PSI - Issue 32

A.S. Shalimov et al. / Procedia Structural Integrity 32 (2021) 230–237 Shalimov A.S., Tashkinov M.A. / Structural Integrity Procedia 00 (2019) 000–000

232

3

variables (Berk, 1987; Cahn, 1965):

N

1

π ϕ

 

  

∑

( )

cos 2 

f x

c

i k x

=

+

(1)

i

i

A

N

1

i

=

Here, x is position of radius vector, A is size of RVE, N sets the number of harmonics, wave phases i ϕ are evenly distributed on the interval [ ) 0, 2 π , i k are wave directions. The coefficient i c is randomly selected. Different phases of a representative volume are determined by assigning points of space according to the following conditions: the point belongs to phase 1 if ( ) f x ξ < and phase 2 if ( ) f x ξ ≥ , where the parameter ξ defines the separation surface. The example of resulting open-cell geometry model is presented on Fig. 2a.

а

b

Fig. 2. a) RVE geometry with pores volume fraction 68%; b) Meshed RVE with pores volume fraction 68%

All RVEs had dimensions of 200x200x200nm and different volume fraction of porous phase. A discretization algorithm was applied to create a finite element mesh based on the obtained RVE geometry. Four-point tetrahedral elements are used for meshing. The finite element mesh for the geometry model in Figure 2b. Three mechanical models for the matrix phase were studied: elastic formulation, elastoplastic formulation and elastoplastic formulation with damage accumulation implemented as a property degradation procedure. For each geometry model, all three material models were applied. Gold was chosen as the matrix material with the following elastic constants: Young's modulus 48 E GPa = , Poisson's ratio 0.44 ν = (Bargmann et al., 2016; Li et al., 2019). For the elastoplastic formulation, with the elastic limit 96 y MPa σ = , the properties of the plastic zone were taken from the hardening curve graph presented in the paper (Li et al., 2019). For the elastoplastic formulation with damage, a customized UMAT subroutine was used. Using UMAT, the model incorporates some internal state variables ij D that characterize the development of microstructural damage: ( ) C D σ ε = . Their value increases when the degradation criterion is met, while the components of the stiffness matrix are underestimated. In this article the value 0.9 D = was chosen. The maximum principal stress failure criterion was taken as follows: max σ σ ≥ , where 190 max MPa σ = (Lee et al., 2007). All structures were subjected to tension or compression load applied in 2-direction (Y-axis) in displacements of 6 u nm = . The bottom face was rigidly fixed.