Prediction and examination of the impact of the raster angle on the orthotropic elastic response of 3D-printed objects using a novel homogenization strategy based on the real clustering of RVEs

The distinct process of layer-by-layer 3D printing generates an anisotropic distribution of mechanical properties. The use of computational models to predict the mechanical properties of 3D-printed objects made by fused filament fabrication (FFF) has received little attention despite a large literature on experimental approaches. The primary objective of this study is to investigate the application of multi-scale computational models in order to achieve precise predictions of the mechanical properties of observed components. A numerical homogenization method is used in combination with clustering algorithms, notably the “K-means” algorithm, to adjust the predicted behavior of the parts to the actual properties resulting from this process as a function of the variation of the raster angle. This method analyzes samples’ internal structure at the micro- and mesoscales. K-means algorithms classify observations to create representative volume elements (RVEs) that match the reel morphologies of the interlayer cavities. The data is utilized to create micromechanical models that calculate effective orthotropic constants based on filament orientation; the derived constants are then used to develop macroscale numerical models that simulate the mechanical response of 3D-printed samples subjected to tensile stress. In summary, the findings suggest that employing the homogenization technique is a reliable approach for forecasting the elastic behavior of 3D-printed elements. Moreover, it is imperative to utilize existing models, such as homogenization utilizing Green’s functions or homogenization based on ideal geometry-material models, in order to obtain an initial approximation of the elastic response of 3D-printed components. Furthermore, the methodology employed in this study, which combines the homogenization process with intelligent clustering algorithms, effectively minimizes the error between numerical simulations and experimental findings. This, in turn, improves the development of precise predictive models that accurately represent the elastic properties of structures fabricated using FFF. This methodology has the potential to be implemented across all materials utilized in FFF manufacturing. This study presents trustworthy prediction laws that enable the designer to conduct a quicker iterative analysis and select the ideal printing process parameters based on FE analysis in order to create high-quality 3D FFF-printed components.


Goals and motivation
The personalized approach of additive manufacturing gives it several benefits over traditional manufacturing, including the ability to produce complex internal and/or external structures, homogeneous or heterogeneous structures, in addition to a significant increase in raw materials [1].These advantages and the latest advances in 3D printing technology have made it possible to produce lightweight components with intricate geometries in many industries (automotive, aerospace, biomedical, etc.).To achieve performance criteria with the least amount of material, optimal densities are used, for example, to create 3D-printed cellular mesh architectures [2,3].
Likewise, as with other manufacturing techniques, the quality of components produced through additive manufacturing technology is contingent upon the constraints inherent to the process and the imprecision of the machinery.Microscopic deformations at a small scale are usually the cause of crack formation and sudden alterations in the mechanical characteristics of components.Nevertheless, the presence of a robust bond within the filament bonding area leads to delayed crack propagation and enhanced durability of the component.These crucial bonding regions often span a few tens of microns, making it challenging to accurately assess the mechanical properties within these zones using conventional macro-scale mechanical experiments.Similar to the manufacturing process of composite materials, the determination of mechanical properties at core-skin interfaces is typically a complex task.A study was conducted to investigate the nanoindentation measurement of local viscoelastic properties in the interphase region of a sandwich composite.The properties examined included creep compliance and Young's relaxation modulus, with a particular focus on their time-dependent behavior.The study's findings indicate that the nanoindentation technique is a valuable and efficient method for evaluating the mechanical properties and bonding integrity within the interphase region of sandwich composites [4].An alternative approach is proposed in a recent study to enhance the interfacial shear strength of carbon fiber-reinforced polymer composites.This method involves the utilization of carbon nanotube (CNT) sheets, which are wrapped around individual carbon fibers.Nanoindentation tests with both experimental and computer-based methods were used to measure the interfacial shear strength of composites with and without carbon nanotube (CNT) sheets.The results show that adding CNT sheets, especially when they are wrapped around carbon fibers, can increase the interfacial shear strength of composites by a large amount.This improvement holds significant potential for enhancing the overall performance of such composite materials [5].In addition, the optimization of resin absorption is of utmost importance due to its significant influence on the stiffnessto-weight ratio of sandwich composites.It is worth noting that there is a specific range of resin absorption that is suitable for each type of core-sandwich composite.Moreover, the failure mechanism of sandwich composites is contingent on the stiffness of the core and the texture of the surface.Introducing grooves and perforations on the surface of the core can enhance the flow of resin and enhance resistance against delamination in the region where the skin and core interface [6].
Within the realm of additive manufacturing, numerous studies have been carried out in the past to compare designed and printed parts in terms of dimensional accuracy [7,8], roughness [9,10], other manufacturing criteria [11,12], and even material attributes [13,14].The aim of this paper is the fused filament fabrication (FFF) process.During this process, the filament goes through fundamental phase transformations that change not only its geometry but also its mechanical properties.Furthermore, the parameters and orientations of construction have a big impact on the performance of the final product.Three-dimensional FFF additive manufacturing produces parts with heterogeneous, anisotropic, or orthotropic distributions of material properties since an identical model printed at two distinct deposition angles will result in two parts with differing mechanical properties.Numerous studies and experiments have been carried out [13] to predict the material properties and understand the effects of various construction parameters on the behavior of final components.Test specimens have been printed in shapes that adhere to established standards (ASTM D7264 [15]).
The correctness of the design model for printed components depends on its capacity to accurately represent both the geometrical characteristics of the model and its mechanical properties.In this perspective, three distinct scales may be used to evaluate the material attributes (Fig. 1).The microscale generates the phase transitions of the material, which facilitate the deposition of the material layer by layer at the mesoscale; these layers will be fused and solidified to build the final part with effective properties at the macro-scale.The computational substrates that enable a direct mechanical analysis of the printed component are necessary for the transition of Fig. 1 From the filament to the final part through the three scales (micro, meso, and macro) 3D printing from the prototyping stage to the industrial world.The evaluation and prediction of the effective properties of the printed material are the goals of this work.

