1 Introduction

Over the last few decades, additive manufacturing (AM) has become a novel alternative in the manufacturing sector, introducing a unique approach to producing mechanical components along with various benefits. Continuous research is being conducted to enhance the printing processes for diverse materials, including metals, ceramics, polymers, and composites [1, 2]. A key benefit of the AM process is its ability to fabricate parts with intricate shapes and varied densities, thereby minimizing material usage without sacrificing the strength and integrity of these components [3]. Particularly for polymer-based parts, there is an emphasis on balancing the advantage of lightness with the requisite mechanical robustness to make them suitable for more challenging applications [4]. Commonly, processes like fused filament fabrication (FFF), selective laser sintering (SLS), and stereolithography (SLA) are preferred for the reliable construction of these polymer components [5].

During the printing phase, the internal configuration of the parts is crucial for maintaining mechanical properties, reducing weight, and minimizing printing time [6]. This involves a strategic choice between opting for a fully dense infill, which, although resource-intensive in terms of material consumption and fabrication time, thereby increasing the cost, ensures superior mechanical robustness. Alternatively, employing infill patterns (IP) can replicate specific mechanical attributes while enhancing the part’s overall lightness [7, 8]. IP, integral to the optimization of strength within additive manufacturing, denote the internal lattice or grid-like structures within a 3D-printed component. The selection from a diverse array of available infill geometries, each imparting unique mechanical characteristics such as stiffness and tensile strength, is crucial [9]. The challenge lies in identifying the most appropriate pattern that aligns with the desired mechanical properties and application-specific requirements. A suboptimal choice in infill configuration can lead to inefficiencies in material usage, extended production timelines, and potentially compromised functional performance of the final printed part [10, 11].

To accurately evaluate the mechanical integrity of components fabricated via additive manufacturing techniques, it is imperative to conduct extensive experimental investigations. These investigations are aimed at elucidating the relationship between the inherent material properties and an array of process-specific parameters [12, 13]. In the scientific literature, there has been substantial research dedicated to ascertaining the mechanical attributes of polymers processed through AM modalities [14]. Lubombo and Huneault [8] investigated the effect of 3D printed PLA infill patterns on the mechanical properties. They found that by simply altering the IP or increasing the number of perimeter shells while maintaining the same density, the material’s stiffness doubled and its strength increased by up to 82% and 84%, respectively. In the work of Forés-Garriga et al. [15], it was investigated the role of the IP on the weight reduction and mechanical properties of PEI ultem vi FFF. The results analysis establishes a relationship between the infill configuration and mechanical performance, taking into account both intra-layer and inter-layer bonding. A 3D waiving IP was investigated by Yao et al. [16]. According to their findings, when compared to parts produced by layer-to-layer deposition in 3D weaving, the anisotropy within the layers’ mechanical properties is notably reduced, dropping to 10.21% and 0.98%. LaleganiDezaki and MohdAriffin [17] investigated the effect of combined IP on the mechanical properties of FFF parts. Additionally, samples built at 0° orientation, both in the flat and on-edge directions, exhibited the strongest layer adhesion and highest quality. Conversely, samples built at perpendicular orientations, such as 60° and 75°, demonstrated weaker adhesion and were the most fragile in both flat and on-edge orientations. In summary, the study found that increasing the build orientation angle leads to a decrease in strength.

