Issue 49

H. Berrekia et alii, Frattura ed Integrità Strutturale, 49 (2019) 643-654; DOI: 10.3221/IGF-ESIS.49.58

 ES D   2





 

H p p 

F f

(6)



0

2 1

S: is a constant characterizing the damage, depends on the material and the temperature. H: Heavyside function. 0 p : is the accumulated plastic strain, when the damage is zero (the threshold of damage)

where :









s 

0 0 H p p   If: 0 H p p   If:  

eq

with:

p p 

0

0

The normality rule provides the evolution laws of:

 

F E  

p



(7.a)

 0 . F D P H p p S      



  

(7.b)

f   

f

λ: is the plastic multiplier, determined by the consistency condition

0

Where :

0 p p  if D = 0)

P: is the accumulated plastic strain (

With:

1 3

2 3

 

  

p ij ij 

    

(8)

The damage evolution law (7.b) can be written:

S    If

p p 

D

(9)

0

When replacing Y by its value of (4) we obtain:

   

2        

2 S D  eq

 

2 1



     



  

H eq





D

(10)

3 1 2

 2 1



2

3

 

 

In summary, the constitutive laws written for Newton's numerical method [16]:

e

p

ij ij     

3 2        D ij p ij

ij 



1       e  

Si:

f   

kk

ij 

f

0

ij

1     1 D

D

eq

With:

2

2 v D R S    eq

p p 



If:

(11)

0

650