Overview and contribution
This work suggests a new way to model and predict the effective (macroscopic) material properties inside components made by the fused filament fabrication (FFF) process through homogenization.This technique permits the transition from a known specific geometry at a finer scale with multi-phase material characteristics to a simplified "effective" geometric domain at a larger scale with single-phase material characteristics (solid).Typically, a cuboid performs as the effective domain, and the material characteristics are chosen based on the average relationship between stress and strain on that cuboid.Although the representative volume element (RVE) must be carefully chosen and characterized, it is frequently difficult to specify geometry and material properties at the micro and mesoscales, making the homogenization of printed components a challenging task.This paper examines the problem in three steps.
The initial stage focuses on modeling the RVE (mesoscale) to produce the final part's anisotropic properties through homogenization.Based on the planning of the manufacturing process in the form of a G-code, this study will focus only on the variation of the deposition angle and its effect on the material properties.Creating a representative volume element (RVE), which reflects with precision the anisotropic, or orthotropic characteristics of the constructed structures and the properties of the heterogeneous distribution, is a crucial step.In this approach, data clustering [16] and, more specifically, K-means algorithms [17] classify the various shapes of the voids between the filaments (Fig. 2).This useful mesoscale geometry-material model's implicit classification can be assessed on demand for additional processing.And this is the first contribution to this paper.
The second stage includes evaluating the results with experimental and numerical results as well as analyzing the influence of the RVE's shape on the material's effective properties.
As the second contribution of this article, the validated results will be used in the third step to predict with a reasonable level of uncertainty the various properties of the material as a function of the deposition's angle.

Fused filament fabrication (FFF)
There are numerous flaws produced by the additive manufacturing (FFF) technique.Differential cooling of the component causes warpage.A dimensional error brought on by distortion and shrinkage as a result of the thermoplastic cooling at differing rates.When the filament is extruded, the layers may not adhere well together.Recesses between layers caused by the filament's oval shape, the nozzle's downward push, and the partial solidification of the other layers result in a concentration of stresses where cracks may occur when the part is subjected to loads.Anisotropic and orthotropic features are the outcome of these many flaws.
The transition from an isotropic raw material to an anisotropic material generates differences between the properties of the designed model and the real properties of the printed part.In this regard, several studies have focused on the one hand to comprehend the impact of process parameters on the behavior of printed components.Experimental investigations and numerical analyses (Table 1) are the two categories that might be used to further classify these studies.In the paragraphs that follow, we shall present a synopsis of some of these works.

Numerical analysis
A restricted quantity of numerical analyses has been undertaken to assess the structural performance of parts produced using fused filament fabrication (FFF) technology.An investigation employing finite element microstructural modeling and homogenization techniques to compute the effective orthotropic properties of the printed components and subsequently verify their accuracy through experimental measurements aims to analyze and simulate the mechanical properties of printed parts, specifically focusing on tensile strength, shear strength, and flexural strength.The research will consider various process parameters, including raster angle, build orientation, and infill density [13]; the findings indicate that the homogenization technique is a dependable and effective method for replicating the mechanical Fig. 2 Optical microscope image of a cross-section of anilox layers showing the shape of individual filaments after deposition, where w and h represent the width and height of the filament [13] characteristics of 3D-printed components.A comparable study offers an analysis and numerical simulation of the structural performance of samples produced using the fused filament fabrication (FFF) process, wherein the infill density values are varied.The researchers employ numerical methods in order to investigate the impact of process parameters on the mechanical characteristics of fused filament fabrication (FFF) components fabricated from polylactic acid (PLA) material.The researchers conduct a comparative analysis between the outcomes of tensile tests and finite element simulations, subsequently deriving conclusions based on their findings.The objective of this study is to forecast the structural efficacy of fused filament fabrication (FFF) components through the utilization of numerical models [18].Another study presents a novel methodology for modeling and estimating the effective elastic properties of components produced through fused filament fabrication (FFF), a widely used additive manufacturing technique.The methodology comprises of two primary phases: initially, a proficient mesoscale geometry-material model of the printed structure is formulated by utilizing the process plan and measured material properties; subsequently, the mesoscale model is homogenized through the application of Green's function method in order to derive the effective macroscale elasticity tensor.The validity of the proposed method is established through a comparative analysis of the computed results with those obtained from the finite element method and physical tests.The study's findings indicate that the two-stage approach, as proposed, demonstrates efficacy in modeling and estimating the effective material properties of structures produced through fused filament fabrication (FFF) printing.The methodology exhibits adaptability and can be employed in various additive manufacturing techniques featuring distinct patterns of material deposition [19].A different search introduces a computational homogenization technique for materials fabricated through 3D printing, utilizing a reduced order model known as mixed transformation field analysis (MxTFA).The proposed methodology has the capability to accurately account for the influence of the microstructure and the elasto-plastic characteristics of the printed material on the overall response.The periodic microstructure of the 3D-printed material is represented by employing a unit cell that consists of a fiber and voids.The stress and inelastic strain fields are estimated through the utilization of self-equilibrated modes within each subset of the unit cell.The methodology is employed on various unit cells and loading scenarios, and subsequently compared with experimental data and finite element analysis outcomes.The methodology exhibits commendable precision and effectiveness in forecasting the mechanical characteristics of materials produced through three-dimensional printing.The proposed methodology incorporates the consideration of microstructural characteristics, such as fiber orientation and void density, in order to analyze their impact on the overall constitutive behavior at a macroscopic level.
The proposed method has the potential to decrease both computational time and memory storage requirements in comparison to traditional non-linear finite element analyses [20].

