Garud multi- linear models for unbalanced uniaxial or NP loadings, thus allowing the prediction of ratcheting. Contrary to the Mróz or NLK multi-surface formulations, the two-surface kinematic hardening model does not adopt a rule to explicitly calculate the surface translation direction between surfaces i  2 and i  3 (respectively the bounding and failure surfaces in the two-surface formulation). The continuous definition of the modulus P from Eq. (9), as opposed to the piecewise-constant P  P i from the Mróz multi-surface formulation, introduces a non-linear component in the two- surface model that allows it to reasonably predict uniaxial and multiaxial ratcheting under constant amplitude loading. However, the quality of such ratcheting predictions strongly depends on the non-linear kinematic hardening rule employed to define the translation direction of the yield surface [8]. In practice, non-linear kinematic models should be preferred over two-surface models for cyclic unbalanced loadings that cause ratcheting or mean stress relaxation. Due to its computational simplicity, the two-surface model has been widely adopted for the prediction of the deformation behavior of metals under monotonic and constant amplitude loadings. The need for only M  2 moving surfaces makes it attractive to model pressure-sensitive or even anisotropic materials, whose elaborate yield function would result in a high computational cost in Mróz or NLK multi-surface formulations with M >> 2 . Even though two-surface models are not the best option for unbalanced or variable amplitude loadings, they can provide excellent results for monotonic loading applications, such as in sheet metal forming [23].