The rapid evolution of 3D printing technologies has necessitated the integration of advanced simulation tools to optimize design and manufacturing processes. Among these tools, finite element analysis (FEA) stands out due to its capability to accurately model and analyze complex geometrical structures [18, 19]. Mercado-Colmenero et al. [20] presented a combined numerical and experimental investigation into the uniaxial compression properties of PLA (polylactic acid) manufactured using FFF technology, specifically focusing on product specifications. The results demonstrate that the model, which applies a polynomial function, closely aligns with experimental outcomes, displaying a minimal error of only 0.36%, in contrast to the previous model that used a constant Young’s compression modulus and exhibited a significantly higher error of 4.27%. Using FEM, Forés-Garriga et al. [21] investigated the mechanical response of AM cell patterns and the influence of cell density on printing time and compression strength. The authors have conducted a detailed analysis of the excellent elastic performance of Sparse infill and derived analytical expressions to calculate the elastic moduli in three orthogonal directions. Using fluid-filled structures, Soe et al. [22] performed experimental and FEA studies on the dynamic compression behavior of FFF parts. One part was produced completely hollow, containing only air, while another was 70% filled with water, up to half its height, and a third was entirely filled with water. The findings indicate that closed-cell AM structures filled with fluid may reduce acceleration upon impact. Additionally, SPH and fluid-based air-cavity computational techniques were effectively used for simulating water and air, respectively [23]. Gonabadi et al. [24] explored the impact of raster angle, build orientation, and infill density (ID) on the elastic behavior of 3D printed components through finite element microstructural modeling and homogenization FEM methods. The authors found that by using FEM, they were able to predict the elastic properties, although there is a need to develop more advanced models for samples with IP [25]. An experimental and numerical investigation on the influence of vertical strut configurations on the compressive behavior of 3D printed polymer lattice structures was investigated by Fadeel et al. [26]. Both experimental findings and finite element modeling indicate that the presence of vertical struts enhances the strength of lattice structures, with the specific energy absorption (SEA) being dependent on the lattice geometry rather than its mass. Finally, Hmeidat et al. [27] examined experimentally and by fine element analysis, how IP affect the strength and stiffness of 3D printed topologically optimized geometries. The study utilizes a benchmark topology optimization problem and fused filament fabrication to evaluate how different infill patterns affect the strength, stiffness, and failure behavior of 3D printed structures. The results show that the choice of IP leads to notable differences in the stiffness and failure load of the printed objects. This highlights the crucial role of selecting appropriate infill patterns in the design process to achieve optimal performance in 3D printed structures.

The literature review highlights that, although there are studies relevant to the mechanical behavior of solid 3D printed and lattice structures, to the best of the authors’ knowledge, there is a lack of studies integrating FEM and experimental analysis on the compression of samples with different infill density values, printed using FFF. For that reason, a comprehensive experimental and numerical work was carried out, by studying the effect of infill density on the compression behavior of polymer specimens in a wide range of infill density values, spanning from 0% (specimen with top and bottom surfaces as well as walls) to 80%. The results indicate that the adopted FE approach provides highly accurate results, especially for infill density values up to 65%, and can be efficiently used as a reliable tool for the prediction of the mechanical behavior of printed parts.

1.1 Literature review

In order to prove the necessity and importance of the current work, a concise review of the relevant literature was performed. Various authors have carried out studies based on FE models in order to predict the mechanical behavior of FFF-printed specimens. Based on the work of Paul [28], only a small amount of works have been carried out regarding FE modeling of FDM compared to other processes, mostly on mechanical properties characterization, and even these models often lack realistic features. The existing works can be categorized into works which model the properties of printed parts indirectly (either by a simplified geometric model or by micromechanical model), works which model lattice structures and works which model the internal structure of the printed specimens explicitly by taking into account the infill pattern, infill density, and wall thickness, such as the model which is presented in this work. As our work focuses on the compressive behavior of printed specimens, most of the works which will be discussed are relevant to the compression test of FFF-printed specimens.

Regarding the first category of models, Torre et al. [29] conducted FE numerical simulations for the compressive behavior of PLA in which the material was modeled as elastoplastic isotropic material based on experimental results. Mercado-Colmenero et al. [30] used an elastic model for the simulation of FFF-printed parts, based on experimental results about material properties. Mercado-Colmenero et al. [31] also simulated a special structure of a printed construction element using a linear isotropic elastic model. Athale et al. [32] modeled the 3D printed material by an orthotropic elastic-anisotropic plastic material. Bandinelli et al. [33] also took into consideration the anisotropy of printed material properties for their FE model of PA12-CFRP composites compression. Garcia-Granada [34] carried out simulations for compression of printed specimens under dynamic loading and compared the use of a foam constitutive model and an elasto-plastic one, created based on experimental results. Kerekes et al. [35] adopted the Gurson-Tvergaard model for the mechanical behavior of the printed specimens, which was fitted based on the experimental results. Bhandari et al. [36] created a special bimodular elasto-plastic material for FE simulations of FFF-printed parts, including different properties for tensile and compressive behavior. The part was modeled as a shell with stacked layers of orthotropic laminae for which the mechanical behavior was determined by laminate analysis. Pastor-Artigues et al. [37] also raised the issue of bimodular behavior of printed PLA specimens, indicating that a modified constitutive model is required to model their behavior by numerical models.

