1 Introduction

3D printed concrete (3DPC) is an additive manufacturing (AM) method that offers new opportunities in the building construction industry to reduce onsite labor costs and onsite construction time, and minimize job site health and safety risks (Ko and Kuo 2019). Compared to cast concrete structures, 3DPC can be used to create complex, free-form structural elements quickly and accurately without the use of formwork at a range of scales (Xiao et al. 2021). Such elements can be structurally informed through various methods of optimization (Plessis et al. 2021) that can reduce the total amount of volume printed while maintaining equal or similar critical load capacity as unoptimized geometries. However, 3DPC is primarily used in the design of compressive-only structures because of the material’s low tensile strength, as exhibited in many cementitious and sedimentary structures that do not possess additional reinforcement. Furthermore, it is difficult to integrate vertical rebar systems (for added tension) into the printing process because of the logistical challenge of the extruding end-effector colliding into any rebar higher than the current print layer. As such, 3DPC buildings and building components are often vertical or near vertical extrusions of the first layer of deposited material with very small-to-no overhang between layers.

The limits of the overhang are not only affected by the properties of the extruded material, but also by the geometry being printed and its tool path design. For example, it is easier to print a doubly curved dome as compared to a canted wall with the same overhang angle, because doubly curved shells evenly distribute all compression and tension forces through the shell’s cross section, thereby eliminating almost all bending moments by sustaining the external and gravity loads through membrane stresses (Joedicke 1963). Biologically, double curvature can be found in cellular solids, such as wood, sponge, honeycomb, trabecular bone, and foams, all of which can be found throughout nature, but also manufactured through primarily isotropic materials, such as polymers, metals, ceramics, and glass (Gibson 2003). Structures with these materials are often derivative of triply periodic minimal surfaces (TPMS), which can often be further derived from various polyhedrons. TPMS are non-self-intersecting surfaces that have three lattice vectors that remain invariant (unchanged) per each translation vector. Because most TPMS have a polyhedral base (Sadoc and Charvolin 1989), they can easily be parameterized as clusters (Al-Ketan and Abu Al-Rub 2021). Depending on the cellular structure as well as the mechanical properties of the material, these types of cellular solids are typologically either bending-dominant structures or stretch-dominant structures (Gibson et al. 2010). The gyroid is a bending-dominant TPMS that exhibits high stiffness relative to the unit cell (bounding box or volume fraction) when subjected to compression loading (Downing et al. 2021; Maskery et al. 2018). Whereas stretch-dominant TPMS exhibit high stiffness when subjected to tension loading. Compared to other TPMS typologies, a gyroid can more effectively distribute compression loading from each Cartesian orientation of the cubic unit cell (Maskery et al. 2018). Despite the gyroid’s geometric ability to efficiently distribute compressive loads, it has yet to be broadly applied in 3DPC building applications. In part, this is due to the difficulty of effectively printing the steep sinusoidal overhangs, inherent to the doubly curved, cellular nature of the gyroid TPMS (Downing et al. 2021). Consequently, the employment of the gyroid has been limited to small and nano-scale AM applications such as tissue engineering, thermal management devices, and soft robotics (Al-Ketan and Abu Al-Rub 2021).

To cope with the overhang issue inherent to gyroid geometries, the researchers of this paper have (a) developed a method to parametrically define the gyroid TPMS through Non-Uniform Rational Basis Spline (NURBS)-based modeling which was used to (b) conduct a series of geometric density studies and (c) develop a continuous tool path. The continuous tool path was necessary, because (d) the mortar pump used for experimentation in this setup cannot be paused during the printing process to prevent unwanted hardening of material inside the end-effector. Additionally, the NURBS-based modeling method was used to (e) parametrically modify the degree of overhang for layer-by-layer printing in a series of physically prototyped overhang tests. One of the overhang prototypes was duplicated three times for (f) a series of compression tests. Given the material constraints and equipment defined in the Methods section of this paper, this research demonstrates a method to additively manufacture gyroid concrete structures with a continuous tool path.

