A multiscale-based methodology for the fatigue failure analysis of additively manufactured lattice structures applied to the four-point-bending test

A multiscale-based methodology for the fatigue failure analysis of additively manufactured lattice structures applied to the four-point-bending test Antonio Coluccia , Giorgio De Pasquale * Smart Structures and Systems Lab – Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy ARTICLE INFO Keywords: Additive manufacturing Lattice structures Homogenization RVE Lightweight ABSTRACT The utilization of lattice structures has become increasingly widespread across various technological applications, encompassing both functional structures and materials designed for structural integrity. These structures are extensively employed in biomedical engineering, aerospace, and automotive industries, as well as in heat exchangers and thermal management systems. Recent research on the integration of lattice structures into primary load-bearing components, particularly in aerostructures, is gaining prominence. Specifically, fatigue resistance is one of the most critical requirements for materials intended for structural applications, and this is particularly valid for aerostructures, where standards are more rigorous and severe. In this study, a fatigue failure analysis methodology specifically developed for lattice structures is applied to a 316L steel lattice beam subjected to four-point bending under fully reversed loading (R= − 1). The proposed methodology employes different techniques both related to metamaterials and conventional analysis of structures subjected to alternate cyclic loads: homogenization, a technique largely used to synthetize mechanical properties of multi-phase repetitive structures, allows the model to be computationally efficient. Classical multiaxial Sines fatigue criterion has been used to compute the equivalent stress state of the critical de-homogenized lattice cell. The investigation also includes experimental validation tests and a comparative analysis of the results with existing literature data. 1. Introduction In recent years, additively manufactured lattice structures have garnered significant attention across various technological research fields, particularly in structural engineering and functional materials. Owing to their outstanding specific mechanical properties—initially investigated by Ashby [1,2], several decades ago, prior to the advent of additive manufacturing (AM)—lattice structures hold great potential in lightweight engineering applications. These structures belong to the class of cellular solids, characterized by periodic unit cells and a high degree of porosity, often resembling the macroscopic architectures of natural materials such as wood and sponges. One of the most significant and frequently exploited properties of lattice structures is energy absorption [3–5]. Bio-inspired and naturemimicking lattice topologies are continuously studied to further understand and enhance their energy absorption potential [6]. Alongside conventional topologies, auxetic lattices and honeycomb structures exhibit remarkable performance in compression [7] and impact resistance [8]. Beyond topology, various lattice parameters play a crucial role in determining their mechanical properties. Factors such as cell size, strut/wall thickness [9], and the direction and intensity of the density gradient [10] significantly influence structural behavior. In recent years, advances in AM, particularly the powder bed fusion – laser beam with metals” (PBF-LB/M), have set new benchmarks for design freedom, enabling the fabrication of a wide variety of complex lattice topologies. As previously stated, lattice structures can be employed in a wide range of applications. In biomedical engineering, they are extensively utilized for the fabrication of custom implants and prostheses using AM, owing to their favorable mechanical properties and excellent osseointegration capabilities. In particular, the mechanical interactions between bone and lattice structures have been the focus of numerous investigations [11–13], alongside a substantial body of literature * Corresponding author. E-mail addresses: [email protected] (A. Coluccia), [email protected] (G. De Pasquale). Contents lists available at ScienceDirect International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue https://doi.org/10.1016/j.ijfatigue.2025.109122 Received 7 February 2025; Received in revised form 5 June 2025; Accepted 20 June 2025 International Journal of Fatigue 200 (2025) 109122 Available online 21 June 2025 0142-1123/© 2025 Published by Elsevier Ltd. dedicated to novel design approaches [14]. From a functional perspective, lattice structures are also highly effective in applications requiring thermal insulation [15] and heat conduction optimization [16]. Moreover, their use in structural applications is particularly relevant in aerospace engineering, where stringent lightweight requirements make them an attractive solution. In [17], both the energy absorption and thermal management properties of lattice structures are leveraged in the design of an aircraft anti-icing system. Additionally, periodic lattice-like unit cell structures are being explored in the space sector for various applications. Examples include their potential use in Whipple shield designs for spacecraft protection against micrometeoroid impacts [18] and the fabrication of cylindrical spacecraft panels [19], where fatigue behavior is a critical factor. Computational tools play a crucial role in lattice structure research, with the finite element method (FEM) being widely employed, often in conjunction with specifically developed analytical models, to simulate both the structural and functional behavior of these architectures. Lattice topologies exhibit a high degree of versatility, as their geometries can be optimized to achieve specific performance objectives. A significant portion of topological optimization studies focuses on enhancing energy absorption and impact resistance. In [20], an optimization strategy aimed at improving lightweight design and energy absorption is presented, integrating a neural network trained for the automated generation of FEM models, which are subsequently evaluated to identify the optimal design parameters. Computationally aided optimization processes can also be employed to enhance properties beyond energy absorption. For instance, in [21], optimization techniques are applied to improve stiffness and deformation modes, while in [22], thermal properties are the primary focus. Beyond optimization, computational tools serve various other purposes. In [23], computational design is utilized to develop finite mechanisms for shape-morphing lattice structures, while in [24], neural networks are employed to predict uncertainties in lattice strut diameters. A specific computational approach that relies on FEM for its implementation is homogenization. This technique, commonly used for multi-phase composite materials and adopted in this study, determines equivalent mechanical properties based on a representative volume element (RVE). Homogenization has been widely applied in fatigue behavior investigations [25,26]. Given the widespread implementation of lattice structures in various fields requiring structural integrity—such as aerospace, biomedical, and mechanical components—their fatigue behavior is a critical aspect that must be thoroughly considered. The body of literature on lattice fatigue is rapidly expanding, with numerous studies dedicated to this topic. The fatigue and quasi-static properties of PBF-LB/M Ti-6Al-4V lattice structures have been extensively investigated for both truss-based [27] and triply periodic minimal surface (TPMS) lattices [28]. Studies indicate that gradient lattices exhibit superior mechanical performance compared to uniform ones, and, in general, surface-based lattices outperform truss-based structures. Several studies have analyzed the influence of lattice characteristics on fatigue behavior. In [29], the effects of relative density and cell topology on fatigue performance were examined, alongside the impact of mean stress on the Goodman equation. The findings indicate that higher relative densities lead to improved fatigue resistance, and that the exponent in the fatigue curve equation remains independent of cell topology. The effects of relative density [30] and lateral loading [31] on fatigue performance have been investigated using gyroid lattices as benchmark structures. The influence of load direction on fatigue behavior is examined in [32], where tensile cyclic loading is applied to diamond lattices with different orientations. Fatigue crack propagation [33] and fracture toughness [34] in octet lattices have been analyzed under varying structural orientations, utilizing FEM to estimate stress distributions at fracture limits. Investigations have extended beyond metallic lattices to include stereolithography-manufactured polymeric octet lattices subjected to compressive fatigue loading, with a focus on the effects of structural orientation [35]. The response of Schwarz lattice structures to different cyclic fatigue loading conditions—considering both constant and variable stress ratios (R)—has been studied in [36]. Additionally, topological optimization strategies, such as the introduction of rounded fillets to enhance the fatigue performance of BCC lattices [37], have demonstrated improvements in fatigue strength. The literature on the fatigue properties of lattice structures also explores computational tools as a means to predict fatigue behavior. A parametric computational approach, which considers factors such as the number of unit cells, relative density, and topology, has been employed in [38] to predict fatigue resistance and lifespan. In [39], a numerical framework based on cascading failure of lattice layers was proposed to predict the fatigue life of lattice structures, demonstrating good agreement with experimental results. Additionally, a computational model utilizing the finite cell method was developed in [40] to estimate the fatigue life of asbuilt samples, with the goal of understanding how production-induced defects impact performance. Previous works by the authors have contributed to the analysis of fatigue failure in lattice structures by examining material properties [41–43] and developing numerical models for failure prediction [44–47] with extensions to applications in aerostructures [48]. In this work, a fatigue analysis has been conducted on a lattice structure using a specific methodology, which has previously shown promising results when applied to beams subjected to bending [26]. In this study, the methodology has been extended to analyze a fully reversed (R= − 1) four-point-bending lattice beam. The approach begins with the homogenization of the lattice unit cell, followed by simulation to assess the strain distribution in the most critical section of the sample. Finally, de-homogenization is applied to retrieve the original stress fields. The results obtained are validated through an experimental campaign, where both the sample and the mechanical setup have been specifically designed for this test. The lattice samples were fabricated using PBF-LB/M with 316L stainless steel. Fatigue life prediction is achieved by comparing the experimental S-N curve obtained from testing with scaled S-N curves derived from tests on traditional 316L steel samples. 