<figure><img src="image_1.png" /></figure>

# Microstructure and compressive behaviour of PLA/PHA-wood lattice structures processed using additive manufacturing

Tao Liu<sup>a, b</sup>, Ji-hong Zhu<sup>a</sup>, Weihong Zhang<sup>a</sup>, Sofiane Belhabib<sup>c</sup>, Sofiane Guessasma<sup>a, *</sup>

## ARTICLE INFO

### Keywords:

- Lattice structures

- Fused filament fabrication

- Compression behaviour

- Microstructure

- X-ray micro-tomography

## ABSTRACT

This study investigates the microstructure and compressive behaviour of PLA/PHA-wood lattice structures manufactured using fused filament fabrication (FFF). The primary objective is to evaluate the effects of defects, such as porosity and surface roughness, on the mechanical properties of these lattice structures. X-ray micro-tomography (XRT) and finite element analysis are employed to compare CAD models with real lattice structures, offering insights into defect-induced performance deviations. The novel approach integrates experimental and numerical methods to better understand damage accumulation in porous structures. The methodology includes uniaxial compression testing and image processing for microstructural characterization. Results reveal significant differences in relative density and stress concentration due to manufacturing defects. In particular, 3D printed lattice structures display typical cellular material behaviour with a primary bending deformation mechanism. X-ray tomography reveals that process-induced porosity generate stress heterogeneity, which is not captured from CAD-based model resulting in an overestimation of the stiffness. The study concludes that accounting for these defects can be quantified by shifting the target relative density to compensate lack of performance in 3D-printed lattice materials.

## 1. Introduction

Lattice structures are defined in the literature as objects that are periodic in nature, continuously repeating unit cells that interconnect in three dimensions [1,2]. In terms of cellular architecture, three categories can be defined for lattice structures: randomised, periodic, and pseudo-periodic [3]. In every category, the thickness of the cell wall may remain consistent or change, resulting in either homogeneous or heterogeneous lattice structures.

Lattice structures have a broad range of applications and are highly regarded as lightweight structures [4] in various engineering scenarios, including engine brackets, racing cars, and structural components of aerospace [5-7]. Lattice structures are also known as good shock absorbers and thermal barriers because of the amount of distributed voids [8,9]. They are also considered for biomedical applications [10], as well as metamaterials [11,12].

The mechanical properties of lattice structures, including stiffness, strength, and energy absorption, have been explored in various studies, with much of the research focusing on the relationship between the

lattice geometry and its mechanical performance [13,14]. One of the most prominent contributions to this area is the work by Gibson and Ashby, who established scaling laws for cellular materials that relate their mechanical properties to relative density [14]. Their models demonstrate that lattice structures' stiffness and strength can be scaled with the cube of the relative density, assuming bending-dominated behaviour in open-cell structures. This scaling law has become a cornerstone in the design of lightweight structures, guiding the optimization of mechanical performance by varying density and geometry. Fan et al. [15] employed the equivalent continuum method to investigate the mechanical properties of lattice truss materials. They demonstrated that complete stress-strain curves could be modelled using an incremental approach, wherein the nonlinear behaviour is approximated through a series of linear segments. Xue et al. [16] explored the mechanical performance of aluminium-based auxetic lattice structures, showing that polymer filler reinforcement significantly enhances stiffness, compressive strength, and energy absorption capacity. Liu et al. [13] examined the impact of nonuniform hollow struts on the mechanical properties of lattice structures designed using a deep learning

<footnote><sup>a</sup> INRAE, Research Unit BIA UR1268, Rue Geraudiere, F-44316 Nantes, France</footnote>

<footnote><sup>b</sup> Laboratory of Engineering Simulation & Aerospace Computing-ESAC, Northwestern Polytechnical University, 710072 XiAn Shaanxi, China</footnote>

<footnote><sup>c</sup> Nantes Université, CNRS, GEPEA, UMR 6144 F-44000 Nantes, France</footnote>

<footnote>* Corresponding author.
E-mail address: sofiane.guessasma@inrae.fr (S. Guessasma).</footnote>
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** (a) Morphology of as-received PLA/PHA-wood determined by X-ray micro-tomography, (b) Overview of the printing process and the final rendering of the 3D printed lattice structures.</figcaption></figure>
<table><caption>**Table 1**

