Issue 33

M.A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 33 (2015) 357-367; DOI: 10.3221/IGF-ESIS.33.39 359 Except for the failure surface, all other hardening surfaces can translate as the material strain-hardens, as shown in Fig. 1(right). The centers of the hardening surfaces (which are circles in the 2D example from Fig. 1) move as the material plastically deforms and hardens, because they are successively pushed by the inner surfaces. The radii r i of the various hardening surfaces are equal to the stress levels associated with the plastic strains  xpli that delineate the multi-linear representation of the stress-strain curve, fitted to properly describe the stress-strain  behavior of the material. The difference between the radii of each pair of consecutive surfaces is defined as  r i  r i  1  r i . In principle, all hardening surfaces radii r i may change during plastic deformation as a result of isotropic and NP hardening effects. The backstress vector    , which locates the current yield surface center 1 c s      , can be decomposed as the sum of up to M surface backstresses 1    , …, M    that describe the relative positions 1 i i i c c s s           between centers of the consecutive hardening surfaces, see Fig. 2. Note that the length (norm) i    of each surface backstress is always between 0 i     , if the surface centers i c s   and 1 i c s    coincide (as in an unhardened condition), and i i r      , if the surfaces are mutually tangent (a saturation condition with maximum hardening). The main rules in the evolution of these yield and hardening surfaces are: (i) they must translate as rigid bodies when the point s   that defines the current deviatoric stress state in the E 5s space reaches their boundaries, to guarantee that such stress point is never outside any surface. In other words, neither the yield surface nor any hardening surface translate unless the stress state is at their boundary ( |s s| i c i r      ) trying to move outwards; and (ii) the hardening surfaces cannot cross through one another, therefore they gradually become mutually tangent to one another at the current stress point s   as the material plastically deforms. Figure 2 : Yield, hardening, and failure surfaces in the deviatoric space for M  3 , showing the backstress vector    that defines location of the yield surface center 1 c s      and its components 1    , 2    , and 3    that describe the relative positions between the centers of consecutive surfaces. In the Mróz multi-surface formulation, the outermost surface that is moving at any instant is called the active surface , denoted here as the surface with index i A . Any changes in the stress state that happen inside the yield surface are assumed elastic, not resulting in any surface translation as long as 1 1 |s s| c r      , therefore no surface would be active in this case and thus i A  0 . Each surface is associated with a generalized plastic modulus P i ( i  1 , 2 , …, M  1 ), which altogether define a field of hardening moduli . The value of P is then chosen as the P i from the active surface i  i A . Fig. 3 illustrates two consecutive surfaces i and i  1 ( i  1 ), with radii r i and r i  1 in the E 5s 5D deviatoric stress space. Assuming that the