Apart from the works in which the FFF-printed material is modeled by a specific material model, there are a few works which focus on the use of micromechanical models, whose characteristics are extensively discussed in the work of Bol and Šavija [38]. For example, Gonabadi et al. [39] developed an RVE model to study the micromechanical behavior of FFF-printed specimens with interbead voids. Using the RVE model and homogenized properties, they simulated cases with different layer height, layer width, and overlap and determined the response of the specimens in each case.

Another category of FE simulations of the mechanical behavior of FFF-printed parts is related to lattice structures. Lattice structures are usually open-cell structures with a special geometry, aiming for improved mechanical behavior. In the relevant literature, a wide range of lattice structures were simulated, such as BCC [40, 41], BCC-Z [42, 43], cubic [44], honeycomb [44,45,46,47,48,49], bioinspired, such as bamboo [44, 47] or other structures [47, 50], gyroid [51], and auxetic geometries [48, 49]. In these studies, the exact geometry of the lattice was explicitly modeled, whereas the material model was usually elasto-plastic [44, 45, 47, 49, 51], isotropic-hardening plastic [40, 41], or based on the experimental stress–strain curves [42, 43, 50].

However, as it was aforementioned, additional features of FFF-printed parts, which directly affect their mechanical behavior, were not captured in these simulations, such as the walls, bottom, and top layers, as the lattice structures in these simulations are not considered to be part of the internal structure of a specimen. Thus, a third category of models exists, in which geometric models of the exact internal geometry of specimens are used. In the relevant literature, it can be seen that the research conducted, especially on compressive behavior of printed parts with detailed geometrical models, is rather limited. Podroužek et al. [52] carried out experimental and numerical studies on different 3D infill patterns such as gyroid, Schwarz D, and Schwarz P using an exact geometric model of the specimens and a non-linear material model based on the experimental stress–strain relationships. Sunny et al. [53] developed a special model for printed structures with specific infill patterns, in which they took into consideration the exact infill pattern geometry and density, whereas the effect of entrapped gas in the cavities was also modeled. Sukindar et al. [54] also created exact geometric models for compressive specimens using three different infill patterns and three different infill density values, from 30 to 70%. Finally, Guessasma et al. [55] adopted a different methodology in order to take into account the internal structure of printed specimens, by varying the Young’s modulus appropriately in the computational domain so that the differences in stiffness due to overlapping filaments or voids and other weak points could be included in the model. From the literature review, it can be concluded that there is a rather limited amount of works conducted in the specific topic, and especially, a comprehensive study of the effect of infill density on the specimen strength has not yet been carried out. Thus, the present work can be proved important in this scientific field as it can clarify the evolution of mechanical behavior for a wide range of infill densities and be later used as a reference for future studies based on different infill patterns or other process parameters’ values.

2 Materials and methods

2.1 Experimental study

In this study, we conducted compression tests on acrylonitrile styrene acrylate (ASA) samples, which were produced using FFF with varying infill densities and patterns. As a copolymer, ASA combines three distinct monomers, each contributing specific properties to the material. Acrylonitrile (C3H3N) provides chemical resistance and thermal stability [56], styrene (C8H8) enhances the polymer’s rigidity and processability [57], and acrylate, typically butyl acrylate, significantly improves the weather resistance and impact strength [58]. Specifically, we created compressive test specimens measuring 12.7 mm in diameter and 25.4 mm in height, in line with the ASTM D695-15 standard [59]. In the compression tests, the displacement rate was 5 mm/min. Given that this displacement rate value is usually considered to be related to quasi-static conditions, it is not necessary to adopt a transient model, especially with an explicit solver, as it will be later explained in the next subsection. These specimens were designed with infill densities of 20%, 50%, and 80% and featured two distinct IPs: honeycomb (hexagonal) and gyroid. For each set of conditions, we fabricated five identical samples following the prescribed standard, orienting them vertically during printing. Additional printing parameters are detailed in Table 1, while Fig. 1 illustrates the specific IPs used in the printed samples. In order to further prove the reliability of the numerical model, a second series of experiments was carried out with infill density values of 0%, 35%, and 65%, and the results were subsequently analyzed and discussed.

