PSI - Issue 61

Ahmet Arda Akay et al. / Procedia Structural Integrity 61 (2024) 138–147 A.A. Akay et al. / Structural Integrity Procedia 00 (2024) 000–000

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• STEP 4 : Using the opening displacements [ u r ] calculated in STEP 3, the current interface condition of each interface node pair is determined. – STEP 4.1 : Depending on the condition of the node pair determined in STEP 4, the interface stress σ int andmoduli k int = σ int / [ u r ] are calculated. – STEP 4.2 : For each node pair at the interface, the check for equilibrium state is achieved by using the relation given below.

[ u r ] − [ u r - hist ] [ u r ]

e =

> e 0 ,

(13)

where e 0 and u r - hist are defined as the reference error tolerance for iterations to stop and the di ff erence of normal displacements in the previous iteration, respectively. – STEP4.3 : Even if the equilibrium conditions given in Equation (13) are satisfied for one single node pair, the solution returns to STEP 3. If none of the interface node pairs satisfies Equation (13) or the number of iterations reaches the designated number of maximum iterations N , the solution will continue with STEP 2.

3. Numerical Examples

The BEM-based homogenization of a heterogeneous RVE with a bilinear interface model is considered for two studies from the literature. The results of the proposed BEM-based homogenization approach are compared with the literature in the following subsections.

3.1. First Model Problem

The study presented by Wu et al. (2020) is a boundary element method-based study, in which the interface model presented in Figure 2 is used. The investigated RVE is given in Figure 4. Three models having inclusion diameters ( d ) of 40 µ m, 60 µ m, and 80 µ m are analyzed. For all the models, the volume area fraction of the inclusions ( ϕ f ) is kept constant at 0.2, so the edge length ( a ) of square RVE changes for each model. Each edge of the square boundary of the RVE is divided into 10 elements and the interface is divided into 60 boundary elements. The material and the interface properties used in the study are given in Table 1.

Table 1: Material and interface properties

Matrix

Inclusion

Interface

E 0 [MPa]

ν 0 [-]

E 1 [MPa]

ν 1 [-]

σ max [MPa] k σ [MPa /µ m] ˜ k σ [MPa /µ m]

1

0.4

150

0.3

0.02

1

0.02

Fig. 4: Investigated 2-phase representative volume element

In this study, the RVE given in Figure 4 is investigated under equibiaxial macroscopic strain loading for three di ff er ent diameters. The e ff ects of three classical boundary conditions (uniform traction (TBC), linear displacement (UBC), and periodic displacement (PBC)) on the homogenized stress response are studied. In Figure 5, the macroscopic stress versus macroscopic strain (black curves and y-axis on the left) and the iteration number versus macroscopic strain (red curves and y-axis on the right) curves obtained for the three di ff erent diameters of a representative volume element are presented. As seen from the curves, the results of three di ff erent boundary conditions obtained in this study are con sistent with the results presented in the literature in which only the periodic boundary condition (PBC) is considered. It is also observed that the highest iteration number is reached at the load step, where the softening starts.