The process-structure-property relation in short-ﬁber reinforced composites made in Fused Filament Fabrication (FFF)remainsinadequatelyunderstood, muchtrial- and-error and extensive testing is required to use these materials for load-bearing applications. As a consequence, the largely empirical design process has hindered the adoption of this technology, notably due to the lack of reliable structural analysis capability. In order to surpass the limitations of mechanical property prediction using simpliﬁed artiﬁcial microstructures, this work demonstrates the decisive advantage of using geometries obtained directly from imaging of printed specimen instead. The analysis of µCT images is performed via a purpose-built extraction tool called OpenFiberSeg, yielding profound insight into the process- structure relation. The use of real microstructures is shown to considerably improve the mechanical behavior prediction capability via dual-scale FFT-based homoge- nization, bringing relative error margins below 5%, for full anisotropic description. It also becomes possible to investigate the eﬀect of processing parameters such as nozzle diameter and printing pattern on morphological properties and on mechanical behavior, revealing the magnitude of the spatial variation of local properties. The combination of experimental and simulation enables insight that is not accessible to either alone. Original imaging data and source code are made publicly available. 1. Introduction Fused ﬁlament fabrication (FFF) is an additive manufacturing (AM) method by which complex parts are built layer-by-layer by the deposition of semi-molten polymer ﬁlaments through numerical control. As opposed to injection or compression molding, the process requires no molds or tooling, which leads to fewer design restrictions, low cost per unit in small batches, customizability and simplicity of manufacturing [1–4]. By employing short ﬁber reinforced polymer (SFRPs) ﬁlaments made from high-performance polymers such as poly ether-ether-ketone (PEEK) mixed with short carbon ﬁbers (CFs), parts made through FFF exhibit weight-speciﬁc properties giving them the potential of replacing metallic components in ﬁelds like the aerospace or automotive industries [1, 4, 5]. In addition, by controlling the orientation of reinforcements, ⋆This research project is funded by the National Science and Engineering Research Council of Canada and Safran Tech. ∗Corresponding author ∗∗Principal corresponding author facundo.sosa-rey@polymtl.ca ( Facundo Sosa-Rey); martin.levesque@polymtl.ca ( Martin Lévesque) orcid(s):0000-0003-1140-0635 ( Facundo Sosa-Rey) Sosa-Rey et al.: Preprint submitted to Elsevier Page 1 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology the mechanical, thermal and electrical properties of parts can be engineered for speciﬁc purposes [1, 3, 6]. However, before FFF parts can be used for load-bearing applications, structural analysis tools must be signiﬁcantly improved [3, 7, 8]. The lack of understanding of the relation between processing parameters and mechanical properties makes existing tools insuﬃciently reliable [1, 7, 9], rendering the design process largely empirical [3, 10]. The main problem resides in the fact that morphological characteristics are inﬂuenced ﬁrst by the properties of the base materials and the processing parameters [1, 11–13], which lead to localized gradients in material properties and contribute to suboptimal part consistency [14–17]. Fiber length and orientation distributions, local ﬁber volume fractions and the formation of porosity are all aﬀected by processing conditions, and exhibit local variation within parts [6, 8, 11, 12, 18, 19]. In addition, parts made by FFF exhibit structural features at several distinct length scales [1, 7], some of which are absent in molded parts [13]. Understanding the link between these microstructural features and the resulting mechanical properties is a crucial, and yet-unsolved, challenge [3, 9, 10, 18, 20, 21]. Generally,threedistinctscalesarerelevantinthecontextofSFRPsmanufacturedbyFFF.Themicroscale is where the constituents (matrix, ﬁbers, intra-bead voids) are explicitly considered [7, 11]. Then, the part geometry scale, or macroscale, is always present and