Energy Absorption Properties of 3D-Printed Polymeric Gyroid Structures for an Aircraft Wing Leading Edge

Abstract

Laminar flow offers significant potential for increasing the energy efficiency of future transport aircraft. At the Cluster of Excellence SE²A—Sustainable and Energy-Efficient Aviation—the laminarization of the wing by means of hybrid laminar flow control (HLFC) is being investigated. The aim is to maintain the boundary layer as laminar for up to 80% of the chord length of the wing. This is achieved by active suction on the leading edge and the rear part of the wing. The suction panels are constructed with a thin micro-perforated skin and a supporting open-cellular core structure. The mechanical requirements for this kind of sandwich structure vary depending on its position of usage. The suction panel on the leading edge must be able to sustain bird strikes, while the suction panel on the rear part must sustain bending loads from the deformation of the wing. The objective of this study was to investigate the energy absorption properties of a triply periodic minimal surface (TPMS) structure that can be used as a bird strike-resistant core in the wing leading edge. To this end, cubic-sheet-based gyroid specimens of different polymeric materials and different geometric dimensions were manufactured using additive manufacturing processes. The specimens were then tested under quasi-static compression and dynamic crushing loading until failure. It was found that the mechanical behavior was dependent on the material, the unit cell size, the relative density, and the loading rate. In general, the weight-specific energy absorption (SEA) at 50% compaction increased with increasing relative density. Polyurethane specimens exhibited an increase in SEA with increasing loading rate, as opposed to the specimens of the other investigated polymers. A smaller unit cell size induced a more consistent energy absorption, due to the higher plateau force.

1. Introduction

Laminar flow control through active boundary layer suction shows great potential for more energy-efficient aviation by means of drag reduction [1]. Numerous configurations for such drag reduction of an aircraft wing have been explored. One approach is the use of plasma actuators that generate momentum in the air, postponing the boundary layer separation [2]. One concept involves the principle of natural laminar flow (NLF), which can be achieved using a smooth surface and a targeted profile design. Another approach is based on laminar flow control (LFC), which uses active boundary layer suction [3]. The combination of NLF and LFC has been investigated at the Cluster of Excellence SE²A—Sustainable and Energy-Efficient Aviation—for an electrically powered regional aircraft. The hybrid laminar flow control (HLFC) concept incorporates LFC on the leading edge of the wing and NLF on the rear part of the wing. This novel extended hybrid laminar flow control (xHLFC) concept is based on active suction through a suction panel consisting of a sandwich structure, with a micro-perforated skin and a triply periodic minimal surface (TPMS) structure as a core on the rear part of the wing [4]. The proposed suction panel could potentially reduce drag by up to 45.5% [5].

Previous studies have also investigated active suction on the leading edge of the wing. The tailored skin single duct (TSSD) concept [6] uses a multi-layered, metallic outer skin, which provides a pressure drop. Schrauf et al. [7] conducted initial flight tests with an HLFC system on the vertical tail plane of an Airbus A320. The system consisted of a simple double-skin structure with the inner and the micro-perforated outer skin connected by stringers. The results of the first tests indicated that the turbulent transition was delayed due to active or passive suction. Lobitz et al. [4] investigated an integrally, additively manufactured sandwich suction panel consisting of a gyroid core and a micro-perforated outer skin for xHLFC in the rear part of a wing. The study compared different materials and gyroid configurations regarding the loading on the wing and the suction panel. The material Nylon11CF was identified as the optimal material, in terms of light weight considerations. The adhesive and erosive effects of atmospheric pollutants, such as aerosols, organic matter, and rain, were also discussed. Adhesion of organic matter on the wing surface may cause an increase in drag. Frequent washing may be needed, since a smooth surface is essential for HLFC systems. This may reduce the benefit of the fuel savings with HLFC, depending on the mission [8]. Clogging of the micro-perforations may also counteract the benefits of HLFC, since the major part of debris is located at the leading edge. In the same manner, erosion mostly occurs at the leading edge part of the wing [9]. Additionally, the accumulation of ice on the leading edge can alter the geometry of a wing and cause drag. Therefore, anti-icing techniques, such as application of heat or anti-freeze agents, need to be considered. This is more of a system design problem than a technical obstacle for HLFC [10].

Positioning the suction panel at the leading edge necessitates the imposition of additional requirements on its core structure. All parts oriented in flight direction are susceptible to bird strike damage, which makes bird strike a design-driving issue [11]. Cihan Tezel et al. [12] conducted a numerical study on different design options of a wing leading edge, concluding that a sandwich leading edge structure with an aluminum skin and a cellular core was the lightest design fulfilling the given requirements for bird strike resistance.

Based on these findings, a new approach for a novel wing leading edge for transport aircraft was investigated. This approach combines the benefits of active suction on the wing, which is characterized by an increase in efficiency through drag reduction, and a lightweight sandwich structure. A schematic representation of the wing leading edge with a sandwich structure and incorporating a TPMS-based core resistant to bird strike is provided in Figure 1.

