Enhancing Compressive Performance of Gyroid Structures Through Evolutionary Design ^†

Abstract

Gyroids are fascinating natural structures characterized by their local minimum surface area with equal periodicity in all three spatial directions, making them continuous and isotropic. Such intricate patterns have led to various lightweight innovative constructions. This article proposes a novel approach to enhance their compressive performance via edge modification, taking inspiration from biomimicry, specifically following a plant-growth algorithm. Later, a patch surface was generated using these edges, while maintaining the same aspect ratio. The 3D-printed prototypes were tested under compressive load and further validated using FE analysis. The results provide good evidence for modified gyroids being superior, as compared to conventional gyroids.

1. Motivation

Gyroids belong to a fascinating family of structures known as triply periodic minimal surfaces (TPMSs) and are of significant interest to physical scientists, biologists, and mathematicians due to their unique properties. A minimal surface, such as a soap film, is characterized by its local minimum surface area. This means that any small section of the surface will have the smallest possible surface area among all surfaces with the same boundaries. Thus, a minimal surface will have a zero-mean curvature. The term ‘triply periodic’ refers to the equal periodicity in all three spatial directions, which makes these surfaces continuous and isotropic. Over the past century, numerous surfaces have been classified within the TPMS family, and they can be found in a wide range of systems, such as block copolymers, nanocomposites, micellar materials, lipid–water systems, cell membranes, etc. [1,2]. Gyroids, one of the family of TPMSs, exist in various biological structures, e.g., butterfly wing scales, bird feathers, etc., [3,4,5] where depending on the occurrence, the intricate patterns of gyroid structures provide characteristic colors, strength, flexibility, and necessary insulation.

Although the mathematical model of the gyroid structure was developed back in the 1970s [6], the complexity of the structure hindered its practical applications in engineering. However, with the recent developments in Additive Manufacturing (AM) technologies and design software, engineers are now able to design and manufacture such geometries up to intricate details [7]. As a result, numerous studies have demonstrated the viability of gyroid structures as a superior alternative for applications requiring high energy absorption, mechanical strength-to-weight ratio, permeability, etc. This has led to the development of innovative applications, including lattice structures in sandwich panels for automotive and aerospace industries, as well as various biomedical applications such as bone implants, intervertebral devices, etc. [8].

Here, the objective was to enhance the compressive performance of such structures while keeping the basic aspect ratio intact. This is achieved by decomposing the gyroid surface into its basic constituent surface element and then rebuilding the surface through edge modification following biomimicry, specifically following a plant-growth algorithm [9]. The modified gyroid was then reconstructed via reverse engineering, and subsequent tests were conducted to evaluate the performance of such a pseudo-gyroid geometry in contrast to a conventional gyroid structure generated under similar conditions.

2. Design Methodology

To extract the basic constituent surface of a gyroid unit cell, geometrical operations inside the Grasshopper 1.0.0008 parametric tool within the Rhino 5 solid modeling domain were conducted. Figure 1 and Figure 2 describe the step-by-step deconstruction process.

Once edges were extracted as different curves, the next step was to modify these curves. For this, an analogy between the auxin–light relationship and force–form relationship was evaluated and an evolutionary programming was assigned to find the optimal forms of the curves under planar loads. The methodology, briefly described in Figure 3, has been established in previous papers [9,10], and this work serves as an extension. Once all edges were redefined, a new patch surface was generated using reverse engineering, as shown in Figure 4.

3. Materials

Once surfaces were generated, a computational FE analysis replicating a compressive loading scenario was performed using a Karamba 3D plug-in within the parametric modeling domain with identical isotropic material definition. The base of the surfaces was constrained in all directions, while representative compressive impact loads were applied on the top, and the corresponding scaled mesh deformation factor ξ [9] was calculated as illustrated in Figure 5.

Surface FE simulations, as shown above, revealed that the modified surface outperformed the standard gyroid surface under the specified boundary conditions. To confirm these findings, identical thicknesses were assigned to both surfaces, and various prototypes were generated via 3D printing. Two distinct printing processes were employed, utilizing different materials:

• ▪ For FDM [11] printing, standard PLA [12] material was used.
• ▪ For SLA [13] printing, Formlabs Rigid [14] resin was used.

The cube’s dimensions bounding the generated geometry were also varied, with different scales and infill percentages. In some test cases, printed plates were attached to the top and bottom of the geometries to ensure even load distribution. Additionally, the unit cells within the bounding volume were modified in certain instances. For simplicity, gyroid and modified gyroid geometries were coded as G and M, while each variation was assigned a unique number. The ranges of these variations are summarized in

Table 1. Detailed description of prototypes printed for compression testing with corresponding variations.

Variations	d	% Fill	Unit Cell Combinations	Support Plates	Printing Method/Material
G_1	30 mm	10	1 × 1 × 1	No	FDM/PLA
M_1	30 mm	10	1 × 1 × 1	No	FDM/PLA
G_2	30 mm	10	1 × 1 × 1	Yes	FDM/PLA
M_2	30 mm	10	1 × 1 × 1	Yes	FDM/PLA
G_3	30 mm	10	2 × 2 × 2	No	FDM/PLA
M_3	30 mm	10	2 × 2 × 2	No	FDM/PLA
G_4	40 mm	15	1 × 1 × 1	Yes	SLA/Formlabs Rigid
M_4	40 mm	15	1 × 1 × 1	Yes	SLA/Formlabs Rigid
G_5	20 mm	30	2 × 2 × 2	Yes	SLA/Formlabs Rigid
M_5	20 mm	30	2 × 2 × 2	Yes	SLA/Formlabs Rigid

.

4. Results

The results of the compression tests as documented in Figure 6 and Figure 7 below reveal that for each test case, the modified gyroid structures absorb higher load at failure compared to their conventional counterparts.

5. Summary

The results of the compression tests demonstrate that the modified pseudo-gyroid structure performs better than its conventional counterpart in all test scenarios. A closer examination of the results also confirms the assumption that the application of support plates improves the structure’s performance under compression loads. While the modification process yields an improved geometry for withstanding compression, it is essential to note that the resulting surface deviates from the continuous, smooth nature of a typical gyroid surface. Thus, it is suitable to refer to them as pseudo geometries, given that a discrete surface of this nature does not occur naturally. Nevertheless, the method described here can be a valuable tool for generating high-performance geometries tailored to specific applications and manufacturing methods, such as additive manufacturing.