is the main focus of mechanical design. Lastly, the FFF process imparts an intermediate, or mesoscale, level of structuring that is determined by the printing inﬁll pattern selected, and the characteristics of the exterior part shell. Even with the simplest of inﬁll patterns, i.e., all passes aligned with each other, and 100% volume occupation, each bead being round at the nozzle tip and forced into a rectangular section leaves voids at the corner of this rectangle, which form a periodic array of neatly aligned cavities throughout the part [1, 21, 22], contributing to part anisotropy [13, 23–25]. All these scales must be encompassed by an attempt to account for the morphology-property relation of AM SFRPs [7]. 2. Background 2.1. Morphological characterization: ﬁber properties Predicting the mechanical stiﬀness of SFRPs requires a description of their microstructure, often generated from statistical observations of real materials. Fiber length and orientation distributions are commonly obtained as follows. The elliptical cross-section of individual ﬁbers can be obtained from scanning electron microscopy (SEM) images of polished samples. The orientation of individual ﬁbers in three- dimensional (3D) space are computed by measuring the angle those ellipses make with the printing direction and their ratio of minor to major axis [11, 26, 27]. Fiber lengths can be measured under optical microscopy or SEM after they are separated from the matrix, either though pyrolysis [2, 11] or dissolution [26]. However, Sosa-Rey et al.: Preprint submitted to Elsevier Page 2 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology those procedures are time-consuming and suﬀer from repeatability issues [27], and accuracy is reduced for ﬁbers almost perpendicular to the image plane [28]. Further, SEM images are two-dimensional (2D) rather than volumetric, and thus cannot account for property variation at diﬀerent depths in the material. In addition, the lengths and orientations aren’t measured at once on the same ﬁbers, therefore the SEM images method does not produce a full-ﬁeld description including positions in 3D space, and the correlation between length and orientation is unknown [27]. Some studies have measured this correlation, more so for carbon ﬁbers than for glass ﬁbers because of their higher average aspect ratio [11, 29, 30]. It is possible to obtain data at several depths, by repeatedly polishing the specimen to produce a 3D rendering [27, 31], or use confocal microscopy to image beyond the surface [32]. These statistics can be used directly in analytical modelling [18, 26, 33, 34] or artiﬁcial geometries can be tailored to have similar average properties [2, 11, 35–38]. These geometrical descriptions, or representative volume elements (RVEs), are a volume description of a sample of the microstructure which is small enough so modelling is achievable, but rich and complex enough to be representative of the variability of the real morphology (for strict deﬁnitions, see [39, 40]). The generation of such RVEs is complex and computationally costly, especially when large ﬁber aspect ratios and volume ﬁlling fractions vfare required [2, 11, 37, 41]. As a consequence, the vfconsidered only rarely reach 25% [38], whereas the vfin SFRPs are commercially available at 30% and can reach as much as 40% [29, 42]. A commonly used method of microstructure generation is random sequential adsorption (RSA) [14, 38, 43–45], in which inclusions (ﬁbers and/or pores) are seeded into a volume one at a time, checking for interference with existing inclusions, and randomly repositioned in case of interference, until the desired volume fraction is reached. Fiber segments that extend beyond volume boundaries are repeated on opposite faces of the volume, ensuring geometric periodicity [38, 41]. Notably for large aspect ratios, this method has been modiﬁed in several ways to enable higher volume fractions [9, 11, 45]. Other methods draw from molecular dynamics modeling, in which ﬁbers have velocities and collide with each other while increasing in volume, until the desired vf is reached [41, 46, 47]. A few studies attempted generating artiﬁcial microstructures with a speciﬁc set of orientation and lengths distributions, in