Cellular structures can be classified into two categories: stochastic (foams) and periodic (lattices). Lattice structures can be optimized for end-user products or applications in a topological manner and include honeycombs and TPMSs. TPMSs are a subcategory of minimal surfaces that locally minimize the surface area within a given boundary [13]. TPMSs are defined by implicit equations consisting of a combination of sine and cosine functions, making TPMSs periodically duplicable in all three dimensions. The first TPMSs described in the literature were the diamond and primitive surfaces by Schwarz [14] in 1890. Multiple other TPMSs were introduced by Schoen [13] in 1970. One of them was the gyroid surface. The description of the gyroid surface is presented in Equation (1):

$$\Phi \left(x,y,z\right)=\cos \left({\lambda }_{x}x\right)\sin \left({\lambda }_{y}y\right)+\cos \left({\lambda }_{y}y\right)\sin \left({\lambda }_{z}z\right)+\cos \left({\lambda }_{z}z\right)\sin \left({\lambda }_{x}x\right)=c,$$

(1)

with

$${\lambda }_i=2\pi {\displaystyle \frac{n_i}{L_{i,tot}}},\mathrm{for}i=x,y,z.$$

An isosurface is generated at all points where the equation is equal to the constant c [15]. The constant c modifies the geometry of the gyroid [16]. A typical value for this parameter is zero. The total length of the structure in each dimension, designated as L_i,tot, and the unit cell size in each dimension, designated as n_i, are employed to define the periodicity parameter, designated as ${\lambda }_i$. Two distinct methodologies may be employed to generate a structure from the surface: The extrusion of material in a perpendicular direction to the surface generates a solid network structure. Extruding the material in both directions perpendicular to the surface generates a sheet network structure [17]. Numerical investigations by Li et al. [18] indicated that a sheet network unit cell tends to be isotropic, while a strut-based unit cell tends to show anisotropic behavior. Furthermore, the sheet-based gyroid was found to be theoretically more suitable for energy absorption than the strut-based gyroid [18]. The behavior and mechanical properties of gyroid structures are contingent upon the material, unit cell size, manufacturing process, and relative density [17]. The relative density ${\rho }^{\ast }$ is defined as the density of the cellular structure ${\rho }_G$ divided by the density of the solid material ${\rho }_S$:

$${\rho }^{\ast }={\displaystyle \frac{{\rho }_G}{{\rho }_S}}.$$

(2)

In general, lattice structures deform, when under macroscopic loading, through a combination of bending and stretching of the struts or sheets [17]. For the majority of lattice structures, the force-displacement response commences with an elastic region, which is succeeded by the initial stage of plastic deformation. Stretching-dominated structures display a softening after the elastic region, followed by a force plateau until densification. Bending-dominated structures exhibit a force plateau following the elastic region, without softening [19]. Oftentimes, it is observed that the plateau force exhibits oscillatory behavior due to the geometry of the lattice [17]: In sheet network gyroid structures, collapse of the layers can be seen [20,21,22].

The mechanical properties of a cellular structure are dependent on its relative density ${\rho }^{\ast }$ and its geometry. This relation can, in principle, be described by a power law of the following scheme [23]:

$${\displaystyle \frac{P_G}{P_S}}=C·{\left({\displaystyle \frac{{\rho }_G}{{\rho }_S}}\right)}^n.$$

(3)

The ratio of the physical property of a gyroid structure P_G to the physical property of the solid material P_S corresponds to a geometric constant C times the relative density ${\rho }^{\ast }$ to the power parameter n. The parameter n is dependent on the structure’s behavior.

A comprehensive overview of experimental and numerical investigations of sheet-based gyroid structures is presented in [17]. Experimental studies were conducted to investigate the mechanical behavior of such structures under compressive loading. Maskery et al. [24] explored the compressive behavior of cubic gyroid specimens made of Al-Si10-Mg with an edge length of 18 $\mathrm{m}$$\mathrm{m}$, unit cell sizes of 3 mm, 4.5 mm, 6 mm and 9.5 mm, and a relative density of 30% under a quasi-static compressive strain rate of 5 × 10⁻⁴ s⁻¹. The specimens exhibited three distinct modes of failure, depending on the unit cell size: A successive collapse of cells in planes perpendicular to the loading was observed for cell sizes of 4.5 mm and 6 mm. The second failure mode was indicated by brittle fracturing of the cell walls. A crack propagated with its main component in the plane parallel to the loading direction. This phenomenon was observed for larger unit cell sizes. The third failure mode was characterized by the formation of a diagonal shear band, which occurred exclusively for the smallest unit cell size. Other studies have investigated the mechanical behavior of gyroid structures of different materials and low strain rates, including Al-Si7-Mg0.6 at a strain rate $\dot{\epsilon }$ of 10⁻³ s⁻¹ [22] and stainless steel at a strain rate $\dot{\epsilon }$ of 10⁻³ s⁻¹ [20]. Additionally, the mechanical behavior of PA2200 at a strain rate $\dot{\epsilon }$ of 10⁻² s⁻¹ was investigated in [21], while [18] examined the behavior of a polymer at a strain rate $\dot{\epsilon }$ of 4.6 × 10⁻³ s⁻¹.

