Compression and energy absorption characteristics of short fiber‐reinforced 2D composite lattices made by material extrusion

With the advent of additive manufacturing, lattice structures have been of increasing interest for engineering applications involving light‐weighting and energy absorption. Several studies have investigated mechanical properties of various lattices made up of mostly unreinforced polymers and lack numerical analysis for reinforced lattice structures. In this paper, mechanical response of short fiber reinforced lattice structures under compression is investigated through experiments and numerical simulations. Three different 2D lattices namely, square grid, honeycomb, and isogrid along with their rotated counterparts were fabricated using 15% wt carbon fiber‐acrylonitrile butadiene styrene and experimentally evaluated through uniaxial compression testing up to 30% strain. As simulations on fiber‐reinforced lattices under large compressive strains are rarely performed and published, finite element models accounting for fiber orientation induced anisotropic mechanical properties and geometrical imperfections were developed to predict the stress–strain characteristics up to 30% compressive strains. The stress–strain curves predicted from the numerical simulations matches well with the experimental responses for various lattice geometries. Various failure mechanisms such as thin strut buckling, contact within the lattice, and fracture of struts under large deformation were investigated. Analyzing energy absorption characteristics of these lattices revealed that the honeycomb structures in horizontal configuration exhibits superior energy absorption capability.


INTRODUCTION
2][3][4] These structures can be adapted taking inspiration from nature or can also be engineered to obtain material-independent responses (metamaterials). 4,5Among the different lattices, honeycomb structures tend to have superior energy absorption capabilities due to the nature of their plateau stress, wherein the stresses remain constant even for large strains. 68][9] However, customizability, design flexibility, cell size and resolution are all limited in these processes.Additive manufacturing (AM) enables fabricating unique, customized, lattice structures with a variety of materials and with short lead times compared to conventional processes.
Material extrusion AM processes such as fused filament fabrication (FFF), are one of the most widely used AM techniques. 10Recent developments in FFF have enhanced the material pool to include fiber reinforced composites, 11 metal-infused polymers, 12 and electrically conductive polymers. 13][19][20] Lattice structures including honeycomb structures have been studied and verified using FFF systems.Bates et al. 21tudied 3D printed polyurethane honeycombs that showed outstanding energy absorption (0.3-0.4 J/cm 3 /cycle) when subjected to compressive loads repeatedly.The same authors also later reported graded honeycombs structures to exhibit higher energy absorption capabilities when compared to ungraded structures. 22Ajdari et al. 23 found that hierarchical honeycombs can be stiffer than the traditional lattice designs (i.e., non-hierarchical, uniform honeycomb structure) depending on the size ratio between the small cell and the large cell.The hierarchical structures were fabricated from an FFF system using acrylonitrile butadiene styrene (ABS).The elastic modulus and strength of 3D-printed honeycombs were investigated by Hedayati et al. 24,25 using three different approaches: experiments, finite element (FE) simulation, and analytical models based on Euler-Bernoulli and Timoshenko beam theories.For strength prediction via the FE simulation and the analytical approach, the yield strength of the material was used as one of the base parameters.On the other hand, Quan et al. studied auxetic honeycomb structures printed using continuous fiber reinforced composites. 26Their study reveals that continuous fiber reinforced composites exhibit twice the stiffness and energy absorption of the printed pure PLA honeycombs.However, printing using continuous fibers is challenging at large scale due to difficulties associated with hardware and typically these materials require special-purpose printers.Alternatively, short fiber composites used in this study are commercially available and are easy to print on readily available small scale (desktop) printers without any hardware modifications.Such short fiber composites have also been used to print large scale structures such as Shelby Cobra car chassis using Big Area Additive Manufacturing at the Oak Ridge National Laboratory. 27,28[29][30][31][32][33][34][35] A few studies discussed the use of continuous carbon fiber (CF) reinforced thermoplastic lattice trusses between face sheets in sandwich structures. 36,37However, printing lattice structures with short fiber reinforced composites can potentially offer advantages such as simultaneous improvement in strength, energy absorption properties, and the stability of crush properties in terms of a stable stress-strain behavior.For instance, mechanical properties of short Basal fiber reinforced PLA and PCL honeycomb-type lattices have been studied experimentally by Sang et al. 38 Experiments show that supplementing PLA matrix with PCL elastomer improves energy absorption by ∼50%.Nevertheless, numerical modeling of fiber-reinforced lattice structures under large compressive strains are rarely performed.The challenges include accounting for the fiber orientation, nonlinear behavior, and capturing various failure or fracture mechanisms at large strains.
In this paper, compression characteristics of three different short fiber reinforced lattices, namely, square gird, honeycomb, and isogrid in two different configurations were investigated.Honeycomb structures are known to exhibit excellent energy absorption capabilities due to their ability to undergo large compressive strains at nearly constant stress level and absorb a large amount of energy without creating high stresses.Isogrid structures are of interest due to their lightweight and superior stiffness responses while square grid is simple, easily manufacturable structure with 90 • symmetry that is useful in many engineering applications.First, lattices were printed using 15% wt short CF reinforced ABS material through FFF process.A FE-based numerical model that accounts for anisotropic material behavior and initial geometric imperfections was developed to predict the mechanical response up to a large compressive strain of 30%, including experimental validation.The energy absorption characteristics of the three different lattices in two orientations were evaluated to reveal the effects of lattice geometries and orientations under compressive loads.
The paper is organized as follows: In Section 2, AM of various lattice structures, experimental set up for quasi-static compression, and numerical simulation framework is entailed.In Section 3, mechanical responses of lattices from experiments are compared with the predictions from numerical simulations.The energy absorption characteristics of different lattices is discussed.The conclusion is presented in Section 4.

