Issue 70

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

The scalar d λ is the plastic multiplier, which is equal to the rate of the accumulated equivalent plastic strain. Given that p d ε represents the plastic strain rate, it can be demonstrated using Eqn. 23 and the von Mises equivalent stress eq σ , that the plastic multiplier is equivalent to the rate of the accumulated plastic strain:

   

   

   

2 3d λ S 3d λ S

3 S S

2





dp

d λ

(26)

3 2 σ

2 σ

2

σ σ

eq

eq

eq eq

dp d λ 

(27)

The plastic multiplier d λ in Eq. 22 is determined by using the consistency condition, to obtain the following expression:

 

 

 

   



f

f

f

f R R p

R



dp d σ   

dp 0 , and 

R p  

(28)

d σ



σ

p

σ

p

Eq. 26 can be written as:



dR d

eq

Nd σ H d λ 0 and H  

 

(29)

P

P

p eq

dp d ε

Using Eq. 17 and Eq. 19 in Eq. 22 leads to:



f

  1 C 



d σ d λ 

d ε

(30)





C T N :

After pre-multiplying both sides of Eq. 30 by

T T N C d ε N d σ N C d λ N   T

(31)

T

T N C d λ N

(32)

eq N C d ε H .d ε P 



The following expression is obtained for the plastic multiplier:

N:C.d ε

p eq  

(33)

d λ d ε

N:C:N H 

P

p d ε d λ .N  into Eq. 17, the elastoplastic tangent modulus is derived

Finally, by substituting the expression of the plastic part

as:



  N:C N:C N:C:N H  



ep C C

 

(34)

P

At last, the updated stresses ( σ ) at the conclusion of the time step ( ∆ t) can be expressed as follows:

n 1 n σ σ d σ   

(35)

The subscript (n+1) indicates the values corresponding to the end of the time increment.

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