PSI - Issue 25

Umberto De Maio et al. / Procedia Structural Integrity 25 (2020) 400–412 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

405

6

− + = − n n .

(7)

( ) i V  is given as follow: ( , ) ( , ) ( ( , ) )( t t t + − − = − − u X u X F X I X X . + −

Definitively, the periodic boundary condition on

) (8) Imposing the Eq. (8) at the boundary of the RVE, this condition leads to an anti-periodic traction field and to a periodic fluctuation field as expressed in the following: ( , ) ( , ) R R t t + − = − t X t X , (9) ( , ) ( , ) R R t t + − = w X w X . (10) In order to take into account for frictionless self-contact phenomena between opposite crack surfaces, the following conditions at each point X u of the upper crack surfaces are introduced:

    

l

u X u X +

( , ), ) 0 0 u t t 

h

(

( ) ( )

.

(11)

u

u

t t h 

on

 



R

( ) i

u

l

u X u X +

( , ), ) = 0 u t t

(

R

0 u h = and

0 l h = describe the deformed upper and lower crack surfaces, denoted by u  and

As shown in Fig. 2,

u u u R i  = T n n is the normal nominal contact reaction on the upper crack surface, where R

u n represents the

l  , while

( )

( ) u

u

R i R = T n r represents the nominal contact reaction. For sake of brevity, only the

deformed outward normal, and

equations governing the rate equilibrium solution of the microstructure are reported in the following:

      

T

0

Div

( ) in on V i B 

=

R

+

−

t

t

= −

R

R

( ) i

(

)

u

u

u R

u R

, X X l

u

( ) X

( ) X

( ) = 0 and X

u

u

u

if

0,

0,

0

= 0



=













n

R n

R n







C

C

C

C

(12)

  

  

  

  

u

l

u

l

dS

dS

dS

dS

(

)

( ) i R    − = dS + r r l R R r  u

( ) i

( ) i

( ) i

l

u u n

l l n

, X X l 

u

0

+

+

=





R

R

R

dS

dS

dS

C

l l   R

l l n n

l

on



C i

( )

where ( ) n u X denotes the projection of the displacement rate along the normal

l n to the deformed lower crack C X X , R r is the nominal contact

( ) l

( ) u

( ) X

n X X u −

u

u

represents its jump at a contact point pair ( , ) u l

surface,

C  =

n

n

reaction rate, defined as ( ) R R i = r T n . To derive the infinitesimal stability and non-bifurcation conditions of the equilibrium configuration, the following contact surface integral is examined: ( ) ( ) ( ) ( ) ( ) + l u c i c i l l u u rate c R i R i L dS dS           =       r u r u . (13) representing the virtual work of the contact reaction rate acting on the present contact interface in a virtual displacement * ( ) A ,   u F F . Using Eq. (12) 4 the contact surface integral becomes:

   

  

  

  

  

 

l

l

u

dS

dS

dS

dS

( )





l r u R 

r

u

( ) i

( ) i

( ) i

l

l

l l n

l

u u n

u u dS

L

dS









 =

−

+

+

=

R

rate c

( ) i

R

R

( ) i

( ) u i

dS

dS

dS dS  

l

u





c i

c i

( )

( )

(14)

   

   

  

( ) i dS dS dS dS    ( ) i l u





l r u R 

( ) n i dS u  l l

l

l

dS



−

−

 

 

 

( ) i

R

l

u

dS

dS

dS

dS



c

( ) i

( ) i

l

l





c i

c i

( )

( )