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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ℎ =2 2 ( ) − √( 2 − 2 ) =2 2 ( ℎ )−ℎ√( 2 −ℎ 2 )

(1)

(2) Where is the filament radius, and ,ℎ are the horizontal and vertical offsets from the filament centers to the bonding edges, respectively. The total porosity area within the RVE is computed by subtracting the areas of bonded and unbonded filament regions from the total RVE area : = − (4 + 2 ( ℎ + )) (3) where, = ℎ . Accordingly, the analytical expression for the porosity fraction is given by: = (4) This geometric model captures the influence of incomplete inter-filament bonding and void formation on the overall porosity distribution. It provides a framework for parametric studies focused on the effects of filament radius and bonding region dimensions, both horizontal and vertical, on the spatial arrangement of solid and void phases within the RVE. These analytical expressions are valid for the case where the filament radius r > b > h (Zone III, Fig. 3) . The analytical model adopts a 2D approximation by considering only cross-sectional areas rather than full volumes, justified by the invariance of the geometry along the third dimension, that of the filament deposition direction. Numerical homogenization of a Representative Volume Element (RVE) is performed using ABAQUS to evaluate porosity and inter- filament bonding effects on effective Young’s modulus, while establishin g a framework for future characterization of complex properties (e.g., Poisson’s ratio, bending behaviour, thermomechanical properties). The homogenized finite element results are inherently aligned with the Voigt upper bound due to enforced kinematic uniformity. These results are subsequently compared with predictions from the Mori – Tanaka method. 2.3.1 Numerical homogenization framework using FEA C3D8R elements (8-node linear hexahedral elements with reduced integration) were used to discretize the geometry. This element type offers a balance between computational cost and efficiency making it suitable for capturing the mechanical response of the RVE, including the effects of inter-filament bonding regions. Displacement constraints (u x =0, u y =0, u z =0) were enforced on x -, y -, and z -normal faces, respectively, with the bottom surface fixed in x - and y - directions aligned with the sample axes within the cartesian coordinate system (x, y, z). Kinematic coupling constraint enforced rigid interaction between the reference point (RP) and the top face, with a prescribed displacement Δ z = 0.1 mm was applied to the reference point RP to impose the axial strain = in the depth ( = 0.2 . The homogenized effective modulus was calculated following the approach by (Paux et al. 2023) in the z-direction and is given by Eq. (5) : ℎ = 1 ⟨ ⟩ (5) Where: ⟨. ⟩= 1 ∫(. ) ; is the local stress in the z-direction, is the imposed axial strain, and is the total volume of the RVE. This approach accounts for stress redistribution due to stiffness heterogeneity, void geometry, and interfacial degradation. 2.3. Homogenization models for porous microstructures