Liu et al. give an overview of studies on the impact behavior of additively manufactured metals and structures in [25]. In their survey, less than 40 studies on the dynamic behavior of lattice structures were found. Of the 40 studies, only a few addressed sheet-based gyroid structures. In their study, Li et al. [26] examined the mechanical behavior of cubic gyroid structures made from 316 L stainless steel with an edge length of 20 mm, five unit cells per dimension, and a relative density of 30% under four different strain rates, ranging from 1 × 10⁻³ s⁻¹ to 8.5 × 10⁻² s⁻¹. Lu et al. [27] analyzed the mechanical behavior of different 316L stainless steel cubic gyroid specimens with an edge length of 20 mm and a relative density of around 12% under loading velocities ranging from 5 ms⁻¹ to 50 ms⁻¹. Ramos et al. [28] performed low-velocity impact tests on cubic Al-Si10-Mg gyroid specimens with different loading velocities.

However, as presented, not much research has been carried out on the dynamic behavior of polymeric gyroid structures subjected to impact loading. Therefore, in the present study, experimental tests were conducted on various gyroid configurations of different additively manufactured polymeric materials. For application as a lightweight structure and the suction of the boundary layer flow, the relative density was restricted from 7% to 23%. This study aimed to help fill the gap in research on polymeric gyroid structures under dynamic loading. Research on HLFC deals, to a great extent, with the aerodynamic effects of the applied technique. This study is intended to address aircraft safety in a specific HLFC approach using a sandwich panel structure with a gyroid core.

The outline of this present study is as follows. The investigated materials, the manufacturing process, and the methods employed to generate the gyroid surface are presented. This is followed by the results and a discussion of the compressive tests under quasi-static and dynamic loading, where the failure behavior as well as the plateau force and the weight-specific energy absorption of the specimens are discussed. A brief summary of the study’s findings is presented in the conclusion section.

2. Materials and Methods

2.1. Materials

The aim of this study was to investigate the influence of the material, as well as the gyroid configuration, on energy absorption performance. Accordingly, a variety of polymeric materials were selected for testing purposes. Due to the gyroid structures’ geometric restrictions, the specimens had to be additively manufactured using different techniques. Two 3D-printers were utilized for the manufacturing of specimens: A Formlabs Form 3+ and a Formlabs Fuse 1+ 30 W from the company Formlabs (Somerville, MA, USA). Both are capable of manufacturing a range of materials, including Formlabs Standard Resin, Formlabs Polyurethane Rigid 650 (PU Rigid 650), and Formlabs Nylon 11. Standard Resin and Polyurethane Rigid 650 are both liquid resin-based polymers, while Nylon 11 is a powder-based polymer.

Standard Resin is a thermoset plastic characterized by a rather stiff and brittle material behavior, and utilized for the fabrication of prototypes. Polyurethane Rigid 650, on the other hand, is highly flexible and has a great elongation at break. The two resin-based materials are manufactured using stereolithography (SLA) technology. The liquid resin was locally partially cured using a laser. Following the printing process, the Standard Resin components were washed in an isopropanol bath to remove any residual resin from the desired geometry. Subsequently, the curing of the parts was completed under UV light at a temperature of 60 °C.

Polyurethane Rigid 650 is a material that has recently become accessible to additive manufacturing processes. However, the manufacturing process is more challenging than with other materials. It is beneficial to manufacture it in a low-relative-humidity environment with a relative humidity below 5%, due to its tendency to interact with water. Polyurethane Rigid 650 parts were washed in a propane-1,2-diyl diacetate (PGDA) bath and cured after the printing process in an oven at a temperature of 46 °C and a relative humidity of 70%.

Nylon 11 is a bio-based thermoplastic characterized by its high ductility. It is employed for the fabrication of prototypes and end-user products that are flexible and impact-resistant. Nylon 11 was manufactured using a Formlabs Fuse 1+ 30 W powder 3D-printer with selective laser sintering (SLS) technology. In this process, the powder was locally melted by a laser. After the printing process, the residual powder was mechanically removed from the desired geometry.

The material properties of the utilized materials are presented in

Table 1. Material properties after curing according to manufacturer.

