PSI - Issue 66

Daniele Gaetano et al. / Procedia Structural Integrity 66 (2024) 478–485 Author name / Structural Integrity Procedia 00 (2025) 000 – 000

481

4

through the well-known additive decomposition ( , ) t

:

( ) t = + x X F X w X

( , ) t

 



.

(5)

l

n

( ) X

( ) X

u

w

=

−

( F X X u

)

l − 

n

n





c

c

The rate version of the cohesive law can be instead defined as:

l t t = + +

l n n t . nch sch

t

(6)

Rch nch

For more details about the rate equilibrium problems and homogenized rate response, as well as the contact cohesive interface modeling, here summarized for the sake of brevity, please see (Greco et al., 2024). 2.2. Microscopic stability analysis The exact stability functional can be defined as follows:

( ) i dS dS dS dS   

  

(

)    X F w ,









(7)

R

l l n R

l

l 

l

l

( , ) F w C =

(i) dV  − w

t

S

dS

w dS

dS





+

+





w

( ) i

( )

( )

R

n i

Rch

ch i



c

( ) i

l

l

l

B









( ) i

( )

( )

( )

c i

c i

ch i

c

It has been defined over the set of admissible fluctuation rates: ( )  1 ( ) # ( ) 0 if 0, 0 if , u i n R n c c H V u u    =  =   F F w



*

.

(8)

u

( ) l on and c i

u

0 = 

A





( )

R

c i

The positivity of the stability functional guarantees the stability of the equilibrium configuration. The corresponding critical load level can be found as the critical stretch ratio cs t associated with the lowest zero eigenvalue  :

  

    

  



( , ) / F w w w S  

dV

.

(9)

( ) inf F

min

0

 =



(i)

k N 

( ( ) A k V  * N

B

0

#)( ,

)

w x

F F

=

( ) i

Starting from Eq. (7), two simplified versions of the proposed stability functional have been introduced. Under the hypotheses of small cohesive surface relative deformations, allowing large interface displacements, the first one (named Simplified I) can be obtained by neglecting the term associated with the relative surface deformations (third term in Eq. (7)): ( )   ( ) ( ) ( ) ( ) ( ) ( , ) , c l l i c i ch i R l l l l (i) R i Rch ch i B S dV dS dS     =  −  +      F w C w n t X F w w  . (10) The second one (Simplified II), instead, can be obtained by also neglecting the terms related to the surface rotations at the contact/cohesive interfaces (second term in Eq. (7)). In this case the hypothesis of small displacement at the cohesive interface has to be assumed: ( )   ( ) ( ) ( ) ( , ) , l i ch i R l (i) Rch ch i B S dV dS  =  +     F w C w t X F w  . (11) For more details about the proposed formulations, please refer to (Greco et al., 2024).