2. Methodology The fatigue analysis methodology developed in this work combines several techniques, including FEM-based linear homogenization, dehomogenization, and the application of a multi-axial stress fatigue criterion. The method has been specifically designed for lattice structures and, more broadly, cellular solids. It offers notable advantages in terms of both computational efficiency and accuracy in identifying failure locations. A graphical representation of the steps involved is provided in Fig. 1. 2.1. Homogenization Once the geometry of the component to be analyzed is defined and the lattice structure composing the component is designed, the first step of the methodology, homogenization, can be applied. This process has been extensively described in a previous study [26]. Homogenization facilitates the transition from the mesoscale to the macroscale, with the initial focus on defining the general mechanical properties of the component or sample under fatigue loading, treating the lattice as a homogeneous medium. Additionally, homogenization improves computational efficiency. The mechanical properties derived from this step are then used to define the equivalent orthotropic material properties of the medium representing the lattice within the component or sample, including elastic moduli, shear moduli, and Poisson’s ratios. 2.2. Identification of the critical cell Cyclic loads applied to the component under consideration must be separated into a mean load and an alternating load. Prior to applying the A. Coluccia and G. De Pasquale International Journal of Fatigue 200 (2025) 109122 2 second step of the methodology, both the mean and alternating loads are treated as static load cases in a static FEM simulation. For sections of the FEM model where the lattice is represented by the equivalent bulk material, each element corresponds to a single lattice cell. The boundary conditions are determined by the specific cases being analyzed, as will be detailed in the following section, where the method will be applied to two distinct case studies. Upon completion of the simulation, the strain tensor norm, as defined by Eq. (1), can be calculated for each element of the equivalent material representing the lattice. ‖ ε ‖ =  ∑ i∑ j ε ij⋅ ε ij √(1) [ ε ] = ⎡ ⎣ ε xx ε xy ε xz ε yx ε yy ε yz ε zx ε zy ε zz ⎤ ⎦(2) The most critical cell, interpreted as the most critical element of the homogenized material, is determined based on the maximum value of ‖ ε ‖. The components of the strain tensor associated with the critical cell, as evaluated in expression (2), are retained, as they will be used in the de-homogenization step. It can be assumed that, considering this critical cell-element as the most loaded within the lattice structure—and thus the weakest link of the entire component—it will exhibit the lowest fatigue life. Consequently, this cell is considered the origin of fracture. As a result, the fatigue failure analysis can be confined to this smaller region of the component. 2.3. De-homogenization The next step involves the de-homogenization of the critical cell. This process begins with the strain tensor (2) corresponding to the critical cell. A static FEM model of the RVE with the actual lattice geometry is defined, in the same way the model used for homogenization is generated. The individual elements of the strain tensor are then used to define the strain field to be applied to the de-homogenized FEM model. The application of such strain components is performed though the same steps used for homogenization, therefore through the application of constraint equation, aimed at the reproduction of the single strain components; but, while for homogenization six simulations with unitary strain are performed, de-homogenization is based on a single simulation where the combination of tension/compression strains and shear strains is considered as the loading case for the model. This process allows the lattice cell to be subjected to the same loads as it would experience if part of a larger component under specific loading conditions (as applied in the second step), while focusing on a single cell rather than a more complex structure. In addition to facilitating the focus on the most critical part of the component, this approach offers significant advantages in terms of computational efficiency. The primary advantage of dehomogenization is the ability to retrieve the original stress state of the critical cell, starting from a homogenized medium. A flow chart following different steps of de-homogenization is provided in Fig. 2. Given the complex geometry of lattices, multi-axial stress components can be computed and used to apply various fatigue failure criteria. Regarding this process, literature offers different examples of dehomogenization applications: some are similar to what has been performed in this work [49], and others have the purpose of enhancing multiscale topology optimization results [50]. 