Additive manufacturing
In this study, three different 2D lattice structures extruded in 3D were investigated, namely, square grid, honeycomb grid, and an iso grid; each oriented in two directions as shown in Figure 1. Figure 1A,B shows a square grid with the lines along (A) 0   /−45 • directions.Figure 1C,D shows a honeycomb grid (C) having two among the six sides of each hexagon directed along the vertical direction, and (D) having two among the six sides of each hexagon directed along the horizontal direction.Figure 1E,F shows an iso grid having one of the three lines directed along the vertical direction, and (F) having one of the three lines directed along the horizontal direction.
All samples were printed on a commercially available desktop printer (MakerGear M2) using 15% wt% CF reinforced ABS (CarbonX™ CFR-ABS, 3DXTech).Sample designs were sliced using a concentric infill pattern ensuring two beads across each thin wall.The toolpaths for various lattice structures were obtained from Simplify3D.The bed temperature was set to 100 • C and the nozzle temperature was set to 230 • C. The diameter of the nozzle was 0.5 mm and thus the layer thickness is 0.25 mm.The dimensions of each sample were 62.5 mm × 62.5 mm × 30 mm in X, Y , and Z directions respectively, as shown in Figure 2A.Samples printed using all six lattice designs as shown in Figure 1 had the same overall   dimensions as well as a similar weight of ∼54 g to ensure consistent experimental conditions.All the lattice structures are printed with sufficient number of unit cells and edge walls at four surfaces were included to ensure that any defects or breakage of struts at the edges does not significantly affect the compression response.Furthermore, printing path was selected to minimize defects at the junctions between different unit cells of the lattice as described in Habib et al. 39 (see Figures 3 and 4).Studying the effect of printing toolpath on resulting fiber orientations and hence material properties is beyond the scope of the current study.

Compression testing
The samples were placed between the platens such that the compressive load was applied in the vertical direction, that is, perpendicular to the build height direction, as shown in Figure 2B.A crosshead speed of 1.27 mm/min, equivalent to a strain rate of 0.00034/s (=0.02/min) was used for all the tests, which is the commonly recommended strain rate for a quasi-static compression test as per ASTM D6264 standard. 40Both simulations and experiments were performed with the overall strain up to 30%.A strain level of 30% provides ample opportunity for significant deformation and fracture to occur in the specimen, enabling the capture and analysis of both the elastic and post-elastic responses.Furthermore, simulation of higher strain amplitudes would require a longer timeframe, which presents a challenge due to the severe nonlinearity of the mechanical response and small time increments involved in explicit FE simulations used in the study.

