PSI - Issue 33

Andrea Pranno et al. / Procedia Structural Integrity 33 (2021) 1103–1114 Author name / Structural Integrity Procedia 00 (2019) 000–000

1107

5

1

   , X F +  hk mn I 



  F



(8)

R

R

hk w dV mn

C

C



( ) i

ijhk

ijmn

V

( ) i

V

( ) i

hk w  is the incremental fluctuation field and hk mn I

mh nk    is the identity tensor.

where

3. Microscopic stability analysis The microscopic stability analysis is performed on a generic unit cell assembly ( ) i V of a periodic nacre-like composite material by superimposing an additional displacement field on the RVE equilibrium configuration. As reported in (Greco and Luciano, 2011), the incremental form of the classical stability condition (Hill, 1965) allowed to determine the microscopic stability condition which is based on the positive definiteness of the stability functional ( , ) S F w  :

( , )  F w C w X w X    [ ( )]   ( ) N k V R 

0

S

dV



(9)

( ) i

( ) i

Definitively, if the eigenvalue ( )  F :



    

    

    

[ ( )]   C w x w x   ( ) R

dV

N k V

( ) inf  F

min

0,



(10)





( )   

( ) w x w x 

dV

1

N

k N H k U  

(

)

w 

( ) # i

N k V

taken over all the admissible incremental periodic fluctuations ( ) w X  on the ( ) i V ensemble of unit cells, remains strictly positive, the composite material in the deformed configuration results to be stable. Thus, the microscopic critical load factor c  is taken as the first load parameter giving the minimum eigenvalue ( ( )) c   F equal to 0. 4. Macroscopic stability analysis In (Geymonat et al., 1993) has been demonstrated that the onset of primary instability characterized by long wavelength, in heterogeneous materials subjected to monotonic loading processes, can be detected by monitoring the classical macroscopic stability condition (coinciding with the strong ellipticity condition of the homogenized tangent moduli tensor). Contrarily to the microscopic stability analysis, the macroscopic one is performed on a unit cell assembly ( ) i U of a periodic nacre-like composite material. Specifically, the macroscopic stability condition is strictly related to the positive definiteness of the acoustic tensor 0 0 ( ) R ih ijhk j k Q C n n  n at the given stress state. The onset of macroscopic instability is found when the lowest eigenvalue ( ( ))   F , taken over all unit vectors m and n , become equal to 0:

0 || ||=|| || 1 ( ( )) min { ( R c m n     F

) C m n m n   

} 0, 

(11)

where c  is the macroscopic critical load factor and 0

R C is the nominal instantaneous moduli tensor defined by

means the following relation:

1

.

(12)

R

R

C

F F C



0

ijhk

jm ln imkn

det

F