Printing parameters used in this study.</caption><thead><tr><th>Support</th><th></th><th>Printing</th><th></th><th>Extruder</th><th></th></tr></thead><tbody><tr><td>Support density</td><td>0.2</td><td>Extruder temperature</td><td>210 °C</td><td>Filament diameter</td><td>1.75 mm</td></tr><tr><td>Support margin</td><td>0.5 mm</td><td>Bed temperature</td><td>25°</td><td>Retraction distance</td><td>1.3 mm</td></tr><tr><td>Support spacing</td><td>0.4 mm</td><td>Travel speed</td><td>150 mm/s</td><td>Retraction speed</td><td>25 mm/s</td></tr><tr><td>Support angle</td><td>68°</td><td>Z-axis travel speed</td><td>23 mm/s</td><td>Infill density</td><td>100 %</td></tr><tr><td>Support layer thickness</td><td>0.2 mm</td><td>Sample orientation</td><td>Building// thickness</td><td>Nozzle diameter</td><td>0.4 mm</td></tr><tr><td></td><td></td><td>layer height</td><td>0.2 mm</td><td>Layup</td><td>+45°/-45°</td></tr></tbody></table>

strategy, concluding that optimized structures with relative densities between 0.25 and 0.35 outperform the benchmark lattice.

Extensive research has focused on the manufacturing of lattice structures. Traditional methods such as investment casting, expanded metal sheet, metallic wire assembly, or snap fit have been commonly employed [17]. However, these conventional approaches have limitations when it comes to accommodating complex part designs resulting from processes like topology optimization. Various additive manufacturing (AM) techniques have been explored for the production of lattice structures, including stereolithography, fused filament fabrication (3 F), selective laser melting, and electron-beam melting [2, 18–20]. These techniques utilize feedstock materials in the form of powder, filament, or liquid. However, the manufacturing process, whether it involves laser/beam scanning or filament deposition, can introduce defects that lead to significant disparities between the CAD (Computer-Aided Design) model and the actual part [21,22]. Common defects in lattice structures include incomplete cell walls, inner porosity within the struts, deviations in cell wall thickness, and undesired waviness [23]. These defects significantly reduce stiffness and strength, while also causing stress concentrations that can result in unexpected failures or shorter part lifespans [24–28].

This study employs a combined experimental and numerical approach to gain further insights into the implications of defects in lattice structures manufactured using fused filament fabrication (3 F). Defect analysis is carried out using X-ray micro-tomography (XRD) and compared to CAD models. The finite element method is utilized to predict the behaviour of the lattice structures, considering either the CAD or XRD models as input geometries. A comparison is made between defect-free and defect-induced predictions using constitutive laws that account for the development and accumulation of damage in the cellular structure. The findings from both models are thoroughly discussed to derive meaningful conclusions regarding the influence of defects on the mechanical behaviour of lattice structures. The knowledge gained from such an analysis can help in optimising the design of lattice structures considering the defects induced by AM.

<table><caption>**Table 2**

Lattice structures considered in this study.</caption><thead><tr><th>Geometry</th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Relative density (%)</td><td>0.2</td><td>0.3</td><td>0.4</td><td>0.5</td><td>0.6</td></tr><tr><td>Cell size (mm)</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Cell number (–)</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td></tr><tr><td>Cell wall size (mm)</td><td>1.07</td><td>1.35</td><td>1.61</td><td>1.86</td><td>2.11</td></tr></tbody></table>

## 2. Experimental layout

The feedstock material utilized in this study is a composite consisting of a weight content of 30 % recycled pinewood fibre and 70 % PLA/PHA biosourced material, obtained from the FDBI company and marketed as ColorFabb woodfill. This material is primarily used for prototyping and 3D design in various consumer products, offering a wood-like texture in eco-friendly items due to the inclusion of wood fibres in a bio-based matrix. Common applications include decorative objects like figurines, sculptures, and art pieces that imitate natural wood textures, as well as furniture prototypes and custom home décor, such as picture frames, lamps, and decorative panels.

