PSI - Issue 66

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

480

3

2. Theoretical setting Let us consider a composite periodic microstructure described by a Representative Volume Element (RVE). In the initial undeformed configuration, the RVE incorporates contact and cohesive interfaces (Greco, 2013). The chosen RVE, according to the classical homogenization theory, can be made of a single unit cell or by an assembly of unit cells in order to capture microscopic instability phenomena. The defined RVE is then driven along a macro strain loading path by prescribing boundary conditions depending on the macroscopic deformation gradient, defined by the classical macro-to-micro coupling relations, in which variables referring to the macroscale are over-signed:

1

1





( ) t

( ) t

F

 x n

T

t

X

,

(1)

dS

dS

=

=



( ) i

( ) i

R

R

(i)

V

V

( ) i

( ) i

V

V





( ) i

( ) i

where ( ) i V and ( ) i V  represent the domain and the boundary of the RVE, on which acts the traction vector of the first Piola-Kirchhoff stress tensor R t . The tensor product is indicated by  , whereas the outward normal related to the boundary of the RVE is indicated by ( ) i n . The mapping point of the current and the initial configuration are expressed by means of x and X , respectively. It is important to note that the whole deformation process is described by the time-like parameter t . As previously introduced, the RVE contains internal contact and cohesive interfaces. Once a cohesive point pair is defined as a pair of material points in the same initial configuration, the displacement jump vector across the cohesive/contact surface can be decomposed into normal and tangential component vectors, with reference to the current unit normal to the lower cohesive surface l n .

   

( l −  n 1 n n n l  = 

(2)

)

=

s 



l

Accordingly, the normal and the tangential cohesive traction vectors are defined as:

(    = −  t t t nch Rch l sch  =

l

n

(3)

) t 1 n n l

Rch

In the present work, the reversible cohesive traction-separation law (Tvergaard, 1990), used to describe the behaviour of both cohesive and contact interfaces, has been improved and employed in a finite formulation setting. 2.1. Rate equilibrium problem The governing equations of the RVE quasi-static rate response are obtained by appropriate asymptotic expansion of finite equations (for more details see (Greco, 2013) and reference therein). At the generic contact pair the rate contact conditions, which implies the non-penetration condition, can be given in the following forms. The projection to the lower deformed surface l n is defined by means of the subscript n:

0

( ) X

u





n





c

0

u R n u

( ) X

=



(4)

 



c

0 and

( ) X

0 if

0

u R

u

u R

u



=

=









R n



c

The considered normal displacement rate jump can be written as a function of the fluctuation field rate ( ) w X ,