2.4. Application of the Sines criterion for the equivalent stress evaluation In the previous investigation [26], the simple yet effective fatigue failure criterion employed was the Crossland criterion [51], which is widely used for high-cycle fatigue [52] and multi-axial stress fields. However, this criterion is limited in its ability to account for specific loading configurations, particularly when both mean and alternating stresses are present simultaneously, as highlighted in the literature [53]. Although no static preload is applied in the case study, the Sines criterion [54] has been considered to provide a more versatile tool for broader applications. This criterion can be summarized in expression (3):  J2,a √+ 3 √ σ D σ u σ H,m≤ τ D(3) where the square root of the stress tensor second invariant  J2,a √and the mean hydrostatic pressure σ H,m are defined as in expressions (4) and (5).  J2,a √= σ eq,a= ( σ 1,a− σ 2,a)2+( σ 1,a− σ 3,a)2+( σ 2,a− σ 3,a)2 2 √(4) σ H,m= σ eq,m= σ 1,m+ σ 2,m+ σ 3,m 3(5) σ u is the ultimate static stress of the material considered, while σ D and τ D are respectively the fully reversed axial and torsional fatigue limits. Subscript m refers to the mean state of stress of the load case and subscript a to the alternate state. The formulation of the Sines criterion shares many features with the Crossland criterion: both are based on the first and second invariants of the stress tensor, which are equivalent to the Von Mises and hydrostatic stresses, respectively. Additionally, both expressions account for both uniaxial and torsional fatigue limits. The main difference between the two criteria is that the Sines criterion incorporates the hydrostatic stress Fig. 1. Steps of the fatigue failure methodology developed. A. Coluccia and G. De Pasquale International Journal of Fatigue 200 (2025) 109122 3 evaluated at the mean stress. Beyond the application of the criterion itself, which determines whether the component fails under the considered loading conditions, the equivalent stresses for both the alternating and mean loads can be evaluated (4) and (5). The formulations of both criteria and the definition of equivalent stresses facilitate the reduction of a complex multi-axial stress field to a simpler uniaxial case. For instance, equivalent stresses can be used to define a working point on the Haigh diagram, as well as to calculate an alternating stress for the construction of a W¨ ohler diagram. 2.5. Comparison with bulk material and scaling coefficients Anticipating the results presented in the next section, the alternating stress levels derived from lattices in the W¨ ohler diagram are significantly lower compared to those obtained from the bulk material. In general, the entire bulk material diagram must be scaled to derive the lattice diagram, including the fatigue strength, as illustrated in Fig. 3. The correct fully reversed axial fatigue limit can be scaled using the following Eq. (6): σ corr D= σ DCLCFCS= σ DC(6) Scaling coefficients account for various effects: load case (CL), surface finish (CF), and scale effect (CS). Theory at the basis of these assumptions can be found in [55]. Therefore, the lattice can be treated as a general case of the bulk material used to manufacture the structure, in a similar way to treat lattice as a component rather than a material. Given the possibilities that lattices offer in term of design freedom and development of new topologies using the proposed methodology, obtaining information about the fatigue behavior of these structures, starting from bulk material data, can be a useful tool for early stages of design (prototypes and design of experiments) and in the optic of mapping the mechanical properties of different types of structures made of the same origin material, while limiting costs on testing and manufacturing. As mentioned, these effects would also influence the case of a static preload, and thus the mean stress. Consequently, the working envelope defined by the bulk material Haigh diagram is modified and constrained, as shown in Fig. 4. In the diagram, the curve is represented for N= 6•106, which is conventionally set as the infinite life threshold. The modification of the working envelope (green area) results from the change in the inclination of the Goodman line. Once the mean stress and alternating stress are defined, it is possible to determine the curve passing through the working point, with the scaling also applied. This curve provides information about the component’s life, defined by a specific number of cycles Ni, since, in the case of an applied static preload, the W¨ ohler diagram alone would not be sufficient, as it is evaluated for a fixed σ m. Fig. 2. Steps for the application of de-homogenization. Fig. 3. Representation of the W¨ ohler diagram scaling from bulk material to lattice. A. Coluccia and G. De Pasquale International Journal of Fatigue 200 (2025) 109122 4 3. Four-point-bending test The previously described analysis methodology has been implemented for the four-points-bending case study. Materials considered, sample design and the experimental campaign will be described in the following paragraphs. 