The material is supplied in the form of a wire cartridge, with filaments typically having a diameter of 1.75 ± 0.05 mm. The documented glass transition temperature for this wood-based filament is 55 °C. According to former Differential Scanning Calorimetry (DSC) results, the material exhibits a glass transition temperature of approximately 57 °C and a melting temperature of 152 °C [29]. Further information about the composition of the filament is achieved by X-ray microtomography from synchrotron radiation (ESRF beamline BM5, Grenoble, France). The acquisition parameters include an energy of 75 kV, a detector-to-sample distance of 1396 mm, a scan range of 360°, and 5000 radiographic images, 41 reference images, and 40 dark images. The detector resolution is 2048 × 1400 pixels, with a count time of 15 ms and a latency of 15. The voxel size is 3.2 µm. Tomograms are acquired using a back-projection reconstruction algorithm, and a Paganin filter is applied to enhance phase retrieval in phase-contrast images. The typical tomogram resolution is 598 × 596 × 339 voxels, covering a diameter of 1.8 mm and a height of 1 mm. Image analysis is performed using ImageJ software from NIH (USA). The morphology of the as received filament is shown in Fig. 1a.

The supplier recommends a printing temperature range of 195 °C–220 °C, along with a heated printing bed set between 50 °C and 60 °C. Furthermore, the suggested printing speed for this material falls within the range of 40–100 mm/s.

The FFF process is carried out using a MakerBot Replicator 2 printer, controlled by the MakerBot© Desktop software. A comprehensive set of printing parameters can be found in Table 1. The geometries and main structural characteristics of lattice structures are shown in Table 2. These are body-centered cubic (BCC) cell known for bending dominated behaviour (see Table 2).

Fig. 1b shows the main stages of FFF fabrication of lattice structures and final rendering of the parts.

Microstructural characterisation of lattice structures is undertaken using X-ray micro-tomography equipment with a source energy of 230 kV, an operated voltage of 60 V, a current intensity of 480 µA, and an angular step of 0.35°. The 3D images are acquired with a voxel size of 24 µm, which represents 45 to 88 times the cell wall thickness. The resolution of the images ranges from 0.86 × 10⁹ to 1.06 × 10⁹ voxels, which corresponds to maximum physical dimensions of 24 × 24 × 25 mm. Image processing is undertaken using open source ImageJ software. As part of the image processing, image segmentation is performed to
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Overview of the testing configurations. (a) PLA/PHA-wood filament tensile testing configuration, (b) compression testing of lattice structures, (c) deformed lattice structures as a function of loading level up to the densification (ε).</figcaption></figure>

separate the background and porosity from the solid phase. A threshold value of 50 in a range of 256 grey levels is selected as it allows a clear separation between the phases. Granulometry technique is used to retrieve the size distribution of features of interest as well as 3D labelling

technique to derive the pore and solid phase connectivity.

Mechanical characterisation is conducted on both filament and 3D printed lattice structures. Data about the intrinsic properties of the wood filaments are derived from tensile experiments to evaluate the amount of
<figure><img src="image_5.png" /><figcaption>Fig. 2. (continued).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Illustration of boundary conditions for the simulation of the compression behaviour of 3D printed lattice structures using in finite element computation.</figcaption></figure>

stiffness reduction of the lattice structures. Tensile testing of the filaments is adapted from the standard ASTM E8/ISO 6892-1 used for metal wires [30]. Tensile mechanical tests of PLA/PHA-wood filaments are performed on a universal machine from Zwick Roell with a length filament of 44 mm tested under a displacement rate of 5 mm/min (Fig. 2a). The use of small samples (20 mm in length with a cell size of 5 mm) is justified by the constraints needed to reach a densification stage with approximately 70 % height reduction. Testing larger samples requires a universal machine with a higher load cell capacity. The 3D printed lattice structures undergo uniaxial compression testing on the three orthogonal faces (planes XY, XZ, and YZ) to explore their mechanical behaviour at different porosity contents (Fv). Testing conditions on these materials are derived from the standard ISO 604 related to the compressive properties of plastics [31]. A 100 KN MTS machine is employed to apply the loading, maintaining a constant cross-head displacement rate of 5 mm/min (Fig. 2b). The dimensions of the compression stage are adjusted to ensure full contact with the deformed sample at all stages of loading (Fig. 2b). The height of the sample is reduced by up to 50 % from its initial height to allow for the appearance of damage during densification (Fig. 2c). Fig. 2b provides a visual representation of the conducted tests, where optical recording is considered to monitor local displacement. The measured force and crosshead displacement data are converted to their respective engineering stress

and strain values. These data are related to the density of the lattice structures.