addition to inﬁll fractions. Tian et al. proposed a ﬁber growth methodbywhichtheywereabletoreach25%ofinﬁll[14,45],butonly19%wasachievedwithanaspectratio of 10 [38]. Tang et al. achieved vf= 38%at a mean aspect ratio of 15, but only for random orientations [9]. Using a modiﬁcation of the RSA algorithm, they generated microstructures (including ﬁbers and voids) in which ﬁbers conformed to an orientation tensor obtained from a computed micro-tomography ( µCT) scan of a real CF polyamide-12 (PA12) specimen. By generating RVEs with varying characteristics, they Sosa-Rey et al.: Preprint submitted to Elsevier Page 3 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology demonstrated that ﬁber alignment has a pronounced eﬀect on tensile properties: an RVE with an average deviation of 18 °shows a tensile modulus 2.67 larger than one with random orientation. Nasirov et al. [2] used artiﬁcially generated RVEs with vfup to 10%with a constant ﬁber length of 60µm, with either uniform (random) orientations or aligned with the print direction. This microstructure was homogenized ﬁrst, then embedded in a mesotructure of unidirectional or 0◦-90◦printing pattern (each layer has aligned beads, and subsequent layers are perpendicular to each other). For their mesostructure description, the authors considered idealRVEs of beads modelled as rounded rectangles and actualRVEs, generated according to mesotructures observed in microscope images of physical specimens. They compared the tensile moduli predicted by simulation to tensile testing measurements performed on carbon ﬁber reinforced polylactic acid (PLA), and encounter up to 20% error in unidirectional conﬁguration, and up to 40% error for 0◦-90◦, using actual mesotructure RVEs (error was higher for ideal mesotructures). Another study from the same group considered aligned carbon ﬁbers in polycarbonate (PC), and an ideal mesostructure, and reported 3.5% to 36% error in tensile modulus prediction [22]. These studies suggest the mesoscale porosity has a signiﬁcant inﬂuence on eﬀective properties, notably in making the material anisotropic, even when considering a pure (non-reinforced) isotropic polymer [13]. They also highlight the fact that real microstructures are not perfectly aligned, which has a decisive impact on eﬀective properties. Thenon-uniformnatureofrealmesoscaleporosityandtheprobablespatialvariationinmicroscaleproperties can also cause discrepancies between those models and reality. The correlation between those attributes is also not accounted for. Those factors explain the rather large error encountered when comparing predictions to experimentally measured tensile moduli. Volumetric characterization based on imaging methods such as µCT has also been reported for SFRPs [8, 19, 27, 29, 48–50]. Obtaining microstructures this way is technically challenging, requiring specialized post-processing that is computationally costly, but the microstructures produced are detailed and high ﬁdelity to physical specimen can be achieved [27, 29, 48]. To the authors’ knowledge, however, material data produced in this manner has only rarely [51] been utilized directly for mechanical properties prediction. 2.2. Eﬀect of processing on microstructure The rheological phenomena by which ﬁber alignment and lengths are aﬀected by processing has been studied from modelling and expeximental perspectives. Using ﬂow simulations, most studies [33, 52] utilize a so-calledweaklycoupledapproach,inwhichﬂuidﬂowissolvedﬁrstwithoutconsideringthepresenceofﬁbers. Then, a ﬁber orientation tensor is inferred from ﬂow proﬁles via the Advani and Tucker’s equations [53]. Sosa-Rey et al.: Preprint submitted to Elsevier Page 4 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology The weakly coupled assumption is considered valid for vf<10%, beyond which ﬁber-ﬁber and ﬁber-matrix interactions induce strong coupling. These models predict a change of ﬁber alignment at nozzle tip. Lewicki et al. [6] performed ﬂow simulations using discretely modelled ﬁbers and epoxy resin, with a vf= 6%and aspect ratio of 50. They found that alignment is promoted during the extrusion phase, most notably close to the walls. Zhang et al. [10] performed a coupled ﬂuids/discrete ﬁber modelling of extrusion in