3.1. Materials and sample configuration For the selection of the lattice topology for the manufactured samples, the all-face-centered-cubic (afcc) cell was chosen, as shown in Fig. 5. Lattice topologies within the family of face-centered cells typically exhibit high stiffness due to their stretching-dominated behavior. Specifically, the afcc geometry has a Maxwell number equal to 0, indicating a stretching-dominated structure with no self-stress states. It is important to note that the Maxwell number is a simple indicator, not a categorical threshold between the different ways loads are absorbed by the structure. Therefore, although the overall component may exhibit a high degree of stiffness, individual lattice struts may still exhibit a mixed behavior, transitioning between stretchingand bending-dominated elements. This results in a complex stress configuration that may require a more detailed analysis. The equivalent relative density of the samples with the prescribed dimensions is ρ =0,17. The material used for the production of the samplesis AISI 316L steel. Properties of this material have been considered on the basis of an investigation on AISI 316L manufactured via PBF-LB/M [56], with a Young’s modulus of E =200 GPa, a shear modulus of G =75.18 GPa, and a Poisson’s ratio of ν =0.33.The lattice structure section, along with the two specimen heads on the sides to facilitate load application (see Fig. 5a), were printed using the Concept Laser M2 Cusing machine, with the holes in the specimen heads oriented upwards. The printing parameters adopted for the fabrication of the samples via PBF-LB/M are based on the findings reported in [57], which presents a comprehensive investigation on the influence of printing parameters on the mechanical properties of 316L steel lattice structures. Trial production tests have been performed to optimize laser power and scanning speed in conjunction with lattice geometry, in order to enhance printability, reduce thermally induced stresses during fabrication, and mitigate residual stresses upon cooling. In the present work, the following printing parameters were employed: laser power of 300 W, scanning speed of 1 m/s, laser spot size of 90 μ m, layer height of 80 μ m, and hatch spacing of 90 μ m. In the following production steps, the samples underwent ultrasonic cleaning to remove any residual powder, with no heat treatment applied (given the optimized printing parameters). A total of 20 samples were fabricated. Process parameters have been set this way, in particular aiming for a not so high volumetric energy density (around 70 J/mm 3 ) in order to limit powder accumulation, being one of the main reasons for geometrical deviations and excessive roughness. An innovative sample configuration is presented for the four-pointbending test, as shown in Fig. 6. In this configuration, only the central portion of the beam—the region subjected to the highest (and constant along its length) bending moment—is made of lattice, while specimen heads present the holes where load transmission plugs are inserted (red features in Fig. 6b). The dimensions of the lattice section are 57 ×21 × 18 mm, with 19 ×7 ×6 cells in each direction. Bulk steel beams, called U-beams, are clamped to the specimen heads, allowing the sample to be elongated on both lateral sides until the two external points where the sample will be constrained (green features in Fig. 6b). These two additional U-beams are essential to allow the bending moment to increase from zero (at the external constraints) to its maximum, coinciding with the lattice section; this feature can be appreciated in the bending moment over length axis diagram of Fig. 6b. Another key feature of this sample configuration pertains to the load application method. To enable a fully reversed (R=− 1)loading condition, the sample must be capable of deflecting both upwards and downwards. The momentum diagram in Fig. 6b indicates in fact that both positive and negative momentum are obtainable. To achieve this, Fig. 4. Representation of the modified working envelope in the passage from bulk to lattice. Fig. 5. Afcc cell lattice dimensions and topology. A. Coluccia and G. De Pasquale International Journal of Fatigue 200 (2025) 109122 5 load transfer is accomplished through plugs rather than simple contact points. The two eyelets at the extremes of the U-beams facilitate the insertion of two plugs, which secure the sample in place; the distance between these two points is 194 mm, as depicted. The holes placed on the specimen heads are designed for the insertion of load-transmitting plugs. 3.2. Experimental setup For the experimental tests, the Baldwin SF-01-U vibrating system was employed. A custom mechanical test setup, shown in Fig. 7, was designed and fabricated specifically for the four-point-bending test. Two bulk steel beams are responsible for the load transmission and constraint systems. The upper beam, mounted on the fixed head of the machine, is connected to the external point of the sample, ensuring no displacement in the vertical direction. Similarly, the lower beam, connected to the vibrating head of the machine, allows the two internal points of the sample to be cyclically loaded by the machine’s eccentric mass. It is the lower head shown in the bottom part of Fig. 7 that vibrates while the upper head of device is completely fixed. No static preload was applied. The beams are connected to the sample via plugs inserted into the holes, as shown in Fig. 8, where the same setup is depicted with a missing support. This design allows the sample section to freely rotate. The plugs are secured in position by screws. A special feature has been applied to the eyelets where the external plugs are inserted. This modification addresses the fact that the machine cannot detect sample failure until the limit switch is activated by the moving head, while still allowing for fully reversed load tests. As highlighted in a detailed view in Fig. 8, the plug is positioned slightly inward from the extreme of the eyelets, enabling the plug to slip within the eyelets when sample collapse occurs, thereby activating the switch. 