### 3. Modelling technique

Implementation of XRT images of 3D printed lattice structures in a finite element model involves converting the voxels in the images into structural elements. These elements represent the solid phase and are meshed regularly using a voxel-to-element conversion process. The structural elements have cubic shapes defined by eight nodes, and each node has three displacement components (UX, UY, and UZ). To evaluate the impact of microstructural details and representative elementary volume, both the size of the samples and the resolutions are varied. Cubic volumes are selected from the central region of the imaged sample, with an increased size ranging from 5 to 18 mm. The size of the model varies from $8 \times 10^6$ to $56 \times 10^6$ degrees of freedom (dofs), depending on the resolution (voxel size ranging from 24.00 µm to 240 µm) and the relative density.

To assess the compression behaviour of 3D printed lattice structures, an isotropic elastic material model is employed. The PLA's Young's modulus (EY) and Poisson's coefficient (ν) are incorporated based on experimental data and the technical sheet provided by the supplier.

The modelling approach involves a quasi-static linear model that enables the determination of the stiffness of 3D printed lattice structures by utilizing microstructural information. In this model, tension is simulated along the three main directions (Fig. 3). For instance, a loading in X-direction is conducted by restricting all lateral displacement components (UY = 0 and UZ = 0) at both ends (x = 0 and x = L, where L represents the length of the sample). A prescribed displacement value (UX = U < 0) is applied to all nodes located at x = L, while the same displacement component (UY = 0) is fixed for nodes at the other end (x = 0). The same loading conditions are applied for Y and Z directions.

The elasticity problem is solved using the preconditioned Conjugate Gradient (PCG) solver, and the overall Young's modulus is predicted by analysing the reaction force in the loading direction at the loaded end (x, y, z = L). Additionally, stress fields are derived to determine the influence of the microstructure on stress heterogeneity.

To handle the computational demands of the extensive models employed, high-performance computing resources are utilized. These resources consist of CPUs operating at 4.4 GHz and 1 terabyte of RAM. However, due to the complexity of the simulations, the computation duration can last up to 60 min per load increment.
<figure><img src="image_7.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Comparison between CAD and XRD for a lattice structure exhibiting a solid volume content 20 %. (a) Perspective view with three highlighted planes in the CAD model. (b) Cross-section views compared at three locations.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 5. Comparison between linear relative densities of XRT and CAD models as a function of depth in (a) building and (b) lateral directions.</figcaption></figure>

# 4. Results and discussion

## 4.1. Microstructural results

Fig. 1a presents the morphology of the as-received PLA/PHA-wood filament obtained through X-ray micro-tomography. The PLA and wood particles show only slight variations in grey levels, making it difficult to distinctly separate the two phases. By applying a combination of thresholding and flooding techniques, the total filament volume is determined to be 2.19 mm³. Additionally, the filament contains approximately 25 % porosity, with a connectivity of 81 % as measured through 3D labelling of the pores. The average porosity size is around 35 µm, with a size distribution width of 27 µm, determined using 3D granulometry.

Fig. 4a compares the rendering of the CAD and XRD for a lattice structure containing a solid volume fraction of 20 %. Although the same structural characteristics are depicted for both CAD and XRT images, a rough finishing surface is a main difference derived from the perspective views. Further exploration of microstructural defects reveals significant variation in the cross-section views (Fig. 4b). These are related to the presence of hollow structures within the junctions, alteration of the solid phase and the presence of porosity inside the cell walls.

Fig. 5a depicts the linear relative density in the building direction

<figure><img src="image_10.png" /><figcaption>Fig. 6. (a) Comparison between the desired and obtained density of lattice structures based on weight and XRT measurements expressed as a solid volume fraction. (b) Cell wall volume distribution in lattice structures as a function of the relative density.</figcaption></figure>

corresponding for both XRT and CAD models of a lattice structure containing 80 % of intended porosities. The differences in peak values indicate a lower density achieved in the real part compared to the intended one in the CAD model. A closer look at the joint (Fig. 4), where most of the porous structures is revealed provides a clue about the finite resolution limits to populate the joint with a solid material. In addition, porosity within the filament themselves adds more to the difference. This difference can be quantified by integrating both XRT and CAD density profiles within the same range in the building direction. The difference of the area under the curves ranks this difference as 22 % of underestimation of the relative density in the real part.