Experimental investigations
An experimental investigation presents a research focusing on the phenomenon of inter-layer failure in fused filament fabrication (FFF) 3D printing of polylactic acid (PLA) material.Inter-layer failure is a prevalent defect that significantly diminishes the tensile failure strength of the printed objects.The researchers performed tensile experiments using various layer thicknesses and printing angles, and observed the distribution range and characteristics of inter-layer failure.Additionally, a novel generalized strength model has been proposed, which is founded on the principles of stress invariant and transverse isotropy.This model aims to predict the tensile failure strength associated with inter-layer failure.The model was validated using empirical data and demonstrated high levels of accuracy and applicability.The researchers discovered that inter-layer failure exhibits certain patterns, including an expansion of distribution range in relation to layer thickness, a reduction in tensile failure strength with printing angle, and the presence of a generalized maximum tensile failure strength and a gradually varying range [21].Alternative experimental research investigates the anisotropic material behavior of 3D-printed composite structures through the utilization of material extrusion additive manufacturing (AM) technology.This article examines the influence of various factors, including printing strategy, build orientation, layer thickness, layup order, and material composition, on the mechanical properties and failure modes of printed components when subjected to different loading conditions.The article additionally presents a comparison between the experimental findings and the classical laminate theory, highlighting the impact of the printing strategy on the functional part's performance.The primary outcomes of the mechanical examination conducted on 3D-printed components employing various printing methodologies.It indicates that the build orientation of the model plays a crucial role in influencing the flexural properties, failure behavior, and overall performance of the printed parts.Additionally, it is indicated that the thickness of the layers, the order in which they are laid up, and the composition of the materials all have an impact on the ultimate material characteristics of the components.The statement implies that it is imperative to consider the printing strategy and build orientation of the crucial cross-sections of models in order to guarantee the durability and reliability of the produced components [22].Another study examines the impact of process parameters, specifically printing temperature and speed, on the mechanical properties of 3D-printed polylactic acid (PLA) lattice structures.The investigation employs digital image correlation (DIC) and scanning electron microscope (SEM) techniques to analyze and evaluate the outcomes.
The study fabricates tensile samples and lattice structures using the fused filament fabrication (FFF) technique, followed by conducting experiments to evaluate their tensile and compressive properties.The study examines the load-displacement curves, stress-strain curves, displacement fields, strain fields, and fracture morphologies of the specimens.The study's findings indicate that the most   Fig. 5 Uniaxial and pure shear strain state; a longitudinal strain mode; b, c transverse strain modes; d, e, f shear strain modes in the XY, YZ, and XZ planes, respectively [13] that the processing parameters exerted varying effects on both the dimensional accuracy and mechanical properties.The researchers reached the conclusion that the temperature of extrusion, thickness of layers, percentage of infill, and pattern of infill all had notable impacts on the dimensional accuracy and mechanical properties of parts produced through FFF.However, it is important to note that there was no universally optimal combination of values that achieved the desired outcomes for both objectives simultaneously.Additionally, it was discovered that the FFF technique typically resulted in the production of components with dimensions that were larger than those specified in the computer-aided design (CAD) model.Furthermore, it was observed that the magnitude of the dimensional error exhibited a positive correlation with the nominal dimension.The authors proposed that achieving improved dimensional accuracy necessitated a reduction in extrusion temperature, a decrease in layer thickness, a lower infill percentage, and the adoption of a hexagonal infill pattern.Conversely, attaining higher strength and stiffness required an increase in extrusion temperature, optimization of layer thickness, utilization of a triangular infill pattern, and a higher infill percentage [25].

Comparative and validation
To approve the approach proposed in this document, a comparative study is made between the results obtained by FEA simulations of this approach and the results achieved by simulations and mechanical tests performed by Gonabadi et al. [13].