2 State of the art

AM with concrete is often used in the design and construction of compressive-based structures, such as walls, columns, or vaults because of the material’s inability to span large distances. Naturally, it is difficult for 3DPC assemblies to exhibit steep overhangs or cantilevers during the layer-by-layer printing process because of both its mechanical and rheological properties. This often limits 3DPC to vertical (or near vertical), extruded printed structures, especially in full-scale built projects. Additionally, the integration of rebar cages into these systems demonstrates an added logistical challenge during the extrusion process. However, starting in 2014, there is precedent by the authors and others to use 3DPC as formwork for cavities that are then filled with concrete rebar cages (Paul et al. 2018; Jipa and Dillenburger 2022; Lok et al. 2024). Additionally, others have begun to integrate tension cables into prefab 3DPC post-fabrication (Vantyghem et al. 2020; Akbarzadeh et al. 2024). Despite these advancements of casting concrete into 3DPC or post-tensioning in 3DPC, AM with concrete is limited by the degree of overhang and cantilever that is possible during the printing process. In a series of prefabricated bespoke columns, a 20° maximum limiting overhang was demonstrated based upon a series of fabrication tests (Anton et al. 2021). Research by others have demonstrated that concrete corbelled arches can be printed with a 25° overhang (Duarte et al. 2021). AM with concrete does not always need to extrude the material perpendicular to the ground plane or the layer below; rather, it can also be extruded at an angle based on the friction coefficient between extruded layers. In the fabrication of the 3DPC masonry arch bridge, Striatus, the researchers determined a maximum overhang angle of 25° between the first and last layers printed to be the limiting angle (Dell’Endice et al. 2023). Using a similar robotic print method, others have demonstrated that corbelled arches can be printed with a cementitious material with a general overhang angle of 37° and a local maximum overhang of 60° at the top of the arch (Carneau et al. 2019). The same group of researchers also demonstrated that 45° overhangs could be printed for domes with a square base as part of their investigation into printing Nubian vaults with 3DPC (Carneau et al. 2019). In additive manufacturing, Nubian vaults are created by printing a mound of material to create a slope (around 30°–45° off of the horizontal) from which subsequent extruded material can be continuously added to create a barrel vault, and thus, each layer of the vault is supported by the previous layer and the catenary arch shape the current layer (Duarte et al. 2021; Carneau et al. 2019; Darweesh et al. 2024). A gyroid with uniform x, y, and z scaling has an overhang of 45°, making it difficult and previously unreported to be constructed in concrete AM (Liu et al. 2022). Another TPMS, the Schwartz-D, or Diamond, discovered by Alan Schoen, was recently 3D printed as a series of prefab 3DPC sections that were post-tensioned into a concrete canopy, Diamanti, with a maximum overhang angle of 35° (Akbarzadeh et al. 2024).

As mentioned before, the gyroid has some geometric structural advantages (Gibson 2003; Gibson et al. 2010; Downing et al. 2021; Maskery et al. 2018) making it an attractive investigation for AM with concrete. To address the overhang issue of the gyroid geometry, others have 3D printed PLA molds and then cast concrete to develop a concrete gyroid (Nguyen-Van et al. 2020). Additionally, researchers at Autodesk have used fiber-reinforced polymer (FRP), to 3D print a cantilevering bridge with a gyroid infill (Wakefield 2022). In addition, others have used parametric models to simulate geometric density tests for various TPMSs by changing the thickness of the digital mesh through gradient transformations (Al-Ketan and Abu Al-Rub 2021; Zhang et al. 2017). Changing the thickness of the gyroid mesh assumes that one can calibrate the control of material flow rate relative to the travel speed of the nozzle. In the research presented in this paper, these variables were left constant.

3 Materials and methods

The gyroid and other similar cellular solids can be found in nature as efficient, bending-dominant structures (Plessis et al. 2021; Gibson 2003). Because the gyroid TPMS effectively distributes stress under compression loading (Downing et al. 2021), this research both investigates and presents the methods whereby the gyroid geometry can be employed in AM with concrete. Section 3.1 describes a surface-based modeling method used to construct a single gyroid unit. This method was then applied to a series of geometric density studies based on the dimensions of the toolpath (Sect. 3.2). Next, in Sect. 3.3, the design of a continuous tool path was based on the materials and equipment described in Sect. 3.4, to conduct a series of overhang tests (Sect. 3.5). One of the samples from the overhang tests was reproduced in a group of three for compression testing; the findings of such are reported in the results of the paper.