The same density profiles are plotted along the lateral direction as shown in Fig. 5b. Larger peaks are depicted for XRT but the underestimation of the relative density still holds in this case but with a lesser extent of 12 %. Overall differences in relative densities for the other lattice structures are further explored through the comparison between the predicted and measured relative densities. Measurements are done according to two different methods: the first one is based on weight assessment and the second is directly extracted from XRT. Fig. 6a shows that both methods agree in depicting the underestimation of the relative densities for all lattice structures. In the range between 20 and 60 % of intended relative densities, the mismatch can be quantified through the linear function
<figure><img src="image_11.png" /><figcaption>Fig. 7. Magnified view of the cell wall junctions highlighting a perspective view, surface plot of the inner porosity and a cross-section view of a segmented plane at the middle of the junction as a function of relative density (a) 30–40 %, (b) 50–60 %.</figcaption></figure>

$$V_F(\%)_{EXP} \approx -3 + 0.82 \times V_F(\%)_{CAD} \quad (1)$$

Where the rate of 0.82 represents the average mismatch between the expected and the obtained relative density.

In order to quantify further the effect of process-generated porosity on the lattice structure, the volume distribution of cell walls in XRT for all studied relative densities are shown in Fig. 6b. All lattice structures exhibit the same asymmetric gaussian distribution within the range from 24 µm up to 1.6 mm. There are some differences that highlight the effect the porous structure within the cell walls. Indeed, for most of the relative

<figure><img src="image_12.png" /><figcaption>Fig. 8. Compression loading results of lattice structures (a) stress – strain curves as a function of the relative density (Fv) for a loading direction aligned with the building direction (b) quantification of deformation stages according to a set of five engineering constants.</figcaption></figure>

densities up to 50 %, the cell wall distribution is shifted towards low values. This shift is explained by the discontinuity in solid phase within the cell wall, which reflects both a limited resolution for printing the cell walls and the porosity within the filaments. At the highest relative density of 60 %, the cell wall distribution widens and stretches towards large values, which indicates that the solid phase is more continuous for thicker cell walls. The average cell wall thickness goes from 360 µm to 578 µm when the relative density increases from 30 % up to 60 %. Fig. 7 shows magnified views of the cell wall junctions, which highlights the differences in the inner porosity structures. These views confirm the cell wall size distributions in Fig. 6, as the large amount of inner porosity is found to be consistent with relative densities lower than 60 %. Despite the presence of inner porosity, the solid phase connectivity is ensured at a rate of 99.8 % as attested by the ratio between the largest feature over the total solid phase in the lattice structure.

## 4.2. Mechanical results

The 3D printed lattice structures exhibit a typical response of an open-cell cellular material due to the presence of large porosity content (Fig. 8a). The ranking of the curve depends on the porosity volume content but all lattice structures follow the same trend. Indeed, three main stages are identified (Fig. 8b): elasticity, plastic collapse and densification. The initial stage elasticity-plastic stage is characterized by the slope (EY) representing Young’s modulus, yield stress (σY), yield
<caption>**Table 3**</caption>

Summary of main engineering constants ($E_Y$: Young's modulus; $\sigma_Y$: Yield stress; $\varepsilon_Y$: Yield strain; $\tau$: Compression strength; $\omega$: Plateau width; $\sigma_M$: minimum plateau stress; $\psi$: densification modulus) extracted from the compression loading of 3D printed lattice structures as a function of the relative density $F_V$.

<table><thead><tr><th>V<sub>F</sub>(%)</th><th>E<sub>Y</sub> (MPa)</th><th>σ<sub>Y</sub> (MPa)</th><th>ε<sub>Y</sub> ( - )</th><th>τ (MPa)</th><th>ω ( - )</th><th>σ<sub>M</sub> (MPa)</th><th>ψ (MPa)</th></tr></thead><tbody><tr><td>20</td><td>3.5</td><td>0.06</td><td>0.190</td><td>0.23</td><td>0.451</td><td>0.17</td><td>15.2</td></tr><tr><td>30</td><td>19.4</td><td>0.80</td><td>0.044</td><td>1.03</td><td>0.376</td><td>0.64</td><td>44.6</td></tr><tr><td>40</td><td>51.7</td><td>1.51</td><td>0.032</td><td>2.14</td><td>0.468</td><td>1.61</td><td>73.8</td></tr><tr><td>50</td><td>79.2</td><td>2.48</td><td>0.033</td><td>3.50</td><td>0.509</td><td>3.07</td><td>153.4</td></tr><tr><td>60</td><td>175.6</td><td>5.18</td><td>0.033</td><td>4.50</td><td>0.52</td><td>3.80</td><td>281.3</td></tr></tbody></table>