FE microstructural model of tensile test specimens
Using the ANSYS software, the elastic behavior of the materials in the current investigation was homogenized using finite elements based on the mesoscale filament structure.The isotropic properties of PLA were calculated using Bollard-type tensile clamps [14,29] as E = 3500 MPa and v = 0.35, and these values were applied as the input data for the finite element models.The form of each filament and the overlapping region that was identified under the microscope are both visible in Fig. 2. Using a calibrated optical microscope, the filament's height (h) and width (w) were determined to be 0.2 mm and 0.4 mm, respectively.These observations allowed the development of a more realistic and accurate geometric model for FE of the mesostructure of the test samples printed by the FFF 3D.In the validation step, the input data for the finite element models were the homogenized orthotropic characteristics of PLA as a function of the raster angle variation.Figure 4 roughly describes the deposition angle configurations.

Macro-scale FE modeling based on homogenization
To describe the tensile specimen design for various printing process parameters, the macro-scale FE model uses the orthotropic properties of the RVE.Boundary conditions are included in the internal design of the RVE in the FE model.The RVE micro-models (Fig. 6) are separately subjected to each of the six deformations (Fig. 5) generally adopting the periodic boundary conditions.This provides effective elastic orthotropic engineering constants for the RVE which are then used as input to the FE simulation.Notably, neither explicit microstructural modeling nor neither finite element homogenization techniques take into account the viscoelastic or plastic behavior of PLA materials in the constitutive behavior of the material.

The constitutive material behavior of 3D-printed specimens
In this comparative research, the material behavior in the finite element stress analysis is taken into consideration by evaluating the constitutive behavior of printed specimens at three tensile deposition angles.
The nine elastic constants of the orthotropic constitutive equations are composed of three Young's moduli (E x , E y , and E z ), three Poisson's ratios (v xy , v xz , and v zy ), and three shear moduli (G xy , G xz , and G yz ).The stress-strain relationship is described as follows for an orthotropic material: where S is the compliance matrix: A numerical homogenization method was employed to calculate the various compliance matrix coefficients. (1)

Homogenization process
A representative volume element of the material with repeating unit cells RVE (Fig. 6) is taken into account for the numerical homogenization analysis [30].By assuming that the stored strain energy in the heterogeneous RVE volume (V rve ) is the same as that of a homogeneous RVE, the effective properties of heterogeneous materials may be determined using the homogenization approach.The following formula is used to compute the stored strain energy of the heterogeneous RVE volume (V rve ): Similarly, the corresponding homogeneous RVE's strain energy is defined as follows: where ij and ij are obtained by averaging the local stress and strain fields across the RVE volume (V rve ) and strain over the RVE volume, respectively (V rve ).
By setting periodic boundary conditions on the RVE and inserting Eq. 1 into Eq.4, it is possible to identify the compliance matrix components (orthotropic elastic constants) in Eq. 2. This is achieved by applying the boundary conditions for the uniaxial (Fig. 5a, b, and c) and shear deformation (Fig. 5d,  e, and f) states.For each deformation state in Fig. 5, the FE outputs are used to calculate the volume average stress, strain, and total strain energy, which are then applied to calculate the numerical prediction of the orthotropic elastic properties. (3) When a displacement is applied to the RVE's surface, boundary nodal forces are generated at the impacted boundary surfaces.As a result, the amount of stress corresponding to the applied strain (applied displacement divided by the length of the RVE) is calculated by dividing the area of the affected surface by the reaction force, which is represented by F in Eqs.6 and 7.As a consequence, Young's modulus and shear modulus are computed as illustrated in Eqs.6, 7, and Fig. 7.By calculating the transverse strain and dividing it by the applied strain, Poisson's ratio may also be obtained.
where F* is the sum of the nodal forces of the front surface along the x-axis and F** is the sum of the nodal forces of the upper surface along the x-axis.

RVE based on the idealized filament geometry
The first objective of this paper is to generate a geometrymaterial model to simulate this manufacturing process.At each instant, the printer nozzle deposits an element of manufacturing volume (EMF) whose primitive shape can (6) be represented as a simple super elliptic with dimensions W, H, and L. The length L represents the sweep of the EVF along the G-code-imposed extrusion tool path (Fig. 8).
The homogenization process proposed in [13] starts with an isotropic (E = 3500 MPa) base material (PLA) and ends with an effective orthotropic material whose mechanical properties change depending on the deposition angle.This process uses an idealized RVE.The difference between the results of this model and those of the corresponding experiments carried out varies between 6 and 15.4% (E x = f (Angle of deposition)).
Such an idealized model [13,18] is necessary to make a first approximation of the geometry-material model at the mesoscale and to get ideas about the effect of process parameters on mechanical behavior.On the other hand, most of the differences between mechanical and designed performance stem from the process planning stage; therefore, the idealized inter-layer recesses as well as the contact areas between the layers modeled by this ideal model have a much larger impact on the actual properties of the homogenized material These process imperfections, such as warpage, cannot be modeled by an idealized model.This model is limited for the estimation of the total volume of the basic material when the distance between the filaments is different from the width of the extruded filament; in fact, the solidification of the layers is not a conservative function of the volume [19].It is obvious that the idealized model is restricted and cannot generate accurate results because of the arbitrary inter-layer orientations and recesses inside the printed part.

