PSI - Issue 28

Saiaf Bin Rayhan et al. / Procedia Structural Integrity 28 (2020) 1892–1900 Author name / Structural Integrity Procedia 00 (2019) 000–000

1894

3

2. Finite Element Modeling Technique For a periodic unit cell, 6 load cases are considered: 3 tensile tests (X, Y, and Z) and 3 shear tests (XY, YZ, and ZX). A corresponding macroscopic strain is applied in each case, and reaction forces in the boundary faces of the RVE are used to assemble the stiffness matrix. Then the engineering constants are extracted as follows.

0 0 0 0 0 0 0 0 0

11 D D D D D D D D D 12 21 22 0 0 0 31 32

x    z y                          xy yz zx         

x    z y                 xy yz zx       

13

23

(1)

33

0 0

D

44

0 0 0 0

0

D

55

0 0 0 0 0

D

66

and all other strains are set to zero, the first column of the stiffness

If the strain in the X-direction is fixed to

0.01

x  

matrix is obtained as follows.

11       21 D D D

x    y        

31 1 0 0.01                          0 0 z xy yz zx   

(2)

Assuming the RVE occupies the volume 

, on the faces normal to the X-axis, enforce as follows.

y      

0,

0, L L

0,

L

x

z

 u L y z u y z L u L y z u y z u L y z u y z             , , , , , , 0, , 0, , 0, , x x x y x y

x

(3)

z

x

z

Similarly, on the faces normal to the Y-axis, enforce as follows.             , , ,0, , , ,0, , , ,0, x y x y y y z y z u x L z u x z u x L z u x z u x L z u x z   

(4)

Besides, for Z-Axis, enforce as follows.

 z u x y L u x y u x y L u x y u x y L u x y         , , , , , , x z x y z y z z

, ,0 , ,0 , ,0

(5)

In addition to these periodicity conditions, rigid body motions must also be prevented. This is done by enforcing

( z u a po with x u a po with y u a po with z int int int ( ( x y

0) 0 0) 0 0) 0

     

(6)

To compute macroscopic stresses, the forces on the top faces are integrated. Let us consider x  . The force in the X-

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