		Material
Property	Unit	Formlabs Standard Resin [29]	Formlabs PU Rigid 650 [30]	Formlabs Nylon 11 [31]
Young’s modulus	$\mathrm{G}$$\mathrm{Pa}$	$2.8$	$0.67$ ± $0.06$	$1.6$
Ultimate tensile strength	$\mathrm{M}$$\mathrm{Pa}$	65	34 ± $3.4$	49
Elongation at break	%	$6.2$	170 ± 17	40
Density	g cm−3	$1.18$	$1.16$
Notched Izod	J m−1	25	375	71

, according to the manufacturer [29,30,31].

2.2. Gyroid Configuration

Different geometrical gyroid configurations were investigated in this study. Cubical gyroid samples with an edge length of 50 $\mathrm{m}$$\mathrm{m}$ were designed. The wall thickness of each configuration was used as a manufacturing restriction contingent on the relative density and unit cell size. The lowest wall thickness could be realized using the Standard Resin material and the SLA printing process. All other manufacturing processes and materials were constrained by a minimum wall thickness of approx. $0.5$ $\mathrm{m}$$\mathrm{m}$. The wall thickness of a gyroid structure can be approximated according to Ramírez et al. [32] by

$$d\left(L_i,{\rho }^{\ast }\right)={\displaystyle \frac{{\rho }^{\ast }·L_i^{0.966}}{2.568}}$$

(4)

with the wall thickness d, the unit cell size L_i, and the relative density of the gyroid ${\rho }^{\ast }$. This is described to be valid only for wall thicknesses between $2.5$ $\mathrm{m}$$\mathrm{m}$ and $11.5$ $\mathrm{m}$$\mathrm{m}$, but also provides a good approximation for lower wall thicknesses.

In this study, four relative densities—7%, 12%, 17% and 23%—were considered. The unit cell sizes were 7 mm, 10 mm, 14 mm and 20 $\mathrm{m}$$\mathrm{m}$ (s. Figure 2). Subsequently, the number of unit cells per side was 7.1, 5, 3.6 and 2.5. The resulting wall thickness for each combination could be approximated using Equation (4) and are listed in

Table 2. Approximate calculated wall thicknesses for different unit cell sizes and relative densities according to Equation (4).

Wall Thickness/mm		Unit Cell Size/mm
		7	10	14	20
Relative density/%	7	0.179	0.252	0.349	0.492
	12	0.306	0.432	0.598	0.844
	17	0.434	0.612	0.847	1.196
	23	0.587	0.828	1.146	1.618

. Representative figures of the specimens with different unit cell sizes and a constant relative density of 23% are shown in Figure 2. The different relative densities at a constant unit cell size of 20 mm of Nylon 11 are shown in Figure 3. Due to manufacturing restrictions and difficulties, not all combinations of unit cell size and relative density were manufactured for all desired polymeric materials.

The process of generating and manufacturing the specimens was as follows: An .stl file was generated in Matlab R2022b with parameters for the total edge length, unit cell size, relative density, and resolution. The total length of the cubic specimens was defined in the parameter L. The unit cell size L_i was set to the according value. The relative density was defined by the thickness parameter t, which applied a thickness to the isosurface according to Equation (5) [24,28]:

$$\begin{array}{cc}\hfill \Phi \left(x,y,z\right)& ={\left(\cos \left({\lambda }_{x}x\right)\sin \left({\lambda }_{y}y\right)+\cos \left({\lambda }_{y}y\right)\sin \left({\lambda }_{z}z\right)+\cos \left({\lambda }_{z}z\right)\sin \left({\lambda }_{x}x\right)\right)}^2-t^2\hfill \\ \hfill & =0.\hfill \end{array}$$

(5)

The resolution is defined as the number of points per unit cell in which the equation is evaluated. A high resolution equals a more precise depiction of the gyroid structure but requires more computational power and storage. The generated .stl file was imported into the slicer software PreForm 3.40.1 by Formlabs. Here, the 3D-printer, as well as the material, were selected. The layer thickness was chosen and the supporting structures were generated. Multiple gyroid structures could be manufactured at the same time. The .form file was then imported into the 3D-printer. The manufacturing process is described in Section 2.3.1.

2.3. Methods

2.3.1. Manufacturing of Specimens

Cubic gyroid specimens were manufactured using a variety of polymeric materials as described previously. All specimens were produced through additive manufacturing, as described in Section 2.1. The layer thickness of the specimens manufactured by SLA was 50 $\mu$$\mathrm{m}$ and small wall thicknesses could be realized. Specimens manufactured by SLM had a layer thickness of 110 $\mu$$\mathrm{m}$.

A squared glass-fiber-reinforced plastic plate with a thickness of $1.5$ $\mathrm{m}$$\mathrm{m}$ and an edge length of 53 $\mathrm{m}$$\mathrm{m}$ was adhesively joined to the top and bottom side of each specimen as a sandwich skin plate, to ensure uniform load transfer into the gyroid structure during the testing. The structural epoxy adhesive EC-9323 B/A by 3M Company (St. Paul, MN, USA) [33] was used. The adhesive was applied to the plate. Thereafter, the plate was placed on the gyroid specimen. To ensure the bonding of the plate with the gyroid structure, some weight was placed on top of the specimen. The adhesive was cured at a temperature of 70 °C for a duration of two hours.