FFF to study the phenomena of nozzle clogging. They used their modelling and µCT data of CF ﬁlled PA6 to propose nozzle geometries and ﬁber characteristic to prevent nozzle clogging and promote ﬁber alignment. On the experimental front, Chacón et al. [54] studied the eﬀect of nozzle diameter on the tensile and strength properties of CF reinforced polyamide-6 (CF-PA6) specimens, along with porosity generation and geometric accuracy. They found that mechanical properties increased with a larger nozzle diameter, and that porosity is lowest for large diameter nozzles (measured up to 0.8mm). However, to the author’ knowledge, there hasn’t been numerical prediction of mechanical properties taking nozzle diameter as a variable. Extrusion of SFRPs is known to cause ﬁber breakage, which attenuates their mechanical properties [6, 12, 55]. Limited data exists on this phenomenon, whether experimental [12, 29] or simulated [10], as the computational requirements to model this phenomenon would be prohibitive [3]. 2.3. Mechanical property prediction: homogenization Homogenization is the process of estimating the eﬀective properties of a heterogeneous solid. There are generally two families of methods by which this process is carried out: mean-ﬁeld and full-ﬁeld methods. Mean-ﬁeld, or analytical, methods use statistics about the size and orientation of inclusions along with their material properties to estimate those of the composite solid. Such methods include the Mori-Tanaka [56, 57], the self-consistent scheme [58], Lielen [59] and others, see [36]. These methods all assume ellipsoidal inclusion shapes and uniform spatial distribution of orientations, sizes and local density [60]. Further, the interaction between inclusions is possible only to a limited extent (usually for vf<15%) [36, 60]. It has been suggested that these simpliﬁcations make mean-ﬁeld methods unsuitable for modelling complex microstructures such as high-vfSFRP [38], notably at high property contrasts and aspect ratios [36]. Recent progress in full-ﬁeld or volumetric approaches have produced more accurate predictions [2, 11, 36, 37, 60], though at a much higher computational cost. In full-ﬁeld methods, a unit cell representing the microstructure (the RVE) is subjected to external strain, and the local stress response is computed at each point in space by numerical resolution of the constitutive laws. The eﬀective response is obtained by volume averaging of the local responses [18] (see Appendix A for details). The chosen boundary conditions (BCs) Sosa-Rey et al.: Preprint submitted to Elsevier Page 5 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology directly aﬀect the convergence rate towards true eﬀective properties [40, 45], with periodic BCs converging the fastest [39]. Themostwell-knownoffull-ﬁeldmethodsistheFiniteElement(FE)method,whichhashadsomesuccess in the context of homogenization of SFRPs [11, 14, 38, 45, 51]. However, the FE method requires a suitable mesh representing the microstructure geometry, whose construction becomes increasingly challenging and computationally costly with high vf[11, 38]. For high ﬁber content and aspect ratios, the meshing can fail altogether due to numerous contact points between ﬁbers [14, 61]. Mesh-free methods oﬀer a means to circumvent this problem. One such method was developed by Moulinec and Suquet [62], where the elasticity equation is expressed in a so-called Lippman-Schwinger form, and resolved iteratively with the help of the Fast Fourier Transform operation (FFT) (see Appendix B for details). In the FFT method, the microstructure is described by an orthogonal grid (a 3D array), which means this method has the decisive advantage of being able to use data from 3D-imaging techniques directly [20]. The absence of meshing also lowers the memory requirements, which combined with the computational eﬃciency of the FFT method, enables the resolution of large and realistic microstuctures [60, 63]. Recent developments have extended the original Moulinec and Suquet method to allow convergence in presence of inﬁnite phase contrast (as for porous materials), which was a limitation of the original formulation [64]. The AMITEX software implements the FFT method with these improvements, along with distributed computing techniques that allow performance gains on multi-CPU architectures [65, 66]. When a material exhibits structuring phenomena at dissimilar length scales, such as in SFRPs made by FFF, attempting toconsider the smallest features atthe largest scale is oftenunfeasible[20, 67]. Sommireddy et al.