PSI - Issue 41

Giorgio De Pasquale et al. / Procedia Structural Integrity 41 (2022) 535–543 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

538

4

Table 1. Summary of boundary conditions applied on the RVE for the calculation of the stiffness matrix components: Ui terms represents the displacement degrees of freedom (DOF) of the nodes of the RVE in the system ( x 1 , x 2 , x 3 ) , “0” = constrained DOF, “/” = free DOF. 1 ̅ = 1 2 ̅ = 1 3 ̅ = 1 4 ̅ = 1 5 ̅̅̅ = 1 6 ̅̅̅ = 1 U 1 U 2 U 3 U 1 U 2 U 3 U 1 U 2 U 3 U 1 U 2 U 3 U 1 U 2 U 3 U 1 U 2 U 3 = − − 1 / / 0 / / 0 / / 0 / / / 0 − 1 / − 1 0 = 1 / / 0 / / 0 / / 0 / / / 0 1 / 1 0 = − / 0 / / − 2 / / 0 / 0 / − 2 / 0 / − 2 / 0 = / 0 / / 2 / / 0 / 0 / 2 / 0 / 2 / 0 = − / / 0 / / 0 / / − 3 0 − 3 / − 3 0 / / / 0 = / / 0 / / 0 / / 3 0 3 / 3 0 / / / 0 For each static analysis, one strain component is set to the unity, with reference to the imposed boundary displacement conditions. The strain components of the RVE are described as follows: RVE face 1 st analysis 2 nd analysis 3 rd analysis 4 th analysis 5 th analysis 6 th analysis

{ 1 ( 1 , 2 , 3 ) − 1 (− 1 , 2 , 3 ) = 2 1 10 1 2 ( 1 , 2 , 3 ) − 2 (− 1 , 2 , 3 ) = 2 1 20 1 3 ( 1 , 2 , 3 ) − 3 (− 1 , 2 , 3 ) = 2 1 30 1 { 1 ( 1 , 2 , 3 ) − 1 ( 1 , − 2 , 3 ) = 2 2 10 2 2 ( 1 , 2 , 3 ) − 2 ( 1 , − 2 , 3 ) = 2 2 20 2 3 ( 1 , 2 , 3 ) − 3 ( 1 , − 2 , 3 ) = 2 2 30 2 { 1 ( 1 , 2 , 3 ) − 1 ( 1 , 2 , − 3 ) = 2 3 10 3 2 ( , 2 , 3 ) − 2 ( 1 , 2 , − 3 ) = 2 3 20 3 3 ( 1 , 2 , 3 ) − 3 ( 1 , 2 , − 3 ) = 2 3 30 3

(5)

Every edge belongs at the same time to two faces, and since only one constraint equation can be established for a geometrical feature, edges need their own set of conditions, defined by Eqs. (6): { ( 1 , 2 , 3 ) − (− 1 , − 2 , 3 ) = 2 1 0 1 + 2 2 0 2 ( 1 , − 2 , 3 ) − (− 1 , 2 , 3 ) = 2 1 0 1 − 2 2 0 2 ℎ = 1,2,3 { ( 1 , 2 , 3 ) − (− 1 , 2 , − 3 ) = 2 1 0 1 + 2 3 0 3 ( 1 , 2 , − 3 ) − (− 1 , 2 , 3 ) = 2 1 0 1 − 2 3 0 3 ℎ = 1,2,3 { ( 1 , 2 , 3 ) − ( 1 , − 2 , − 3 ) = 2 2 0 2 + 2 3 0 3 ( 1 , 2 , − 3 ) − ( 1 , − 2 , 3 ) = 22 0 2 − 2 3 0 3 ℎ = 1,2,3 Similar consideration is valid for the corners, which need their own conditions: { ( 1 , 2 , 3 ) − (− 1 , − 2 , − 3 ) = 2 1 0 1 + 2 2 0 2 + 2 3 0 3 ( 1 , 2 , − 3 ) − (− 1 , − 2 , 3 ) = 2 1 0 1 + 2 2 0 2 − 2 3 0 3 (− 1 , 2 , 3 ) − ( 1 , − 2 , − 3 ) = −2 1 0 1 + 2 2 0 2 + 2 3 0 3 ( 1 , − 2 , 3 ) − (− 1 , 2 , − 3 ) = 2 1 0 1 − 2 2 0 2 + 2 3 0 3 ℎ = 1,2,3 Equations (4), (5) and (6) gives all the constraints necessary to the static analyses representing the homogenization process. From the results, the components of the average stress field ̅̅̅ are obtained and by means (6) (7)