RVE based on the real geometry of the recesses
It is typically very difficult to predict the elastic properties of the material printed by the process.This is because, on the one hand, the stiffness of the split filaments in the adjacent bonding zones changes significantly, and, on the other hand, the bond quality depends on the arbitrary diffusion of the interlayer cavities [31].
The combined complexity of these physical phenomena limits the possibility of accurately predicting the material characteristics of models with idealized geometry.The idea of the proposed approach is to incorporate the actual profiles of the gaps into the homogenization process to generate effective properties.The classification process used to determine the shape of the real profiles of the RVEs is based on the image of the cross-sections.
In order to achieve accurate convergence of the K-means algorithm, it is essential to perform monochromatic image processing as a preliminary step (Fig. 9b).The next step is to segment the image using the recess locations, which produces a base with N observations (Fig. 9c).Subsequently, this base is partitioned into K distinct clusters.The entire image shows the complete effective domain of the measurement, and each observation represents the effective domain of the RVE and is included in the class with the closest centroid.

Classification based on K-means algorithms
As previously mentioned, the classification based on K-means unsupervised learning algorithms consists of grouping N observations (Fig. 10) into K classes, of which each observation belongs to the closest class.This method is mainly based on three parameters [17]: -The number of classes "K." -The initial values of the centroids of the classes {C j /j = 1,…, K}. -The measures of the Euclidean distances between the centers of the observations {X i /i = 1,…, N} and the centroids of the classes (C j ).
The process of classifying observations based on their numerical characteristics [32] is illustrated in the following flowchart (Fig. 11).

The optimal choice of the total number of classes
The number of classes K is the most crucial parameter in the classification process by the K-means algorithm; the elbow method [33] is used to determine the optimal number K; and the silhouette score [34] is generated to evaluate the quality of the classification according to K.

Elbow method
The elbow method [33] is an empirical method that measures the cohesiveness of a class grouped with data that are similar to each other.The process of this technique is to calculate the variance (Eq.9) or the sum of squared errors within each class (WSS).
The optimal number of clusters K is chosen when there is a sharp decrease in variance, which results in an angular structure in the form of an elbow in the graph.However, this point is not always clear-cut.However, the quality of a class will not be determined by this method.Figure 12 shows that there are two optimal values (K = 4 and K = 7), which causes ambiguity in deciding which is the best value of K.
The silhouette scores This method [34] allows for the evaluation of the quality of the coherence of the observations; the silhouette score (Eq.10) measures the similarity of an a(i): the average distance of observation i from the observations of the same class (division by |C I | − 1 due to the exclusion of the distance d (X i , X i ) in a(i), Eq. 11).b(i): the minimum average distance of observation i from all other points in neighboring classes (Eq.12, C J ≠ C I ).
Equation 10 can be written as follows: The coefficients s(i) are included in the range [−1; 1].a(i) measures how cohesively an observation fits into its class; hence, a small value of a(i) indicates that the ith observation fits in strongly with its class.The separation, b(i), is measured; a high value of b(i) indicates that the observation i is not coherent with the nearby class; consequently, the closer s(i) is to 1, the more accurate the classification; and finally, the overall average of the coefficients, s(i), must be maximal to determine the best value of K.
The representative curve (Fig. 13) reaches its maximum when K equals 4, thus the elbow method shows that the optimal values of K are 4 and 7.This confusion is resolved by the silhouette score, so K = 4 is the optimal choice for the classification process used.

Classification results
The implementation of the optimal number of classes (K = 4) in the classification process by the K-means algorithm [17] led to a consistent classification of the observations of the RVE profiles.
The features {X i } of each observation are compressed into 2D vectors by principal component analysis [29] to facilitate visualization of the distribution of observations.( 13)  Figure 14 clearly illustrates the existence of four distinct classes; each class contains a number of observations (headcount).A statistical study is made according to the results of the classification: first, a ranking of the distribution of observations in descending order is performed, and then each class is represented by its headcount, its frequency f(i), and its cumulative frequency F(i).
Figure 15 shows that class N°3 is the dominant one with a frequency equal to 33.33%.On the other hand, the cumulative frequency trend tends to be linear; therefore, the distribution of the observations of the RVE profiles in our basic sample is also linear.Each category of RVEs has a substantial impact on the structure in its entirety and, consequently, the properties of the whole effective domain.

Design of RVEs
Obtaining the four classes results in the creation of the 2D models of the representative RVE profiles for each class and the construction of the 3D RVEs by extrusion of a length L of the appropriate 2D profiles; these 3D models are then orientated according to the required angle of deposition (orientation concerning the X axis).During the homogenization stage of the effective domains of the REVs, a periodic mesh with adaptive standards towards the edges is employed.
To establish the elastic characteristics of the effective domains of the RVEs of each class, the generated representative RVE models (Fig. 16) are implemented one by one as input geometries in the homogenization process, using PLA (E = 3500 MPa, v = 0.35) as the base material.The flexibility matrices of each class are denoted by [S i ], i = 1,..., K.

