Issue 59

A. Houari et alii, Frattura ed Integrità Strutturale, 59 (2022) 212-231; DOI: 10.3221/IGF-ESIS.59.16

 d p r

( )

 d r

  

  

( )

 d r

( )

2 ( ) r dE r

( )





p

p



r

 dr

1

(9)

    ( )









 ( ) ( ). ( ) r v r r r 

 r

( ) r

r r



E r

( )

dr

dr

dr

E r

To solve this differential equation, one uses the criterion of plasticity determined by the flow surface and is given according to the criterion of Von Mises by:

      ( , ) 0 f R eq r

(10)

 ( , ) f R : is the yield function,  eq : is the equivalent stress Von Mises of FGM and  r : represents the radius of the yield surface. If  ( , ) f R is negative then the behavior of the material is elastic, if not  ( , ) f R remains zero, there will be plastic flow. From Eqn. (11), we define the equivalent Von Mises stress  eq and the deviator (S) by:

       2 2 r z r     z    

2





1

  2 . 3 S S eq

,

(11)





eq





2

   

 

2

 

         

r

z

3

                   S r S z

      r z

2

(12)

 

S S

3

      z r

2

3

 

The evolution of the plastic strain  p d is governed by a normal flow law of the plasticity criterion:



f

p

p



f

S

2

 

d

(13)







 d N N ,



d



 

  3

eq

p de , the term  p d can be written as follows:

As a function of the tensor

p p

   3 2 d p

(14)

dp

de de

where N is the gradient of the yield function with respect to the stress tensor,  p d is the (scalar) equivalent plastic strain rate. The isotropic hardening function  ( ) r of the FGM will be described as:

   ( ) r

 r R p y and ( ) ( ) 0

(15)

 ( ) . R p H p

where  0 ( ) y r is the initial yield stress of the FGM and ( ) R p is the isotropic hardening. To obtain the plastic multiplier  d , it is therefore sufficient to express the law of evolution of the variable ( ) R p as a function of the flow variables. Using the Prandtl-Reuss law to determine the evolution of the variable in our case in uniaxial tension, the equality   ( , ) 0 f R can be summed up in the form:

   ( ) R p

p

and

(16)

   ( ) r

 ( ) y r R 0

 ( )

H p d

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