Finite element analysis
The FE simulations were performed on all the lattice structures shown in Figure 1 to evaluate the stress-strain and failure characteristics using Abaqus/Explicit.The lattice geometries were discretized using general purpose linear brick elements with reduced integration (C3D8R) to mitigate the shear locking and volumetric locking emanating from incompressible nature of plastic strains.The resultant FE mesh of these lattice structures consists of ∼29-35 k elements.A minimum of four elements are ensured across the width of each cell to minimize hour glassing effect and generation of artificial energy. 41The geometric features and the generated meshes of the lattices are shown in Figure 3 where lattice dimensions were selected such that relative density of the lattice was constant at around 50%.As described by Ashby, behavior of lattice structures in post-elastic regime is a result of three competing mechanisms: plastic yielding at the joints, buckling of the struts, and fracture of the struts near joints. 42Following this description, a material model consisting of linear orthotropic elasticity, Hill plasticity, and fracture is considered in present study to account for relevant failure mechanisms.Due to the preferential fiber orientation induced during AM along the deposition direction, [43][44][45][46] orthotropic elasticity is considered by assigning fiber orientation in Abaqus to account for anisotropy in effective properties.The G-code used to 3D print each lattice is taken from slicer and converted into deposition path coordinate data. 47The fiber orientation directions were then assigned to each FE based on these deposition paths as shown in Figure 4A.The material properties were calibrated using previous experimental results [48][49][50] and summarized in Table 1.Since referenced experiments were performed directly on AM-made composites, the properties used in present study represent "homogenized" behavior of matrix, fibers, pores, and interlayer/intralayer voids.9][50] Since shear data was not available the shear ratios R 12 , R 23 , and R 31 were assumed to be equal to unity.Experimental and theoretical observations have documented that the failure in the lattice structures studied in present study occurs due to bending, buckling, and fracture of struts depending on geometrical features. 42Therefore, it was expected that tensile stresses and anisotropic yield ratios would play leading role in determining the mechanical response of the lattices.Global contact is introduced to capture the self-contacts that occur within the lattice structure during compression through hard contact condition in the normal direction and a penalty friction with a coefficient of 0.27 in the tangential direction. 50 major failure mechanism for lattice structures under compression is buckling of the struts due to small geometric imperfections introduced especially during manufacturing.Modeling such geometric imperfections is critical to ensure   onset of buckling while performing failure analysis of strut-based lattice structures.Thus, a frequency or modal analysis was performed on different lattice structures to determine the structural modes for buckling.The deformation obtained from the first 10 modes is superposed as small imperfections into the initial geometry by multiplying modal displacements with a small weight (10e-5).The modes obtained from frequency analysis for square grid geometries are illustrated in Figure 4B.
Next, the quasi-static compression simulations were performed on the lattice structures with geometrical imperfections using Abaqus/Explicit (refer Figure 4C).The explicit dynamic analysis is employed here to ensure stable numerical results considering that the lattice structures undergo large compressive strains, self-contact, and fracture.Fracture was introduced into the model using a VUSFLD subroutine that allows element deletion based on the maximum principal stress criterion with a maximum strength value of 65 MPa observed in experiments.During explicit simulations, mass scaling is used to ensure reasonable stable time increment for performing quasi-static compression tests on various lattice structures. 51The build orientation for lattices is shown in Figure 5A.The applied compressive load and boundary conditions are illustrated in Figure 5B,C.Rigid surfaces were placed on the top and bottom of the samples to closely simulate the deformation behavior between the two compression platens of the universal testing machine.The rigid surface on the bottom was fully constrained while the rigid surface on the top was free to translate in the x-direction.Given that the lattices here are 2D geometries extruded in 3D, a thickness of 30 mm is considered, and deformation is fixed in thickness direction to prevent any buckling or deformation in the out-of-plane direction (refer Figure 5C).The structure was loaded to 30% strain in 1 s by applying a prescribed displacement to the top rigid surface.