strain ($\varepsilon_Y$), and compression strength ($\tau$). The plasticity collapse stage is captured by the minimum plateau stress ($\sigma_M$), and width ($\omega$). The densification stage is also captured by its slope ($\psi$) and width ($\varepsilon_d$). These parameters are all correlated to the relative density ($F_V$) (Table 3).

The use of the Gibson-Ashby scaling law shows the following correlation between the elasticity and density of the 3D lattice structures

$$E_Y = E_{Y0} \times \varphi \times (F_V)^n \quad (2)$$

where $E_{Y0}$ is Young's modulus of the dense material. Assuming that the prefactor $\varphi = 1$, equation (6) takes the explicit form

$$E_Y = E_{Y0} \times (F_V)^{3.70} \quad (3)$$

Where the exponent $n > 1$ shows that the lattice structure is governed by a mixture of bending and uniaxial deformation.

The same scaling law is applied for the yield and compression strength, which leads to the equations

$$\sigma_Y = \sigma_{Y0} \times 0.94 \times (F_V)^{3.24} \quad (4)$$

$$\tau = \tau_0 \times 0.39 \times (F_V)^{2.44} \quad (5)$$

Where $\sigma_{Y0} = 28$ MPa, $\tau_0 = 49$ MPa refer to the intrinsic properties of the filament. It is worth mentioning that reported modulus and tensile strength of filament of unfilled PHA/PLA is 1.3 GPa and 47.4 MPa [32]. The tensile strength is maintained in the wood filled composite through the presence of wood particles, which can provide mechanical strength up to 60 MPa. However, the reinforcement effect is diminished due to the porosity within the filament. As a result, the overall strength reflects a balance between the wood particles and the porosity.

The yield strain ($\varepsilon_Y$) follows the opposite trend as an exponential decay function

$$\varepsilon_Y = \varepsilon_{Y0} \times 4.02 \times (F_V)^{-2.86} \quad (6)$$

where $\varepsilon_{Y0} = 0.0203$ refers to the yield strain of the filament.

The minimum plateau stress ($\sigma_M$) and the densification slope ($\psi$) are found to increase with the increase of the relative density similarly to the case of Young's modulus ($E_Y$), yield stress ($\sigma_Y$), and compression strength ($\tau$). The evolution is captured using the same nonlinear function of the form

$$\sigma_M = 24.79 \times (F_V)^{3.01} \quad (7)$$

$$\psi = 1350 \times (F_V)^{3.09} \quad (8)$$

where data related minimum plateau stress and densification slope of the filament are not relevant in this case.

Also, it is found that the minimum plateau width ($\omega$) and the densification width ($\varepsilon_d$) have no particular trend with respect to the relative density.

The results derived from equations (2)–(8) indicate that the stiffness of the 3D-printed lattice structures is predominantly influenced by a bending deformation mechanism rather than uniaxial deformation, as suggested by the large exponent values in equations (2) and (3) and reported in Ref. [33]. This can be attributed to defects in the filament and issues during the manufacturing process of the lattice structure,

which activate a less demanding deformation mechanism.

The same goes for the elastic collapse stress, which for a typical regular open cell structure account both for an exponent of the order of 3/2 [34]. In this case, both yield and compression stress (i.e., collapse stress) display a higher exponent, indicating fully plastic behaviour of the cell walls. This allows plastic hinges to form more extensively, depending on the porosity level within the cell walls.

