Tensile properties of sheet-based triply periodic minimal surface lattices fabricated using vat polymerization

Abstract

Additive manufacturing enables the production of lattice structures with tailored mechanical performance. Among these, triply periodic minimal surface (TPMS) lattices have emerged as promising architectures for lightweight and high-strength applications. However, while their compressive behavior has been extensively studied, their tensile behavior remains largely unexplored. This study examines the bulk tensile properties of sheet-based TPMS lattices fabricated via digital light processing, focusing on Gyroid, IW-P, and Primitive topologies at various relative densities. Novel functionally graded Gyroid specimens were first tested to determine the number of unit cells required to capture the bulk tensile response of TPMS sheet-based lattices. The functional grading design of the tensile specimen ensures that premature failure does not occur at the grip and lattice interface. The results show that the tensile properties reduced with the increase in number of unit cells until it is plateaued at 6 × 6 unit cells along the cross-sectional area of the specimen. The bulk tensile responses of different TPMS topologies at different relative densities show that the tensile stiffness exceeds the compressive stiffness, whereas the yield strength is higher in compression than in tension. Among the examined topologies, IW-P exhibited the highest stiffness and strength at higher relative densities, while Primitive performed better at lower relative densities, with Gyroid showing intermediate behavior across the covered range of relative densities, under both uniaxial tension and compression. The results establish quantitative relationships for tension and compression asymmetry in TPMS lattices and provide guidance for designing TPMS-based lattice architectures optimized for tension-dominated applications.

Introduction

Additive manufacturing (AM) has become a key technology in producing porous structures, enabling the creation of tailored components with intricate shapes that traditional methods struggle to achieve. This includes the printing of auxetic metamaterials^1,2,3 which are a class of architected materials that exhibit a negative Poisson’s ratio, i.e. they expand laterally when stretched and contract when compressed. Common auxetic topologies include chiral and anti-chiral structures⁴, re-entrant structures⁵, auxetic foams⁶, rigid and semi-rigid rotating structures⁷, and origami systems⁸. The introduction of auxetic features into cellular structures enhance their energy absorption, indentation resistance, shear stiffness, and fracture resistance⁹. In addition, AM allows the printing of non-auxetic lattice structures (typically higher stiffness and strength than auxetic materials) that have shown promising applications in various engineering fields, such as crash energy absorbers^10,11,12,13; scaffold design for tissue engineering^{14,15,16,17,18}; acoustics^19,20; sandwich panels in the aerospace field^21,22,23,24. Non-auxetic lattices are typically higher in stiffness and strength and usually provide simpler.

Triply periodic minimal surfaces (TPMS) are mathematically-derived intriguing lattices that are both self-supporting and non-intersecting. These surfaces exhibit periodicity in three independent spatial directions and possess minimal area properties, meaning they locally minimize the surface area of the designated boundary so that the average curvature at each point on the surface is zero^25,26. TPMS can be further categorized into ligament/skeletal/solid-networks or sheet-networks²⁷. Solid-network TPMS is generated by considering one of the subdomains divided by the minimal surface to be a solid, whereas the other subdomain is considered to be a void. On the other hand, sheet-network TPMS is generated by solidifying the volume enclosed by two minimal surfaces evaluated at two different level-set constants, where the level-set constant is the specific value of the level-set approximation function at which the interface or contour is defined. Various investigations^28,29,30,31 have found that sheet-based TPMS lattices outperform their solid-based TPMS lattice counterparts, i.e. same topology, in terms of mechanical properties at the same relative density.

One of the most important aspects of cellular materials is defining their bulk mechanical properties, which are the macroscopic properties of cellular materials obtained when enough number of unit cells are considered within the cross-sectional area (perpendicular to the loading direction) of the testing specimen³². In terms of the compressive properties of TPMS lattices, Abueidda et al.³³ performed uniaxial compression experiments on polymeric Primitive sheet-based TPMS lattice cubes with various number of unit cells (from 1 to 5), and have found that at least a 4 × 4 × 4 lattice cube is needed to obtained the bulk mechanical properties of the Primitive lattice. Similarly, Maskery et al.³⁴ performed a numerical investigation on the Diamond sheet-based TPMS lattice using ABAQUS finite element software and have found that four unit cells are enough to obtain a converged elastic modulus, i.e. the bulk elastic modulus. Overall, the bulk compressive properties of various TPMS lattices have been heavily investigated already both experimentally and numerically^{28,33,35,36,37,38,39,40,41}. In terms of physical experiments, two studies have investigated uniform TPMS lattices in uniaxial compression, focusing on sheet-based TPMS cubic samples of Schwartz Primitive, Schoen’s I-WP, and Neovius³³, and Gyroid³⁶. The elastic modulus and the ultimate strength of these four sheet-based TPMS lattices were investigated for a range of relative densities and results indicated that Neovius lattice possesses the highest properties in compression, followed by Gyroid, I-WP and Primitive, in descending order. Similarly, Al-Ketan et al.²⁸ investigated uniform TPMS lattices based on both sheet-based and strut-based Diamond and Gyroid TPMS lattice topologies. The uniaxial modulus and yield strength were investigated under uniaxial compression, for a range of relative densities, and results indicate that sheet-based TPMS lattices are superior to their strut-based counterparts, where Diamond topology possesses higher properties than Gyroid in both TPMS categories. Furthermore, Afshar et al.³⁵ and Zhang et al.³⁷ investigated TPMS lattice cubes under uniaxial compression tests, using both uniform and functionally graded lattices. Afshar et al.³⁵ investigated uniform and linearly graded lattices in terms of relative density, for both Primitive and Diamond strut-based TPMS lattices. Zhang et al.³⁷ investigated uniform and functionally graded lattices in terms of relative density, lattice topology and both together, considering Schwartz Primitive, Schoen’s I-WP, Gyroid and Diamond.

On the other hand, the bulk tensile properties have not been investigated as much, as initially these lattices were mainly limited to applications where they were mainly loaded under uniaxial compression, such as in sandwich panels used in aerospace applications. In addition, designing and testing the compression specimens is easier than the tensile specimens, where the latter can have stress concentration regions at the transition region between the TPMS lattice and the solid parts if not well designed that can lead to early failures. However, the applications of cellular lattices are no longer limited to ones where the lattice is solely under uniaxial compression but are desired in various applications where they undergo different loading cases, such as uniaxial tension, shear and multiaxial loading. For example, in the biomedical field, lattices are highly desired and implemented in the fabrication of prosthetics and implants that are designed to withstand such loading cases. Interestingly, only few studies have investigated the tensile properties of TPMS lattices. Alsalla et al.⁴² found that tensile samples failed around the interface between the solid endplate and the lattice, while assessing the impact of construction direction on the tensile strength of 316 L stainless steel Gyroid structures. This emphasizes the importance of designing proper tensile specimens with a smooth and low stress concentration transition between the solid end and the lattice. Park et al.⁴³ conducted uniaxial tensile experiments to investigate the tensile properties of Neovius and I-WP sheet-based TPMS lattices printed by laser power bed fusion (LPBF) 3D printing technology using CoCrMo alloy powder. 1 × 1, 2 × 2, and 5 × 5 unit cells were used within the cross sectional to print various samples of each topology, but each being at a different relative density. If the relative density was kept constant with different number of unit cells, then a conclusion could have been made on the number of unit cells needed to obtain the bulk properties of these topologies. Results showed that Neovius lattice is superior to the I-WP lattice in terms of the yield strength, elastic modulus and elongation. In addition, it was found that the yield strength decreased and the elongations increased as the size of the unit cell increased. Cao et al.⁴⁴ conducted uniaxial tensile experiments to investigate the tensile properties of Shwartz Primitive and Gyroid sheet-based TPMS lattices printed by fused filament fabrication 3D printing technology using biodegradable polymer composite filaments based on polybutyleneadipate-co-terephthalate (PBAT) and polylactic acid (PLA). The tensile specimens used consist of uniform and functionally graded lattices in terms of relative density with 2 × 2 unit cells across the cross sectional area. Findings showed that the two uniform TPMS structures showed similar yield and strength values while the functionally graded Primitive lattice showed higher strength values than the functionally graded Gyroid lattice. Another study⁴⁵ conducted uniaxial tensile experiments on five sheet-based TPMS lattices (Diamond, Gyroid, Fischer, I-WP, and Primitive) at 30% relative density only, with 2 × 2 unit cells across the cross sectional area. The lattices were printed by LPBF using 316 L stainless steel. Recently, Araya et al.⁴⁶ conducted uniaxial tensile experiments to investigate the tensile properties of Gyroid sheet-based TPMS lattice printed by LPBF using Ti–6Al–4 V alloy powder. The study considered five different relative densities ranging from 10% to 50%, without keeping the number of unit cells constant. Based on the available studies, there is a clear need for more investigations on the bulk tensile properties of TPMS lattices across a range of relative densities and base materials (specially polymer-based materials). In fact, there is a clear need to investigate the number of unit cells needed to obtain the bulk tensile response of these lattices, to properly assess their capabilities under tensile loading cases.

In this study, Gyroid sheet-based TPMS lattice is considered, and an investigation is conducted into the number of unit cells needed to obtain the bulk tensile properties, at a constant relative density. Once that is established, the bulk tensile properties of Gyroid, I-WP, and Primitive sheet-based TPMS lattices at different relative densities are investigated and compared to their compressive counterparts, to have a proper understanding of the difference between the bulk tensile and compressive properties of these lattices.

Materials and methodology

In this section, the specimen preparation and testing are discussed, including the specimen design approach, specimen fabrication and the testing apparatus.

Specimen design

In this study, the Schoen Gyroid (Fig. 1a), Schoen I-WP (Fig. 1b), and Schwarz Primitive (Fig. 1c) sheet-based TPMS lattices were investigated which are defined using the following level-set approximation functions:

$$- c\left( {x,y,z} \right) < \phi \left( {x,y,z} \right) < + c\left( {x,y,z} \right)$$

(1)

where $\phi \left( {x,y,z} \right)$ defines a minimal surface defined for each topology as follows:

Fig. 1

Representation of (a) Gyroid, (b) I-WP, and (c) Primitive sheet-based TPMS lattices, showing a 3 × 3 × 3 tessellated structure and one unit cell structure of each topology.

$${\phi _{Gyroid}} = \sin X\cos Y + \sin Y\cos Z + \sin Z\cos X$$

(2)

$${\phi _{I - WP}} = 2\left( {\cos X\cos Y + \cos Y\cos Z + \cos Z\cos X} \right) - \left( {\cos 2X + \cos 2Y + \cos 2Z} \right)$$

(3)

$${\phi _{\Pr imitive}} = \cos X + \cos Y + \cos Z.$$

(4)

In Eq. (1), c is the isovalue (i.e., level-set constant) which is a constant value in case of a uniform relative density or is a function of the cartesian coordinates in case the relative density is functionally graded. In case the functional grading is along one direction only (for example along x coordinate), then c is solely defined in terms of that coordinate (that means c(x)). In Eqs. (2)–(4), $X = 2\pi x/{L_x}$, $Y = 2\pi y/{L_y}$, $Z = 2\pi z/{L_z}$, where ${L_x}$, ${L_y}$, and ${L_z}$ are the unit cell sizes in the x, y, and z directions, respectively. The lattice structures were generated using MSLattice 1.0 software⁴⁷ which is an open source TPMS lattice generator that provides a user interface to easily generate these lattices using any structure and unit cell dimensions, at the desired relative density. In addition, it can produce both uniform and functionally graded lattices, with customizable cell size and relative density gradients. The tensile specimen proposed in this study is presented in Fig. 2a, where the middle section of the specimen consists of TPMS lattices at a uniform relative density (25% initially), followed by functionally graded lattices (in terms of relative density) on both sides, from 25% to 100%, joining the lattice to the solid grips at the ends of the specimen, as demonstrated in the zoomed-in view. The functionally graded section is used to ensure a smooth transition between the uniform lattice section and the solid grips, to prevent early and abrupt failure at the transition region. In addition, fillets at the solid grips are used to reduce any stress concentration in that region, to ensure failure occurs at the uniform lattice region. The dimensions of the specimen are controlled by the maximum possible grip size available for testing, which ensures that maximum number of unit cells can be added along the cross-sectional area of the specimen, while meeting the minimum resolution of the 3D printer. The uniform and functionally graded lattice parts were modeled in MSLattice separately, then were joined together along with the solid grips, after which the model was re-meshed all together as a stereolithography (STL) file.

Fig. 2

(a) Configuration of tensile specimen used in this study, with a zoomed-in view of the functional graded section. (b) Printing and testing orientation of all specimens used in this study.

Using the 25% relative density Gyroid lattice and a unit cell size of 2.5 mm, the number of unit cells within the lattice sections, across the cross-sectional area of the specimen, were varied from 1 × 1 to 7 × 7 (Fig. 3) to investigate the number of unit cells required to obtain the bulk tensile properties, where the dimensions of the specimens are listed in Table 1. In later sections, the same approach was followed to design tensile specimens at different relative densities of 13%, 18%, and 35% (Fig. 4a,c). Similarly, the tensile specimens of IW-P and Primitive were prepared at the four relative densities considered, as shown in Figs. 5a–d, 6a–d, respectively. In terms of the cubic compression samples discussed in later sections, these were simply modeled in MSLattice as STL files and then printed with no support. These are presented in Fig. 4d–g for Gyroid, Fig. 5e–h for IW-P, and Fig. 6e–h for Primitive. Furthermore, two replicates of standard solid tensile samples (Fig. 4h) were designed to investigate the bulk mechanical properties of the constituent resin, following ASTM D638 Type IV standard.

Specimen fabrication

The specimens were fabricated through Digital Light Processing (DLP) AM technology⁴⁸, specifically using Halot Ray resin 3D printer (Creality 3D Technology Co., Ltd., Shenzhen, China). The printer has a build volume of 198 × 123 × 210 mm (width × depth × height) and 30 × XY resolution with LCD screen resolution of 6 K. The material used was fast curable resin supplied by the printer manufacturer, which has a density of 1.20 g/cm³ at 25 °C. All the specimens were printed directly on the build plate, i.e. without any support material, in the orientation shown in Fig. 2b. The same printing parameters were used for all specimens, which were: initial exposure of 50 s; bottom layer count of 6 layers; layer thickness of 0.05 mm; layer curing time of 6.5 s; rising height of 8 mm; motor speed of 3 mm/s; light-off delay of 4 s. After printing, all the specimens were washed for 30 min using Isopropyl Alcohol (IPA) 99.9% in Form Wash (Formlabs Inc., Massachusetts, USA), then cured for 60 min in Form Cure (Formlabs Inc., Massachusetts, USA) without any heating. For each specimen design, two replicates were printed for a total of 62 specimens (2 solid specimens of standard design, 14 tensile specimens for investigating the number of unit cells, 8 tensile and compression specimens for four different relative densities (13%, 18%, 25%, and 35%) for each topology.

Table 1 Dimensions of the tensile specimens based on the number of unit cells used.

Fig. 3

Latticed tensile specimens with Gyroid sheet-based TPMS lattice at 25% relative density with varying number of unit cells. (a) 1 × 1, (b) 2 × 2, (c) 3 × 3, (d) 4 × 4, (e) 5 × 5, (f) 6 × 6, and (g) 7 × 7 unit cells.

Fig. 4

Latticed tensile samples with 6 × 6 unit cells (along the cross sectional area) of Gyroid sheet-based TPMS lattice at (a) 13%, (b) 18%, and (c) 35% relative densities. Cubic compression samples with 6 × 6 × 6 unit cells of Gyroid sheet-based TPMS lattice at (d) 13%, (e) 18%, (f) 25%, and (g) 35% relative densities. (h) Standard tensile sample. In all subfigures, the left column presents the STL designs while the right column presents the actual printed specimens.

Fig. 5

Latticed tensile samples with 6 × 6 unit cells (along the cross sectional area) of I-WP sheet-based TPMS lattice at (a) 13%, (b) 18%, (c) 25%, and (d) 35% relative densities. Cubic compression samples with 6 × 6 × 6 unit cells of I-WP sheet-based TPMS lattice at (e) 13%, (f) 18%, (g) 25%, and (h) 35% relative densities.

Fig. 6

Latticed tensile samples with 6 × 6 unit cells (along the cross sectional area) of Primitive sheet-based TPMS lattice at (a) 13%, (b) 18%, (c) 25%, and (d) 35% relative densities. Cubic compression samples with 6 × 6 × 6 unit cells of Primitive sheet-based TPMS lattice at (e) 13%, (f) 18%, (g) 25%, and (h) 35% relative densities.

Mechanical experiments

Axial quasi-static tests were performed using Instron 5969 universal testing machine with a 50 kN load cell (Instron, Massachusetts, USA). Both tension and compression tests were conducted at a strain rate of 0.001/s. In the uniaxial compression tests (Fig. 7c), the strains were measured using the testing machine crosshead, which is known to yield accurate results in such tests. However, in the uniaxial tension tests (Fig. 7a,b), a non-contacting video extensometer (Instron AVE2, Massachusetts, USA) coupled with a 2D Digital Image Correlation (DIC) system ‘DIC Replay’ (Instron, Massachusetts, USA) was used to measure the strains. To perform DIC measurements, the lattice structures were first coated with white spray paint, followed by a speckle pattern using black spray paint which increases surface contrast and eventually allows proper tracking of the strain within the lattice structure. Imaging was conducted at an approximate frequency of 5 Hz, i.e. around 5 image captures per second, ensuring high-resolution capture of the sample’s deformation during the uniaxial tensile tests.

Results and discussion

In this section, results are split into two parts. The first part presents and discusses the results of the investigation of the number of unit cells needed to obtain the bulk tensile results of Gyroid lattice. In the second part, the results from the investigation of the effect of relative density on the bulk tensile and compression properties are presented and discussed for all three topologies, i.e. Gyroid, I-WP, and Primitive.

Fig. 7

Experimental set-up of the mechanical experiments for (a) standard tensile specimens, (b) latticed tensile specimens, and (c) lattice cubes under compression. Demonstration of completely failed specimens after testing: (d) standard tensile specimen, (e) latticed tensile specimen, (f) and compression cube.

Investigation of the number of unit cells

Initially, the stress-strain response of the standard tensile specimen was investigated, and the response is shown in Fig. 8a. An image of one of the fractured specimens is presented in Fig. 7d which shows that failure occurred within the gauge length of the specimen. The calculated elastic modulus, yield strength, and ultimate strength values of the resin are 2740 ± 35 MPa, 36.8 ± 0.14 MPa, and 53.0 ± 0.3 MPa, respectively. Furthermore, the average mass of the two replicate specimens is measured to be 7.26 ± 0.02 g, where it compares very well with the theoretical mass of the specimen that is calculated as 7.22 g using the density of the resin and the theoretical volume of the design obtained through ANSYS SpaceClaim 2024/R1 software. The percentage error between the actual and theoretical masses is about 0.5%, indicating a very high print accuracy.

As previously discussed, it is essential to figure out the number of unit cells needed to obtain the bulk tensile properties of Gyroid lattice. The experiments were conducted on tensile specimens with 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5, 6 × 6, and 7 × 7 unit cells across the cross sectional area with uniform relative density of 25%. The size of one unit cell was maintained as 2.5 mm in all samples, meaning that as the number of unit cells increases, the size of the specimen increases as well. Initially, the dimensions of all the specimens were measured and confirmed to match with the software-generated models with high accuracy. Given that the seven designs all have the same unit cell size of 2.5 mm, the unit cell sheet thickness is the same for all. By using Alicona InfitniteFocus (Bruker Alicona, Chicago, U.S.A) optical surface roughness measuring device, the unit cell wall thickness of 25% relative density Gyroid is viewed and measured, and it is compared to the theoretical unit cell wall thickness of the design, as shown in Fig. 9c. It is found that the measured unit cell wall thickness has an error of less than 0.43% of the theoretical value of the design. The above measurement confirms that the specimens were printed properly, conforming with the minimum resolution of the printer. Also, an image of one of the fractured specimens is presented in Fig. 7e which shows that failure occurred within the gauge length of the specimen. The engineering stress-strain response of each specimen is presented in Fig. 8b, where one response out of the two replicates of each design is shown. The results show good replicability of each test, and clearly show that as the number of unit cells increases, the elastic modulus, yield strength, and ultimate strength all decrease, except for the specimen with 1 × 1 unit cell, which shows lower elastic modulus and strength values than the 2 × 2 unit cells specimen. This can be accounted to the difficulty in setting up the 1 × 1 unit cell specimen for testing, given its very thin size and delicate structure. The average elastic modulus, yield strength, and ultimate strength values of each design is recorded and presented for comparison in Fig. 10a–c, respectively. Based on these figures, it can be seen that the elastic modulus, yield strength and ultimate strength values plateau at 6 × 6 unit cells, where the 6 × 6 and 7 × 7 unit cells specimens show the same elastic modulus value and very close yield and ultimate strength values. This trend can be justified by investigating the surface area to volume ratio of the gauge length of the specimen that consists of the lattice at uniform relative density of 25%. The surface area to volume ratios of 1 × 1 to 7 × 7 unit cells are: 11.33, 10.66, 10.44, 10.32, 10.26, 10.21, and 10.18, respectively. This indicates that as the number of unit cells increase, the surface area to volume ratio decreases, which in turn yields lower modulus and strength values. Moreover, by investigating the percentage difference in the surface area to volume ratio, it is noted that the percentage difference decreases as the number of unit cells increase, where the differences from two to seven unit cells are: 5.9%, 2.1%, 1.1%, 0.58%, 0.49%, and 0.29%. This indicates that as the surface area to volume ratio difference becomes very small with increasing number of unit cells, the behavior of the lattice becomes that of a material at the macroscale, which in turn yields the bulk response of the lattice. These findings conclude that 6 × 6 unit cells are needed to obtain the bulk tensile properties of Gyroid lattice, which is more than the four unit cells needed to obtain the bulk compression properties of TPMS lattices^33,34. Moreover, it is important to highlight the difference in using samples made of 2 × 2 unit cells (commonly used in literature) and 6 × 6 unit cells, where the difference in strength can reach more than 40%. This conclusion was applied to the other TPMS lattices considered in this study as well to obtain their bulk properties, as discussed in the next section.

Fig. 8

Stress-strain response of (a) standard solid tensile specimen, (b) tensile specimens composed of Gyroid 25% relative density with different number of unit cells. Stress-strain response of (c) Gyroid, (e) I-WP, (g) Primitive tensile specimens and (d) Gyroid, (f) I-WP, (h) Primitive compression specimens with 6 × 6 × 6 unit cells at different relative densities.

Fig. 9

Demonstration of the unit cell wall thickness values of the software-generated design (left) and the printed specimens (right) for Gyroid sheet-based TPMS lattice at relative densities of (a) 13%, (b) 18%, (c) 25%, and (d) 35%. The wall thickness images and measurements on the actual specimens were conducted using Alicona InifiniteFocus optical surface roughness measuring device.

Fig. 10

(a) Elastic modulus, (b) yield and (c) ultimate strength values of tensile specimens composed of Gyroid lattice at 25% relative density with different number of unit cells. (d) Elastic modulus, (e) yield, (f) ultimate/plateau strength, and (g) SEA of tensile (6 × 6 unit cells) and compression specimens (6 × 6 × 6 unit cells) composed of Gyroid lattice at different relative densities.

Investigation of the effect of relative density

Based on the findings of the previous investigation, at least 6 × 6 unit cells are needed within the cross sectional area to obtain the bulk tensile properties of sheet-based TPMS lattices. Thus, all tensile specimens were designed with 6 × 6 unit cells (using 2.5 mm unit cell size) at different relative densities, which are 13%, 18%, 25%, and 35%, with two replicates of each. The Gyroid tensile specimens are shown in Fig. 4a–c (25% relative density with 6 × 6 unit cells was already modeled and tested in the previous section), the I-WP specimens are shown in Fig. 5a–d, and the I-WP specimens are shown in Fig. 6a–d. Similarly, four cubic compression specimens were designed with 6 × 6 × 6 unit cells at different relative densities, which are 13%, 18%, 25%, and 35% for each topology, where each unit cell has a size of 2.5 mm (Fig. 4d–g for Gyroid, Fig. 5e–h for I-WP, amd Fig. 6e–h for Primitive). The compression specimens have the same dimensions and are identical to the uniform lattice section of the tensile specimens. Initially, the unit cell wall thicknesses of the new relative density designs of Gyroid are shown in Fig. 9. Overall, it is found that the error between the actual and theoretical unit cell wall thicknesses is approximately 1.65%, 1.24%, 0.43%, and 2.48%, for the relative densities of 13%, 18%, 25%, and 35%, respectively.

The stress–strain responses of the tensile and compression tests at each relative density for Gyroid are presented in Fig. 8c,d, respectively. As expected, the elastic modulus, yield strength, and ultimate/plateau strength values increase as the relative density increases from 13% to 35% in both sets of tests. The average values of the elastic modulus, yield strength, ultimate/plateau strength, and specific energy absorption (SEA) values of each design are recorded and presented in Fig. 10d–g, respectively. SEA is calculated as the area under the load-displacement graph divided by the mass of the lattice. For the tensile specimens, the end point of SEA calculation is the failure point, and the mass of the specimen is the mass of the uniform lattice section within the specimen. For the compression cubes, the mass used in the SEA calculation is simply the mass of the cube, while the end point of SEA calculation is the strain level of 0.5, which seems to approximate the densification point of all tests in a good manner (Fig. 8). Based on Fig. 10d, the elastic modulus is higher under uniaxial tension than under uniaxial compression for all relative densities, except for 35% relative density, where it seems that at higher relative densities, the elastic modulus is higher under uniaxial compression than under uniaxial tension. Moreover, the percentage difference in values at lower relative densities seem to be higher than at higher relative densities. On the other hand, based on Fig. 10e, the yield strength values under uniaxial compression are higher than under uniaxial tension for all relative densities, where the percentage difference increases as the relative density increases. In terms of the ultimate strength in uniaxial tension and plateau strength in uniaxial compression, the comparison between these properties is not valid as they represent different lattice properties. However, it is interesting to highlight that the change in ultimate strength under uniaxial tension is more consistent than the change in the plateau strength under uniaxial compression between different relative densities. This is clearly observed between the values at 25% and 35% relative density in Fig. 10f. The SEA values are presented in Fig. 10g where it is clear that SEA in compression is way higher than SEA in tension, given the brittle fracture of Gyroid under tension across all relative densities. SEA values of Gyroid increase as the relative density increases under both uniaxial tension and compression. In addition, an image of one of the tested compression specimens is presented in Fig. 7f which justifies the densification seen in the stress-strain responses of uniaxial compression specimens.

The stress–strain responses of the tensile and compression tests at each relative density for IW-P are presented in Fig. 8e,f, respectively. As expected, the elastic modulus, yield strength, and ultimate/plateau strength values increase as the relative density increases from 13% to 35% in both sets of tests. The average values of the elastic modulus, yield strength, and ultimate/plateau strength values of each design are recorded and presented in Fig. 11a,c,e, respectively. Based on Fig. 11a, the elastic modulus is relatively similar under uniaxial tension and compression for 13% and 18% relative densities but becomes significantly higher under uniaxial tension for 25% and 35% relative densities. On the other hand, based on Fig. 11c, the yield strength values under uniaxial compression are higher than under uniaxial tension for all relative densities, where the percentage difference increases as the relative density increases, just like the Gyroid results. In terms of the ultimate strength in uniaxial tension and plateau strength in uniaxial compression in Fig. 11e, both properties seem to follow the same trend with the increase in relative density. However, it is important to note the stress oscillations seen in the response of IW-P at 35% relative density under uniaxial compression (Fig. 8f), where few layer collapses have been observed within the plateau region, before the beginning of densification. SEA results in Fig. 11g show similar trend to that of Gyroid in both uniaxial tension and compression, where SEA values and the percentage difference increases as the relative density increases.

The stress–strain responses of the tensile and compression tests at each relative density for Primitive are presented in Fig. 8e,f, respectively. As expected, the elastic modulus, yield strength, and ultimate/plateau strength values increase as the relative density increases from 13% to 35% in both sets of tests. The average values of the elastic modulus, yield strength, and ultimate/plateau strength values of each design are recorded and presented in Fig. 11b,d,f, respectively. Based on Fig. 11b, the elastic modulus is relatively similar under uniaxial tension and compression where it is slightly higher in tension for 13% relative density and slightly lower for 18% relative density. However, for 25% and 35% relative densities, the elastic modulus is significantly higher under uniaxial tension than compression. On the other hand, based on Fig. 11d, the yield strength values under uniaxial compression are higher than under uniaxial tension for all relative densities, except for 13%, where under uniaxial tension, Primitive slightly has a higher yield strength than under uniaxial compression. In terms of the ultimate strength in uniaxial tension and plateau strength in uniaxial compression in Fig. 11f, both properties increase with the increase in relative density. However, it is important to note that unlike Gyroid and I-WP, Primitive shows a very small increase in ultimate strength from 25% to 35% relative density under uniaxial tension. Also unlike Gyroid and I-WP, Primitive shows a decline in SEA values under uniaxial tension as the relative density increases, as shown in Fig. 11h, which is attributed to the significant decline in ductility as the relative density increases (Fig. 8g). Under uniaxial compression, SEA of Primitive increases as the relative density increases, but with a slight plateau between 18% and 25% relative density, before increasing again at 35% relative density.

Fig. 11

Elastic modulus of (a) I-WP and (b) Primitive. Yield strength of (c) I-WP and (d) Primitive. Ultimate/plateau strength of (e) I-WP and (f) Primitive. SEA of (g) I-WP and (h) Primitive. The plots include results from tensile (6 × 6 unit cells) and compression specimens (6 × 6 × 6 unit cells) at different relative densities.

Generally, results indicate that for all three topologies, the elastic modulus is higher under uniaxial tension than under uniaxial compression, except for Gyroid at 35% relative density and Primitive at 18% relative density. Conversely, for all three topologies, the yield strength is higher under uniaxial compression than under uniaxial tension, except for Primitive at 13% relative density. In terms of the ultimate strength in tension and plateau strength in compression, the values increase with an increase in relative density for all three topologies.

Power law fit

Using the elastic modulus, yield strength, and ultimate/plateau strength values of the latticed tensile and compression specimens at different relative densities, a log-log scale graph of each topology is produced versus the relative density. Then, these fits are fitted with a Gibson-Ashby scaling power law fit⁴⁹ in the form of $\varphi /{\varphi _s} = C\rho _r^n$, where $\varphi$ and ${\varphi _s}$are the mechanical properties of the cellular structure and the constituent solid (i.e. elastic modulus, ultimate strength or plateau strength), respectively, ${\rho _r}$ is the relative density, and C (scaling coefficient) and n (scaling exponent) are fitting constants. It is important to highlight that the constituent solid properties were calculated and reported in the previous section as 2,740 MPa, 35 MPa, and 53 MPa for the elastic modulus E, yield strength $\sigma ^Y$, and ultimate strength ${\sigma ^{ut}}$, respectively. The power law fits of each topology (uniaxial tension and compression) are presented in Fig. 12 and the fitting constants are reported in Table 2. It is important to note that the plateau strength is normalized with the ultimate strength of the constituent material as no compression test was conducted on the constituent solid material. Architected cellular materials can display stretching-dominated, bending-dominated, or mixed-mode deformation behaviors, depending on the internal structure’s morphology and the loading conditions⁴⁹. The Voigt maximum theoretical limit for the stiffness and strength of a lattice structure corresponds to a scaling coefficient and exponent values of C = n = 1⁵⁰, which is desirable for maximizing stiffness and strength. The power exponent n generally represents the nature of the dominant deformation mode and can be used to categorize cellular materials according to their suitability for energy absorption applications (bending-dominated mode) or for lightweight structures that require high stiffness and strength (stretching-dominated mode). Typically, the structure is considered to undergo stretching-dominated deformation when n $\rm{\approx}$ 1 in terms of stiffness and strength, and bending-dominated deformation when n $\geq$ 2 (stiffness) and n $\rm{\geq}$ 1.5 (strength), and mixed deformation mode for a value in-between⁵¹. Based on the fitting constants of the elastic moduli reported in Table 2, it can be concluded that Gyroid and Primitive sheet-based TPMS lattices exhibit mixed deformation mode under uniaxial tension loading, while I-WP sheet-based TPMS lattice exhibits bending dominated deformation mode. In terms of uniaxial compression loading, Primitive still exhibits mixed deformation mode while Gyroid and I-WP exhibit bending dominated deformation mode. In terms of yield strength, Gyroid and Primitive exhibit mixed deformation mode under uniaxial tension, while I-WP exhibits bending dominated deformation mode, where these results align with the results of the elastic moduli. On the other hand, all three topologies exhibit bending dominated deformation mode under uniaxial compression in terms of the yield strength.

In addition to the scaling power law fits of this study, results from Araya et al.⁴⁶ are added in Fig. 12a,b, which is the only available study found that investigated Gyroid lattices under uniaxial tension for a range of relative densities, manufactured from Ti–6Al–4 V alloy powder. However, it is important to highlight that the authors did not report the experimental elastic modulus and yield strength of the consistent solid material upon testing, hence these values were obtained from the material supplier’s (SLM Solutions, Lübeck, Germany) datasheet ($E_s$ = 124 GPa and $\sigma_s^Y$ = 935 MPa). The two studies show very comparable power exponents n for tensile loading (n = 1.35 and n = 1.36 in this study and n = 1.41 and n = 1.32 in Araya et al.⁴⁶, in terms of elastic modulus and yield strength, respectively). But this is not the case for the scaling coefficients C of elastic modulus and yield strength, as C = 0.55 and C = 0.60 in this study while C = 0.93 and C = 1.09 in Araya et al.⁴⁶, respectively, where the latter exhibits high elastic modulus and yield strength values at the same relative densities.

Even though the constituent solid material and the defects risen from the different 3D printing processes can account for these variations, it is highly likely that the differences in these results are due to the number of unit cells used. The number of unit cells varied between 3 × 3 and 6 × 6 unit cells, based on the relative density, in the work of Araya et al.⁴⁶, which according to the current study, 3 × 3 unit cells yield a much higher elastic modulus value than the 6 × 6 unit cells needed to obtain the bulk response of the lattice (Fig. 4a). In fact, this is the case in Fig. 12a,b, where the scaling coefficient reported in literature is higher than the one found in this study. This highlights the importance of using sufficient number of unit cells to investigate the bulk tensile properties of lattices to avoid overestimating or underestimating their performances, and to have an accurate comparison between the tensile properties of different lattices reported by different studies.

Table 2 Fitting constants of the power law scaling fits of elastic modulus, yield strength, ultimate/plateau strength, and SEA of Gyroid, IW-P, and primitive under uniaxial tension and compression.

By observing the power law fits of the three topologies in Fig. 12a, Gyroid exhibits the highest tensile elastic modulus up until 20% relative density, after which I-WP starts to exhibit significantly higher tensile elastic modulus. In terms of the elastic modulus in compression, I-WP exhibits the highest modulus, followed by Gyroid then Primitive, for relative densities above 15%. Below that value, the three topologies exhibit very close modulus values with Primitive being the highest, followed by Gyroid then I-WP. The yield strength fits in Fig. 12b show that the yield strength of I-WP is highest again above 20% relative density under uniaxial tension. However, unlike the elastic modulus, Primitive exhibits higher yield strength than Gyroid under uniaxial tension through the range of relative densities considered. For uniaxial compression, I-WP exhibits the highest yield strength above 18% relative density, followed by Gyroid then Primitive, where Gyroid exhibits higher yield strength than Primitive throughout the range of relative densities. At 13% relative density, I-WP exhibits the lowest yield strength value. Moving towards the ultimate strength in uniaxial tension (Fig. 12c), it is interesting to note that below 22% relative density, Primitive exhibits the highest ultimate strength, followed by Gyroid than I-WP. However, above 22–23% relative density, Primitive exhibits the lowest ultimate strength while I-WP exhibits the highest ultimate strength. Figure 12d shows the plateau strength in uniaxial compression where Gyroid generally outperforms Primitive along the entire range of relative densities (they show very comparable values at 13%). However, I-WP exhibits the lowest plateau strength at lower relative densities. Above 16% relative density, I-WP exhibits higher plateau strength than Primitive, and above 20% relative density, I-WP exhibits higher plateau strength than Gyroid. At last, the SEA results in Fig. 12e shows that both Gyroid and I-WP exhibit almost similar SEA under uniaxial compression that exceeds that of Primitive. However, under uniaxial tension, I-WP exhibits higher SEA than Gyroid at higher relative densities and vice versa at lower relative densities. On the other hand, Primitive exhibits the highest SEA at lower relative densities and then the lowest SEA at higher relative densities, where its SEA decreases with the increase in relative density due to significant decrease in its ductility under uniaxial tension at higher relative densities.

Fig. 12

Scaling power law fits of (a) elastic modulus, (b) yield strength, (c) ultimate strength, (d) plateau strength, and (e) SEA in log-log plots of Gyroid, I-WP, and Primitive sheet-based TPMS lattices. The current study utilizes 6 × 6 unit cells while the literature study⁴⁶ utilizes 3 × 3 unit cells across the cross-sectional area of the tensile specimens.

The above results can be attributed to the layer-by-layer collapse seen in I-WP at 13% relative density (Fig. 13a), compared to the shear bands seen in both Gyroid (double bands) and Primitive (single band) lattices at the same relative density (Fig. 13b,c), where the layer-by-layer collapse comparatively leads to lower stiffness and strength. However, at higher relative densities (i.e. ≥ 25%), I-WP mainly shows more of a uniform global compression represented as barreling, which explains the increased stiffness and strength, as seen in Fig. 13b. In addition, this barreling effect is seen in Gyroid at higher relative densities (Fig. 13a), along with the double shear bands, which leads to higher stiffness and strength than Primitive lattice, but less than that of IW-P. Primitive lattice does not show any barreling effect even at high relative densities (Fig. 13c), but only shows double shear bands.

Fig. 13

Deformation patterns of (a) Gyroid, (b) I-WP, (c) Primitive sheet-based TPMS compression specimens at 13% and 25% relative densities, at various strain levels.

Conclusions

This study focuses on the tensile properties of sheet-based TPMS lattices manufactured from fast curing resin through Digital Light Processing (DLP) 3D technique. The tensile sample proposed consists of lattices at uniform relative density at the gauge length, followed by functionally graded sections, up to 100% relative density, and solid grips at both ends, to ensure that failure occurs within the uniform relative density section.

Initially, an investigation was conducted on the number of unit cells needed within the cross-sectional area of Gyroid specimen, to obtain the bulk response of sheet-based TPMS lattices. Fixing the unit cell size to 2.5 mm, various tensile specimens were printed with different number of unit cells, ranging from 1 × 1 to 7 × 7 unit cells, where the higher the number of unit cells, the larger is the specimen size. Uniaxial tests with DIC were conducted and results showed that as the number of unit cells increased, the elastic modulus, yield strength and ultimate strength values decreased (with the exception of one unit cell model), until reaching 6 × 6 unit cells which showed very similar values (converged values) as the 7 × 7 unit cells. In fact, the difference in strength between using samples made of 2 × 2 unit cells and 6 × 6 unit cells reached more than 40%.

Based on the previous conclusions, latticed tensile and latticed compression cube specimens of Gyroid, I-WP, and Primitive were manufactured with 6 × 6 unit cells at various relative densities of 13%, 18%, 25%, and 35%. Uniaxial compression tests were conducted to simply demonstrate the difference in the performance of the lattices under uniaxial tension and compression loading. Results from both loading cases for all topologies confirmed that as the relative density increases, the elastic modulus, yield strength and ultimate/plateau strength increases. Overall, the findings show that for all three topologies, the elastic modulus tends to be greater under uniaxial tension compared to uniaxial compression, with the exceptions observed in the Gyroid at 35% relative density and the Primitive at 18% relative density. These findings expand the potential of implementing sheet-based TPMS lattices into applications where tensile stresses are expected, rather than the typical sandwich panel applications where only uniaxial compression stresses are mainly expected. In contrast, the yield strength is generally higher in uniaxial compression than in uniaxial tension, except in the case of the Primitive at 13% relative density.

Gibson-Ashby scaling power law fits were produced for the tensile and compression results for all topologies where results show that across the relative density range considered in this study, I-WP generally outperforms the other topologies in both tensile and compressive stiffness and strength at higher relative densities, while Primitive exhibits competitive or superior performance at lower relative densities. Gyroid typically shows intermediate properties, outperforming Primitive in compressive strength but being surpassed by I-WP above approximately 20% relative density. Furthermore, comparison with literature confirms that using insufficient number (lower number) of unit cells can lead to higher elastic modulus values that do not represent the bulk uniaxial tensile modulus of Gyroid lattice, which in turn can lead to unexpected results when implemented in various applications.

Building on the present findings, future work will focus on expanding the tensile characterization of lattice structures to a wider range of lattice topologies and possibly base materials, to accurately capture their bulk mechanical properties under uniaxial tension. Furthermore, extending this analysis to other multi-axial loading conditions will support the development of predictive models and design guidelines for lattice structures.