Experimental results
Figure 5 shows the stress-strain profile of 0/90 square grid sample subjected to quasi-static compression testing.As observed, for this lattice orientation the stress-strain profile had a regular oscillating pattern.Each oscillation (peak and valley) in the profile was the result of an individual row collapsing.For example, the first valley resulted from the collapse of the second row from the bottom (see Figure 6a 1 ).At the third valley (see Figure 6a 2 ), the mid-section was pushed out due to an upper row collapse, which caused a significant drop in the stress.The last picture in the series (Figure 6a 3 ) shows the collapse of multiple rows.The collapse of each individual row was attributed to the buckling of the vertical lines of the corresponding row in the 0/90 square grid.
For the 45 • /−45 • square grid, the stress-strain profile demonstrated an oscillating pattern, as shown in Figure 7.However, the overall stress in the 45 • /−45 • grid was much lower than that in the 0/90 grid.The oscillations observed corresponded to the collapse of rows along the diagonal, which mostly occurred in the 45 • direction (upper right to lower left) in Figure 7.During the compression test, after the first collapse of a diagonal row, subsequent collapses of adjacent diagonal rows occurred in the same direction.Because of the row-by-row collapse, both Figures 5 and 6 show oscillatory stress-strain behaviors.However, the 0/90 square grid shows more prominent periodic peaks and valleys of the stress and higher overall stress than the −45/45 square grid.This comparison between Figures 6 and 7 shows the effect of load direction on the stress-strain profile for the same infill shape.The third configuration, honeycomb grid, showed a drastically different failure pattern compared to the two square grid configurations.The most noticeable difference was the absence of periodic peaks and valleys in the stress-strain profile, as observed in Figure 8. Also, for the same honeycomb structure, changing the loading direction from vertical to horizontal influenced the compressive response.The vertical honeycomb grid showed a high initial peak, followed by a series of cascading failures along the 45 • direction (c 1 ).After the first collapse along the diagonal, subsequent collapses occurred at the adjacent cells in the same diagonal direction, similar to the failure observed for the 45/−45 square grid samples.On the other hand, the horizontal honeycomb showed a different failure pattern.Here, the first collapse occurred in the −30 • direction, but some of the subsequent failures occurred along different directions.The overall stress-strain curve of the horizontal honeycomb grid shows a very stable response.Papka and Kyriakides studied uniaxial and biaxial crushing of honeycombs and concluded that the location of initiation of collapse is influenced by geometric imperfections present in the materials, and therefore failure patterns varied between experiments. 52,53However, plateau stress and the extent of the plateau stress were consistent in all the experiments.It is interesting to note that in studies by Papka and Kyriakides 52,53 on a lattice fabricated by packing cylindrical tubes for a vertical honeycomb arrangement, the stress peak does not appear in the stress-strain curve.Instead, the stress-strain curve shows almost steady response similar to horizontal honeycombs but with low-amplitude peaks and valleys.Habib et al. 39 also studied the compressive response of a 3D-printed honeycomb lattice and found that the stress-strain curve from a vertical honeycomb had an initial peak (X2 in figure 12 of Habib et al. 39 ), whereas the stress-strain curve from a horizontal honeycomb showed a stable response (X1 in figure 12 of Habib et al. 39 ).The initial peak from a vertical honeycomb was more pronounced for a lattice with a high relative density ( = 0.26).Therefore, the failure is not perfectly aligned along the 45 • diagonals as compared to the F I G U R E 8 Stress-strain profiles of a honeycomb grid in vertical direction (left) and in horizontal direction (right).For the same structure, changing the orientation results in a dissimilar stress-strain profile.Horizontal honeycomb (right) shows a stable stress.horizontal honeycomb.Hence, the horizontal honeycomb maintains a relatively constant stress level even at large deformation, which makes it more efficient in energy absorption and exhibits higher energy absorption compared to other lattices considered in this study.