Table 1 FFF process parameters’ values
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Fig. 1
figure 1

Test specimen geometry used in the computational model for each case: a 20% infill density, b 50% infill density, c 80% infill density

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2.2 Numerical model details

In this study, it is intended to create a realistic model for the simulation of the behavior of the FFF-printed specimen under compressive loading. Thus, special attention is paid on different aspects of the numerical model which can guarantee the accurate representation of the compression test procedure. The simulations were carried out in ANSYS Workbench 2023 R2 software.

At first, regarding the geometry of the model, the exact geometry of the compression test specimen should be created. An important consideration for creating the actual geometry is to take into account the geometry of the actual printed specimen based on the relevant ISO standard, which is produced after the slicing procedure and is not 100% solid. As in the current work, three different infill density percentages were selected, and three separate models were developed, as can be seen in Fig. 1. Thus, the main modeling idea is to treat the specimen as a part with the properties of the solid material but with the exact printed geometry, in contrast to other studies which include a plain cylindrical geometry with experimentally derived material properties taking into consideration the infill density.

All cases which are considered in the present work are based on the hexagonal or honeycomb IP, and also, the wall thickness of the specimen is taken into account, based on the data obtained from the slicer software for the FFF process, as can be seen in Fig. 1.

The full computational model is depicted in Fig. 2, including the test specimen, as well as the two plates representing the actual platens of the testing machine. In order to reduce the simulation cost given the fact that the model is axisymmetric, only half of the compressive specimen and the upper and lower plates will be modeled. The two plates are simplified to two square-shaped plates, with a side length of 20 mm and a thickness of 1 mm.

Fig. 2
figure 2

The geometry of the computational model: a geometry of the upper and lower plate, b side view of the model, c the actual symmetric model used in the simulations

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Regarding material modeling, as can be seen in Fig. 3, apart from the ASA compressive sample, the plates are modeled as rigid material with properties equivalent to structural steel, available in the software database. ASA is modeled as an elastic–plastic material with the plasticity taken into account by a nonlinear isotropic hardening power law, which is presented in Eq. 1:

$$\frac{{\sigma }_{\Upsilon}}{{\sigma }_{0}}={\left(\frac{{\sigma }_{\Upsilon}}{{\sigma }_{0}}+\frac{3G}{{\sigma }_{0}}{\varepsilon }^{pl}\right)}^{N}$$
(1)

where σΥ represents the current yield stress, σ0 represents the initial yield stress, G is the shear modulus, εpl is the accumulated equivalent plastic strain, and N is the hardening exponent.

Fig. 3
figure 3

Materials assigned to different parts of the model

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The data for this material model were obtained from the relevant literature, as well as material databases (MATWEB) [21].

The boundary conditions of the model are mainly applied on the upper and lower plates, as can be seen in Fig. 4. A displacement boundary condition is applied on the upper plate in order to simulate its actual gradual movement during the compression test, and a fixed support boundary condition is applied on the bottom surface of the lower plate in order to restrict its movement, as it occurs also during the actual experiment. The contact between the rigid plates and the deformable compressive specimen is assumed to be frictional with friction coefficient value of 0.1. The compressive test specimen is selected as contact body and the two plates as target bodies. Moreover, a frictional contact is defined for the test specimen, as it is expected that due to the deformation, a self-contact condition may occur.

Fig. 4
figure 4

Boundary and contact conditions of the computational model

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The computational mesh used for the simulations included 3D solid elements, both 20-node hexahedral (Hex20) and 15-node prismatic (Wedge15 wedge elements). The length of the elements was 0.5 mm in the first two cases, whereas it was 0.6 mm for the case of 80% infill density, as it was determined by the mesh independence study which will be presented in Section 3.1.