[51] attempted it for predicting tensile properties, using a single RVE with a simpliﬁed description of both micro and mesostructure, with error margins of 16 to 26%, depending on load direction. Multiscale techniques are used instead, where the constitutive equations at the macroscale are substituted by the output of the boundary value problem computed at the microscale. Doing so allows the combination of diﬀerent methods for each scale, drawing on the strengths of each. For instance, the eﬃciency of FFT method applied to the microscale can be combined by the ability of FE at the macroscale to account for arbitrary part geometry and BCs [60, 63]. For the multiscale method to be valid, however, a suﬃcient scale separation must exist between the each scale: the typical length at which the smallest features vary, or ln, must be small when compared to the RVE size used to homogenize those features, or Ln. That RVE size must also be small when compared to the features at the scale above it, or ln+1, and the RVE size used to model those, orLn+1[68]. This can be expressed as ln<<L n<<l n+1<<L n+1. Sosa-Rey et al.: Preprint submitted to Elsevier Page 6 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology 3. Methods and materials We investigated the process-structure-property relation of SFRPs made by FFF. As shown in Figure 1, specimens of CF reinforced PEEK were manufactured using unidirectional (0◦-0◦), 0◦-90◦and 90◦-90◦ patterns and an intermediate nozzle diameter. Unidirectional specimens were also fabricated for 2 additional nozzle diameters, one smaller and one larger. As shown in Figure 2, for each scenario, both the microscale and mesoscale morphologies were characterized through µCT imaging. Component extraction was then performed using OpenFiberSeg, a special-purpose tool developed by the authors [29]. This tool enables the identiﬁcation of individual ﬁbers, as well as the intra-bead (microscale) and inter-layer (mesoscale) porosity. Sub-volumes where no mesoscale features are present are located, which serve as representation of the intra-bead microstructure itself. This microscale description C(x)(stiﬀness as a function of position in space) was then processed with the AMITEX software to obtain an homogenized eﬀective stiﬀness at the microscale, ˜Cmicro, which was used in turn to obtain the homogenized stiﬀness at the mesoscale, or ˜Cmeso. The stiﬀness predicted by dual-scale homogenization was then validated by comparing with experimental data and contrasted with predictions made with simpliﬁed artiﬁcial microstructures. Figure 1: Schematic representation of the printing patterns used to produce the test specimens. For all nozzle diameters, the 0◦-0◦or unidirectional along the longitudinal axis was studied. For the 0.4mmnozzle, additional specimens using the 0◦-90◦and 90◦-90◦pattern were also considered. 3.1. ASTM D638 Type 4 specimen preparation Commercial ﬁlament Tecaﬁl ™PEEK VX 30CF from Ensinger was placed in a vaccum oven for 24h at60◦C and 17kPa. Figure 1 shows an ASTM D638 Type 4 dogbone specimen that was printed in an AON3D ™Mark 2 industrial 3D printer using 3 diﬀerent nozzle diameters (Volcano model from E3D, diametersD= 0.25, 0.4 and 0.8mm). The slicing parameters utilized in the Simplify3D ™slicer are shown in Table 1. The layer height and extrusion multiplier were set empirically by attempting to minimize inter-layer porosity, with a printing speed of 30mm/s. In addition, the following temperature settings were common to all specimens: nozzle temperature: 410◦C, bed temperature: 180◦C, and chamber temperature: 120◦C (maximum allowable in AON Mark 2). A total of 6 specimens were produced for each nozzle diameter, Sosa-Rey et al.: Preprint submitted to Elsevier Page 7 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology Figure2: Schematicﬂowchartofthedual-scaleFFT-homogenization.a) µCTimagingdataofphysicalspecimen,performed with large ﬁeld of view