### 4.3. Finite element predictions

Fig. 9 shows typical displacement and stress fields of the considered lattice structures at different resolutions in the loading direction $z$, corresponding to the building direction. Within the range of $8.4 \times 10^6$ to $56 \times 10^6$ dof, displacement field in $Z$ direction (UZ) captures the intended compression loading in the building direction. Furthermore, stress heterogeneity (S33), encompassing both negative and positive stress values, is also accounted for in all the resolutions under consideration. Higher accuracy is achieved for a resolution higher than $18 \times 10^6$ dof, which corresponds to an average element size of 200 µm. Fig. 9b confirms that resolutions lower than $18 \times 10^6$ dof results in relative error in the stiffness prediction above 5 % except for $F_V = 60$ %. For most of the lattice structures, the relative error on stiffness prediction evolves linearly with the model resolution. To ensure sufficient accuracy of finite element predictions, all computations are performed with a resolution of $55.5 \times 10^6$ dof, for which the residual error for stiffness prediction is below 0.5 %.

Fig. 10a shows the effect of the sampling ratio of lattice structures on the predicted stress field for a loading performed in the building direction. As the size of the cropped volumes increases the surface effects decreases more significantly when the relative density is large. Similar alternations of negative and positive levels are depicted for all sampling ratios. Denser lattice structures experience more uniaxial deformation compared to airy lattice structures, which combine more bending with uniaxial deformation. Fig. 10b shows the trends of RVE analysis for three relative densities, where the scaled sampling volume represents the percentage of the cropped volume compared to the original lattice structure taken at the resolution of $55.5 \times 10^6$ dof. The evolution of the error on stiffness prediction in the loading direction well show rapid and linear decrease of the error as the crop size increases for dense lattice structures. A nonlinear trend is predicted for lattice structures exhibiting a low relative density. In most of the cases, the minimum RVE size determined under a criterion of 5 % of variation in stiffness is only achieved for a sampling ratio around 96 % from the original volumes. In other words, XRT-based finite element computation that accounts for the process-induced porosity shows that the mechanical RVE size is too close to the original size of the studied lattice structures, which does not allow to conclude on the stability of the compression performance below 20 mm.

Fig. 11a shows the von Mises stress distributions of XRT models by considering the following parameters for the solid phase material properties: $E_0 = 1156$ MPa, $v_0 = 0.3$. These counterplots are to be compared with the CAD-based model homologues in Fig. 9. Significant differences in terms of stress heterogeneity is depicted as a result of the real microstructural defects identified in Fig. 4. Indeed, process-induced porosities act as stress concentrator sand they are found to be
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Effect of resolution on finite element predictions of lattice structure compression performance. (a) Nodal solutions showing displacement and stress field as a function of the model size, (b) relative error in stiffness as a function of the number of degrees of freedom (dof).</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 10. RVE analysis showing the effect of sample size on finite element predictions of lattice structure compression performance. (a) Nodal solutions representing the stress component S33 fields as a function of the crop size, (b) relative error in stiffness as a function of the sampling percentage.</figcaption></figure>

responsible for the stress distribution mismatch, which reaches a value of 1 % according to the stress levels extracted from the profiles shown in Fig. 10b. Indeed, this figure also shows that smoother stress variations are obtained in the case of low relative density lattice structure for XRT-based models. These variations are due to the microstructural mismatch between the real part and the CAD model. This mismatch becomes lesser for larger relative densities. The plot of the relative Young's modulus as a function of the relative density for all considered 3D printed lattice structures shows that there is a large gap in ranking between the XRT and CAD based models (Fig. 11c). The XRT model seems to fit better the experimental results. The use of the Gibson-Ashby scaling law shows the following correlations between the elasticity and density of the 3D lattice structures for both XRT and CAD-based models

$$XRT : E_R = E_Y \times \rho^{3.72} \qquad (9)$$

$$CAD : E_R = E_Y \times \rho^{2.86} \qquad (10)$$

The exponents for both experimental (equation (7)), and XRT-based model agree with a small deviation of about 0.5 %. Both predict a dominant bending mechanism as the exponent is larger than 1. However, a smaller exponent is predicted for the CAD-based model indicating an overestimation of the uniaxial deformation leading to an upper ranking compared to EXP and XRT model and a deviation of the exponent by 23 %.

