Issue 33

M.A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 33 (2015) 357-367; DOI: 10.3221/IGF-ESIS.33.39 362 wrong Mróz and Garud predictions are both qualitatively and quantitatively dependent on the number of surfaces adopted in the model, without any clear convergence. To correctly predict the stress-strain history in unbalanced loadings, P and i v   must be coupled, in addition to introducing non-linearity in the surface translation equations, generating the non-linear kinematic (NLK) models discussed next. The first non-linear kinematic (NLK) hardening model was proposed by Armstrong and Frederick in 1966 [4]. Their original single-surface model did not include any hardening surface, but their yield surface already translated according to a non- linear rule. Chaboche et al. [5] significantly improved Armstrong-Frederick’s model capabilities by indirectly introducing the concept of multiple hardening surfaces. As demonstrated by Ohno and Wang in [10], this improvement allowed the NLK formulation to use the same representation of the hardening state as the one in the Mróz multi-surface formulation, which includes one yield surface, M  1 hardening surfaces, and one failure surface, all of them nested within each other without crossing. Therefore, the improved NLK models also adopt a multi-surface formulation, using a “coupled procedure” based on non-linear translation rules. However, unless otherwise noted, the denomination “multi-surface model” is traditionally associated in the literature with the Mróz “uncoupled procedure” based on multi-linear translation rules and piecewise-constant generalized plastic moduli P i [8]. In summary, despite their significant differences, both Mróz and NLK approaches can be represented using the same multi-surface formulation. As in the Mróz models, instead of defining these surfaces in the 6D stress or deviatoric stress spaces, in this work a 5D reduced order deviatoric stress space E 5s is adopted, using the Mises yield function to describe each surface. In the multi-surface version of NLK models, the backstress    that locates the center of the hardening surface is also decomposed as the sum of M surface backstresses 1 2 , ,..., M          , that describe the relative positions 1 i i i c c s s           between the centers of consecutive surfaces, exactly as in the Mróz multi-surface formulation. From these relations, the center of the yield surface ( i  1 ) or of any hardening surface ( 2  i  M ) can be written as the 5D vector 1 2 2 1 ... c i i i i M M M s                                (5) Once again, the length i    of each hardening surface backstress is always between 0 (in the unhardened condition) and its saturation value  r i (a maximum hardening condition when these surfaces become tangent, see Fig. 3), while the failure surface is always centered at the origin, i.e. 1 0 M c s     . As usual, the hardening surfaces cannot pass through one another, remaining nested at all times, since | | i i r      . But during plastic straining in the NLK multi-surface formulation, the yield and all hardening surfaces do translate, as opposed to the Mróz formulation, where all surfaces outside the active one would not move. The NLK models do not use an “active surface” concept, since all hardening surfaces become active during plastic straining. The yield and hardening surfaces from the NLK models behave as if they were all attached to one another with non-linear spring-slider elements, causing coupled translations even before they enter in contact, i.e. even for | | i i r      . Therefore, any hardening surface translation causes all surfaces to translate, usually with different magnitudes and directions, even before they become tangent to each other. Pairs of consecutive hardening surfaces i and i  1 may eventually become mutually tangent at the saturation condition | | i i r      , when their respective translations have the same magnitude and direction 1 0 i i c c i d ds ds            , so when there is no surface backstress variation for this pair. So, plastic straining causes increments 0 i d     in all surface backstresses, except for the ones from the saturated surfaces. Quantitatively, these surface translation rules state that, during plastic straining, , if | | 0, if | | i i i i i i i p v dp r d r                       (6) for the yield ( i  1 ) and all hardening surfaces i  2, …, M , where (2 3) | | pl dp de     is the equivalent plastic strain increment, i v   is the surface translation direction vector (in the 5D space E 5s ), and p i is a generalized plastic modulus coefficient that must be calibrated for every surface, used in the calculation of P .