for the mesoscale, and at high resolution for the microscale. b) Constituent extraction using specializedsoftwareOpenFiberSeg.c)TheresultofFFT-homogenizationatthemicroscaleservesasinputforthemesoscale homogenization. Table 1 Printing parameters used for each specimen type D, nozzle Layer Extrusion Extrusion Printing diameter (mm) height (mm) width (mm) multiplier pattern 0.25 0.10 0.22 0.95 0◦−0◦ 0.40 0.20 0.42 1.10 0◦−0◦,0◦−90◦,90◦−90◦ 0.80 0.25 0.80 1.01 0◦−0◦ using longitudinal or 0◦-0◦(all passes are along the longitudinal axis of the specimen, i.e., the 0◦direction). Using the 0.4mm nozzle, 6 additional dogbones were made using a 0◦-90◦pattern (layers alternate printing passes along the long and short axis) and a 90◦-90◦pattern (all passes perpendicular to the dogbone axis). Filaments spools were kept in the heated chamber for the duration of printing. Additionally, for each of the nozzle diameters considered, 3 specimens were made by extrusion in free- space, meaning the printhead is not pressed against the table. These specimens enable the characterization of the microstructure at the bead scale, excluding all phenomena related to the deposition onto a part. 3.2. Tensile testing The specimens were loaded at a rate of 5mm/min using a MTS Insight Material Testing system and a MTS uniaxial extensometer model 634_25F-25. For each loading sequence, the strain was obtained directly from the calibrated extensometer readings, and the stress was computed by dividing the loading force by the specimen cross-section area (measured from the average of 4 caliper measurements along the narrow section of the dogbone). The linearly elastic range for each specimen was identiﬁed in the following manner. Sosa-Rey et al.: Preprint submitted to Elsevier Page 8 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology A ﬁrst-degree polynomial was ﬁtted by the least-square method to the entire stress-strain data. The ﬁt was then performed iteratively, each time removing the point at the highest stress from the data, until the relative error between the ﬁtted line and the experimental curve reached 1%. This segment of the data was considered within the linear regime, and the tensile modulus for each specimen is the slope of the ﬁtted line. 3.3. Volumetric characterization 3.3.1. Tomographic data acquisition Using a specimen of roughly the same physical dimensions as the scanning ﬁeld of view (FoV) minimizes the noise contributed by material outside the FoV and helps achieve the highest quality imaging data. To ensure that individual ﬁbers are diﬀerentiated and tracked, at least 10 pixels [8, 29] for the typical ﬁber diameter of 7−10µm are required for carbon ﬁbers, which leads to a pixel size of 0.7−1.0µm. The Zeiss XRadia ™520 micro-tomographic scanner has a cylindrical FoV with a diameter and height of ∼1000pixels, which at that pixel size allows for a diameter and height of 0.7−1.0mm of physical specimen volume. In each specimen, a roughly cylindrical section of diameter 1−1.5mm was manually carved from the narrow part of the dogbones and mounted on the specimen holder. The following scanning parameters were used: source power: 7W, source voltage 80kV, distance from source to specimen: 9.1mm, distance from detector to specimen: 23.0mm, optical magniﬁcation: 10 ×, exposition time: 1.25s, number of projections: 3200, and no ﬁlter (i.e., using air attenuation). This conﬁguration yields a pixel size of 0.764µm, and a total scan time of 2 h 20 m per specimen. Two separate FoVs were chosen inside each specimen to account for local variations in properties. In total, 4 FoVs were scanned at high resolution for each printing conﬁguration: 2 FoV per specimen and 2 specimens per conﬁguration. The free-space specimens, along with a section of feedstock 1.75mm ﬁlament, were also scanned in this manner. Exceptwhenusingthenozzlewith D= 0.25mm,thegrid-likeprintingpatternislargerthanthediameter ofthehighresolutionscans.Toenabletheanalysisofthosemesoscalefeatures,rectangularsectionsmeasuring ∼4mm on one side were also produced and scanned at a larger FoV (meaning at a lower resolution). Here, the scan resolution was reduced considerably since we are mainly interested in the larger mesoscale