An important consideration was also the choice of the suitable solver for the model, which could be a static structural, transient structural, or explicit dynamic structural one. Based on preliminary simulations, it was shown that the use of static or transient structural simulation did not provide considerable difference and also the use of explicit dynamics was not necessary given that the mechanical tests are conducted under quasi-static conditions, with relatively low crosshair speeds (namely 5 mm/min), so that the use of an explicit dynamics model with element deletion would lead to higher complexity of the model without a definite improvement of its accuracy, as additional parameters would be required to be determined. Thus, a static structural solver was employed with the option to take into account large deformations by recalculating the structure stiffness, as it is expected that large deformation will occur in several regions of the model. In Section 3.1, the accuracy and stability of the solver will be further discussed.

3 Results and discussion

Following the completion of simulations based on three distinct experimental cases, force-extension curves and deformation outcomes for the compressive test specimen were obtained. In this section, the findings of the mesh independence study for each model will be initially presented, subsequently followed by a detailed examination and discussion of the simulation results.

3.1 Mesh independence study

The initial step in the simulation process was to create an optimal mesh that balances mesh quality and computational efficiency in order to use the computational resources more effectively while ensuring the reliability and accuracy of the results. To achieve this, the element mesh size was varied within a reasonable range, and a thorough mesh independence study for each of the three models was conducted, which featured different test specimen geometries. The effect of the size on the simulation results was investigated in a carefully selected range in order to provide a meaningful comparison.

For the model with an infill density of 20%, the results of the mesh independence study evaluated by the force-extension curve are presented in Fig. 5. The variation of the results obtained by meshes of different sizes is not considerably high but noticeable. Apart from the accuracy of the results, there is another important reason for selecting an appropriate mesh, as it is proven that the use of coarser meshes leads to higher instabilities of the numerical solution and finally the simulation is aborted much earlier than the desired displacement value due to issues such as high local deformation of elements. On the other hand, the use of a finer mesh can ensure the stable completion of the simulation in order to achieve a result comparable to the experimental one, but a too-fine mesh increases the computational time. More specifically, based on the results of Fig. 5, it can be seen that although in some cases, a coarse mesh (with a size of 0.9 mm) can lead to relative stable simulations, higher stability can be mainly achieved when the mesh size is below 0.6 mm, and especially with a mesh size of 0.5 mm, it is possible to simulate the compression test with an extension value up to 16 mm.

Fig. 5
figure 5

Force-extension curve from mesh independence study simulations for 20% infill density

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As can be directly seen in Fig. 6, the element quality of the selected mesh is sufficiently high, with most elements having a quality of 0.85 and higher (with 1.0 being the optimum value).

Fig. 6
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Mesh quality for the model of 20% infill density

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In the case of the model with 50% infill density, as can be seen in Fig. 7, the mesh independence study results also indicate that the use of coarse meshes not only reduces the accuracy of the model but is also related to stability issues, as the model with element size 1 mm becomes unstable before the extension reaches the value of 10 mm, whereas models with a finer mesh can eventually be stable up to 12, 13, and 14 mm, especially in the case of mesh size of 0.5 mm.

Fig. 7
figure 7

Force-extension curve from mesh independence study simulations for 50% infill density

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As can be seen in Fig. 8, the element quality of the selected mesh is rather high, with most elements having a quality of 0.9 and higher, something that proves the capabilities of the developed model.

Fig. 8
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Mesh quality for the model of 50% infill density

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In the case of the model with 80% infill density, as can be seen in Fig. 9, probably due to the higher resemblance to a totally solid model and the considerably lower amount of voids, the results of the mesh independence study show that most of the models are stable for extension values over 15 mm and the finest model can reach up to 16 mm. The selected mesh size was 0.6 mm, as further reduction of the mesh size did not lead to a considerable difference of the results.

Fig. 9
figure 9

Force-extension curve from mesh independence study simulations for 80% infill density

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As can be seen in Fig. 10, the quality of the elements of the mesh is relatively high, with most elements having a quality between 0.7 and 0.88. The reason for the lower quality of elements in comparison to the two previous cases can be mainly attributed to the more complex geometry of the model, as it includes multiple small voids, with the corresponding elements having a less ideal shape. However, it is not expected that the accuracy of the results will be compromised, as the use of a finer mesh did not show a considerable difference of the obtained results.