2.3.2. Testing

The experimental testing was conducted with two distinct loading rates. They were here constituted as quasi-static compression and dynamic crushing, and abbreviated as QS and Dyn, respectively:

Quasi-Static Compression

The cubic gyroid specimens were subjected to quasi-static compression using a universal test machine from the company ZwickRoell GmbH & Co.KG (Ulm, Germany). The test machine was equipped with a load cell with a capacity of 50 $\mathrm{k}$$\mathrm{N}$ and a resolution of 50 $\mathrm{N}$. The test process was displacement controlled with a loading rate of 2 $\mathrm{m}$$\mathrm{m}$ min⁻¹, which resulted in a nominal strain rate $\dot{\epsilon }$ of 6.67 × 10⁻⁴ s⁻¹.

The test setup is depicted in Figure 4a. The cubic gyroid specimen was positioned between two parallel metallic planes that were connected to the test machine. The displacement of the two stamps relative to each other was recorded using the optical measurement tool ARAMIS from ZEISS (Oberkochen, Germany).Images were recorded every ten seconds by an external camera. Two light sources were utilized to illuminate the specimen.

Dynamic Crushing

The dynamic crushing was conducted using a drop tower. The drop tower comprised a double-sided guided drop sled. The total mass of the drop sled including the load cell and the impactor was 16 $\mathrm{k}$$\mathrm{g}$. The maximum drop height was six meters. A load cell with a maximum load of 160 $\mathrm{k}$$\mathrm{N}$ and a resolution of 160 $\mathrm{N}$ was attached between the drop sled and the circular impactor. The experiment was recorded with a high-speed camera capable of capturing images at up to 40 k fps. Two light sources were utilized to illuminate the test setup. The load and the displacement were recorded at a rate of 49 k samples s⁻¹. The test setup is depicted in Figure 4b. The specimen was positioned on the metallic base plate. The drop sled impacted the specimen with a velocity of approx. 3 ms⁻¹, resulting in a nominal strain rate of 6 × 10¹ s⁻¹.

3. Results and Discussion

This section presents and discusses the results of the quasi-static compression and dynamic crushing testing of different gyroid configurations of different polymeric materials. Additionally, a short discussion on the additive manufacturing is presented.

3.1. Manufacturing of Specimens

Additive manufacturing of gyroid specimens has advantages and disadvantages. One advantage is the possibility of manufacturing complex geometries. Furthermore, the print quality achieved with a small layer thicknesses is conducive to isotropic behavior. However, one disadvantage of using SLA and SLS technologies is the necessity of the removal of residual material. The liquid resin that is attached to the structure after the printing process can only partially be washed out, as the movement of the solvent is typically insufficient to reach all inner parts of the structure. The removal of residual powder from the inner structure is more challenging, due to the necessity of mechanical intervention. As the unit cell sizes decrease, specimens become increasingly susceptible to this issue. This complication results in an increase in the mass of specimens, which consequently leads to an increase in their relative density, opposed to their originally desired density. This issue could be circumvented by employing fused deposition modeling (FDM) additive manufacturing. Nevertheless, this additive manufacturing process cannot achieve a printing quality as good as other technologies. For instance, the layer thickness and the achievable minimum wall thickness are much larger.

3.2. Standard Resin

Force–displacement curves of the Standard Resin gyroid structures are presented in Figure 5. Each figure illustrates the influence of varying unit cell sizes at a constant relative density under quasi-static and dynamic loading conditions. The shadowing depicts the standard deviation of the numerous tested specimens. In general, a typical cellular structure’s force–displacement curve can be characterized as follows: An initial linear elastic increase in force is observed, until the peak force is reached, after which a force drop occurs, due to a local failure such as a crack, to a plateau force that remains nearly constant until the densification phase. The local failure can lead to the global collapse of a layer of the gyroid structure. This indicates that the structure is undergoing a stretching-dominated behavior of the walls [17]. Furthermore, it can be observed that the initial maximum force increases with increasing unit cell size. Nevertheless, a smaller unit cell size allows for a more consistent energy absorption following the first initial force peak, due to a greater constant plateau force until densification. The energy absorption E_abs is defined as the area under the force–displacement curve:

$$E_{abs}={\int }_0^{s}F\left(s\right)ds.$$

(6)

Moreover, the Standard Resin gyroid specimens exhibited, in general, greater forces under quasi-static loading conditions compared to the same specimens under dynamic loading conditions. The measured forces showed a positive correlation with the relative density. This phenomenon has also been described by [23]. Specimens with larger unit cell sizes tended to have greater peak forces, yet they also demonstrated a proclivity for collapse.