porosity, notthesmallintra-beadpores,northeﬁbers.Theparametersusedarethesameasabove,exceptfor:distance from source to specimen: 21.1mm, distance from detector to specimen: 9.0mm, optical magniﬁcation 4 ×, exposition time: 0.4s, number of projections: 1600. This yielded a pixel size of 4.69µm and a total scan time of 40 min. One such FoV was produced for each nozzle diameter and print conﬁguration. As the specimen preparation for µCT scanning is destructive (a small section is carved from the dogbones) all imaging was performed after tensile testing. Sosa-Rey et al.: Preprint submitted to Elsevier Page 9 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology Figure 3: Segmentation of mesoscale porosity and deﬁnition of subvolume for homogenization. a) Slice of tomographic data showing a rectangular specimen printed according to the 0◦-0◦pattern and a 0.4mmnozzle. Horizontal printing layers are visible, leaving mesoscale porosity in dark. b) The inter-layer porosity identiﬁed by OpenFiberSeg is highlighted in light blue. c) Cumulative sum of all porosity voxels in the out-of-plane direction. The grid-like arrangement of printing passes is revealed in sharp contrast. The red rectangle is the chosen subvolume for homogenization. 3.3.2. Component extraction (OpenFiberSeg) The tomographic scans were processed with the help of OpenFiberSeg, an open-source software built for the purpose of identifying the constituents in the volume (matrix, pores and individual ﬁbers). With this software,theuserﬁrstmanuallylabelsasmallpatchofdatatoidentifytheconstituents.Thislabellingisthen fed to a machine learning algorithm to identify the remaining pixels in the entire dataset. On a slice-by-slice basis, the pixels containing ﬁbers are automatically grouped into connected regions, and the centroids of those regions are tracked across 3D space to form neatly aligned chains, which are used to infer the presence of individual ﬁbers. Thanks to this extraction, it is possible to build a voxel-based (regularly-spaced volume elements) representation of the microstructure for a speciﬁc specimen. Porosity is labelled using classical image processing methods such as the Canny edge detection algorithm. As can be seen in Figure 3 a), at this length scale, the large pores belonging to the mesostructure and the smaller intra-bead pores can be discriminated according to their size. Each pore size is estimated by the number of connected pixels it contains, then a size threshold ( ∼1%of total FoV volume) was chosen such that only mesoscale porosity is kept. Examination of Figure 3 b), in which the pores above the threshold are in shown in light blue serves to ascertain that all the mesoscale porosity has been labelled. 3.3.3. Subvolume deﬁnition, micro and mesoscale For the purpose of homogenization, one or more rectangular subvolumes inside the cylindrical FoV need to be identiﬁed. These combined subvolumes form the RVE for a particular printing conﬁguration. The FoV must be cropped for several reasons. First, the rectangular subvolume must lie entirely within the material. Then, for the case of mesoscale homogenization, the subvolume dimensions should respect the Sosa-Rey et al.: Preprint submitted to Elsevier Page 10 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology periodic structure formed by the evenly spaced printing layers and extrusion beads, since only a handful of periods is included. Lastly, for the purpose of dual-scale homogenization, the microscale structure containing ﬁbers/matrix/intra-bead pores must be isolated from the mesoscale pores, to not skew the results by including those twice. The shape and size of porosity made by the priting pattern is irregular and the grid-like pattern is not readily visible on a single slice. To reveal this pattern more clearly and to identify suitable subvolumes within a FoV, the following method was used. Across the entire volume, the pixels containing pores were assigned the value 1, and all other pixels the value 0. Then, we took the cumulative sum of all 2D slices in the volume along a reference direction (for instance, the out-of-plane direction in the Figure 3 b), creating a 2D map with high values where the mesoscale porosity in concentrated, and low values otherwise, as shown in Figure 3 