PSI - Issue 33

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

1108

6

5. Numerical results The nacre-like composite microstructure investigated is depicted in Fig.2. It is developed by arranging reinforcing platelets (black gray areas) and soft matrix (light gray areas) in a brick-and-mortar microstructure. L and H denotes the length and the high of the unit cell, respectively. The platelets volume fraction is defined by the relation:

(

)

L H

(13)

p p

,

v



f

(

)( L H H H   p i p

)

i

where L p = 20 mm denotes the platelets length and H p denotes the platelets height. H i is the thickness of the soft matrix interphase and the aspect ratio / p p w L H  is defined as the ratio between L p and H p . The material constitutive law assumed is the neo-Hookean hyperelastic model which is characterized by the following energy density function:

1 2

1 2

(14)

2 ln( ) , J

(tr( ) 3) C 

ln( ) J

W



 







where  and C denote the initial shear modulus and the right elastic Cauchy-Green tensor, respectively; while J and  denote the Jacobian of the transformation and the first Lamé parameter (governing the material compressibility), respectively. The incompressible behavior of the microconstituents is imposed by assuming 1000    . The shear modulus contrast / p m k    is defined by the ratio between the initial shear moduli of the stiff platelets p  and the soft matrix m  .

Fig. 2. Repetitive unit cell of a periodic nacre-like composite material in which, the black gray areas denote the stiff inclusions and the light gray areas denote the soft matrix interphase.

The nacre-like composite materials is subjected to a uniaxial loading process along the X 1 direction, in a large deformation framework and under plane strain conditions. The macroscopic deformation gradient tensor ( )  F is imposed by means of the periodic boundary conditions and it assumes the following form:

(15)

1

( )

,



F

1 e e 

2 e e 

  





1

2

where  denotes the macroscopic stretch ratio, 1 e is the unit basis vectors in the X 1 direction, and 2 e is the unit basis vectors in the X 2 direction. It is worth noting that, the imposition of ( )  F equal to 1   with reference to the loading direction X 2 was necessary to maintain the incompressibility condition during the uniaxial deformation process. The commercial software COMSOL MULTIPHISICS v5.6 has been employed to perform the numerical simulation, and two homemade subroutines were written in COMSOL Application Builder environment to perform