The compressive characteristics of the horizontal honeycomb lattice structures are unique because of inconsistent directional collapses during the compression as shown in Figure 9.The initial crack occurred in the −30 • direction which is the same angle as the attachment direction from one hexagon to the neighboring hexagons as shown in Figure 9A.The next failure occurred with a slightly higher angle as shown in Figure 9B.The third and fourth failures occurred in a positive diagonal direction (+45 • ) and a negative diagonal direction (−45 • ), respectively (Figure 9C,D).The diagonal directions are the maximum shear stress direction for a solid under uniaxial compression, and it is interesting to find diagonal collapses from a horizontal honeycomb lattice that does not have any intrinsic diagonal (±45 • ) row of cells or struts.Therefore, unlike square grid pattern and vertical honeycomb pattern where collapsing rows stack upon each other, rows collapse at various locations and angles in the case of horizontal honeycomb.Similar results were observed in the studies by Habib et al. (figure 8 of Habib et al. 39 ) for 3D printed polymeric honeycombs, where horizontal honeycombs were crushed, and rows/cells collapsed randomly across the specimen under various angles while stress stayed near constant up until the densification region.
The experimental results of the vertical and horizontal isogrid samples are given in Figure 10.The vertical isogrid showed a similar profile to the vertical honeycomb grid, as there was a high initial peak followed by a large stress drop and a gradual stress build-up.On the other hand, the horizontal isogrid showed a drastically different profile when compared to the horizontal honeycomb grid.The initial failure occurred along a −60 • direction.The most noticeable difference in the compressive response of the horizontal isogrid lattice is that the initial fracture occurred due to a tensile loading along the horizontal struts instead of compressive loading or buckling, which led to a very high initial stress peak.Both vertical isogrid and horizontal isogrid show localized buckling sites in addition to row-by-row buckling.
The initial stress peak is highly affected by the strut directions in the lattice which also affects the initial stiffness.Chopra 54 investigated the effect of load directions on the modulus of various types of lattice structures using an analytical approach.Figure 3.9 of Chopra 54 shows the stiffness of 13 different lattice structures in all directions.Among the 13 lattice shapes, four different shapes that are relevant to this study were selected and re-plotted in Figure 10.A point on a contour profile represents the magnitude of the stiffness (=distance between the origin (0, 0) and the point) and the direction of the load.The direction of the load is the angle between the distance vector and either the horizontal axis (=y-direction) or the vertical axis (=x-direction), depending on the reference axis.For a square grid (indicated as green), the stiffness is high in the two struct directions (x-direction and y-direction) and very low in all the other directions.In our experiments, the 0/90 square grid showed a very high initial peak, whereas the −45/45 square grid showed a very low initial peak.In Figure 10, both isogrid (indicated as red) and honeycomb grid (indicated as blue) show consistent stiffnesses in all directions and isogrid shows much higher stiffness than the honeycomb grid.In our experiments, the initial peaks from horizontal honeycomb and vertical honeycomb were similar to each other.The initial peaks from the two (i.e., horizontal and vertical) isogrids were higher than those from the two honeycombs.The vertical isogrid showed a higher initial peak than the horizontal isogrid in our experiments, which is attributed to the tensile fracture instead of buckling failure.
Figure 11 also includes a diamond shape to show the effect of non-balanced struct directions.In composite manufacturing, typically, a laminate with 0/60/−60 fiber layups is considered as a quasi-isotropic stacking with equal numbers of plies (therefore, fibers) in each direction.Isogrid has 0/60/−60 directional lines with equal amount, and honeycomb grid also has 0/60/−60 directional struts with equal amount in a discontinuous fashion.The geometry of the diamond shape in Figure 11 is similar to the isogrid except that every other horizontal line is missing in the diamond shape.By elimination every other horizontal line, the stiffness changes from a circular profile (consistent stiffness in all directions) to a snowman shape profile (high stiffness in the horizontal direction, but low stiffness in the vertical direction).
Although lattice structures were manufactured using small-scale AM technologies in this study, the mechanical properties exhibited by a given structure with a chosen base material are scale-independent theoretically.However, differences in the manufacturing process itself could lead to different mechanical properties of lattice structures at mid-scale and large-scale AM, affecting their compression performance.6][57][58][59] However, detailed investigations into the effect of process parameters of large-scale AM on mechanical response of lattice structures is out of the scope of the current study.