4. Results and discussion Results obtained from the application of the methodology described in chapter 2 to the proposed case study and its relative experimental campaign will be shown and commented on in the following chapter. Finally, a comparison with bulk material data from literature is performed and results discussed. 4.1. Application of the methodology to the four-points-bending case study The samples for the four-point bending tests feature a single configuration, with a uniform lattice area, as described in the previous section. Homogenization has been applied to the afcc cell, as shown in Fig. 5. Based on a sensitivity analysis, the mesh size used for the homogenization simulations is set to 0.12 mm, using linear order tetrahedral elements. The sensitivity analysis has been based on the computation of the maximum Von Mises stress of the unit cell and using explorative loading conditions (0,001 strain in the x direction), while decreasing the mesh size, until the stabilization of the stress value occurs on a level that does not present excessive variation. Going below 0,12 mm results in an excessive deviation, while going beyond the chosen value defines a downward trend. Results can be appreciated in Figs. 9 and 10, where the maximum Von Mises stress over element size and relative contours are shown. Considering the results of the sensitivity analysis, linear order elements are sufficient for this analysis, also considering that simulations with element size equal to 0,1 mm and Fig. 6. Picture of the additively manufactured lattice section of the sample with specimen heads to let plugs to be inserted for the load transmission (a) and picture of the whole sample with mounted U-beams for the four-point-bending tests (b). Based on the picture, the evolution of the minimum and maximum bending moments is shown along the x axis (being the specimen length axis). The green features represent the fixed points, while the red features indicate the loaded points. Dimensions are not to scale and are given in millimeters. A. Coluccia and G. De Pasquale International Journal of Fatigue 200 (2025) 109122 6 0,12 mm can last up to 2–3 h also depending on the boundary conditions applied (which in these cases consist of a complex system of constraint equations). The final value chosen for the simulations could be found at the beginning of the asymptote, allowing for an efficient, yet accurate analysis. The equivalent orthotropic material properties are presented in Table 1. The material properties obtained from homogenization are identical for each direction considered, due to the shape of the afcc cell, which has three planes of symmetry. Once the medium material is characterized, simulations involving the features of the entire sample can be performed. A visual representation of the model is shown in Fig. 11. The developed model is defined in a three-dimensional space, using the same exact dimensions previously described and visible in Fig. 6. The model has been generated using no contacts, therefore in one single part. Two different material models have been used: linear elastic isotropic for the bulk steel (blue in Fig. 11), with the properties mentioned in the previous chapter, and linear elastic orthotropic for the lattice (purple in Fig. 11), with the properties shown in Table 1. Regarding the different material models used for the description of the mechanical properties, it can be noted how homogenization allows the model to identify the stiffness mismatch occurring at the bulk-lattice interface. In fact, the model described includes traditional elements (discretizing the massive ends) and homogenized elements (for the lattice region), which replicates the effective stiffness of each portion. Then, the stiffness transition at the interface is properly represented. In the next chapter it will be reported the local stress concentration correlated to the density transition. The mesh size of the homogenized section is set to 3 mm (equal to the cell edge length), while a 1.5 mm mesh size is applied to the rest of the model. Elements used are linear order blocks (and prisms for round zones). The boundary conditions (BCs) applied to the sample simulation are illustrated in the same figure. The external points, which fix the sample in place, are subjected to two different constraints. This design allows the sample to bend while minimizing any additional axial load that could arise during the bending process, but not due to bending itself. Given the design of both the sample and the grips, as discussed in the previous chapter, the surplus axial load is minimized. This feature is also present in the experimental setup, where external plugs are free to move in the horizontal direction in the eyelets, but not in the vertical one. External contact points are enabled to slide on the plugs, permitting the rotation of the corresponding section, without inducing additional axial solicitation. All three translational degrees of freedom (DOFs) are constrained for the left external