The weight-specific energy absorption (SEA) of Standard Resin gyroid specimens over the displacement is presented in Figure 6 for different unit cell sizes at a constant relative density. The weight specific energy absorption SEA is defined as the energy absorption E_abs divided by the total mass of the specimen m_tot before testing:

$$SEA={\displaystyle \frac{E_{abs}}{m_{tot}}}.$$

(7)

A different approach to determining the specific energy absorption is to use only the mass that is involved in the absorption of energy by weighing the specimen before and after testing [34]. In this paper, the first presented approach was used because the force attacked the full surface of the specimen and the full body of the specimen was involved in absorbing the energy. The specific energy absorption increased with increasing relative density. The specimens with smaller unit cells demonstrated a consistent specific energy absorption gradient until the onset of densification. In contrast, specimens with larger unit cells exhibited a higher specific energy absorption in the initial 10% of compaction, followed by a near plateau phase, with minimal specific energy absorption gain.

A closer look at the force–displacement curves of the Standard Resin specimens of a relative density of 7% and 12% is given in Figure 7a,b. Here, it can also be seen that the structures with a larger unit cell size incorporated a greater stiffness, since the elastic slope decreased with decreasing unit cell size, due to the greater wall thickness. In Figure 7c,d, the SEA over displacement of the same specimens is shown. Due to the greater stiffness of the larger unit cells, the initial SEA was greater compared to the smaller unit cell sizes. However, due to the drastic force drop initiated by a first layer collapse or global failure, the specimens with smaller unit cell sizes showed a greater SEA after the initial peak force.

The Standard Resin gyroid specimens tended to exhibit brittle failure, particularly under dynamic loading. A layer-by-layer collapse of the Standard Resin structure cannot be identified in the force–displacement curves, as was reported in [20,21,22]. This can be explained by the brittle behavior of the Standard Resin structure.

3.3. Polyurethane Rigid 650

The force–displacement curves of the Polyurethane Rigid 650 gyroid specimens are presented in Figure 8a for a constant relative density of 23% and varying unit cell sizes under quasi-static and dynamic loading. The curves do not exhibit a distinct initial peak force and a subsequent force drop to the plateau force. The plateau was characterized by minimal force fluctuations, implying the gyroid structure failed in a holistic manner rather than in a layered fashion. This indicates a bending-dominated behavior of the structure of this material. The densification process commenced at displacements between 25 mm and 35 mm. With this material, the force levels increased with increasing unit cell size. However, the discrepancy between a unit cell size of 7 mm and 10 mm was considerably greater than that between a unit cell size of 10 mm and 14 mm. The behavior under dynamic loading showed an increase in the force for all tested unit cell sizes, which was in contrast to the other materials under investigation.

The specific energy absorption over the displacement of Polyurethane Rigid 650 gyroid specimens is presented in Figure 8b. Here, a linear increase in the specific energy absorption can be observed for all specimens with different unit cell sizes, after a first area with near to no energy absorption over the displacement. The specimen with the highest specific energy absorption rate was identified as the one with a unit cell size of 14 $\mathrm{m}$$\mathrm{m}$ under dynamic crushing conditions. In general, the energy absorption and energy absorption rate were higher for dynamic loading than quasi-static loading. The energy absorption curve of the specimen with a 7 $\mathrm{m}$$\mathrm{m}$ unit cell is situated in close proximity to the curves of specimens with the other unit cell sizes but under quasi-static loading.

The elastic behavior of the Polyurethane Rigid 650 gyroid specimens under compressive loading is shown in Figure 9. The stretching of the cell walls in two directions perpendicular to the load can be identified. The high elongation at break enabled the specimens of this material to be compressed to a higher degree of compaction without global failure.

Furthermore, the Polyurethane Rigid 650 gyroid specimens demonstrated an elastic response. Following the removal of the load, the specimens tended to return to their original form, provided that no plastic damage had occurred to the material. Specimens with a unit cell size of 14 mm or greater failed under the propagation of a crack in a plane parallel to the loading direction, as described in [24] as failure mode III.