### 4.4. Discussion of main findings related to the microstructural defects associated with fused filament fabrication

Numerous studies have emphasized the microstructural defects inherent in fused filament fabrication (FFF), such as porosity, incomplete bonding between layers, and surface roughness [35]. Despite considerable research, a definitive solution to completely eliminate these defects remains elusive. Current approaches primarily focus on mitigating their impact through adjustments in the printing process
<figure><img src="image_15.png" /><figcaption>Fig. 11. Finite element results illustrated (a) S33 stress component of CAD-based finite element model for three relative densities at a strain level of 1 %, (b) Stress profiles along the main direction for both CAD-based and XRT-based models (c) Relative Young's modulus as a function of the relative density for XRT and CAD based model compared to the experimental results.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 11. (continued).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 12. Design and production of a technical component using PLA/PHA-wood, incorporating a lattice structure support to manage overhangs.</figcaption></figure>

using for instance design of experiment approaches [36]. For instance, modifying key parameters such as increasing filament overlap, reducing layer height, and raising printing temperatures can help reduce the severity of these defects, but none can fully eliminate them [37,38]. The literature consistently addresses these challenges, with most research concentrating on optimising process parameters to improve mechanical performance. Adjustments like large filament overlaps enhance inter-layer adhesion, while reduced layer heights contribute to smoother surfaces and more uniform structures [38,39]. Additionally, higher printing temperatures promote better material flow and fusion between layers, which can reduce voids and improve the overall strength of printed parts [40]. However, these adjustments offer incremental improvements, and no single approach has yet provided a comprehensive

method to fully eliminate the defects associated with FFF.

In the current study, the issue of porosity in lattice structures remains unresolved because the porosities are multi-scale. This complexity arises from the fact that the filaments used in these lattice structures are composite materials, which inherently contain a degree of porosity within the filaments themselves as shown by X-ray micro-tomography results and reported in Ref. [29]. Fig. 11c indicates that adjusting the relative density could partially compensate for process-induced porosity, particularly by improving stiffness towards the target values of the CAD model. For instance, an approximate 9% increase in relative density is required to restore the relative Young's modulus to 0.1.

This methodology of compensating for porosity can be extended to other mechanical properties, such as strength, by incorporating pre-
existing knowledge about the morphology and spatial distribution of defects. With precise modelling, it is possible to predict and mitigate the effects of these defects. This approach highlights the need for an accurate multi-scale model that can adjust for the presence of microstructural porosities across various scales, ensuring that the mechanical performance remains within acceptable ranges.

Fig. 12 illustrates an example of a part created using PLA/PHA-wood as the feedstock material, with dimensions of 48 × 40 × 40 mm³ and a typical material thickness of 2 mm. The design is a holder for a 3D printing head. In the initial design, the cylindrical holder extends continuously along the height of the part. However, in the final design, material savings were achieved through a finite element model, which necessitated the use of a lattice structure for support. The part, printed at a temperature of 230 °C, demonstrates enhanced load-bearing capacity due to the optimized weight of the lattice structure.

## 5. Conclusions

This study concludes that the 3D printing of lattice structures offer wide possibility of structural design tuning. Lattice structures printed using FFF with various porosity volume fractions exhibit a typical cellular material behaviour with a dominant bending deformation mechanism. Two major FFF manufacturing defects, namely porosity and rough surface are detected via X-ray tomography. The simulation of the compression behaviour demonstrates the overestimation of the CAD-based model compared to the XRT model due to the presence of stress concentrators in XRT models that are not considered in the CAD model. The presence of manufacturing defects significantly compromises the structural mechanical properties, highlighting the need to account for these defects in the design of the lattice structures.

The study emphasizes that porosity in lattice structures, especially in composite filaments, remains a challenging issue due to its multi-scale nature. However, increasing the relative density can partially offset the effects of process-induced porosity, enhancing mechanical properties such as stiffness. This approach requires the use of a multi-scale model to predict the performance gap, which helps mitigate the impact of porosities.

## CRediT authorship contribution statement

**Tao Liu:** Writing – original draft, Visualization, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Ji-hong Zhu:** Writing – review & editing, Validation, Supervision, Software, Resources, Project administration, Methodology, Conceptualization. **Weihong Zhang:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Funding acquisition, Conceptualization. **Sofiane Belhabib:** Writing – review & editing, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Sofiane Guessasma:** Writing – original draft, Visualization, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgement

This work is supported by National key R&D Program of China (2022YFB3402200) and key Project of NSFC (92271205).

## Data availability

Data will be made available on request.

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