Generation of the effective elastic properties of the entire domain
The material properties of the entire effective domain of the sample are computed using two approaches, the first of which is statistical and based on the classification's direct findings.The classes are spread out linearly across the whole domain, and since each class affects the effective material properties, the total domain's flexibility matrix is the linear combination of the flexibility matrices for each class.Hence, the flexibility matrix of the total domain can be written as follows: Table 4 Elastic properties of PLA material according to the approach of this paper and the reference's approach [13] Properties Homogenization EF of this paper Homogenization EF of reference [13] Experience of reference [13] ± 60 Fig. 18 The error between the FE results and the experimental results [13] -[S D ] is the flexibility matrix associated with the entire effective domain.-[S i ] is the flexibility matrix associated with class i.
f(i) is the frequency of class i.
The second technique involves locating the observations of each class inside the overall effective domain (Fig. 17) and then re-homogenizing the entire domain; the input data for the material are the homogenized effective properties corresponding to each class.
The above figure clearly shows that each class contains neighboring observations; this is rational in the sense that the thermoplastic deformation field is arbitrary (not uniform in the entirety of the printed components [31,35]), but it is roughly a locally uniform field.Consequently, it is evident that adjacent observations belong to the same class, and the distribution demonstrates the effectiveness of the suggested classification strategy.( 14)

Results and discussion
To comprehend the effect of the raster angle and to validate this approach, the numerical results of this work are compared with the numerical and experimental results of reference [13], in particular the longitudinal elastic modulus E x .For this purpose, the effective orthotropic properties of the RVEs and those of the entire effective domain are calculated.These parameters are then applied as a starting point for the numerical modeling of the tensile strength as a function of the deposition angle.

Homogenization results
The RVEs employed in this work are generated entirely based on the real topology of the mesostructure, i.e., the microdeformation caused by the process is taken into consideration.The following table (Table 3) illustrates the various components of the flexibility matrices of the classes, as well as those of the entire effective domain.
-D Tf represents the discretized effective total domain whose associated effective properties are computed by Eq. 14.The calculated errors between the effective properties of the total domain estimated by Eq. 14 and those obtained by the re-homogenization do not exceed 1%; therefore, the rest of the calculations will be performed by Eq. 14. (15)

Comparison of outcomes
Before validating the proposed method and figuring out how the angle affects how the parts behave, a comparison is made between our homogenization results based on the real geometry of the inter-layer recesses and the homogenization results based on the idealized geometry of the interlayer gaps, as well as the elastic properties of the material obtained from the mechanical tests [13].
The homogenization results based on classification RVEs are extremely close to the experimentally observed values and are included in the experimental tolerance interval, as shown in Table 4.In contrast, the outcomes obtained from an idealized RVE are distant from the experimental results.This paper's methodology yielded results similar to those obtained from mechanical tests [13].The comparison of the errors between the effective material properties obtained by this technique of homogenization and those generated by idealized RVEs [13] with the experimental measurements of mechanical properties [13] revealed that the errors of the idealized homogenization range between 11.98 and 18.31% for E 2 , E 3 , G 12 , G 23 , and G 13 , and 6.67% for E 1 , whereas the errors of the homogenization of this paper range between 0.43 and 7.13% for E 1 , E 2 , G 12 , G 13 , and G 23 , and 12.69% for E 3 (Fig. 18).

Validation of the results
For engineering and industrial applications, a 10% difference between the simulated and experimental outcomes is deemed acceptable and modest enough.The quantitative evaluation of the accuracy of the proposed approach determined that the error values of this method are generally low, in the sense that five out of six of the properties do not exceed 7.13% sufficiently lower than 10%, except for the transverse elastic modulus E 3 , which presents an error value of 12.69% compared to the average value of the same modulus measured experimentally, but which remains very close to the 10% threshold.In contrast, the homogenization error values of the idealized RVEs [13] significantly exceed the 10% threshold; therefore, our model allows the simulation of the behavior of the parts in tension, as is demonstrated in Fig. 19.Additionally, the effective orthotropic engineering constant E 1 obtained by this study falls within the tolerance range of the experimentally measured value of this component.

