Issue 70

A. Baryakh et alii, Frattura ed Integrità Strutturale, 70(2024) 191-209; DOI: 10.3221/IGF-ESIS.70.11

However, any yield criterion could be reduced to (34) by assuming as  e the yield function with its free term being added, and taking the free term of the yield function as  y . Given this, the Perzyna viscoplastic model could be written in terms of the yield function

1



   

m



( , ) A 

1



(35)

 



 

( ) A

 

y

where

(36)

( , )  

( , ) A A A     ( ) 

e

n

y

and  y ( A ) is the free term of the yield function representing a certain limit of equivalent stress. The set of viscoplasticity parameters now consists of two elements   , m  Ρ 

(37)

The parameter m specifies the material reaction to a strain rate change. The closer it is to zero, the more the material behavior is consistent with plastic straining. The limit m → 0 is equivalent to a perfect plastic behavior, i.e. the transient effects are completely absent. The constraint equation of the plastic multiplier in the system of residuals according to the Perzyna viscoplastic law is written as follows:

m

t         

(38)

( ) A A   

( , )  



0

R



n

y

The main drawback of the Perzyna model also follows from equation (38). In the limit m = 0 the model doubles the yield point  e (  , A ) = 2  y ( A ). Non-associated Mohr-Coulomb criterion As before, we consider the cases when the stress tensor is related to the surface and when to one of the edges. In the first case the system of residuals in matrix notation takes the form:

e

trial     R

D N 0 

  



n

1

   

(39)

m

t          







( , n 



)

(

)

0

R

A

A







y

1

where the yield point  y for the Mohr-Coulomb criterion is written as

(40)

(   A 



) 2 cos c

y

The corresponding Jacobian is

P

    

   

I

1

(41)



m y     t          m

J

N 

1 T

 

Note that the diagonal element in the second row of the Jacobian cannot be determined from the initial approximation   = 0. In order to eliminate this computational uncertainty, we rearrange the equation R  as follows

202