Issue 47

R. Fincato et alii, Frattura ed Integrità Strutturale, 47 (2019) 231-246; DOI: 10.3221/IGF-ESIS.47.18 234 1 (1 ) : e D       D E σ A A wA Aw    (2) Eqn. (2) 2 defines the co-rotational rate for a generic second-order tensor A , being w the spin of the rigid-body rotation. The analytical expression of the plastic strain rate passes through the definition of the normal-yield and subloading surface, here coupled with the isotropic scalar variable D . ˆ ˆ ˆ ˆ ˆ ( ) (1 ) ( ); ( ) (1 ) ( ); , , f D F H f D RF H R R              σ σ σ σ α σ σ σ α α s s s s α (3) The function f is a generic yield-stress scalar function, herein the von Mises criterion is assumed. The similarity ratio R expresses the ratio between the normal-yield and the subloading surface and it characterizes the stress state. For 0 1 R   the stress belongs to the sub-yield state, inside the normal-yield surface. For the stress lies on the normal-yield surface (which coincides with the subloading surface) and it belongs to the fully plasticized state. The expression of the similarity ratio is given in its incremental form during the loading (Eqn.(4) 1 ). Where depends on the material parameter u and the norm of the plastic strain tensor. The term defines the size of an eventual sub-elastic domain, introduced by Tsutsumi et al. [27] to model metallic materials. However, during the elastic unloading, the similarity ratio can be computed by the analytical formula in Eqn.(4) 2 . The symbol “ ” indicates the deviatoric part of the tensor. Fincato and Tsutsumi in [11] defined the similarity ratio formulas for the DSS model.     2 2 2 2 2 2 2 2 ( ) ; ( ) cot during loading 2 1 2 ˆ ˆ ˆ (1 ) 3 during unloading. 2 ˆ (1 ) 3 e p e R R R U R U R u R tr tr D F R D F                                                D σ s σ s s σ s     (4) The consistency condition of Eqn. (3) 2 and the similarity ratio expression of Eqn. (4) 2 require the definition of the similarity center s , the back stress and the isotropic-hardening function F . Briefly, their expressions are reported in the following Eqs. (5)-(7), respectively. 1 1 ˆ ˆ ˆ 1 (1 ) p dF D c H R dH F D                           σ α s D s s s      (5) 2 0 0 1 1 p h H H p F F K H H F h e             (6)   1 ; , 1,... n p i i i i i i C B n N        α D α α α    (7) As mentioned at the beginning of this section the similarity-center movement depends on the norm of the plastic strain rate together with the material constants c and . The rate form of the damage variable will be discussed in the following subsection. The isotropic-hardening function of Eqn. (6) depends on four material parameters h 1 , h 2 , K and H p . The first two constants have the same meaning as in [15], K gives an additional linear contribution, whereas H p was introduced by the authors to model a strain hardening plateau. The kinematic hardening law of Eqn. (7) was introduced by Chaboche [28] to define the back stress rate as the linear combination of N non-linear contributions, regulated by the material constants C i and B i . In the present paper N was set to be equal to two. The DSS model considers an associated flow rule for the generation of the plastic strain rate tensor . Consequently, the plastic deformations are generated along the normal tensor N to the plastic potential at the current deviatoric stress state  σ . 1 R  R  e R ' α  D  p D