3.4. Nylon 11

The force–displacement curves of the Nylon 11 gyroid specimens are displayed in Figure 10 for the unit cell sizes of 14 $\mathrm{m}$$\mathrm{m}$ and 20 $\mathrm{m}$$\mathrm{m}$ for the different relative densities. The curves for quasi-static loading are comparable to those observed in the Polyurethane Rigid 650 specimens: No initial peak force is identifiable, a slightly increased force plateau is observed, followed by a densification process that sets in between displacements of 25 $\mathrm{m}$$\mathrm{m}$ and 35 $\mathrm{m}$$\mathrm{m}$. Once more, this was a regime of a bending-dominated behavior. Nevertheless, the force–displacement curves of the Nylon 11 specimens under dynamic loading follow the curve of the specimens under quasi-static loading only until the end of the elastic slope. Here, an increase in the force compared to the quasi-static loading can be identified. This is followed by a significant decline in force to a value near zero. The specimens exhibited a sudden and catastrophic failure under the sudden application of compressive pressure. This indicates a stretching-dominated behavior. The Nylon 11 gyroid specimens exhibited a brittle behavior under dynamic loading, whereas they exhibited ductile behavior under quasi-static compression loading. This is also reflected in Figure 11, which presents the specific energy absorption over the displacement of the Nylon 11 gyroid specimens of different relative densities and with the unit cell sizes of 14 $\mathrm{m}$$\mathrm{m}$ and 20 $\mathrm{m}$$\mathrm{m}$. The specimens subjected to quasi-static loading can be characterized by a near linear increase in specific energy absorption with displacement. The specimens subjected to dynamic loading exhibited a linear increase in energy absorption until approx. 5 $\mathrm{m}$$\mathrm{m}$. Thereafter, the energy absorption rate underwent a precipitous decline.

The failure behavior of Nylon 11 gyroid structures is presented in Figure 12. Similarly to the force–displacement curve, the elastic behavior was nearly identical under quasi-static and dynamic loading. However, once the maximum force was reached, the specimen subjected to dynamic loading collapsed globally due to restricted deformation (d), whereas the specimen subjected to quasi-static loading failed with a ductile mechanism due to the successive collapse of the layers (b).

The force–displacement curve of the specimens under quasi-static loading also allows the detection of the layer-by-layer failure of the structure. This collapse of the layers can be observed in the waveform in the plateau region, particularly in Figure 10c. Additionally, a distinction can be made between the 3.5 and 2.5 unit cells. The specimen with a smaller unit cell size had more waves or dents in the force–displacement regime. The failure of two layers of the specimen with 2.5 unit cells per side can be identified in the force–displacement curve.

The Nylon 11 gyroid specimens showed minimal variation in response to changes in unit cell size. The specimens with the two unit cell sizes investigated here showed a near identical force–displacement and therefore energy absorption–displacement regime. It is also noteworthy that the specimens subjected to dynamic loading exhibited enhanced specific energy absorption capabilities with a smaller unit cell size.

3.5. Comparison of Materials

The previously presented results of the compression and crushing tests of the different gyroid structures of different materials can be compared using the weight-specific energy absorption SEA, which is defined as the energy absorption E_abs divided by the total mass of the specimen m_tot before testing, and the plateau force F_plateau, which is defined as the average force between a displacement of 10 $\mathrm{m}$$\mathrm{m}$ and 20 $\mathrm{m}$$\mathrm{m}$, according to the German Norm DIN 50134 for compression tests of cellular materials [35]. In order to exclude the influence of the densification phase of the specimens, and because a compaction to such high strains was not desired, the specific energy absorption was analyzed at a compaction of 50%, which was equivalent to a displacement of 25 $\mathrm{m}$$\mathrm{m}$ of the investigated specimens:

$$SE{A}_{50\%}={\displaystyle \frac{E_{abs,50\%}}{m_{tot}}}.$$

(8)

A high plateau force is indicative of an effective energy-absorbing structure. The plateau force for the Standard Resin and Nylon 11 specimens for the relative density ((a) and (c)) and for Polyurethane Rigid 650 specimens for the unit cell size (b) is shown in Figure 13. The Nylon 11 specimens with high relative density under quasi-static loading exhibited the highest plateau force, due to their ductile behavior. In contrast, the same specimens under dynamic loading exhibited much lower plateau forces, with values below 2 $\mathrm{k}$$\mathrm{N}$, due to their brittle behavior.

The specific energy absorption until 50% compaction of all gyroid specimens tested is listed in

Table 3. Weight-specific energy absorption at 50% compaction of Standard Resin gyroid specimens under quasi-static and dynamic loading.

Standard Resin
SEA at 50% Compaction/$\mathbf{k}$$\mathbf{J}$/$\mathbf{k}$$\mathbf{g}$			Unit Cell Size/mm
			7	10	14	20
Relative density/%	7	QS	1.75 ± 0.16	1.54 ± 0.47	1.18 ± 0.21
	12	QS	2.24 ± 0.43		1.68 ± 0.17	1.05 ± 0.38
	17	QS	3.43 ± 0.35	3.40 ± 0.16	2.35 ± 0.54	1.42 ± 0.29
	23	QS	3.66 ± 0.36	4.40 ± 0.32	2.72 ± 0.42
	12	Dyn	1.41 ± 0.52	0.75 ± 0.42	0.73 ± 0.21	0.38 ± 0.27
	17	Dyn	2.50 ± 0.41	1.53 ± 0.51	0.89 ± 0.33	0.45 ± 0.38
	23	Dyn	1.84 ± 0.83	0.93 ± 0.50	0.44 ± 0.26	0.40 ± 0.20

(Standard Resin),

Table 4. Weight-specific energy absorption at 50% compaction of Polyurethane Rigid 650 gyroid specimens under quasi-static and dynamic loading.

