Modeling Three-Dimensional-Printed Polymer Lattice Metamaterial Recovery After Cyclic Large Deformation

Lattice metamaterials have attracted great attention during the recent decade due to their capacity in tunable mechanical properties and counterintuitive properties when compared with natural materials, such as negative Poisson's ratio [1–3] and negative thermal expansion [4–6]. They are usually composed of periodic “unit cells” which derive from ingenuously designed microstructure [7,8]. The periodically arranged design concept is similar to natural crystal lattice structures [9,10]. Lattice structure designs explore light but strong properties such as with octet-truss lattices stacking to face-centered cubic (FCC) configuration possessing linear scaling relationships with better stiffness and low density [10].

Additive manufacturing technology provides a new way to fabricate complicated three-dimensional (3D) designed structures in micro, millimeter, and centimeter scales, which endow more possibilities in creating ordered cellular structure materials. Light-based 3D printers have incomparable ability in processing extremely hollow 3D structures on a micro-scale. Using theoretical model predictions, 3D printing has produced metallic and ceramic-based lattices metamaterials using stereolithography (SLA) or selective laser sintering (SLS) in micro, millimeter, and mesoscales, and those metamaterials demonstrated better mechanical behaviors in some or most of stiffness, toughness, damage-tolerance (in low-density regions), or good flexibility in reconfiguration ability properties [11–15].

There are several preeminent research related to metal and ceramics lattice structures area, while polymer-based metamaterials are rarely covered. Most polymer materials are recognized for their time-temperature-dependent soft, lightweight properties. Polymer glass transition temperature (T_g) has been described based on the free chain motion of polymer molecules. Polymers with T_g far beyond the room temperature are usually strong and brittle due to chain segment immobility. Polymers with low T_g can be easily deformed under low stress. Such discrepancies among polymers increase the difficulty of studying their lattice metamaterial properties.

Here, we studied polymer lattice metamaterial mechanical behaviors, including the linear and nonlinear deformations of polymeric lattice with rationally designed structures, inspired by the body-centered cubic (BCC) crystal lattice, and constituted by nodes (j) and struts (b) to build the unit cell block aligned in the 3D lattice. A commercial desktop stereolithographic (SLA) 3D printer was used to manufacture the BCC lattice structures with two different acrylate-based polymer materials which exhibit similar performance derived from the meta-structure, while with distinctions originating from the base materials divergence. We will regard the lattice structures as a “new material,” by comparing with the previous micro-/nano metamaterials experiences of (power law?, n) relationship between moduli and stiffness with density in the linear region, to explore the material properties and mechanical performances while transitioning from linear to nonlinear large deformation [3,16–21].

The main targets of the current investigation include the following considerations for polymer (acrylate) resin-based 3D-printed BCC lattice metamaterials:

• Determination of modulus and strength power-law scaling (n) with relative density for these metamaterials and compare with the Maxwell stability criterion for BCC (n = 1) and other structures such as open cell foams (n = 2).

• Determination of the mode(s) of (stretch-induced, buckling, etc.) deformation/collapse for these metamaterials and investigate any presence of inhomogeneous inter and intra unit cell strain patterns and load transfer.

• Use digital image correlation (DIC) to visualize linear compression region deformation/collapse behavior for these metamaterials and develop models to represent typical deformation/collapse pattern transformations observed under cyclic compression condition.

• Determine the strain range(s) and mechanism(s) of cell deformation/collapse reversibility for these metamaterials.

We hope that our work will contribute to efforts to control the deformation/collapse behavior of polymer (acrylate) resin-based 3D-printed BCC lattice metamaterials’ recovery after cyclic large deformation. This will allow tunable mechanical properties and customized local patterns in order to protect specific regions, as required in many applications such as in microelectronic and aerospace areas.

In this study, we used Solidworks© (Dassault Systèmes, Velizy-Villacoublay, France) software to generate 3D lattices computer-aided design (CAD) models and sliced with 50-µm thickness in the vertical direction to a series of 2D images sequentially for a 3D printer. Form 2 SLA printers (Formlabs Company, Somerville, MA) with 405-nm ultraviolet laser beam were used for fabrication. After printing, the object was immersed in isopropyl solvent and placed in an ultrasonic environment for removing uncured resin. Figure 1(d) is the image of a 3D-printed lattice.

The raw resin was acquired from Formlabs company and contained photo-initiator, acrylate-based monomer and oligomer which underwent polymerization when exposed to an ultraviolet environment. In order to explore the relationship between base materials and properties of lattices, two different acrylate-based photopolymer resins, “clear resin” (named Clear in following), which is relatively rigid at room temperature with ∼2 GPa elastic modulus (Fig. 2), and “flexible resin” (named Flex in following), which is bendable and soft with around 7-MPa elastic modulus at room temperature. The glass transition temperature of the Clear is around 80 °C based on dynamic mechanical analysis (DMA), and for Flex, it is around −10 °C based on differential scanning calorimetry (DSC).

Scanning electron microscope imaging was carried out using COXEM EM-30 PLUS Desktop SEM at 20 kV voltage. The static images were captured by the rear camera of iphone7; the images from the compression test were obtained through BlueFOX3, Schneider Kreuznach 50-mm lenses using GOM (GOM, Germany) Digital Image Correlation.

We studied the mechanical properties of all lattice metamaterial samples by using uniaxial compression tests with a 10-kN load cell installed on Instron 5966 machine operated at room temperature and 5 mm/min displacement rate. Two compression force directions were used separately in order to avoid orientation discrepancy, i.e., the compression force applied in the plane parallel to the build plate in 3D printing (Fig. 1(b)) was labeled “parallel orientation”, and the direction perpendicular to the build plate the “perpendicular orientation”. Stress and strain were calculated through Intron record. The length of the samples was around 50 mm. The compression tests were video recorded by BlueFOX3, Schneider Kreuznach 50 mm lenses. The local strain distribution mapping was characterized by Digital Image Correlation using the 2017 GOM correlation software. The spot pattern drawing was constructed using a homemade pigment stamp with a dot size of around 3–7 pixels, and the strain sensitivity of the software was typically 50–100 micro-strain. Four sections were selected from the surface in analysis with Secs. 1 and 2 attributed to the lateral lines at the sixth and fourth rows (bottom lines labeled as the first rows), and Secs. 3 and 4 to the vertical line in the sixth and eighth columns (the first columns refer to the first left lines). Then, we plotted the position in the sections with local strains at different moments. The frequency of recording videos was 1 pic/s. The lattice structure modulus was defined by the slope of the initial linear region. We used the compression loading–unloading cycle test using 5 mm/min displacement rate at room temperature and by presetting 60% as the final strain. At least 24 h of wait time was used between each loading-unloading cycle test for samples to relax.

We used finite element methods (FEM) to simulate the deformation of the unit cell. Models of the BCC unit cell generated by the SolidWorks^© were imported to Abaqus^© (Dassault Systèmes, France) for FEM simulations. The compression of the unit cell was modeled by using displacement-controlled analytical rigid body plates. The bottom rigid body plate was fixed to the unit cell and constrained by all displacement and rotation degrees-of-freedom. The upper rigid body plate was used to compress the unit cell through a displacement boundary condition. We used “surface contact with friction coefficient of 0.1 and hard contact” in Abaqus to simulate the interaction between compression discs with surfaces. The unit cell was meshed using ten-node quadratic tetrahedral elements (C3D10) with isotropic elastic material behavior defined according to the mechanical behavior data of Flex. The unit cell had 1146 elements in the analysis.

The T_g of Clear is around 80 °C, and it is around −10 °C for Flex. Hence, the T_g of Clear is above room temperature making it stiffer than Flex. The Clear shows around 1.5–2% elongation at break, with brittle failure at room temperature (Fig. 2). The Flex exhibits elastic deformation before the break with the break elongation of around 35%, which is regarded as rubber-like material (Fig. 2). The tensile modulus was calculated as ∼2 GPa using the linear region slope from Clear tensile samples and ∼7 MPa for the Flex tensile samples. We also printed a 50 × 50 × 50 mm³ cube which has the same size as the lattice sample for compression. The Clear compressive modulus calculated by the slope of the initial linear region of the stress–strain curve was 560.5 MPa for the stress applied in the perpendicular direction and 541 MPa for the parallel direction (Fig. 2(a)). The Flex compressive modulus was 7435 KPa for the perpendicular and 7659 KPa for the parallel directions (Fig. 2(b)).

The scanning electron microscopic (SEM) images in Fig. 3 revealed that the 3D lattices were typical layer-by-layer structures, and the layer thickness was around 50 µm which is consistent with the printing process input. There were some defects in the struts, and the corners of the struts were typically not perfect right angles. During the printing process, the first several layers attached to the build plate are typically built thicker than the others because the startup layers need to be firmly attached to the build plate, and therefore, these layers take longer exposure time to attach to the build plate. The measured relative density for the printed samples was higher than the geometrical calculation (Fig. 4). Such heavier values are attributed to the difficulty in removing all uncured resin sticking to the struts. The uncured resin contains monomers and oligomers which slowly polymerize in laboratory environments.

Our data revealed that the designed BCC polymer lattice metamaterials exhibited deformable and recoverable mechanical behavior after large compressive deformation even though they were constructed using a brittle base material (acrylate-based photopolymer “clear resin”). Under cyclic uniaxial compression, the 0.3 mm t_s lattice was compressed to 60% strain. The first stress–strain response exhibited multiple yield peaks and had higher stiffness with higher yield strength, but during the consecutive loading cycles, the peak strength decreased progressively with continuing compression cycles. At the end of the stress–strain curve plateau of each loading cycle, stress sharply increased at the final densification (Figs. 5 and 6). In the following loading–unloading cycle of each test, the yield strength decreased and the multiple yield peaks faded away in the stress–strain curve. The elastic modulus in the tenth compression cycle dropped to ∼0.6 MPa, which is 0.1% of the base material modulus (560 MPa). Plastic yielding occurred after linear region with initial high strength followed by post-yield softening due to the irreversible deformation of structures during the cold deformation below T_g.

Compressive loading resulted in lattice volume shrinkage due to squeezing of the vertical space (compression force direction) within 10 rows of unit cells. From the in situ images of lattice deformation, it is revealed that the unit cell's shrinkage process was not homogeneous. The rows which collapsed earlier also led to earlier yield corresponding to the initial yield performance (a behavior which will also be discussed with the flex resin later using photographic data). The buckling rows had strain concentration, most likely at the strut junctions and delayed softening (yielding) in other layers. After all of the rows flattened, the frames of the opposing cells touched to result in local densification; therefore, the lattice strength started to increase as this repeated cell collapse process progressed until the overall lattice densification occurred on macroscopic scale. When a sufficient number of cells collapse, the lattice structure has less open space and a transition from collapse to compression begins to occur for some cells possibly culminating in bulk material architectural behavior and irreversible failure for the whole lattice if it is loaded to high levels of stress. When the 60% strain level is reached and loading discontinued, the flattened lattices started to quickly stand up at the moment of force release, then slowly recovered to go back (close) to the original height.

In Figs. 5 and 6, the image of 7 min after tenth cycle compression showed the geometry recovery. We measured sample size in three dimensions before and after the whole cyclic test, the size change was less than 10%. Also, if we compare images from loading before and after the cyclic test for these 0.3 mm t_s samples loaded in both perpendicular and transverse directions, even if they had a few failures like broken or crack struts, they still maintained the original geometry shape. However, the modulus of the tenth compression cycle decreased to 20% of the original modulus (Fig. 7). Nonetheless, perpendicular and parallel orientations of the t_s = 0.3 mm samples had similar moduli, stress–strain response, and recoverable mechanisms with the initial yield strength values having negligible divergence.

The t_s = 0.3 mm clear resin lattice samples showed good recovery after the 60% compression strain, as well as durable mechanical behavior after 10 loading–unloading cycles. For other struts, we observed that the thinner struts with t_s ≤ 0.75 mm displayed shape recoverability (Figs. 8(g) and 8(h)), images compared before and after the 10-cycle compression displayed shape recovery. The t_s of 0.75 mm represented an approximate transition point from recoverable to irrecoverable at 60% strain (Figs. 8(i) and 8(j)). Although from the images, the t_s = 0.75 mm sample seemed to go back to its original shape, there were many fractures and defects after the final compression. Besides, the modulus of tenth cycle was nearly zero indicating that only the geometry recovered but the stiffness was lost, pointing to a lattice weak in fatigue resistance. When struts’ thickness increased to t_s = 0.9 mm (Fig. 8), the lattice sample experienced an irreversible and catastrophic failure.

Thicker cell walls cause higher lattice density, leading to the early arrival of lattice densification. The compression behavior of the densified samples was similar to the bulk material compression and destroyed base materials.

The Flex lattice structure with T_g of the base material below room temperature has good reversibility after large deformation. Strut thickness is again an important parameter affecting cyclic properties. At thickness t_s less than 0.6 mm, and after the first cycle, the modulus value becomes stable in the linear region during cyclic compression, and the stress–strain curve has good repeatability in ensuing cycles. In Fig. 9(a) (t_s = 0.4 mm), the elastic modulus decreased ∼33% after the first cycle. With t_s = 0.6 mm (Fig. 9(b)), the first and second compression cycles had higher plateau and yield stress, but the third to tenth cycles resulted in cracked structural elements. The thinner struts at thickness t_s < 0.6 mm exhibited typical elastic buckling and recovered back to the original shape immediately after load release. When the thickness of struts increased, the lattices failed like disintegrating slag due to base material destruction by the time 60% compression was reached. The probable causes for this behavior can be listed as follows: (1) The higher density led to earlier densification, and the base material failed sequentially under compression. (2) Some solvents left in the structure during the resin removal swell the struts, thus deteriorating their molecular structure, so the base material was easily damaged.

Figure 10 reveals that the soft lattice samples had the plateau stress around 0.65 kPa after yielding. The initial buckling behaviors started at ∼2.5% strain from two (loading) sides of the lattice at linear regions. Buckling spread from the outside to inside layers at relatively small strain levels with the center layers undergoing a higher extent of buckling than the outside layers and the lattice transformed to the hourglass-like configuration. In the hourglass-like configuration, the crossing struts inside the hourglass frame were like the gripper, and the outside struts bent toward the fixed center point. As the compressive displacement increased, the second layer abruptly collapsed as shown in snapshot V in Fig. 10, and then, the upper layers collapsed quickly. This event corresponds to the serrated fluctuation in the stress–strain curve (VII in Fig. 10). Following this sudden flattening of the layers, the stress-strain curve plateaued before densification. When combined with in-situ video, we could conclude that the serrated peaks corresponded to the instantaneous collapse of a layer, and this collapse triggered a relatively long time for the collapsed and the adjacent struts to adjust their pattern in order to accommodate the ensuing increase in displacement and achieve (further) collapse. So, the buckling of the struts could transform the lattice pattern from an hourglass-like model to a twist model during compression as will be illustrated later using finite element analysis. The fluctuating peaks in the stress-strain curve may come from the moment of pattern transition. However, sometimes, the collapse is reversible; we observed a layer that had already buckled at 13% strain but stood up (recovered) again at 30% strain (see snapshots V–VIII in Fig. 10). In the meantime, the upper and/or lower layers were buckling and collapsing. We considered the possible reason that the buckling/collapsing layers were serving to remove compressive loads from the layer which has buckled earlier. This allowed the “already buckled” layer to go up (recover) even when the whole lattice is under compressive deformation. We note that the assumption of “elastic buckling” supports this argument and allows the transition of localized deformations from one layer (or even from one cell) to the other.

We used the stress-controlled compression mode to compare with the displacement-controlled mode we reported above using t_s = 0.4 mm samples. In Fig. 11, the total time from 4% to 65% compression strain was 67 s, but under displacement-control, it was 330 s (Fig. 10). Hence, the compressive displacement increased fast at around 0.65 KPa stress condition (Fig. 11 snapshots) which is equal to the plateau stress of displacement-controlled mode. The sample exhibited slight strut buckling behavior similar to what is observed under displacement-controlled compression at 70 s (at the beginning of the compression). Until 100 s, the unit cells continued to initiate buckling in a very short time without the obvious sequential layer-by-layer collapse densification. The stress control mode showed very consistent and continuous cell collapse even though the pattern transformation occurred fast due to the short duration of displacement.

The in-situ records of the deformation revealed inhomogeneous inter- and intra-unit cell strain patterns. The compressive force transfer at the vertexes that connected with outside and crossing struts led to struts buckling but also allowed adjustments in buckling configuration as affected by the in-plane or out-of-the plane moments. These events led to inhomogeneous strain distributions in unit cells even under small deformations. Consequently, we utilized digital image correlation to monitor local strains and performed FEA to simulate the observed results. Figure 12 shows the schematic for our proposed hourglass and twist mechanism and the corresponding pattern transformation. This transformation starts with a buckling process, and the corresponding pattern transformation is shown in Fig. 12(a). Our FEA simulation of the initial buckled shape occurring within 5.6% strain is shown in Fig. 12(b). This analysis was performed using the stress–strain curve shown in Fig. 11. FEA simulation of hourglass and twist deformation (Fig. 12(c)) within 33.3% strain is shown in Fig. 12(d). We performed compression tests up to 10% global compressive strain using t_s = 0.8 mm. The stress–strain curve (Fig. 13(b)) was in the linear region, and we used DIC to visualize the plane displacements during compression. Sections 1 and 2 were lateral lines, and Secs. 3 and 4 were vertical lines in the plane (Fig. 14(a)). The lateral sections displayed pretty regular strain variation with the position in sinusoidal form (Fig. 14(b)). In the sinusoidal distribution shown, “wavelength λ” was approximately 5 mm, corresponding to the length of one unit cell, and the “amplitude” represented the maximum difference of local strain in one lateral strut of a unit cell. The results reveal that the vertexes took the primary displacement in the vertical direction, and lateral struts approached nearly zero displacements in the vertical direction. In the perpendicular sections, “wavelengths λ” decreased due to the vertical space compression, but vertexes of the unit cells still showed large displacement, and the struts between two vertexes had less displacement in the applied force direction. Nevertheless, there were some points in the struts showing positive displacement due to the vertical struts bending. Also, the vertical sections had only eight repeated periods because the spots on the top and bottom edges were not captured by the camera. The repeated periods had a similar amplitude at the same moment in time which means that the strain distribution in each unit cell was similar. But in one unit cell, the vertexes experienced larger displacement, so the two ends of a strut contributed larger vertical strain; the shape of the struts was bending like a wave at that moment.

The displacement distribution and buckling behaviors in the linear region are very similar between unit cells. However, in one cell, the vertical and lateral struts are different. The vertical struts show obvious bending compared with lateral struts due to the less vertical space. The vertexes or joint points are contributing more displacement during compression with both vertical and lateral struts. The middle part of lateral struts was nearly zero displacements. So, the inhomogeneous deformation of BCC lattice compression derived from the joint movement dominated displacement with the vertexes bearing more stress and energy during the compression loading. We note that these locations are also where nonuniformities exist due to imperfect printing accentuated at inside corners (Fig. 3).

The yield strength and Young's modulus in cellular structures and natural foam materials display the scaling rules as (E/E_s) ∼ (ρ/ρ_s)ⁿ and (σ_s/σ_ys) ∼ (ρ/ρ_s)ⁿ in the linear region [22,25,26], where E and σ_s are the moduli and yield strength of the lattice, E_s and σ_ys are the moduli and yield strength of the bulk. The exponent n is associated with the lattice structure. For typical bending-dominated and stochastic porosity, cellular foams typically have n = 2 or 3. When the beam structures deform predominately through stretch or compression in cell walls and struts, the lattice deformation is regarded as stretch-dominated leading to higher strength and stiffness due to no intrinsic bending deformation of the truss structure. In this case, the ideal stretch-dominated structure scales linearly with (E/E_s) ≈ (ρ/ρ_s) and (σ_s/σy_s) ≈ (ρ/ρ_s). For a pin-joint frame, Maxwell's stability criterion provides the rule to distinguish between the bending-dominated and stretching-dominated. Our BCC lattice satisfies the condition M > 0, meaning that it is a stretch-dominated structure. In previous literature, there are a series of stretch-dominated lattices with relatively high stiffness at low mass density with linear scaling [22,25,26].

For our BCC Clear Resin lattice in linear-elastic deformation, the modulus and strength follow power-law scaling with relative density with E ≈ ρ^*1.57 and σ_y ≈ ρ^*1.86 (Fig. 15) [27]. The traditional foams usually have E ≈ ρ^*2 for open cells. According to the Maxwell stability equation, our BCC lattices should belong to stretch-dominated group and the exponent n = 1. All Flex Resin samples in the linear region have E ≈ ρ^*1.64 (Fig. 16) [27]. Obviously, these results for the Clear and Flex resins are fairly close aside from also being reasonably close to traditional foams usually having E ≈ ρ^*2 for open cells.

Theoretically, the BCC lattices have better performance with low power-law scaling than natural open-cell. Therefore, a discrepancy exists between our experimental results and theoretical stretch-dominated structure scaling prediction of E ∼ ρ*. We offer to explain this deviation partly by the imperfect SLA printing processing. The excess acrylate-based resins were hard to remove from the struts causing the lattice weight to be heavier than it should be, and thus, the apparent density was also higher. So, the slope was higher than expected. Another reason could be that, during the printing process, the geometry was not totally in accord with the CAD model, especially in strut juncture locations. Such deviations are expected to change the struts’ mechanical behavior during loading. This will also affect the power-law scaling and cause it not to agree with the prediction.

In summary, we reported a buckling-induced uneven deformation phenomenon and good reversible performance in 3D-printed polymer BCC lattice metamaterials. The design concept was inspired by a crystal lattice structure. The desktop SLA printer was imperfect in detail fabrication at a small scale. Two different acrylate-based commercial polymers were used in cyclic compression conditions to study the mechanical behaviors. Thinner cell walls were beneficial to recovery, and when the thickness of the struts reached a critical value, irreversible failures occurred. Thus, when the strut thickness increased, the recoverability decreased under cyclic compression.

Among the main findings of this work, we cite the following:

• The polymer (acrylate) resin-based 3D-printed BCC lattice metamaterials that were tested proved not to follow the Maxwell stability criterion, which places them in stretch-dominated group but exhibited modulus and strength power-law scaling with relative density (n ∼ 1.6) falling between BCC structures (n = 1) and traditional foams with open cell structure (n = 2).

• Our in-situ records of the deformation revealed inhomogeneous inter- and intra-unit cell strain patterns. We showed that the compressive force transfer at the vertexes that connected with outside and crossing struts, leading to struts buckling which also allowed adjustments in buckling configuration as affected by the in-plane or out-of-the-plane moments. These events led to inhomogeneous strain distributions in unit cells even under small deformations.

In order to analyze the buckling deformation mechanism described earlier, the softer acrylate-based resin was used to study the struts’ bending deformation. The resulting hourglass and twist models we propose here represent the typical buckling-induced pattern transformation under cyclic compression conditions and provide an opportunity to achieve tunable modulus in a lattice structure and to control local collapse behaviors. Our DIC data visualized the linear compression region and proved that the nodal (joint) points suffer higher strain and energy concentration compared to the rest of the lattice.

We also noticed that cell collapse could be reversible at low strain levels; we observed a layer that had already buckled at 13% strain but stood up (recovered) again at 30% strain while the upper and/or lower layers were buckling and collapsing. This may be because the buckling/collapsing layer(s) were serving to remove compressive loads from the layer(s) which buckled earlier. This allowed the “already buckled” layer to go up (recover) even when the whole lattice is under compressive deformation. We note that the assumption of “elastic buckling” supports this argument and allows the transition of localized deformations from one layer (or even from one cell) to the other.

In the future, it will be desirable to control buckling behavior in order to be able to control and program the pattern transformation. This will allow tunable mechanical properties and customized local patterns in order to protect specific regions, as required for application in microelectronic and aerospace areas.

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent is not applicable. This article does not include any research in which animal participants were involved.

The authors attest that all data for this study are included in the paper.