point, ensuring that the model is completely fixed in the vertical plane and preventing any unconstrained translation in the depth direction, thereby ensuring the convergence of the model. On the other hand, only the Y-direction translational DOF is constrained for the right external point. This feature allows the sample to bend freely. All the BCs described up to now and visible in the front view of Fig. 11, are applied in all the elements present in the depth dimension of the model. Two internal points, responsible for load transmission to the sample, are constrained using rigid elements that simulate a plug. Nodes located on the internal cylindrical surface of the hole are connected to a dependent node using rigid elements (RBE2), where the load is applied. The loads applied in this case study are based on the actual loads used in the experimental campaign, which range from 200 to 450 N, to generate a proper S-N diagram. Fig. 7. Mechanical setup designed for the four-points-bending fatigue test; upper head is fixed, while lower head is connected to the vibrating system; supports connecting both moving and fixed heads to the sample are indicated in the picture, as well as the load cell used to measure the load. Fig. 8. Details of the mechanical setup, highlighting the plugs system for both load transmission and fixing the sample. A. Coluccia and G. De Pasquale International Journal of Fatigue 200 (2025) 109122 7 At this stage, the critical element of the lattice section, corresponding to the critical lattice cell, is identified based on the maximum strain tensor norm criterion (1). The strain tensor norm contour for the 200 N load case is shown in Fig. 12a. The strain tensor norm contour exhibits a perfectly symmetrical distribution relative to the horizontal plane, with the lowest values at the center of the section and higher norms at the top and bottom extremes of the sample. This result validates that the applied BCs effectively introduce the bending load without inducing any axial load. Regardless of the applied load in the homogenized material model, the Fig. 9. Maximum Von Mises stress over element size diagram for the sensitivity analysis. Fig. 10. Von Mises Stress contours of simulations performed for the sensitivity analysis; different element size have been used, respectively (a) 0,1 mm, (b) 0,12 mm, (c) 0,14 mm and (d) 0,16 mm. Table 1 Homogenized material properties extracted from homogenization applied to the afcc cell. E x =E y =E z [MPa] 7115 G xy =G xz =G yz [MPa] 3950 ν xy = ν xz = ν yz 0,3 A. Coluccia and G. De Pasquale International Journal of Fatigue 200 (2025) 109122 8 strain norm distribution remains constant. Apart from the lateral extremes, this distribution is essentially uniform along the entire length of the lattice section, considering the limitations imposed by the resolution of the contour representation. The contour is replotted in Fig. 12b, focusing solely on the elements of the top and bottom lattice surfaces, to illustrate how the norm is distributed in these critical regions. The lattice cell exhibiting the maximum strain tensor norm is shown in Fig. 13. This cell is the fourth one, counting from the beginning of the section in the length direction, and is positioned centrally in the width direction. Consequently, based on this result, it is assumed that this point is where the failure originates. A comparison with the experimental data will be discussed later. In this case, the Sines approach has been employed, as opposed to the previous case study. After performing the de-homogenization, the Sines equivalent alternate stress (5) is computed. The equivalent mean stress is zero, given that no static load has been applied. The equivalent stress for the critical cell has been computed under different load cases. Fig. 14 shows the stress distribution for the 200 N load case. The distribution reveals areas of the cell that are almost unloaded, while other areas appear to bear most of the load. Stress concentrations are evident at both lateral extremes of the Fig. 11. FEM model with the homogenized lattice (purple section) and bulk steel (blue sections). BCs are also detailed in the picture: left external constrained point presents locked DOFs in the x,y and z directions, right external point only present two locked DOFs in the y and z directions; finally, the load transition internal holes are related to a dependent node (where the load is actually applied) via constrain equations linking both x and y DOFs. Fig. 12. Strain tensor norm contour for the whole lattice section of the four-points-bending model (a), and the replotted contour just for the top lattice surface where the maximum strain tensor norm occurs (b), considering 200 N bending load. A. Coluccia and G. De Pasquale International Journal of Fatigue 200 (2025) 109122 9 structures: Preliminary considerations. Mater. Des. Process. Commun. 2021;3(4): 2–7. https://doi.org/10.1002/mdp2.249. [41] De Pasquale G, Luceri F, Riccio M. Experimental Characterization of SLM and EBM Cubic Lattice Structures for Lightweight applications. Exp. Mech. Apr. 2019;59(4): 469–82. https://doi.org/10.1007/s11340-019-00481-8. [42] De Pasquale G, Luceri F, Riccio M. Experimental evaluation of selective laser melting process for optimized lattice structures. Proc Inst Mech Eng, Part E Aug. 2019;233(4):763–75. https://doi.org/10.1177/0954408918803194. [43] De Pasquale G, Luceri F. 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