Polyurethane Rigid 650
SEA at 50% Compaction/$\mathbf{k}$$\mathbf{J}$/$\mathbf{k}$$\mathbf{g}$			Unit Cell Size/mm
			7	10	14	20
Relative density/%	23	QS	1.34	2.00	3.15	2.85
	23	Dyn	2.15	3.06	4.57	4.27

(Polyurethane Rigid 650), and

Table 5. Weight-specific energy absorption at 50% compaction of Nylon 11 gyroid specimens under quasi-static and dynamic loading.

Nylon 11
SEA at 50% Compaction/$\mathbf{k}$$\mathbf{J}$/$\mathbf{k}$$\mathbf{g}$			Unit Cell Size/mm
			14	20
Relative density/%	7	QS		1.48
	12	QS	3.40	4.17
	17	QS	5.57	5.96
	23	QS	7.03	7.96
	7	Dyn		0.67
	12	Dyn	1.24	1.07
	17	Dyn	2.44	1.53
	23	Dyn	2.40	1.66

(Nylon 11). A clear trend of increasing SEA with relative density can be identified for all materials and unit cell sizes. The smaller relative densities incorporated a smaller wall thickness of the gyroid structure. This may have caused local buckling of the thin sheet network and therefore resulted in the collapse of a layer. However, the gyroid structures with smaller unit cell sizes and therefore with smaller wall thicknesses performed better than gyroid structures with larger unit cell sizes. This was due to the higher plateau force and the constant interaction of the layers with the loading. A collapse of a small layer only caused a minimal drop in force compared to the larger decrease in force with larger unit cell sizes.

The specimens subjected to dynamic crushing conditions exhibited inferior performance compared to those subjected to quasi-static loading, with the exception of the Polyurethane Rigid 650 specimens, which exhibited the opposite behavior. Additionally, in contrast to the gyroid specimens of the other materials, the influence of the unit cell size was inverted. This may be attributed to the substantial amount of residual liquid resin that was enclosed in the inner structure of the specimens, which negatively influenced the weight-specific energy absorption. However, the two best specimens of Polyurethane Rigid 650 performed similarly to the best energy-absorbing gyroid configuration of Standard Resin under dynamic crushing conditions. A graphical depiction of the data from the Table 3, Table 4 and Table 5 is presented in Figure 14.

It was anticipated that an increase in energy absorption with relative density would result from the presented correlation in Equation (3). However, an increase in weight specific energy absorption was not anticipated, due to the values being normalized to their mass. This overachievement reduces the significance of small relative densities for lightweight applications.

4. Conclusions

In the present study, different polymeric cubic gyroid specimens of the materials Standard Resin, Polyurethane Rigid 650, and Nylon 11 with an edge length of 50 mm, unit cell sizes of 7 mm, 10 mm, 14 mm and 20$\mathrm{m}$$\mathrm{m}$, and relative densities of 7%, 12%, 17% and 23% were subjected to quasi-static compression and dynamic crushing conditions. The Standard Resin gyroid structures demonstrated a correlation between the relative density, unit cell size, and loading rate. The structure exhibited brittle behavior under dynamic loading, whereas the smaller unit cell demonstrated a greater SEA. Moreover, under dynamic loading, the SEA was not affected by the relative density, in contrast to under quasi-static loading. The unit cell size of the Polyurethane Rigid 650 gyroid specimens with constant relative density exerted a significant influence on the SEA. In contrast to the other materials that were tested, the SEA capability was observed to improve under dynamic loading in comparison to quasi-static loading. The Nylon 11 gyroid specimens exhibited a pronounced dependence on the loading rate. The specimens exhibited a ductile failure mechanism under quasi-static loading and a brittle failure mechanism under dynamic loading. Nevertheless, the unit cell size had a less pronounced impact on the SEA than the loading rate. Comparing the SEA data, the best energy-absorbing gyroid configuration under dynamic loading was that of the Polyurethane Rigid 650 material with a relative density of 23% and a unit cell size of 14 $\mathrm{m}$$\mathrm{m}$.

This study provides an overview of the mechanical behavior of three distinct polymeric additively manufactured uniform cubic gyroid structures subjected to quasi-static and dynamic compressive loading. A profound understanding of gyroid structures under dynamic loading is essential for the development of a novel wing leading edge design that integrates lightweight structures with an HLFC concept for enhanced airborne efficiency. It is of paramount importance to understand and maximize the weight specific energy absorption of sheet network gyroid structures for the realization of a novel multi-functional leading edge design that can improve energy efficiency and reduce carbon emissions for airborne transport.