PSI - Issue 71

A. Kumar et al. / Procedia Structural Integrity 71 (2025) 453–460

457

Shear strain

w 

(19)

23  =

23

In the direction- 2 , the corresponding out-of-plane shear modulus

 

G

=

23

(20)

23

23

cos   −  h 

 

t l   =    

l



G

G

(21)

23

s

h l



 

sin 1 1.5 

 +      

3. Finite Element Homogenization A computational homogenizing method employed the periodic boundary conditions (PBC) to derive corresponding out-of-plane continuum material properties. During homogenization, theRCEwas treated as a homogeneous and isotropic material (Kanit et al. (2006)), and the homogenized structure obtained will be homogeneous and orthotropic in nature. The average stress-strain (Xia et al. (2007)) Where ij  and ij  ,and V are the stress, strains, and volume at the local field of RCE. The predicted constitutive relationship for the homogeneous RCE will be the orthotropic material. whose stress strain relation is represented by     { } C   = (23) [ C ] denotes the stiffness matrix of the 3D material with as many as nine independent material constants. Displacement field can be defined as on the RCE’s boundary (24) Right side first part of the eq. (24) is the displacement field, and the following portion is the periodic function of RCE. If the indices A & B are denoting two opposing places on the RCE, then ( ) A B A B k k i i ik x x u u  − = − (25) FE models may use eq. (25) as a displacement constraint for periodic RCEs since the right part of the equation is consistent for any boundary surface (Kumar et al. (2024); Somireddy et al. (2018); Kumar et al. (2025)). 1 2 3 ( , , )d x x x V 1 ij V   =  ij V , 1 2 3 ( , , )d x x x V 1 ij ij V V   =  (22) , u x x x , 3 x u x x x  1 2 3 1 2 ( , ) ( , ) i i ik k  = +