Fig. 10
figure 10

Mesh quality for the model of 80% infill density

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3.2 Simulations results

After the appropriate mesh was chosen for the three different cases, the respective simulations were carried out and the results were evaluated based on the predicted force–displacement curves as well as the deformed shape of the specimens which were directly compared to the experimental ones in order to verify the accuracy of the proposed model.

In the case of the specimen with 20% infill density of ASA material, in Fig. 11, the predicted force–displacement curve is present and compared to the experimental curves. As five repetitions were performed for each test based on the respective standard, it was considered important to depict all the obtained curves in the same graph in order to take into account the variation of mechanical properties during the comparison. The accuracy of the results in this case justified the fact that the proposed model can be regarded as rather promising, as the shape of the predicted force–displacement curve and its values are considerably close to the experimental ones. Initially, the first part of the curve, corresponding to the elastic deformation and the beginning of yield, is almost identical to that of the experimental curves, with a slight increase in error for displacement values around 4 mm. Then, in the region with displacement values in the range of 7–12, the predicted curve lies between the experimental ones, and finally, the curve is close to the curve corresponding to the first experiment, which shows slightly higher force values. In conclusion, in the case with 20% infill density, a sufficient degree of accuracy can be noted both for the shape of the curve, corresponding to critical stages of the compression test and its values, despite the complexity of this case, given the formation of characteristic lobes, as it will be afterwards explained.

Fig. 11
figure 11

Comparison of simulation and experimental results regarding the force-extension curve

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Apart from the presentation and analysis of the force–displacement results, in order to further exhibit the appropriateness of the developed model, several indicative snapshots for the simulated case of 20% infill percentage of ASA are depicted in Fig. 12, along with the stress–strain curves from both simulation and experiments. As the infill density for this test specimen is rather low, considerable deformation occurs even from the earliest stages of compression, with two almost symmetrical lobes being progressively formed near the upper and the bottom plate, respectively. Finally, when the test is stopped, the deformed specimen mainly consists of the two formed lobes and a small region at the center of the specimen. When this image is compared to the image of a deformed specimen from the experiments, the similarity between the predicted and experimental deformed specimens proves the validity of the proposed model. In general, the mode of deformation of the specimen in this case seems closer to the behavior of thin-walled tubes rather than a solid specimen, given that the core of the specimen is mostly void and the main part of the specimen which withstands the applied loads is actually the wall. More specifically, when the specimen is subjected to compressive loading, the stress distribution becomes non-uniform with higher stresses concentrated in the regions which are in contact with the upper plate of the testing machine. Thus, when the material is deformed, lobes are formed, influenced by the ratio of the specimen diameter to the wall thickness.

Fig. 12
figure 12

Stress–strain curve from the compression tests and simulation, with indicative images of the deformed specimen

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In the case of the specimen with an infill density of 50%, an additional amount of material is included than the previous case, leading to a higher force obtained, as can be seen in Fig. 13. In that case, the proposed model can fairly accurately predict the force–displacement curve, at least up to almost 6 mm of displacement, and then, although the shape of the curve is generally similar to some of the experimental curves, the predicted force values are higher, until eventually the difference between the experimental and predicted curves is again reduce at higher values of displacement. The observed discrepancies between the experimental and predicted curves can be attributed to various factors of the proposed model including the material model, geometrical inaccuracies of the produced specimens, and parameters which were not modeled explicitly such as the layer height. However, the proposed model showed a considerable level of realism, as will be afterwards justified based on the observation of the different stages of deformation during the compression test.

Fig. 13
figure 13

Comparison of simulation and experimental results regarding the force-extension curve

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As in the case of the specimen with 20% infill density, several snapshots of the simulated compression test were also obtained in the case of the specimen with 50% infill density, depicted in Fig. 14, in order to further investigate the compressive behavior of the ASA specimen. In this case, although from the earliest stages of the deformation, there is a tendency for lobe formation near the two ends of the specimen, the existence of a higher amount of material hinders the clear formation of the lobes as in the previous case. Thus, at the final stage of the compression test, although the shape of the deformed specimen demonstrates that two areas of higher diameter are formed above and below the central region of the specimen, the reduction of diameter in the central region and the curvature of the lobes are much lower, in comparison to the deformed shape of the previous case. These observations closely match the actual shape of the deformed test specimen.