Effect of the deposition angle on the printed parts under tensile test
The validity of this study, as well as the consistency between the numerical results obtained and the measured results [13], allowed us to perform the various numerical simulations of the tensile test (ASTM D638-08 [27]).The input data of the material are the effective orthotropic components obtained by our approach, taking into account the deposition angle (0°, 45°, and 90°) (Fig. 20).The plastic behavior is not taken into account in this study, for this reason, tensile simulations were performed for the same loading case (F = constant) on specimens having raster angles (0°, 45°, and 90°) and longitudinal strain fields (Fig. 21d, e, and f) were captured for the same value of normal stress fields (38.5 MPa) (Fig. 21a, b, and c), and the increase of the local strain field from 1.2% (mm/mm) for the 0° angle to 1.9% (mm/mm) for the 90° angle demonstrates the effect of this parameter.In addition, the maximum tensile stress that has been measured in previous studies [13] and  This decrease in maximum tensile stress (stress at fracture) and the resulting increase in strain fields for parts printed at angles ranging from 0 to 90° are mostly attributable to the rising localized at the interfaces between the oriented filaments.
In reality, these interfaces are more likely to exhibit interlayer cracks during component deformation, and filament separations are unavoidable under tensile stress.For components produced at 0°, the tensile stress is parallel to the filament axis; therefore, a ductile fracture mode with considerable plastic deformation is anticipated.For a 90° angle, the applied tensile stresses are perpendicular to the filament axes; thus, a brittle fracture is anticipated due to the abrupt collapse of the interlayer link, and the strain-stress curve will have a linear form followed by a quick fracture.In the intermediate instance, when the angle is less than 45°, the material's tensile behavior tends to be ductile, but when the angle is more than 45°, the material's tensile behavior tends to be brittle.
Figure 22 illustrates the strain-strain curves of the simulated material behavior (PLA) under stress.The results of the finite element simulations based on the methodology described in this document agree with the experimental results.Furthermore, the longitudinal elastic modulus E 1 decreases by 34.4% and 33% for the outcomes of this approach and the experimental data [13], respectively, as the angle increases from 0 to 90°, whereas the decrease estimated by FE simulations based on the homogenization of idealized geometries is only 25%, proving the success of the approach presented in this work to accurately predict the behavior of components printed by 3D FFF slit filament deposition.

Summary and industrial applications
The previous findings demonstrate that the combination of homogenization techniques and K-means clustering algorithms yields effective material properties that reflect the actual mechanical behavior of components produced through the fused filament fabrication (FFF) additive manufacturing process.To enhance comprehension, Table 5 outlines the findings regarding the orthographic properties in three key orientations: 0°, 45°, and 90°.The purpose of this table is to serve as a brief reference tool that aids readers in comprehending the fundamental components of our strategy.
The validation of finite element analysis findings against experimental data is of utmost importance in the field of additive manufacturing (3D printing).This study highlights the significance of such validation in various industries, including automotive, aerospace, and biomedical.By accurately determining the optimal material distribution within a given volume under mechanical constraints, this validation process can lead to substantial cost and time savings in the manufacturing of components and load-bearing structures.The integration and optimization of boundary conditions, material types, and 3D printing process parameters, internal microstructures, fill densities, and layer heights, can be accomplished by employing finite element method (FEM) techniques, such as ANSYS SpaceClaim design tools.In a more precise manner, the automotive industry is actively endeavoring to address the challenges associated with manufacturing expenses and production timeframes by optimizing the utilization of materials in high-volume manufacturing operations.In actuality, a marginal decrease (of a few grams) per individual vehicle, across a cumulative quantity of several thousand units, yields substantial material conservation.The aerospace sector has demonstrated a significant level of interest in the utilization of finite element analysis (FEA) for the purpose of 3D printing.Specifically, there is a focus on employing topological optimization techniques to reduce both weight and expenses.Aircraft that possess reduced mass exhibit decreased energy consumption, resulting in substantial cost savings for an airline.The healthcare sector has exhibited a notable inclination towards design methodologies, specifically in relation to the production of personalized implants.Finite element analysis (FEA) tools designed for 3D printing applications, including lattice optimization tools, offer significant advantages in replicating bone density while simultaneously minimizing the weight of components.Numerous implants integrate lattice structures, thereby providing comparable strength to implants that are designed and produced through traditional methodologies.

Prediction of the orthotropic properties of parts as a function of the deposition angle
Based on the classification of the real profiles of the interlayer recesses, the accuracy of the geometry-material model makes it possible to come up with a scaling rule for predicting how the elastic behavior will change.First, various geometry-material models as a function of the raster angle's variation were generated using the suggested method.Next, each effective orthotropic constant was plotted as a function of angle [0°; 90°].Lastly, polynomial laws of the fourth order were assigned to each orthotropic constant.Table 6 Difference between the orthotropic components determined by FE and experimentally [13] Properties Homogenization FE of this article Homogenization FE of reference [13] Reference experiment [13] Difference between FE [13] and EXP [13] Difference between FE of this article and EXP [ As the angle increases from 0 to 90°, experiments [13] and [21] show that the longitudinal elastic modulus E 1 decreases by 34.4%.The modulus E 2 remains almost constant, whereas the modulus E 3 shows a considerable rise (Fig. 23).The behavior of the elastic moduli E i , = 1, 2, and 3, is predicted by Eqs. 16, 17, and 18, respectively.
-X represents the angle of deposition.
-R 2 represents the coefficient of determination.
According to the shear tests [13], the modulus of G 12 shears by 16% when the deposition angle increases from 0 to 45° and by 15% when the angle increases from 45 to 90°.The modulus of G 23 drops as the deposition angle grows from 0 to 15° and then increases when the angle surpasses 15°; the modulus of G 13 decreases as long as the deposition angle is less than 75° and then increases when the angle is more than 75° (Fig. 24).The following equations describe the behavior of the moduli G 12 , G 23 , and G 13 , respectively: The fish coefficient v 12 declines in a non-significant manner when the angle is less than 45° but becomes increasingly significant when the angle surpasses 45°.As the angle grows from 0 to 90°, v 13 reduces from 0.35 to 0.239, whereas v 23 increases from 0.23 to 0.35 (Fig. 25).These equations define the behavior prediction of the coefficients v 12 , v 23 , and v 13 , respectively:

Conclusion
The subsequent behavior of the printed items is influenced by the many settings of the 3D FFF process.Although an isotropic material such as PLA is used for printing, the structure and mechanical behavior of the printed parts are orthotropic.As a way to quantify the various orthotropic effective components of the softness matrix of printed parts, a geometry-material model based on the classification of inter-layer recesses by K-means algorithms and the approach of multi-scale numerical homogenization is described.
First, this work has demonstrated that idealized geometrymaterial models are constrained and unable to precisely predict the behavior of 3D-printed components; rather, they can only be utilized to generate a first estimate of the behavior.
Then, a geometry-material model based on the real profiles of the inter-layer recesses was developed using the previously described procedure: slicing of the microscopic image based on the inter-layer recess locations to build a base of N observations; classification by K-means algorithms to divide the N observations into K classes; generation of representative 3D models for each class; homogenization of the effective domain of the classes; and re-homogenization of the total effective domain of the test component.
Finally, the outputs of the proposed method are compared to the numerical results obtained [13] and confirmed by experimental testing [13].This comparison determined that the model presented in this study is compatible with the experimental data with errors that do not exceed 10% (Table 6); therefore, this model has demonstrated its success in predicting the behavior of parts subjected to traction for different values of deposition angle (0°, 45°, and 90°); therefore, comprehend the effect of this parameter on the behavior of parts printed by the 3D FFF process.
Although the FE methodology developed in this study can accurately predict the elastic properties of 3D-printed parts, there is a significant difference of 12.69% for the elastic modulus E 3 and 16.67% for the Poisson's ratios ν 12 and ν 13 , primarily due to the effect of the interface bonding between the deposited filaments.Despite the fact that this approach was able to accurately represent the actual morphology of interlayer voids, it is limited in representing the filament bonding zone in the extrusion direction.Thus, the finite element (FE) model assumes a flawless bond between filaments at the interfaces, whereas this assumption does not hold true for 3D-printed components undergoing mechanical testing.The homogenization technique focuses primarily on the selection of a realistic RVE equivalent to the mesoscale and/ or macroscale of the structure, i.e., the more the RVE tends towards the reality of the microstructure, the more the margin of error between the FE simulations and the expected results of the experiment tends towards minimum percentages lower 5%, which is valid for the exploitation to manufacture parts that fulfill the requirements.
The numerical methods developed in this study have demonstrated their ability to predict the elastic properties of 3D-printed structures with high accuracy.Moreover, it is worth noting that the discrepancies in numerical values between finite element simulations and experimental findings reported in previous studies, which relied on idealized geometric-material models, are significantly larger compared to the errors produced by the methodology applied in this research paper.Consequently, the strategy implemented in this study lends a high degree of reliability to the prediction laws obtained.
This work demonstrates the capacity of numerical approaches to predict the elasticity of 3D-printed structures (Section 5).This can result in a considerable decrease in the number of mechanical tests necessary to evaluate the performance of 3D-printed parts, saving both time and money.In addition, the method utilized in this study enables the designer to conduct a faster iterative analysis and pick the optimal printing process parameters based on the finite element analysis (FE) to manufacture high-quality 3D-printed components using the 3D FFF process.
In it is anticipated that additional structural properties will be analyzed, along with an examination of the impact of build orientation and infill density on the structural performance of fused filament fabrication (FFF) parts.Furthermore, the development of a rudimentary multiparametric predictive model will be proposed, which considers the fluctuations in deposition angle, build angle, and eventually infill density.

Fig. 3 a
Fig. 3 a A schematic of the FFF procedure [13], b tensile specimen's raster angle

Fig. 7 Fig. 6
Fig. 7 Prediction of Young's modulus, Poisson's ratio, and shear modulus when the RVE is subjected to displacement in the x direction

Fig. 8 Fig. 9 a
Fig.8The shape of the manufacturing volume element (EMF)

Fig. 13 Fig. 14
Fig.13 The representative curve of the silhouette scores

Fig. 17
Fig. 17 Location of classes in the total effective domain

Fig. 21
Fig. 21 Simulated strain field distributions (d, e, and f) for angles 0°, 45°, and 90°, respectively, captured for the same 38.5 MPa stress field (a, b, and c) for the same deposition angles ) of the localized classes.

Fig. 25
Fig.25 Allures of the fish coefficients v ij , i,j = {1,2,3} as a function of the variation of the raster angle

Table 3
The effective properties of the different classes and the total effective domain and the error between the properties of the discretized total domain (D Tf ) and homogenized total domain (D Th )

Table 5
The mechanical orthotropic proprieties of this approach for different raster angles 0°, 45°, and 90°R