F I G U R E 10
Stress-strain profiles of an iso grid in vertical direction (left) and in horizontal direction (right).Both cases have an initial high peak followed by a stress drop.Horizontal isogrid shows an initial tensile fracture.

Finite element analysis results
Failure behavior of the six lattice configurations predicted by the simulations was compared against experimental findings.The experimental and simulated failure initiation locations in each of the tested configurations are shown in Figures 12-14 for the square grid, honeycomb, and isogrid patterns, respectively.Failure was found to initiate due to tensile stresses that form after the thin walls of lattice structures buckle and start bending.Cracks initiate and propagate across the structure, breaking individual thin walls.This propagation forms a line of collapsed thin walls indicated by red arrows as shown in Figures 12-14.
According to Figure 12, the square grid starts to fail when a horizontal row collapses and the buckled struts start to detach from adjacent cells, although this happens at a slightly different height compared to experiments.On the other hand, the rotated square grid configuration starts to buckle along the diagonal, similar to the behavior observed in experiments.As for the honeycomb configurations as shown in Figure 13, a failure path along the line of roughly −45 • can be observed in the case of vertical honeycombs, followed by the breakage of bucked struts due to high tensile stresses at the junction of adjacent honeycomb cells.The horizontal honeycomb configuration showed failure path under the same angle as the experimental sample.Failure initiation of isogrid lattice configurations is shown in Figure 14.The vertical isogrid model follows the same fracture pattern as the experimental counterpart.However, in the case of the horizontal isogrid, the collapse occurred at a different angle and the mechanism of failure shows more excessive bending in simulation when compared to the experiments.As for the failure mechanism, for horizontal isogrid, experiments showed the horizontal struts to break (detach) by tension.However, in simulation, tensional failure was not the predominant initial failure mechanism, which could indicate discrepancies in strength and elastoplastic properties between the simulation and the actual material used in the experiment.Calibration on the material properties remains as a future analysis.Figure 15 shows the FE models after failure for all six lattice configurations.Comparing these with experimental results, it is evident that the square grid, rotated square grid, vertical honeycomb, and vertical isogrid are similar with respect to the final shape and the characteristic ultimate failure.However, the final shapes of the horizontal honeycomb and isogrid models were different from the experimental results even though the failure initiation was similar.In the case of the horizontal honeycombs, rows collapsed under the same angle and created a stack of collapsed rows thus failing to capture random failure paths observed in experiments.However, relatively high stress can be seen in +45 • diagonal, indicating the possibility of simulations being able to capture experimental observations with the help of more accurate constitutive models and material properties.Deviations in failure paths for horizontal isogrids have already been discussed in the context of Figure 14 and thus are omitted here.
Trends observed in the stress-strain curves from simulation and experimental results are compared in Figure 16, where solid lines indicate curves obtained from experiments and dashed lines the indicate ones from simulation.The reported stress corresponds to effective stress (=total force/surface area of the top surface).Strain was calculated by dividing the displacement of the top plate by the initial height of the sample.Therefore, the stress and strain values reported in Figure 16 correspond to engineering values rather than true values.
In general, 0/90 and 45/−45 square grids showed oscillatory patterns, the horizontal honeycomb grid showed a stable compressive behavior, and the horizontal and vertical isogrids showed high initial peaks.Based on these observations, one could classify the 45/−45 square grids and horizontal honeycombs as bending-dominated structures, and 0/90 square grids, vertical honeycombs, and isogrids as tension-dominated structures. 42This is confirmed by experimental and numerical results in present work, since the 45/−45 square grid and horizontal honeycombs show low elastic slope and then almost constant plateau stress.On the other hand, 0/90 square grids, vertical honeycombs, and isogrids show high initial peak stress and then sharp softening after the peak stress point.However, this softening region does not exhibit constant plateau stress as in theoretical idealizations and show high oscillations for all samples except for horizontal honeycombs.
The failure initiation patterns in the experiments and simulations indicate that bending and buckling of the struts cause the failure in the considered lattice structures.Qiu et al. 60 showed stress-strain results of various lattice structures, where they found that the square grids (0/90 and 45/−45) and the triangular grid showed initial peaks for a small strain deformation as shown in figure 13 of Qiu et al. 60 and a stable stress-strain response for a honeycomb shape.In our study, the stress-strain curves for both square grid and rotated square grid patterns from simulation showed similar trends as the experimental data.Regular oscillations were observed in the stress-strain curves of both square grid samples and numerical results capture experimental results accurately.The stress-strain curves for the honeycomb patterns from the simulations also follow the experimental trends, in general.One of the discrepancies in the horizontal honeycomb results is that the plateau stress level from simulation is higher than that from experiment.A similar discrepancy was also observed in studies reported by Habib et al. (figures 13 and 14 in Habib et al. 39 ) for both vertical and horizontal 3D printed polymeric honeycombs with thickness of 1 mm, except for the fact that numerical plateau stress was lower than the experimental one.As for the vertical and horizontal isogrid lattices, the trend of the F I G U R E 16 Comparison of stress-strain curves from experimental (solid lines) and finite element results (dashed lines).stress-strain curves from the simulation was close to that from experiments except for a different overall magnitude of the peak stress in vertical isogrids.The overestimation of plastic yield strength (or plateau stress) is potentially a result of simple fracture criterion that considers only the maximum principal stress to initiate the failure.However, considering the multiaxial nature of the stress state at the failure susceptible elements, one could incorporate effects of all stress components potentially expediting the onset of failure which would result in lower yield/plateau stress.Moreover, the slope of the numerical stress-strain curve is steeper than the experimental one in all six configurations which means that the values of Young's moduli used in the simulations were higher than the actual values observed in experiments.This could be due to the fact that typically there is a large difference in properties obtained using samples printed with different printers, process parameters or other inputs.Another potential source of errors is that elastoplastic properties were approximated based on tensile testing results given in References 48-50 instead of compression testing.

TA B L E 2
Finally, absorbed strain energy densities were calculated by numerical integration for each pattern, as shown in Table 2.The 0/90 square grid, horizontal honeycomb, and vertical isogrid were found to absorb more energy than the rest of the patterns.Zeng et al. 49 performed compression tests on a 45/−45 square grid, a honeycomb grid, and a 0/90 grid and showed that the honeycomb grid has the highest specific energy absorption among the three lattice structures.In our study, while relative errors between experimental and FE predictions are negligible (0% and 3%) for the square grid patterns, errors for the honeycomb and isogrid samples are considerable (17%, 33%, 25%, and 41%).
The energy absorption of lattice structures was further studied in the context of energy absorption efficiency.The concept of efficiency represents the total absorbed strain energy density divided by the maximum stress at a specific strain value.Mathematically, the energy absorption efficiency is given by where E is efficiency,  is stress,  is strain, and  max is maximum stress until the strain reaches maximum value. 21As for physical intuition of the efficiency measure, one could consider the case when efficiency is equal to 100%, the material then behaves like rigid-perfectly plastic up to the point of 100% compressive strain where sudden densification happens.This is the ideal scenario for energy absorbing material.The energy absorption efficiency computed using Equation (1) compares the response of a real lattice structure to this "ideal" energy-absorbing material.Efficiency was calculated for both the experimental and numerical stress-strain results and reported in Table 2. From the experiments, the horizontal honeycomb had the highest efficiency of 28% at 30% strain, followed by the vertical isogrid.The vertical honeycomb, the square grid, and the 45 • rotated square grid all showed a similar amount of efficiency.In simulation, the predicted efficiency showed a similar trend: highest efficiency for the horizontal honeycomb grid, and three lattices (vertical honeycomb, square grid, and 45 • rotated square grid) showing similar efficiencies to each other.The predicted efficiency values were lower than the experimentally measured energy absorption efficiencies for most of the lattices, which is attributed to a high initial stiffness and a high strength from the simulation.A high relative error from the horizontal isogrid was due to the different fracture pattern between experiment and simulation, resulting in larger difference of the stress-strain curve.