Fig. 14
figure 14

Stress–strain curve from the compression tests and simulation, with indicative images of the deformed specimen

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In the case of the specimen with an infill density of 80%, the volume of voids in the specimen is considerably small, and thus, the material is expected to behave in a way more similar to a solid material, rather than a porous one as in the previous two cases. At first, this is confirmed by the force–displacement curves of Fig. 15, due to the much higher force developed in comparison to the previous two cases. Moreover, the shape of the curve resembles to that of open-die forging with a steep monotonic rise of force as the displacement increases. Although the predicted force–displacement curve generally exhibits a similar behavior to most of the experimental curves during the elastic deformation and initial stages after yield, larger errors occur for displacement values between 6 and 10 mm, and finally, both experimental and predicted curves have the same shape, although the simulation results overestimate the force values. These differences can be attributed to the higher complexity of the specimen geometry, in conjunction with possible defects on the printed specimens, which cannot be explicitly modeled. However, it is proven that the proposed model can provide a precise depiction of the deformation stages of the compression test specimen, according to the experimental test, as will be afterwards discussed in respect to the deformed shape images of the test specimen.

Fig. 15
figure 15

Comparison of simulation and experimental results regarding the force-extension curve

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The observations on the snapshots of the simulation of the compression test for the specimen with an infill density of 80% in Fig. 16 justify the previous assumptions, as it becomes evident that the behavior of this test specimen is more similar to the behavior of a solid specimen rather than the porous FFF-printed specimens of the previous cases. The existence of a higher amount of material leads eventually to the barreling phenomenon, by the formation of a barrel-like deformed test specimen due to the friction between the plates and the test specimen, as it was expected in the case of an open die forging, rather than the lobe formation which was observed in cases with lower infill density, when the specimens were similar to thin-walled tubes, something that is verified in comparison to the shape of the actual deformed specimen.

Fig. 16
figure 16

Stress–strain curve from the compression tests and simulation, with indicative images of the deformed specimen

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3.3 Additional simulations results

In order to further assess the validity of the proposed model, additional experimental test cases were carried out and analyzed and the respective simulations were also performed. These cases include compression tests under different infill densities such as 35 and 65%, as well as the extreme case of 0% infill density, where actually the test specimen is only composed of walls with a geometry resembling to that of a thin-walled tube.

After the appropriate mesh size was determined in each case, based on the exact geometry of the test specimens, shown in Fig. 17, the relevant simulations were carried out. In the case of the specimens with 0% infill density, the specimen is actually a thin tube with thin-walled top and bottom surfaces. Thus, its behavior is expected to be relevant to the behavior of compressed thin tubes. The force–displacement curve obtained from the numerical model, presented in Fig. 18, exhibits an almost identical trend at the first stages of compression, as it is very close to the experimental curves up to the regions of yield, then it exhibits higher force values up to 7 mm of displacement, and finally, the values lie towards the lowest limits of experimental values but are still consistent with most of the experimental curves.

Fig. 17
figure 17

Test specimen geometry used in the computational model for the additional simulations: a 0% infill density, b 65% infill density, c 35% infill density

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Fig. 18
figure 18

Comparison of simulation and experimental results regarding the force-extension curve for 0% infill density

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This behavior of the test specimen is absolutely justified, as the formation of lobes, associated with a non-monotonic increase of force, is an anticipated characteristic of thin-walled tubes. The obtained results from the numerical model exhibit a high degree of accordance with the experimental ones as the predicted force-extension values lie between the lower and higher limit of the experimental curves during most stages of the compression test, thus proving that this approach is able to provide reliable results.

