Issue 33

M.A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 33 (2015) 357-367; DOI: 10.3221/IGF-ESIS.33.39 360 current active surface is i  i A , then according to the Mróz model all outer hardening surfaces do not translate, therefore the increments of the respective backstresses are 1 2 ... 0 i i M d d d                , resulting in 1 i i c c i i ds d ds d               . Figure 3 : Illustration of Mróz, Garud, and Prager-Ziegler surface translation rules used to model kinematic hardening in the Mróz multi-surface formulation in the E 5s space. The Mróz multi-surface formulation assumes that, during plastic straining, all inner surfaces 1 , 2 , … , i A  1 must translate altogether with the active surface i  i A , therefore their centers do not move relatively to each other, resulting in 1 2 1 ... 0 i d d d               . Thus, translation rules in the Mróz multi-surface formulation only need to be applied to the evolution of the backstress i d    of the active surface i  i A , giving 1 , if 0, if c c A i i A ds ds d i i d i i                    (1) Moreover, since these inner hardening surfaces 1 , 2 , … , i A  1 are all mutually tangent at the current stress state s   perpendicular to the normal vector n   , their backstresses are all parallel to n   and have reached their saturation (maximum) values. The kinematic rule for the translation i d    of the active yield surface can be defined from an assumed translation direction i v   . Prager [6] assumed that i v   is parallel to the direction of the normal unit vector n   , i.e. i d    happens at the current stress state s   in such normal direction n   . Ziegler, on the other hand, assumed that i d    happens in the radial direction c i s s     from the surface center [7]. For the Mises yield surface, both Prager’s and Ziegler’s rules result in the same Prager-Ziegler direction i v n     , see Fig. 3, which can be calculated from the normalized difference between the current stress state s   and the yield surface center c i s   : | | c c i i c i i s s s s n r s s                   (2) For Mises materials, the translation direction of Prager-Ziegler’s kinematic rule is then 1 ( ) i i i i v n r r n r             (Prager-Ziegler) (3)