Issue 33

M.A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 33 (2015) 357-367; DOI: 10.3221/IGF-ESIS.33.39

current active surface is i  i A

, then according to the Mróz model all outer hardening surfaces do not translate, therefore







1 c d ds        i   i





i  

...       d d 



i  

i 

d

ds

d

0

the increments of the respective backstresses are

, resulting in

.

c





M

1

2

i

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  . Thus, translation rules in the Mróz multi-surface formulation only need to be applied to the ...       0 d d d  

1  

2 

i  

1

i d   

of the active surface i  i A

, giving

evolution of the backstress



 



ds      

c ds   

 

i i 

d

, if

0, if c i

i 

d

A

(1)

1

i i 



A

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 i c s s s s            s s   r 

c 



 

n

(2)

i

i

i

For Mises materials, the translation direction of Prager-Ziegler’s kinematic rule is then

( i v n r      



)    r n r 

(Prager-Ziegler)

(3)



i

i

i

1

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