In the case of infill density value of 35%, the accuracy of the numerical model is still considerably high, as can be seen from Fig. 19. More specifically, similar to the case with infill density value of 20%, the predicted force–displacement curve for the compression test of the ASA sample is very close to the experimental curves during the early stages of elastic and plastic deformation, and for displacement values over 7 mm, it lies between the experimental curves, indicating a rather consistent trend with most of them. These observations clearly demonstrate the high reliability of the numerical model regarding the compressive behavior of ASA samples, as it is shown that it can predict the expected mechanical behavior sufficiently during all the stages of deformation.

Fig. 19
figure 19

Comparison of simulation and experimental results regarding the force-extension curve for 35% infill density

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Finally, the results obtained from experimental tests and numerical simulations for the compression test of ASA samples with 65% infill density are shown in Fig. 20. In this figure, it can be clearly observed that apart from the initial region of elastic deformation and yield, the predicted curve deviates from the experimental results to a higher degree than in the previous cases, especially after 10 mm of displacement. These results indicate that the numerical model, showing a monotonic increase of force after the yield region, predicts a compressive behavior closer to that of a solid specimen rather than a specimen with voids, which is expected to exhibit lower force values during intermediate stages of deformation. These deviations can be attributed to the potential formation of defect structures in the actual specimens and discrepancies of the printing process due to the more complex geometry of the specimens with higher infill density, which cannot be directly modeled and limitations of the numerical approach which was adopted. However, these results are still important as a reference for further studies regarding the behavior of additively manufactured parts.

Fig. 20
figure 20

Comparison of simulation and experimental results regarding the force-extension curve for 65% infill density

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Based on the results analyzed and discussed in Sections 3.2 and 3.3, a sufficiently deep insight can be obtained into the capabilities and limitations of the FE model regarding the prediction of the compressive behavior of additively manufactured specimens. It is considered important to note that the adopted modeling approach can be efficiently applied for the cases with infill density up to 65%, whereas for higher infill density values, this model consistently exhibits a trend for gradually increasing deviations from the experimental results, mainly showing a higher resemblance to the trend of a solid specimen, rather than the behavior of specimens with voids. For that reason, in the near future, modifications will be carried out in the model in order to be able to predict the compressive behavior even in cases with more complex geometry of the additively manufactured parts such as the ones in the range of infill density values of 65–80%.

4 Conclusions

In the present work, realistic numerical models, taking into account the exact geometry of compression specimens, were developed for the simulation of the compression behavior of ASA polymer specimens fabricated by FFF additive manufacturing. Cases of printed specimens with honeycomb IP and with different infill density values were simulated by finite element models, and after the models were validated by direct comparison to the experimental results, various useful conclusions were drawn.

  • The use of the appropriate mesh size leads not only to increased precision but also increases the stability of the numerical simulation of the compression test, as it was found in every case. When the suitable mesh size was employed, the simulation could be carried out without stability issues, regardless of the level of deformation of the simulated test specimens or their geometry.

  • The proposed finite element model can accurately model the different stages of the deformation of the test specimen with 20% infill density. Although the accuracy becomes lower after the region related to the yield, the simulated results closely match the experimental ones and are able to accurately predict the procedure of lobe formation on the deformed specimen due to its lower density in the core region.

  • The accuracy of the proposed model in the case of the model with 50% infill density can be also regarded as acceptable, as it can simulate the shape of the force–displacement curve although the level of accuracy becomes gradually lower for higher displacement values probably due to the material model, unmodeled factors, and defects occurring during the printing process or the experimental test. However, the deformed shape can be realistically predicted based on the model.

  • In the last case, involving a test specimen with a high infill density, the model accurately predicts a different behavior than the two previous cases, leading to barreling phenomenon and a different shape of the force–displacement curve. Despite the differences in the predicted loads by the proposed model, its capabilities are shown to be rather high, compared qualitatively to the experimental results both regarding the force–displacement curve and deformed test specimen.

  • Experimental and numerical studies on additional cases, such as the ones with infill density values of 0, 35, and 65%, further demonstrated the capabilities of the proposed model, indicating that its accuracy is sufficiently high for infill density up to 65% and then, the accuracy is reduced for higher percentages, especially over 65% due to several reasons related to the complexity of the geometry of these specimens. However, this limitation of the proposed model can be addressed in a future study by adopting a more advanced modeling strategy and including details such as the layer height in the FE model.