PSI - Issue 33

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

1106

4

deformation gradient tensor ( ) F X and it is called first-Piola Kirchhoff stress tensor. The stress tensor R T and the so called tangent moduli tensor R C can be obtained as the first and the second derivative of ( ) W F :

2

W W  

.

(1)

R

,

T

C





R

2

F

F





The deformation process is driven by a time-like parameter t ≥ 0 which increases monotonically. By assuming a small value of t the incremental form of the variables can be considered as rate quantities and thus the following linear relation is adopted to evaluate the rate of the first-Piola Kirchhoff stress tensor R  T :

R    T C F

(2)

[ ], R

where  F denotes the rate of the deformation gradient tensor and

R C is the fourth-order tensor of tangent moduli

which possess the major symmetry R ijhk hkij C C  . Assuming that the nacre-like material is subjected to a quasi-static deformation process, the microscopic stress field ( ) R T X is divergence-free and the thus equation of motion, with reference to the undeformed configuration, is defined as: R

(3)

Div

0.

R  T

The averaging relations coupling the microscopic and the macroscopic scales are reported in the following:

1

1





(4)

, X F (i) dS

T

t

 x n

dS







( ) i

R

R

(i)

V

V

( ) i

( ) i

V

V





( ) i

( ) i

where R R i  t T n is the traction vector of the first Piola-Kirchhoff stress tensor acting on the RVE boundaries in the undeformed configuration   i V  ,  denotes the tensor product and ( ) i n denotes the outward normal at X belonging to the RVE boundary in the undeformed configuration   i V  . The periodic distribution of the geometrical and material properties assumed requires a periodic distribution of the stress and strain fields. This corresponds to impose a periodic deformations constraint on (i) V  : ( )

(5)

(   + x x = F X X  

),



that, from the viewpoint of equilibrium, it leads to assuming the condition of antiperiodic tractions on (i) V  :

(6)

(

) = ( 

) . 

T n

T n



( )

( )

R i

R i

The boundary value problem associated to the microstructural RVE is written, as in the following equation, in a variational form and then solved by means of the finite element method:

( ) i        T w w 0 R dV

(7)

1 H V

( ) # ( ), i

V

( ) i

where 1 ( ) # ( ) i H V is the first order Hilbert space of vector-valued functions periodic over all possible unit cell assembly ( ) # ( ) # N i i V k U  with [0, ] N N k k  and N = 2 for bidimensional problems, w is the fluctuation field at a given macroscopic deformation gradient F . As reported in (Greco and Luciano, 2011), by considering the variational microscopic incremental equilibrium problem induced by F  , the macroscopic constitutive response can be determined in terms of the homogenized tangent moduli tensor ( ) R C F written in component form as in the following: