Effective mechanical properties of additive manufactured triply periodic minimal surfaces: experimental and finite element study

Abstract

Recent advances in additive manufacturing (AM) led to the design and production of geometrical complex items with a plethora of materials facilitating the implementation of topology optimization procedures. Lattice designs are often employed for the topology optimization process providing lightweight structures with also additional advantages, such as high porosity and high surface area to volume ratio. Thus, there is a necessity to analyze these structures, examine their mechanical performance, and provide the computational tools to evaluate them. The current paper investigates the mechanical behavior of four triply periodic minimal surfaces (TPMS), namely the Schwarz primitive, the gyroid, the Schwarz diamond, and the Neovius structures. The structure of each lattice was analyzed and the influence of design-related parameters on the relative density was obtained. In order to study the mechanical response of the TPMS, representative volume elements (RVEs) were fabricated from polyamide 12 (PA12), utilizing the selective laser sintering (SLS) AM technique. Uniaxial quasi-static compression tests were conducted on the RVEs; their mechanical response was acquired and exploited in order to build advanced finite element (FE) models to simulate the overall mechanical behavior of each TPMS structure. Furthermore, these FE models with periodic boundary conditions were employed to extract the mechanical response of specimens that consisted of more unit cells. In addition, the numerical and the experimental data of more complex TPMS lattices were compared and verified the reliability and the accuracy of the FE models. Finally, the Schwarz diamond structure showed the best mechanical performance, in terms of strength and stiffness, slightly better than the Neovius, then followed the gyroid and Schwarz primitive structures, respectively.

1 Introduction

Additive manufacturing, also known as 3D printing, is a manufacturing process that enables the fabrication of products with high geometrical complexity. During the additive manufacturing (AM) processes, the desired item is constructed layer by layer according to the designed 3D model. According to the ASTM F2792-12A standardization [1], eight distinct methods of AM technologies have been developed enabling the usage of numerous feedstock materials, i.e., plastic, ceramic, and metals. The first applications of AM technologies were for rapid prototyping of parts [2]; however, due to the development and the improvement of AM technologies, nowadays, these manufacturing processes are employed for rapid fabrication of tools and fully functional products [3]. There are several examples of AM applications scattered in literature derived from various industries [4, 5].

The development of AM technologies provoked an extensive scientific interest in the topology optimization process, as the geometric constraints were lifted and the engineers were able to design topologically optimized parts with advanced and complex geometries. Moreover, the advance of computational power and the development of proper software provided the necessary tools in order to design advanced lattice geometries, such as the triply periodic minimal surfaces (TPMS) [6]. Thus, in the last decade, a plethora of studies investigated the potential use of lattice structures in topology optimization applications in various fields, such as the automotive, aeronautical, and biomechanical industries [7, 8]. However, the researchers are always concerned about the questionable mechanical response of these structures. In the majority of these studies, an intense degradation of the lattice structure’s mechanical properties was observed which was strongly dependent on the relative density [9].

One of the first and most comprehensive studies about the mechanical performance of lattice structures was performed by Gibson and Ashby [10] that examined the mechanical behavior of 2D and 3D strut-lattices. According to this study, the effective mechanical properties of lattice structures are influenced by the relative density regarding the scaling law: Φ_lattice/Φ_solid = C (ρ_relative) ⁿ. Where Φ_lattice and Φ_solid are the mechanical properties for the lattice and material, respectively. C and n are constants that are determined by the employed lattice structure. Furthermore, ρ_relative is the relative density of the structure. Moreover, the lattice structures exhibit two different mechanical behaviors, namely the stretching-dominated and the bending-dominated, and this is expressed by the value of exponent n [11]. There is a vast number of different lattice geometries; however, according to the existing literature [12,13,14,15,16], the most advanced in terms of functionality and mechanical performance are the TPMS lattice structures. TPMS structures offer the additional advantages of high porosity, high surface area to volume ratio, distinct volume domains useful for industrial applications, such as heat exchangers [17]. Furthermore, the effective mechanical properties of TPMS exhibit moderate deterioration as the relative density descends compared to the other lattices.

The mechanical behavior of TPMS structures has been studied by many researchers utilizing various AM techniques. Sychov et al. [12] studied the mechanical response and the energy absorption of TPMS structures, fabricated via the fused filament fabrication (FFF) technique, pointing out remarkable results. Zheng et al. [13] examined the TPMS manufactured via stereolithography (SLA) technique and concluded that the TPMSs are governed by the scaling law and exhibit either stretching-dominated or bending-dominated behavior. In addition, Abueidda et al. [14] and Maskery et al. [15] investigated the mechanical response of the sheet-TPMS structures manufactured with the selective laser sintering (SLS) method and they applied the scaling law and hyper-elastic finite element (FE) models in order to identify their response. Moreover, Al-Ketan et al. [16, 18] studied 3D strut-structures and TPMS structures concluding that the scaling law equations are the governing formulations for the mechanical performance of all lattice structures. Finally, there is extensive literature that applied various numerical and finite element (FE) analyses in order to detect the most suitable computational method to accurately simulate the experimental mechanical behavior of lattice structures [19,20,21].

According to the existing research, all lattices exhibit a notable deterioration of their effective mechanical properties, the degradation rate obeys the scaling law formulation. Nevertheless, the majority of studies explored the experimental behavior of lattice structures while they utilized FE analysis to explicate their mechanical response. Thus, the objective of this research was to conduct novel theoretical analyses for each of the four examined TPMS lattice structures and propose approximation formulations that relate the 3D design-related parameters, such as unit cell length, wall thickness, and the level-set constant, with the relative density and the effective mechanical properties, such as the mechanical strength and elastic modulus. Furthermore, the current paper investigates the four most widespread TPMS lattice structures, namely Schwarz primitive (SP), gyroid (G), Schwarz diamond (SD), and Neovius (N). All examined specimens were fabricated with selective laser sintering (SLS) AM technique in order to produce samples with high geometrical accuracy and quality due to the lack of support geometry and enhanced mechanical performance [22]. Moreover, this research studied these structures from another perspective by utilizing representative volume elements (RVEs) manufactured and examined their mechanical performance under uniaxial compression loading. Advanced FE models were built exploiting the acquired experimental data from RVE specimens. Then, TPMS lattice structures with more unit cells were manufactured in order to examine the capability of FE models to evaluate their mechanical response. In that way, by exploiting acquired experimental data from RVE lattice structures and applying proper periodic boundary conditions, it is possible to build accurate and reliable novel FE models that can evaluate the mechanical behavior of more complex structures with the same lattices, without being computational time-consuming. Figure 1 portrays the flowchart of this current research approach.

Fig. 1

Flowchart of the current research

2 Material and methods

2.1 Theoretical analysis of TPMS lattice structures

Before the examination of the mechanical response of TPMS lattice structures, it is necessary to perform a theoretical analysis in order to explain their unique geometries and characteristics. First of all, triply periodic minimal surfaces (TPMS) are the periodic geometries that extend at three dimensions (ℝ³) and possess the exclusive topological property of mean curvature (H) equal to 0 at every single point of their surface [23]. More specifically, the curvatures (k₁, k₂) through the principal orthogonal curvature planes at every point of the surface are opposite and equal: H = (k₁ + k₂)/2 = 0. There is a plethora of methods to design TPMS-like unit cells; however, the most widespread and direct method is the utilization of approximation trigonometrical equations that emerge simplifying the following Fourier series (Eq. (1)) [24]:

$$\Phi \left(\mathrm{r}\right)=\sum{_{\mathrm{k}=1}^{\mathrm{K}}}{\mathrm{A}}_{\mathrm{k}}\mathrm{cos}\,[2\uppi ({\mathrm{h}}_{\mathrm{k}}\cdot \mathrm{r})/{\uplambda }_{\mathrm{k}}+ {\mathrm{p}}_{\mathrm{k}}]=\mathrm{C}$$

(1)

The above-mentioned formulation is a generic definition of periodic surfaces and the variant k stands for the reciprocal vectors, r is the location vector in ℝ³, A_k is the magnitude factor, the h_k is the kth lattice vector in reciprocal space, λ_k is the period’s wavelength, p_k is a phase shift, and C is a constant. After processing, different level-set trigonometrical equations of the form: f (x, y, z) = c, could be generated to produce the geometry of TPMS surfaces with high fidelity and accuracy [25]. Hence, the level-set method is implemented in order to design solid TPMS structures where the c is a level-set constant that gives different versions of the same minimal surface for different values. When the level-set constant c is equal to 0, the overall volume of the level-set surface is divided into two equal sub-volumes; if c has a value above or below the 0, the level-set surfaces are leading in a normal direction or opposite to normal, respectively. There are more than a hundred different minimal surfaces [26, 27]; however, in the context of this research, the mechanical response of four TPMS lattice structures was investigated, namely the Schwarz primitive (SP), the gyroid (G), the Schwarz diamond (SD), and the Neovius (N) structure. The level-set trigonometrical functions of these minimal surfaces are listed below in Eqs. (2)–(5).

Schwarz primitive:

$$\text{cos}x+\text{cos}y+\text{cos}z=c$$

(2)

Gyroid:

$$\text{sin}x\;\text{cos}y\:+\:\text{sin}y\;\text{cos}z\:+\:\text{sin}z\;\text{sin}x=c$$

(3)

Schwarz diamond:

$$\begin{aligned}\text{sin}x\;\text{sin}y\;\text{sin}z&+\:\text{sin}x\;\text{cos}y\;\text{cos}z\\&+\:\text{cos}x\;\text{sin}y\;\text{cos}z+\:\text{cos}x\;\text{cos}y\;\text{cos}z\:=\:c\end{aligned}$$

(4)

Neovius:

$$3(\text{cos}x\:+\:\text{cos}y\:+\:\text{cos}z)\:+\:4\;\text{cos}x\;\text{cos}y\;\text{cos}z\:=\:c$$

(5)

According to existing literature [21, 28], two methods could produce solid structures of minimal surfaces and build solid TPMS designs exploiting the functions. The first method produces the skeletal (walled) TPMS structures filling one of the two sub-volumes with material and leaving the other one void. In this case, the relative density of the structure is dependent on the volume of the sub-volume domain to overall volume; thus, the level-set constant c has a significant impact on the value of relative density. Thus, the relative density is equal to 50% when the value of c is 0 and varies when the c is below or above 0, as is shown in Fig. 2a. On the other hand, the second method generates solid TPMS lattice structures exploiting two level-set minimal surfaces with equal and opposite level-set constants and extracting the space between them. In that way, a solid sheet is created regarding the desired minimal surface. It is worth mentioning that as the level-set constant reaches the value of 0 the relative density becomes practically 0. The extracted structures from this method are named sheet-TPMS lattice structures and presented in Fig. 2b. Figure 2c portrays a unit cell from each examined TPMS topology, for an indicative relative density, designed with both strategies, i.e., skeletal (walled) and sheet.

Fig. 2

a Skeletal (walled) strategy for TPMS, b sheet strategy for TPMS, c unit cells from each examined TPMS topology (skeletal and sheet)

Skeletal and sheet TPMS lattice structures have similarities in their geometries and their characteristics; however, sheet lattices offer two additional advantages that could be useful in commercial applications. The first advantage is that the sheet structures provide two separate void volumes within the overall structure rising the surface area to volume ratio. This characteristic enhances the potential use of sheet TPMS in industrial applications, such as heat exchangers and catalysts [17, 29]. The second advantage is that the strategy for sheet TPMS could produce structures with ultra-low relative density (< 10%) with uniform distribution of the material and without discontinues within the structure. Therefore, the current investigation focuses on the examination of the four sheet-TPMS lattice structures of the second row of Fig. 2c, namely Schwarz primitive (SP), the gyroid (G), the Schwarz diamond (SD), and the Neovius (N).

The efficiency of TPMS structures in a real-life application depends on the understanding of their topology and the ability of designs/engineers to regulate their shape and their mechanical performance by adjusting a set of design-related parameters. In the case of sheet-TPMS, the main design-related parameters are three: the length of the unit cell (l), the sheet thickness (t_s), and the level-set constant (c). The most crucial property in lattice structures is the relative density $(\overline{\rho })$ due to the high influence on the mechanical properties through the scaling laws [10]. Thus, this work focuses on connecting the design-related parameters with the relative density via a series of approximation formulas. It is worth mentioning that the value of relative density ($\overline{\rho }$) for sheet-TPMS structures, as illustrated in Eq. (6), derives from the volume (V_lattice) that the lattice structure occupies to the volume of the bounding box (V_bb).

$$\begin{aligned}\overline{\rho }&=\frac{{\rho }_{lattice}}{{\rho }_{solid}}=\frac{\frac{{m}_{lattice}}{{V}_{bb}}}{\frac{{m}_{solid}}{{V}_{bb}}}=\frac{{m}_{lattice}}{{m}_{solid}}\\&=\frac{{\rho }_{material}\cdot {V}_{lattice}}{{\rho }_{material}\cdot {V}_{bb}}\to \overline{\rho (l,{t}_{s})}=\frac{{V(l,{t}_{s})}_{lattice}}{{V(l)}_{bb}}\end{aligned}$$

(6)

where m_solid and ρ_solid are the mass and the density of a solid domain with a volume equal to the bounding box of the structure. Moreover, m_lattice and ρ_material are the mass of the lattice structure and the material’s density. Existing research articles [15, 23] have proved that the ratio between the sheet thickness and the unit cell length (t_s/l) has a severe impact on the relative density of sheet-TPMS structures. According to these studies, the relative density of sheet-TPMS could be calculated with sufficient accuracy (lower than 0.25%) by the following function:

$$\overline{\rho }={G}_{1}\cdot \left(\frac{{t}_{s}}{l}\right)$$

(7)

where G₁ is a constant that is depending on the shape of the minimal surface. It is worth mentioning that the accuracy of Eq. (7) decreases as the relative density exceeds 50% due to the fact that the solid TPMS structures deviate from the original geometry due to the existence of surplus of material. On the other hand, the whole structure of a sheet-TPMS could be modified by changing the absolute value of the level-set constant. Different sets of functions with c and -c as the level-set constants lead to different sheet thicknesses changing the t_s/l ratio and the structure’s relative density. Thus, it is necessary to provide a direct link between the level-set constant and the relative density and the t_s/l ratio, for all the examined structures. In this context, the following numerical equations (Eqs. (8) and (9)) show the linear relation between the above-mentioned variables. G₂ is a constant related to the TPMS’s topology.

$$\overline{\rho }={G}_{2}\cdot c$$

(8)

$$\frac{{t}_{s}}{l}=\left(\frac{{G}_{2}}{{G}_{1}}\right)\cdot c$$

(9)

2.2 Design process of the TPMS lattice structures

The current study examines the mechanical response of four distinct sheet-TPMS lattice structures, namely Schwarz primitive (SP), the gyroid (G), the Schwarz diamond (SD), and the Neovius (N). The selection of the specific structures was performed in order to fulfill certain criteria, such as the useability, the enhanced mechanical performance, and the advanced complexity of the structures. These four TPMSs are included in the most widespread TPMS structures which are available in a plethora of design software. Moreover, according to existing studies [13, 15], these lattices performed remarkable mechanical properties, increased energy absorption rate, and high crashworthiness, characteristics that are necessary for industrial applications and potential replacements for the existing lattice structures (strut-lattices, honeycombs, etc.). Therefore, in the context of this research, the selected structures were designed as cubic unit cells at three relative densities, more specifically at 10%, 20%, and 30%, employing the nTopology™ software. In order to examine the mechanical performance of RVEs and build a FE model for each TPMS structure, the unit cell length was selected, according to ASTM standards for compression testing of rigid plastic and cellular material [30, 31], at 12.7 mm due to the compression tests that were carried out in the experiments’ phase. It is worth noting that in this stage, every specimen had one unit cell in order to examine the mechanical response of the RVEs. Figure 3a–d illustrates the 3D model of the TPMS structures (for the three relative densities), providing the sheet thickness (t_s). Extra cubic specimens (verification specimens), with larger external dimensions, were designed and 3D printed to validate the FE models and evaluate the mechanical behavior of each TPMS structure. These specimens have been designed with the same unit cell length, i.e., 12.7 mm. Furthermore, the outer dimensions of the verification specimens were 25.4 mm in each axis, concerning the standardization. Therefore, each specimen was consisted of 8-unit cells, leading to a 2 × 2 × 2 unit-cell configuration. The relative density of verification specimens was at 20% with the thickness of the sheet regulated correspondingly. Figure 3e depicts the 3D models for the four sheet-TPMS lattice structures as designed in nTopology™ software. All models were converted in fine.STL files, which were additively manufactured with the SLS technique.

Fig. 3

Unit cells of sheet-TPMS lattice structures at three relative densities for the following: a Schwarz primitive, b gyroid, c Schwarz diamond, d Neovius; e verification specimens for each TPMS lattice structure

2.3 Material characterization and 3D printing

The sheet-TPMS lattice specimens were 3D printed with selective laser sintering (SLS) additive manufacturing process and the polyamide 12 (PA12) was utilized as construction material. Table 1 lists the main properties of the feedstock material, which were provided by the manufacturer (Sinterit™, Poland). The raw material was in powder form; hence, it was necessary to conduct a material characterization of the powder in order to identify the size of grains and their morphology, which have a high impact on the final quality of 3D printed items [32]. For this material characterization process, SEM imaging and particle size distribution analysis were employed utilizing a scanning electron microscope, namely the Phenom ProX Desktop (ThermoFischer™, USA) and presented in Fig. 4. In Fig. 4a, the SEM image indicates that the majority of the powder particles have irregular shapes with sufficient circularity. Moreover, PSD analysis (Fig. 4b) evaluated the main particle size at 33 µm (D₅₀) and the size of 90% of all particles lower than 42.5 µm (D₉₀). The SLS 3D printer Lisa Sinterit was employed for the AM process. The layer height was chosen at 75 µm resulting in the highest resolution of the 3D printed parts and the main printer-related parameters were selected based on existing literature and the recommendations of the powder’s and 3D printer’s manufacturer (Sinterit™, Poland). More specifically, the hatching distance was set by default from the manufacturer at 200 µm with 90° rotation of the scan vector in each layer. In addition, a laser beam with a power of 5 W was used coupled with a scan speed of 100 mm/s and beam diameter of 0.4 mm. Furthermore, the temperature of the build chamber was regulated at 177.5 °C in order to facilitate the sintering process. Figure 4c shows a 3D illustration of the SLS process with a superimposed image of the actual AM for the TPMS specimens. It is worth noting that the direction of 3D printing was parallel to the direction of compression. Moreover, in the context of the current study, multi-focus images for each structure were acquired via optical stereoscope Leica DMS 1000 (Leica Microsystems GmbH, Switzerland), in order to qualitative examine the morphology of sheet-TPMS structures.

Table 1 Material properties of PA12 provided by the manufacturer

Fig. 4

a SEM image of PA12 powder particles; b PSD analysis of the powder; c 3D schematic representation of SLS process along with a superimposed image of the actual 3D printing process of the specimens

2.4 Mechanical testing

All sheet-TPMS specimens were examined under uniaxial quasi-static compression at room temperature with three repetitions for each design in order to extract reliable results with the minimum standard deviation. For this process, a universal testing machine (M500-50AT Testometric Company, Rochdale, UK) was utilized equipped with a 500-N load cell for the low-density (10%) samples and with a 50-kN load cell for the rest. The calibration of the machine and the model of 3D printed material were evaluated with the assistance of compression and tensile specimens that were 3D printed and tested under uniaxial compression and tensile loading, respectively. These specimens were designed and examined according to the international standards for tensile and for compression testing [33, 34]. It is worth mentioning that all experiments were performed with a strain rate of 5 mm/min [35, 36].

2.5 Finite element models

The mechanical behavior of all the examined structures was investigated with the ANSYS™ software (ANSYS, Inc., Canonsburg, Pennsylvania, U.S). The explicit dynamic module was utilized and accurately simulated the mechanical response of lattices’ RVEs with dynamic finite element analyses (FEA) in order to capture their large deformations, non-linear material behavior and different fracture modes. Mesh sensitivity analyses were performed according to the literature [37, 38], by convergence studies on normalized elastic modulus in order to achieve mesh-independent results. These studies showed that convergence was achieved with almost 100,000 elements for each RVE model and with 250,000 for each verification model. Tetrahedral elements were utilized with a size ranging between 0.2 and 0.6 mm (average value at 0.4 mm) to fit the complex geometry of sheet-TPMS lattice structures and obtain the abovementioned mesh. Furthermore, periodic boundary conditions (PBCs) were applied for uniaxial loading, based on the existing literature [39, 40]. More specifically, for every boundary node, there was an opposite face with another node with an identical position and strain tensor connected according to Eq. (10) [41] where D_ij is the displacement gradient and l is the periodic length:

$${u}_{i}\left(l\right)={u}_{i}\left(0\right)+{D}_{ij}l$$

(10)

In addition, exploiting the obtained results from RVE’s FEA, finite element models were developed which were capable to calculate with high accuracy the mechanical response of larger and more complex sheet-TPMS lattices, as is further elaborated by the verification experiments in Sect. 3.4. FEA was employed for solid compression and tensile samples in order to determine a material model which includes the special characteristics of the SLS AM process. Furthermore, advanced FEMs were built for each lattice curve-fitting the numerical results from FEA with the acquired data from compression experiments and the material model. Therefore, the effect of the printing process and part geometry is included in the FE models. These FE models achieved accurate simulations both for the elastic section and for a sufficient part of the post-yielding section. Regarding the solid specimens, the simulation of tension and compression reached strains up to 10% and approximately 40%, respectively. According to the simulations of sheet-TPMS lattices, the elastic section was calculated utilizing the yield stresses and the elastic modulus reaching strains up to 5%. On the other hand, the post-yielding region (strain > 6%) was properly fitted with the experimental data employing the 3rd order Yeoh model, which has shown remarkable efficiency for lattice structure applications [42]. The 3rd order Yeoh model, which is known as the reduced polynomial model, describes incompressible materials with large deformations and hyper-elastic behavior, a common characteristic of lattice structures [43]. The current model utilizes the numerical tool of strain energy density (W) and it is calculated by the following equation:

$$\mathrm W\left({\mathrm I}_1\right)=\sum{_{\mathrm i=1}^3}{\mathrm C}_{\mathrm i}\left({\mathrm I}_1-3\right)^{\mathrm i}\;\mathrm{with}\;{\mathrm I}_{\mathrm 1}=\mathrm{tr}(\mathrm C)$$

(11)

where I₁ the first strain invariant and C_i is the material constants that were regulated properly in order to match the nonlinear sections of the stress–strain diagrams for the investigated sheet-TPMS lattice structures.

3 Results and discussion

3.1 Calculation of relative density for TPMS lattices

The theoretical background of sheet-TPMS lattices was described previously and showed that the relative density of a solid sheet-TPMS structure depends on the t_s/l ratio and the level-set constant c, according to Eqs. (7)–(9). These variables are connected with the aforementioned relation and with the two constants, G₁ and G₂. Each sheet-TPMS geometry possesses different values for G₁ and G₂ strongly associated with its unique characteristics, its topology, and the mass distribution within the structure. Table 2 lists the values of G₁ and G₂ constants for each sheet-TPMS lattice structure coupled with the ratio of G₂/G₁ that links the t_s/l ratio with level-set constant c. Moreover, Fig. 5a, b illustrate the quantitative diagrams of the t_s/l ratio and the level-set constant c to the relative density, respectively. It is worth mentioning that the calculation of the constants was performed by curve-fitting the equations to the acquired data from the aforementioned design software.

Table 2 G₁ and G₂ constants for each sheet-TPMS geometry

Fig. 5

Quantitative diagrams of the following: a relative density to the t_s/l ratio, b relative density to the level-set constant c

As it is shown in Fig. 5 and the Eqs. (7) and (8), the relations of the relative density between both the t_s/l ratio and the level-set constant c are linear. From Eq. (9), it is clear that the two variables have also linear dependency with a similar trend; however, due to the non-identical G₂/G₁ ratios, the slopes of the curves in Fig. 5a are slightly different from the corresponding slopes in Fig. 5b. These two figures show that as the t_s/l ratio and the c rise, the relative density increases too. Level-set constant equal to 0 leads to only one level-set minimal surface, disabling the generation of a solid sheet-TPMS for all topologies. Furthermore, after examination of the figures, it is clear that the Neovius structure is the most sensitive in changes of design-related parameters, as small changes in ratio or the level-set constant lead to great changes in the relative density. In that matter, the SP structure follows and then the SD and the G structures, respectively. The influence of the design-related factors on the extracted relative density of the structure is related to the existence of multiple sheets (walls) within the volume domain. Hence, the changes in the relative density regarding the design-related parameters are strongly connected with the ratio of the overall lattice surface area to the overall lattice volume for the same t_s/l ratio or the c, with high surface area to volume ratio leading to high slopes in Fig. 5 diagrams.

3.2 Characterization of the mechanical behavior of sheet-TPMS lattice structures

The examination of the tensile and the compression properties for the 3D printed PA12 material is an essential step before the testing of the lattice structures. These experiments reveal the impact of the AM process, in this case, the SLS method, on the mechanical properties of the feedstock material. Thus, the tests were conducted according to the international standardization for tensile and compression loading on plastic materials [33, 34]. The acquired experimental results showed that the basic mechanical properties of the raw material were similar to the manufacturer’s datasheet, except for the elastic modulus. In detail, the print-out density was measured with a high precision scale at 0.97 g/cm³. Moreover, from both the tensile and the compression tests, the yield strength (σ_yield,s), and the elastic modulus (E_solid) were emerged at 23 ± 0.5 MPa and 940 ± 30 MPa, respectively. Furthermore, the ultimate tensile strength (σ_ultimate,s) was measured at 32 ± 0.5 MPa. Figure 6 illustrates the stress to strain diagrams for tensile and compression experiments from which the above-mentioned mechanical properties can be deducted. In addition, the FE model, which simulates the performance of raw material, was developed based on the experimental results. The accuracy and the reliability of the FE model are also visible in Fig. 6 through the stress to strain diagram of finite element analyses.

Fig. 6

Comparison between experimental and FEA-generated tensile and compression stress–strain curves utilizing Yeoh material model

Knowing the mechanical performance of 3D printed PA12 material and extracting an accurate material model, the next step was the fabrication of representative volume elements for each sheet-TPMS structure utilizing the same material and process parameters with the tensile and compression specimens. After the additive manufacturing procedure, all lattice specimens were cleaned from powder and unsintered particles and prepared for testing employing sandblasting as a post-processing procedure. The sandblast process was conducted using fine plastic grains, recommended by the manufacturer of the printer, with low air pressure to ensure the high quality of the examined samples. The twelve different 3D printed RVEs of sheet-TPMS specimens are illustrated in Fig. 7a–e, coupled with multi-focus images of each structure. Moreover, in Fig. 7e, the accuracy of the SLS 3D printing process is presented and the absence of major defects within the structures’ volume domain is verified.

Fig. 7

Additive manufactured RVEs for the following: a Schwarz primitive (SP) structure, b gyroid (G) structure, c Schwarz diamond (SD) structure and d Neovius (N) structure, e indicative multi-focus images for each structure with an optical stereoscope

Sheet-TPMS RVE specimens were examined under compressive loading in order to evaluate the influence of the relative density on the mechanical response of each structure. All tests were conducted three times in order to extract reliable results and the standard deviations in the values of the mechanical properties. The objective was to calculate the main effective mechanical properties of each structure in the three different relative densities, namely the effective elastic modulus (E_lattice), the effective yield stress (σ_yield,l), and the effective peak stress (σ_peak,l). Table 3 lists the values of the above-mentioned effective mechanical properties for all the examined specimens. As it is presented in Table 3 from the yield and peak stresses, the structure with the highest strength for all relative densities was the Neovius followed by the Schwarz diamond, the gyroid, and the Schwarz primitive, respectively. The different strengths that were observed in different structures for the same densities occurred due to the unique mass distribution within the volume domain in each structure. However, the stiffer structure seems to be the Schwarz diamond with the highest effective elastic modulus followed by the Neovius, gyroid, and Schwarz primitive. It is worth noting that for low relative densities (< 10%), Neovius structures revealed stiffer behavior than Schwarz diamond.

Table 3 Main effective properties of the sheet-TPMS lattice specimens

The three values of the effective mechanical properties for each structure, one for each relative density, revealed a specific pattern on how the relative density influenced each structure. Also, from existing literature [13, 16], it is evident that the effective properties of lattice structures comply with specific scaling laws which differ for each structure, changing the constants of the formulations (Eqs. (12)–(14)). Coupling these two, it is straightforward to create functions evaluating the constants for each structure that calculate the expected mechanical property for a desired relative density. The evaluation of the constants was performed by curve-fitting the diagrams with the experimental data. Moreover, employing Eqs. (7) and (8) on Eqs. (12)–(14), it could develop formulas that connect the design-related parameters, namely the t_s/l ratio and the level-set constant c, with the effective mechanical properties (Eqs. (15)–(17)). This offers the comprehensive advantages of the direct connection between a specific mechanical property and an applied design-related parameter.

$$\frac{{E}_{lattice}}{{E}_{solid}}={C}_{E}\cdot {\overline{\rho }}^{{n}_{E}}$$

(12)

$$\frac{{\sigma }_{yield,l}}{{\sigma }_{yield, s}}={C}_{\sigma y}\cdot {\overline{\rho }}^{{n}_{\sigma y}}$$

(13)

$$\frac{{\sigma }_{peak,l}}{{\sigma }_{ultimate, s}}={C}_{\sigma p}\cdot {\overline{\rho }}^{{n}_{\sigma p}}$$

(14)

$$\begin{cases}\frac{{E}_{lattice}}{{E}_{solid}}={C}_{E}\cdot {{G}_{1}^{{n}_{E}}\cdot \left(\frac{{t}_{s}}{l}\right)}^{{n}_{E}} \quad\;\,\;(15)\\ \frac{{E}_{lattice}}{{E}_{solid}}={C}_{E}\cdot {{G}_{2}^{{n}_{E}}\cdot c}^{{n}_{E}} \qquad\;\;\;\;(16)\end{cases}$$

(15)

$$\begin{cases}\frac{{\sigma }_{yield,l}}{{\sigma }_{yield, s}}={C}_{\sigma y}\cdot {{G}_{1}^{{n}_{\sigma y}}\cdot \left(\frac{{t}_{s}}{l}\right)}^{{n}_{\sigma y}}\quad(17)\\ \frac{{\sigma }_{yield,l}}{{\sigma }_{yield, s}}={C}_{\sigma y}\cdot {{G}_{2}^{{n}_{\sigma y}}\cdot c}^{{n}_{\sigma y}}\qquad\;\;\! (18)\end{cases}$$

(16)

$$\begin{cases}\frac{{\sigma }_{peak,l}}{{\sigma }_{ultimate, s}}={C}_{\sigma p}\cdot {{G}_{1}^{{n}_{\sigma p}}\cdot \left(\frac{{t}_{s}}{l}\right)}^{{n}_{\sigma p}}\;(19)\\ \frac{{\sigma }_{peak,l}}{{\sigma }_{ultimate, s}}={C}_{\sigma p}\cdot {{G}_{2}^{{n}_{\sigma p}}\cdot c}^{{n}_{\sigma p}} \quad\;\;\; \!(20)\end{cases}$$

(17)

As it was mentioned, there are two main mechanical behaviors for lattice structures, the stretching-dominated and the bending-dominated behavior. Both are strongly connected with the values of the exponents in the above-mentioned Eqs. ((12)–(17)). More specifically, when the exponents are equal to one, the specimens show a stretching-dominated mechanical behavior which means that the structures have high stiffness and strength and also exhibit a post-softening effect compromising the energy absorption performance [38]. On the other hand, when a structure has a bending-dominated behavior, the values of exponents are above 2 and the specimen shows more ductile behavior with lower strength. Moreover, a plateau occurs on the stress to strain diagram after the peak strength, which shows the superiority of these structures in terms of energy absorption [10]. The constants C_E, C_σy, and C_σp regulate the scaling laws and are different for each structure with values ranging from 0.1 to 4 [15]. Table 4 lists the values of exponents n_E, n_σy, and n_σp as well as the constants C_E, C_σy, and C_σp for each examined sheet-TPMS structure. It is worth noting that these values were evaluated matching the scaling laws with experimental mechanical response in mega-pascal measuring unit. As it is shown in Table 4, Schwarz primitive structure exhibited bending-dominated mechanical behavior as the exponent for the effective elastic modulus exceeded the value of 2. The other three structures had similar values for this exponent indicating a close to stretching-dominated mechanical behavior, with the values of the exponent around 1.2. In addition, the exponents that concern the strength of each structure revealed that the Schwarz primitive structure had the higher exponents which means that the structure’s strength increases exponentially as the relative density raises. Furthermore, Schwarz diamond and gyroid structures have exponents close to 1 which indicates the almost linear relation between the strength and the relative density. Finally, the Neovius structure has returned an exponent for the effective peak stress below 1 which shows that as the relative density increases the peak strength reveals lower growth.

Table 4 Constants of the scaling laws for each sheet-TPMS structure

Figure 8 illustrates the graphical functions of the effective mechanical properties to the relative density. In detail, Fig. 8a portrays the effective elastic modulus for the examined structures. As it was mentioned in the analysis of the functions’ exponents, Schwarz primitive structure has shown an almost bending-dominated behavior due to the quadratic formula between the elastic modulus and the relative density. Hence, an increment of the relative density led to an exponential rising of effective elastic modulus. The Schwarz primitive structure had the lowest values of effective elastic modulus for relative densities below 32%; however, its exponential growth resulted to become stiffer than the gyroid structure after the 32% of relative density. Furthermore, Schwarz diamond, Neovius, and gyroid structures had higher values of modulus for relatively small relative densities (< 30%) leading to a stiffer behavior through the compression tests. Due to the fact that these structures have almost the same exponent and almost linear diagram, the Schwarz diamond and Neovius structure have similar values for effective elastic modulus with the Schwarz diamond became slightly stiffer as the relative density exceeded 30%.

Fig. 8

Quantitative diagrams of each structure along with experimental data for a Effective elastic modulus, b effective yield stress, and c effective peak stress to the relative density

Figure 8b, c show the diagrams between the effective yield stress and the effective peak stress to the relative density, respectively. The measurements revealed that these two stresses are dependable to each other, leading to similar behavior. However, for the Schwarz primitive and Schwarz diamond structures, there are differences that are easy to observe comparing Fig. 8b, c. These occurred due to the deviation of the nonlinear regions of the structures’ stress–strain diagrams. Schwarz primitive had the lower exponent for effective peak stress compared to the yield stress resulting to have exponential relation with relative density but never overcoming the other structures in terms of peak strength. Schwarz diamond structure revealed a linear relation between the effective peak stress and the relative density compared to the effective yield stress. As far as the effective yield stress is concerned, the Neovius structure exhibited the highest loads followed by Schwarz diamond and gyroid structures, respectively. Schwarz primitive structure has shown a remarkable increase in effective peak stress due to the fact that returned an exponent higher than 1. It is worth mentioning that for relative densities above 40% the morphology of the structures may change as there is an excession of material that influences the selected geometry. For the effective peak stresses, Neovius and Schwarz diamond structures show similar values for relative densities up to 20%. After 30%, the Schwarz diamond structure overcame the Neovius structure. Finally, the Schwarz primitive structure exhibited the lowest effective peak stresses for all relative densities.

3.3 Development of RVE FE models

Finite element models for each lattice structure were developed exploiting the acquired experimental data and the formulations of the scaling laws. Hence, the FE models were curve-fitted on the compression test data and regulated for each relative density corresponding with the scaling laws. In addition, due to the high engineering strain (> 10%) and the nonlinear mechanical behavior of the TPMS lattices, it was crucial to employ hyperelastic FE models capable to simulate properly and accurately the mechanical responses of sheet-TPMS lattices structures. It is worth mentioning that the strains and effective stresses of experiments are compared with the numerical strains and effective stresses, according to existing literature [13, 14, 42, 44].

The hyperelastic 3rd order Yeoh model, also known as the reduced polynomial model, was employed in order to capture the exact mechanical behavior of the examined lattices. The material constants, i.e., the constants of the 3rd order Yeoh polynomial C1, C2, and C3, were modified and curve-fitted with the experimental response of 3D printed PA 12 for structure. In detail, C1 and C3 were ranged from 0.83 to 21.6 for the weakest structure (Schwarz primitive 10%) to 18.11 and 270.7 for the strongest structure (Neovius 30%), respectively. Furthermore, C2 had negative values between − 5.24 and − 92.66 regarding again the above-mentioned structures. Figure 9 presents a comparison between the experimental stress–strain diagrams and the computationally generated data from FE analysis. It is worth mentioning that the stress–strain diagrams were evaluated up to 30% of strain in which the major mechanical features were included, i.e., elastic section, peak strength, plateau, and post-softening effect. Furthermore, Fig. 9 shows the von-Mises stress contours at peak loads of each lattice structure for 30% relative density in order to locate the stress concentration regions, observe the distribution of stresses on the lattice volume domain, and understand the mechanisms of failure during the uniaxial quasi-static compression loading. It is essential to mention that due to the relatively low relative densities of RVE’s and the existence of only one unit cell; in some cases, the structures revealed extensive post-softening effect resulting in insignificant mechanical response beyond 30% of strain until the densification effect.

Fig. 9

Comparison between the experimental and FEA stress–strain curves, along with the equivalent von-Mises stresses for a Schwarz primitive structure, b gyroid structure, c Schwarz diamond structure, and d Neovius structure

In Fig. 9a, the mechanical response of Schwarz primitive structures is illustrated in the form of a stress–strain diagram for all examined densities coupled with the stress contour of 30% relative density at peak strength derived from FEA. Indicatively, in Schwarz Primitive specimens with a relative density of 30%, the peak stress was evaluated to be at 6.97% of the nominal peak stress of the material, with a value of 2.23 MPa. The high accuracy of the FE model was verified by the sufficient curve-fit of FEA results on the experimental stress–strain curve. Furthermore, Schwarz primitive’s equivalent von-Mises contour revealed that the concentration of stresses appeared (orange and red color) at regions with high local curvature leading to a failure of the structure. Figure 9b portrays the mechanical performance of RVEs for gyroid structures. In this lattice structure with 30% relative density, the reduction of the effective peak stress was at 7.89% of the nominal, i.e., 2.52 MPa. The stress concentration was observed at the upper region of the structure and more specifically at the region with high circularity resulting in a local buckling effect of the structure, which led to the fracture of the samples. Moreover, the mechanical behavior of the Schwarz diamond structures is shown in Fig. 9c. Schwarz diamond with 30% relative density revealed the best performance in terms of effective peak stress at 16.09% of material’s nominal peak stress that is 5.15 MPa, due to the uniform distribution of the stresses within the volume domain. The uniform stress distribution inside the volume domain is also visible in equivalent von-Mises stress contour leading to a remarkable strength and plasticity with a strain of almost 17%, as it is also derived from the plateaus at stress to strain diagrams. In addition, this structure seems to concentrate the stresses in axes parallel with the applied force resulting in the formation of stress columns within the structures and when these columns excessed their ultimate strength the structure failed.

Finally, Fig. 9d depicts the mechanical response of the Neovius structures. The Neovius lattices showed remarkable effective peak stress reaching the value of 4.65 MPa at 30% relative density, which is at 14.53% of the nominal value. Neovius structure exhibited similar behavior in terms of stress concentration with Schwarz primitive structure, as both structures concentrated their stresses at regions with high local curvature. However, due to Neovius’ better mass distribution and enhanced topology of specific regions with the cross pattern in the upper and bottom sides of the structure, Neovius has shown better performance in terms of peak strength and effective elastic modulus.

3.4 Verification of the FE models

In the previous subsection, the mechanical response of each lattice structure, with the assistance of RVEs, was investigated and hyperelastic FE models were built in order to simulate their mechanical behaviors with high accuracy and reliability at strain up to 15%, as it was depicted in Fig. 9. However, it is crucial to verify these FEMs by utilizing them at more complex lattice structures, i.e., at the same lattice geometries but with more unit cells within the structures, in order to evaluate their functionality. Thus, four distinct more complex test specimens were designed and fabricated, one for each lattice structure. The additive manufactured verification samples are presented in Fig. 10. As conducted in RVE specimens, verification specimens were undergone sandblasting in order to remove any unsintered particles and clean the structures. Also, it is essential to mention that identical processes were performed for the design, fabrication and post-processing of the verification specimens as the RVE specimens.

Fig. 10

3D printed verification specimens of the four sheet-TPMS lattice structures

When the design and the manufacturing processes of the verification specimens were completed, FE analyses were performed employing the hyperelastic material models and proper boundary conditions that were determined from the RVE’s experiments in order to simulate the mechanical response for each verification specimen. Then, the physical models of samples were undergone uniaxial quasi-static compression tests and the obtained experimental results were compared with the FEA results. These comparisons were conducted in terms of stress–strain diagrams from experiments and FEA, as well as in terms of images at peak loading to deliver optical references for the comparison. Figures 11 and 12 illustrate the abovementioned comparisons of Schwarz primitive and gyroid and Schwarz diamond and Neovius structures, respectively.

Fig. 11

Comparison of experimental data and FEA results for a Schwarz primitive and b gyroid structures

Fig. 12

Comparison of experimental data and FEA results for a Schwarz diamond and b Neovius structures

Figure 11a shows the mechanical performance of the Schwarz primitive structure. Schwarz primitive was reached its peak strength (0.91 MPa) at almost 6% strain with sufficient fitting from the FE analysis up to that point. Comparing the images from compression experiments and the FEA, it is obvious that both the physical model and the digital one revealed similar deformation, and the stresses were concentrated in one half of the structure resulting in an oval shape of the structure’s holes. Figure 11b depicts the same comparison for the gyroid structure. Again, the experimental data were accompanied by the acquired data from FEA until the peak strength (1.88 MPa), presenting good accuracy. In both images of peak stress, the structure was compressed at its upper side, leaving the rest of the structure undistorted. It is worth noting that the ultimate stress was observed at 10.2% strain. Figure 12a, b illustrate the comparison between experimental and FEA results for the verification specimens of Schwarz diamond and Neovius structures, respectively. In these cases, the FE models were more accurate. Schwarz diamond exhibited the most remarkable performance reaching high effective peak stress coupled with a wide plateau area. The extensive plateau area was a result of the increased ductility of the structure. As it is visible in both physical and digital models, the stresses were concentrated at the upper side of the structure and gradually distributed at the rest of the structure leading to a smooth and uniform deformation. Neovius structure revealed the highest effective peak strength at 3.07 MPa among all the structures, as it was expected, for 20% of relative density, at 7.1% strain. The concentration of stresses is observed at the unit cell on the upper side of the structure and more specifically at the regions with high local curvature, as was expected from RVE’s analysis. To conclude, the developed FE models extracted similar results for the mechanical response of the examined sheet-TPMS lattice structure with the experimental compression data. Hence, the FE models derived from analyzing the RVE’s mechanical response are suitable to simulate the mechanical performance of more complex structures with reliability and high accuracy.

3.5 Energy absorption of the TMPS structures

One other aspect of lattice structures that concentrates increased scientific interest is the amount of absorbed energy during the compressive loading. It is evident that especially sheet-TPMS structures reveal a high energy absorption rate due to their high mechanical strength and their extensive ductility [12, 16, 18]. Hence, it is essential to measure the absorbed energy for each examined structure in terms of energy absorption per volume and energy absorption per mass. According to international standards [45, 46], the energy absorption derives from the surface area between the stress–strain curve and the strain axis (x-axis) on the compressive stress to strain diagram. In addition, the exact mathematical formulas are shown in Eq. (18) for energy absorption per volume (W) and Eq. (19) for energy absorption per mass, which is also known as specific energy absorption (SEA). Moreover, Eq. (20) calculates the energy absorption efficiency (W_e) which indicates the percentage ratio between the energy absorption of a lattice structure and the absorbed energy of an ideal absorber [47]:

$$W=\int_{0}^{\varepsilon_{0}} \sigma \left(\varepsilon \right)d\varepsilon$$

(18)

$$\mathrm{SEA}=\frac{1}{{\rho }_{m}}\int_{0}^{\varepsilon_{0}}\sigma \left(\varepsilon \right)d\varepsilon$$

(19)

$${W}_{e}=\frac{W}{{\sigma }_{peak}\times {\varepsilon }_{0}}100$$

(20)

where ρ_m is the density of the construction material (PA12), σ is the compressive stress, ε is the strain, and ε₀ is the upper strain limit; typically, this limit is placed before the densification occurs. In the current study, the recommendation of the ISO was followed [45, 46]; hence, the upper strain limit was selected to be 50%. The stress to strain diagrams of the examined lattice structures was extracted with the quasi-static compression test; thus, the tests were performed in small time-steps with different pairs of stress and strain for each time-step. Almost 3500 time-steps were employed for each stress-to-strain diagram up to 50% strain. Furthermore, the trapezoidal method of numerical analysis enabled the discretization of the integrals into sums of finite values with a specific step for the function. Therefore, Fig. 13 presents the diagrams and the results of energy absorption evaluation for the examined sheet-TPMS lattice structures. In detail, Fig. 13a shows the stress–strain curves for each structure up to 50% of strain. It is obvious that the peak strength has an essential role in terms of energy absorption due to the fact that leads to an increased surface below the curve. Also, the existence of plateau and secondary peaks is crucial for the same reason. Figure 13b portrays the acquired quantitative results from the energy absorption analysis. The energy absorption per volume and the specific energy absorption followed identical trends in all four examined lattices. The lowest absorbed energy, at 187.74 kJ/m³ and 197.62 J/kg, was observed for Schwarz primitive structure due to the ultra-low peak strength (0.9 MPa). On the other hand, despite the fact that the Neovius structure revealed the highest peak strength (3.07 MPa), the intense post-softening effect led to the second highest energy absorption rate, at 857.94 kJ/m³ and 903.09 J/kg. The highest energy absorption was revealed for the Schwarz diamond structure, at 903.91 kJ/m³ and 951.48 J/kg, due to the fact that it had high peak strength at 2.7 MPa but simultaneously exhibited a wide plateau before the post-softening effect occurred. In addition, the gyroid structure showed moderate absorbed energy at 506.66 kJ/m³ and 533.27 J/kg with a mild post-softening effect. Furthermore, the most essential indicator to evaluate the energy absorption rate and the crashworthiness of a structure is the energy absorption efficiency, which compares the structure with an ideal absorber. In Fig. 13b, the values of this indicator are illustrated with red color. According to Eq. (20), it is directly linked with energy absorption per volume presenting similar performance. Schwarz primitive had the lowest efficiency with 41.54%; then, the gyroid was followed with efficiency at 53.89%. Neovius structures’ energy absorption efficiency was slightly higher than gyroid, at 55.95%. Finally, the best performance in terms of energy absorption efficiency was observed in the Schwarz diamond structure, which reached up to 66.96% revealing the high value of this structure for impact applications.

Fig. 13

a Stress–strain curves for all the TPMS structures (2 × 2 × 2) up to 50% strain), b energy absorption per volume and per mass comparison between different lattice structures coupled with the energy absorption efficiency of each structure (up to 50% strain)

4 Conclusions

In the current research, four sheet-TPMS lattice structures, namely Schwarz primitive, gyroid, Schwarz diamond, and Neovius, were designed, manufactured, and investigated under uniaxial compressive testing supported by FEA. This work examined the scientific background of the TPMS lattice structures and conducted a novel theoretical analyses connecting the design-related parameters, namely wall thickness, unit’s length, and level-set constant, with the relative density of the overall structure via approximation equations. Through the theoretical analysis, the linear relation of design-related parameters with relative density was pointed out for all four sheet-TPMS lattices. Moreover, the investigation of examined structure’s mechanical response was performed utilizing 3D printed RVEs for each structure via uniaxial quasi-static compression tests. The results from the RVEs’ test have shown that the Schwarz primitive exhibited an almost bending-dominated behavior with low effective peak strength and scaling law exponents around 2. On the other hand, the rest of the examined structures revealed stretching-dominated behavior with higher ultimate strength and exponents around 1. It is worth mentioning that the exponents and the constants of scaling laws were calculated based on the acquired experimental results. Furthermore, FE models were developed exploiting the experimental data of RVEs for each lattice structure. Then, these innovated models assisted with proper periodic boundary conditions were employed on lattice specimens with more unit cells in order to verify their reliability and their accuracy to evaluate the mechanical response of lattices. To conclude, the best mechanical performance in terms of strength, elasticity and energy absorption was observed by Schwarz diamond and Neovius structures, which demonstrated similar overall behavior. Overall, this article proved that by evaluating the mechanical performance of lattice RVEs, it is feasible to develop FE models that assess the mechanical performance of more complex structures saving computational power and time.