PSI - Issue 37

Iulian Constantin Coropețchi et al. / Procedia Structural Integrity 37 (2022) 755 – 762 Coropetchi et al./ Structural Integrity Procedia 00 (2019) 000 – 000

757

3

For a linear elastic analysis, the stress distribution for each finite element is variable if we do not consider a reduced integration element or stress mediation on the whole element. For simplification, we considered that the stress is constant on each element. As not to differentiate between plane stress and plane strain we considered the Poisson ratio of each material as being 0. Finite element analyses were performed by using the PyAnsys software under an MIT License which is a collection of Python packages that enable the usage of Ansys products through Python (Kaszynski et al., 2021). The main property in our analysis is the effective stiffness of the RVE on X and Y direction. This property can be determined as the ratio between the mean stress and mean strain in the whole analysis domain.

1

1   NE i NE =

V



x   x

i xi

V

E

= =

(1)

1

_ eff x

V



i xi

V

1

i

=

1

NE

1   In relations (1) and (2) are presented the relations for determining the effective stiffness of the structure on X and Y directions. In these equations we have the following parameters: (2) _ eff y 1 1 i yi y i NE y   i yi i V V E V V   = = = =

• ̅ / ̅ y – the mean stress on X / Y direction • ̅ / ̅ y – the mean strain on X / Y direction • V – total volume of the analysis domain • NE – number of finite elements • V i – the volume of element i • σ xi / σ yi – stress on X / Y direction of element i • ε xi / ε yi – strain on X / Y direction of element i

• E eff_x / E eff_y – the effective stiffness of the microstructure in X / Y direction

The objective of our work was to design structures with 50% soft material and 50% hard material that have maximum values for the effective stiffness on X and Y direction reducing, at the same time, the anisotropy of the structure. Because we looked to maximize 2 properties ( E eff_x and E eff_y ) the optimization is considered multi-objective, so we formulated several objective functions that ensured the previously mentioned objective. The formulations that we used for the objective function of the optimization process are presented in the following equations:

1

max(

, E E

)

(3)

F

=



_1

_ eff x eff y _

OBJ

_ eff x E E E −

_ eff y

1

+

(4)

0

_ 2 eff x eff y F E E = + ( ) _ 3 _ _ OBJ eff x eff y F E E = + _ _ OBJ

   

   

_ eff y E E E E

_ , eff x eff y _

min

(5)



_ eff x

The first formulation of the objective function presented in equation (3) takes the maximum between the effective stiffness on X and Y direction and penalizes it with the inverse of their difference so that the penalization is proportional with the difference between them. In this equation, we used E 0 = 1 MPa. The penalization is introduced to filter out the distributions that do not ensure equal values for E eff_x and E eff_y . However, the first formulation has a great sensitivity