c). Here, the mesoscale porosity is revealed unambiguously for the entire volume. A subvolume that is inscribed in the natural periodicity of the mesostructure is thus identiﬁed: the red rectangle in Figure 3 c) encompasses an integer number of layers (starts and ends at mid-layer in the vertical direction) and of passes (starts and ends at bead centers). The same projection procedure was repeated for the case of microstructure homogenization, on the high-resolution imaging data, as shown in Figure 4 for the nozzle with D= 0.4mm and a specimen made with a 0◦-0◦pattern. In Figure 4 a), the input imaging data is shown, with the corresponding material labelling in Figure 4 b). In Figure 4 c), the subvolumes identiﬁed are the largest possible such that they contain only microscale porosity. Figure 4: Segmentation of ﬁbers and microscale porosity and deﬁnition of subvolumes for homogenization. a) Slice of tomographic data of a specimen printed according to the 0◦-0◦pattern with a 0.4mmnozzle, showing horizontal printing layers. b) Fibers and porosity found by OpenFiberSeg highlighted in orange and light blue, respectively. c) Cumulative sum of porosity voxels in the out-of-plane direction. The location of mesostructure porosity across the volume is made obvious, relatively to the single-slice rendering in b). The red rectangles are chosen for homogenization, selected to the maximum size while avoiding mesoscale porosity. Sosa-Rey et al.: Preprint submitted to Elsevier Page 11 of 33 Multiscale FFT-homogenization of AM ﬁber reinforced composites from component-wise description of morphology Table 2 Material properties for base constituents, see [69, 71, 72]. Polymer E(GPa)ν Carbon ﬁbers El(GPa)Et(GPa)νlνtGl(GPa) Victrex 450G PEEK 4.0 0.3855 Toray T300 230 13.4 0.2560.327.3 3.4. Dual-scale homogenization Under the condition of scale separation, the microscale morphology (including ﬁbers, matrix, and the small, inter-layer porosity) can be homogenized by itself if the scale at which this homogenization occurs (a single pass) is large when compared to the dimension of constituent phase change (the size of ﬁbers and pores). Using many subvolumes across several diﬀerent scans enables the estimates given by homogenization to be representative of the variability in the real material. The AMITEX software accepts as inputs a volumetric description containing a number of diﬀerent materials, which can be the organized in zones, for instance for ﬁbers or grains with diﬀerent orientations. For the problem at hand, the isotropic matrix is described by tensile modulus Eand Poisson’s ratio ν. If the axis of the transverse isotropy is /vector e1, and the transverse plane is /vector e2×/vector e3, the ﬁbers are described by longitudinal (along /vector e1) and transverse (along /vector e2or/vector e3) moduliElandEt, respectively, Poisson’s ratios between axes ( /vector e1,/vector e2) and (/vector e2,/vector e3), i.e.νlandνtrespectively, and the shear modulus between axes ( /vector e1,/vector e2), i.e. Gl. The matrix properties were obtained from manufacturer datasheets on Victrex 450G PEEK [69]. The precise carbon ﬁber type is not disclosed, but the most likely candidate are the so-called standard modulus ﬁbers, of which Toray T300 carbon is the most common and cost-eﬀective [70]. The material properties for T300 ﬁbers are taken from [71, 72]. Property values are given in Table 2. Whereas all matrix and porosity voxels can share the same default zone (they are isotropic), the diﬀerent ﬁber orientations are represented using zone descriptors at each voxel, which include a rotated reference frame onto which the stiﬀness tensor is projected. The implementation of the AMITEX software has the number of zones limited to 215= 32768. For large subvolumes containing more ﬁbers than this limit, the orientationangles φandθ(sphericalcoordinates)areroundedtothenearestinteger,andﬁberswithcommon angle pairs are grouped under the same zone identiﬁer, with negligible eﬀect on homogenization output (this was tested on subvolumes below the ﬁber count limit, for which it is possible to not round the angle values). The homogenized stiﬀness tensor for an equivalent solid is obtained by subsequently applying the uniaxial strains in the 3