PSI - Issue 80

Thi Ngoc Diep Tran et al. / Procedia Structural Integrity 80 (2026) 378–391 Thi Ngoc Diep Tran/ Structural Integrity Procedia 00 (2019) 000–000

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The Johnson-Holmquist II (JH2) constitutive model was used to characterize the brittle behaviour of Alumina material (Johnson and Holmquist (1994)). In this model, the strength and pressure are normalized by the strength components of the Hugoniot Elastic Limit (HEL) and can be expressed as follows: ∗ = � ∗ − � � ∗ − � ∗ �, (9) � ∗ = ( ∗ + ∗ ) � (1 + ∗ ), (10) � ∗ = ( ∗ ) � (1 + ∗ ), (11) where ∗ , � ∗ , and � ∗ are the normalized equivalent stress, normalized intact equivalent stress, and normalized fracture stress, respectively. D is the damage variable, ∗ is the strain rate, are material parameters. ∗ and ∗ are the normalized pressure and maximum tensile pressure, respectively. The material parameters of Alumina 99.5% used in this study were adopted from Anderson and Morris (1992) and given in Table 1. Table 1. JH-2 model parameters of Alumina 99.5% / ( � ) G (GPa) A N B M �� �� HEL (GPa) ��� (GPa) K1 (GPa) K2 (GPa) K3 (GPa) D1 D2 3890 190 0.88 0.64 0.28 0.6 0.2 6.57 1.46 1.0 231 -160 2774 0.01 0.7

3. Structural morphology 3.1. Calculating TPMS cross-sectional area

This section provides a numerical algorithm based on an example of TPMS Primitive to extract the shape of unit cell’s cross-section and determine its surface area. The TPMS-based Primitive unit cell was generated using Eq. (1) in a three-dimensional 64×64×64 mesh grid. Although the cross-sectional geometry is the same in all cutting directions perpendicular to the coordinate axes, it is appropriate to examine the cross-sections orthogonal to the vertical loading direction. For this purpose, the unit cell was divided into 64 parallel planar sections oriented along the vertical axis. Fig. 3 shows three typical cross-section examples of a TPMS Primitive unit cell.

Fig. 3 Typical cross-sectional shapes of a TPMS Primitive unit cell.

Points with identical vertical coordinates were sorted and grouped to constitute individual cross-sections. Without any arrangement, the points generated by the mathematical equation were misleadingly connected, resulting in an incorrect area as shown in Fig. 4a. To establish a proper sequence for the points on the boundary of the cross-section, the distances between each pair of points were calculated to identify the closest neighbour. Depending on the cross sectional geometry, the next point on the boundary may not always be the closest one, and the path must somehow get into the inner closed curve since each point can only be visited once (see Fig. 4b). This problem was solved by considering a quarter of the symmetrical cross-section where 0.5ℎ ≤ and 0.5ℎ ≤ , with ℎ×ℎ×ℎ as the sample size. However, the points in the corners were not included in this quarter, leading to incorrect pathfinding as