PSI - Issue 80

R. Salem et al. / Procedia Structural Integrity 80 (2026) 256–268 Rania Salem/ Structural Integrity Procedia 00 (2019) 000 – 000

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2.3.2 Voigt upper bound The Voigt model, shown in green Fig. 6 , assumes uniform strain across all constituents and thus represents the theoretical upper bound for effective stiffness. The corresponding modulus is computed using a rule of mixtures: = + ℎ ℎ + + (6) Where : = 4 ; ℎ = ℎ ; = and = The model systematically overestimates stiffness, particularly at higher porosity levels, due to neglected local strain amplification near pores (Paux et al., 2023). The FE h omogenized Young’s modulus curve coincides with the Voigt upper bound in linear elastic simulations because both approaches enforce uniform strain boundary conditions. Conversely, the Reuss lower bound remains fundamentally invalid for porous systems since voids cannot sustain stress. 2.3.3 Mori – Tanaka Homogenization Scheme The Mori – Tanaka model, represented by the blue dashed line in Fig. 6 , is a mean-field homogenization technique widely employed for estimating the effective elastic properties of heterogeneous media containing inclusions or pores, as described by (Mori and Tanaka 1973). The model assumes ellipsoidal, non-interacting inclusions dispersed in a continuous matrix and incorporates the Eshelby tensor to account for the interaction between the inclu sion and matrix phases. The effective Young’s modulus predicted by Mori – Tanaka is given by: = (1+∑ ∈ {ℎ , , } − + ( − ) ) (7) where: represents the effective property of the composite material calculated using the Mori-Tanaka method, is the filament ABS material Young’s modulus, is Young ’ s modulus of the inclusions (where denotes different phases such as horizontal bonding region ( hbr ), vertical bonding region ( vbr ), and porosity ( p )), is the volume fraction of each inclusion, and is the Eshelby tensor or concentration factor accounting for the shape and interaction effects of the inclusions on the overall structure behavior. Here = 1 3 for spherical inclusions, as derived by (Eshelby 1957). The accuracy of Eq. (7) hinges critically on the assumptions of dilute, non-interacting inclusions within a dominant matrix. In the studied system, the layered structure (HBR/VBR) combined with porosity creates a complex, interconnected microstructure violating these assumptions at higher porosity levels. Consequently, Fig. 6 shows increasing deviation between Mori-Tanaka (blue dashed line) and FE results as porosity rises. These analyses enable to evaluate the limitations of the Mori-Tanaka approach in this particular context of this periodic four-phase structure. Fig. 6 presents four interfacial bonding scenarios governing the effective Young's modulus of porous layered structures. Case 1 ( Fig. 6a ): Ehbr=Evbr=1500 MPa) exhibits the lowest modulus with high porosity sensitivity, indicating weak interfaces limit load transfer. Case 2 ( Fig. 6b ): Ehbr = Evbr = 2500 MPa) achieves maximum stiffness through strong interfacial bonding. Case 3 ( Fig. 6c ): Ehbr = 1500 MPa, Evbr = 2500 MPa) shows intermediate values where vertical bonding dominates under deposition loading. Case 4 ( Fig. 6d ): Ehbr = 2500 MPa, Evbr = 1500 MPa) demonstrates comparable stiffness elevation via horizontal bonding. Overall, these results highlight that within the narrow geometric parameter space of Zone III (illustrated in Fig. 3 ), a remarkably wide range of Young's moduli can be achieved. This significant variability underscores the critical influence of specific geometric parameters on the effective stiffness in homogenization modeling.