CONCLUSION
The current study involving numerical simulations and experiments on short fiber reinforced additively manufactured lattice structures revealed important insights on the effect of lattice geometries on their compression and energy absorption characteristics.The 0/90 square grid pattern exhibited a unique oscillating stress-strain response with peak stress of 11.5 MPa due to sequential buckling of horizontal rows.The horizontal isogrid pattern displayed high initial peak stress of 10 MPa, and irregularities of the stress-strain curve in the post-failure regime.The high strength of these patterns makes them appealing for lightweight structural applications, while the horizontal honeycomb pattern demonstrated low initial stiffness with a steady plateau stress of about 7 MPa, indicating its suitability for energy absorption applications.With a strain energy density of around 2.2 MJ/m 3 , it has the capacity to store the highest amount of strain energy per observed maximum stress, resulting in an energy absorption efficiency of 28%.Key failure initiation mechanisms such as buckling, and fracture of struts have been captured in the FE models using element deletion method.The versatile numerical framework developed in present study can be extended to other material extrusion mid-scale and large-scale AM platforms where composites are often the preferred materials for minimizing geometric distortions due to thermal gradients.

1
Six different 2D lattice configurations grouped by three grid shapes: (A, B) square grids in 0 • /90 • directions and −45 • /+45 • directions, (C, D) honeycomb grids in vertical and horizontal directions, and (E, F) iso grids in vertical and horizontal directions.In order to present the unit polygon shape of the lattice, the pictures in this figure show only a portion of the entire domain.
U R E 2 (A) 3D printed lattice sample and its dimensions, and (B) Quasi-static compression test setup.All six lattice configurations have the same dimensions as in (A).

F I G U R E 4
Schematic diagram for the finite element analysis (FEA) workflow: (A) Assigning anisotropic material properties through additive manufacturing (AM) induced fiber orientation, (B) perturbation (modal) analysis to determine structural buckling modes, and (C) compression simulation on lattice structures with buckling modes modeled as small imperfections.
U R E 5 (A) Print orientation of lattice structure.Load and boundary conditions from (B) the front view and (C) the side view.

F
I G U R E 6Stress-strain profile of 0/90 square grid shape.Each peak/valley indicates a collapse of a row.(a 1 ) the first collapse, (a 2 ) the third collapse, and (a 3 ) final shape.F I G U R E 7Stress-strain profile of 45/−45 square grid shape (b1) the first collapse, (b2) after a few collapses (strain of 0.1), (b3) final shape.All failures of the cells are in the +45 direction, and there is no −45 directional failure.

F I G U R E 9
Compression behavior of horizontal honeycomb lattice structure in sequential order.Red arrows indicate a new collapse of a row and green indicates an old collapse from the previous step.(A) Initial failure occurred in the −30 • direction.(B) The next failure occurred with a slightly larger angle.(C) The third failure occurred in the positive diagonal direction (+45 • ).(D) The fourth failure occurred in the negative diagonal direction (−45 • ).

F I G U R E 11
Modulus polar plot of four different lattice shapes in all load directions (0 • -360 • ): Honeycomb, square, isogrid, and diamond (Re-plotted from figure 3.9 of Chopra 54 ).The original figure includes the stiffness from 13 different lattice shapes. 54F I G U R E 12 Initiation of failure for square grid and rotated square grid configurations.

F
I G U R E 13 Initiation of failure for vertical honeycomb and horizontal honeycomb configurations.F I G U R E 14Initiation of failure for vertical isogrid and horizontal isogrid configurations.
U R E 15 Crushed shapes at large strains from the finite element models for (A) square grid, (B) rotated square grid, (C) vertical honeycomb, (D) horizontal honeycomb, (E) vertical isogrid, and (F) horizontal isogrid.
Material properties used in simulations.
TA B L E 1Note: Values are from References 35-37.a R12, R13, and R23 were not reported in References 35-37 and were assumed to be 1.0 in this study.b Strength value was modified from 50 to 65 MPa in this study.