3.1 NURBS-based gyroid development

Alan Schoen developed the following trigonometric equation (Eq. 1) to approximate a gyroid (Schoen 1970):

$$\text{sin}x\text{cos}y+\text{sin}y\text{co}z+\text{sin}z\text{cos}x=0.$$
(1)

Equation 1 can be used to generate an isosurface in which the resulting gyroid is uniform in all three Cartesian directions: x, y, and z (Downing et al. 2021; Li et al. 2022). Isosurfaces are derived from weighted points in space, such that points with smaller values are culled and points with the highest values are used to define a mesh. The accuracy of the isosurface is determined by the density and number of points within a given volume. Because the mesh developed from the isosurface is an approximation, designing through isosurface methods can limit the number of variables, one can use to parametrically modify the gyroid. Such variables might include non-uniform scaling of the geometry in one direction, twisting, or bending the geometry, or joining the gyroid geometry to other digital geometries. However, others have used the isosurface method to parameterize a change in mesh thickness for gyroid TPMSs (Viswanath et al. 2022). Due to the limitations produced through the isosurface methods, it is better to develop the gyroid through individual NURBS curves and surfaces that can be parametrically arrayed and transformed by an associated local Cartesian cube (unit cell or bounding box). Additionally, designing through surfaces guarantees a uniform mesh construction for conducting a linear finite-element analysis (FEA) (Flores-Jimenez et al. 2022). Figure 1 illustrates the differences in gyroid TPMS meshes modeled through isosurface methods vs NURBS-based modeling methods. The advantages of modeling gyroid TPMSs through NURBS as opposed to isosurfaces have also been reported by others (Flores-Jimenez et al. 2022; Yabanigül and Yazar 2021). However, both papers describe methods that employ a “fundamental patch” to fit an approximate NURBS surface to a region of the gyroid by which the rest of the gyroid can be derived through geometric transformations. This “fundamental patch” is the smallest portion of the TPMS from which the whole gyroid can be constructed using Weierstrass parameterization (Flores-Jimenez et al. 2022; Gandy and Klinowski 2000). The fundamental patch was the base portion of a circum-helix around the central axis of the gyroid TPMS from which a warped mesh grid of 20 quadrilaterals and 5 triangles was used to compute the exact fundamental patch (Gandy and Klinowski 2000). The fundamental patch method described by Gandy and Klinowski illustrates a near exact method of representing the gyroid TPMS through 96 fundamental patches, totaling in 960 mesh faces. However, if the fundamental patch can be described through a single NURBS surface, then the resulting gyroid TPMS can be more accurately modeled through 48 surfaces, and the derivative mesh can be any number of mesh faces according to one’s desired mesh resolution. The naked edges of the mesh derived from NURBS-based modeling would all be planar on one of the bounding box faces; this would not inherently be the case using an isosurface modeling method. Thus, the naked edge of the mesh derivative from an NURBS-based geometry will always be more refined than those derivative of isosurface modeling methods. Section 3.3 illustrates the use of other digital geometry that are joined to the NURBS-based gyroid TPMS to create a continuous toolpath. Joining such geometry together would not be so feasible or accurate had the gyroid been created through isosurface methods leading to patches in the geometry and thereby potential gaps in the resulting tool path.

Fig. 1
figure 1

Gyroid geometry through isosurface modeling methods (left) vs. NURBS modeling methods (right)

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A gyroid TPMS can be constructed through x, y, and z sine curves. A single gyroid unit has a wavelength of 2 \(\pi\) in x, y, and z directions. Thus, an eighth unit of the gyroid has a length of \(\pi\). Figure 2a describes how the base curves of a gyroid unit can be constructed through the following equations in terms of \(\pi\):

Fig. 2
figure 2

Gyroid geometry creation through NURBS modeling

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$$\left[x,\text{ sin} x, 0\right]=x\text{ curve}$$
$$\left[0,y,\text{sin}y\right]=y \text{curve}$$
$$\left[\text{sin}z, \pi , z\right]=z \text{curve}.$$
(2)

The above equations were used to define points in space, which were then used to create interpolated curves (Fig. 2a) using Grasshopper, an algorithmic modeling and coding interface for Rhino3d (Rutten 2022; Robert McNeel Associates 2022). The resulting curves are then rotated around the bounding box center by 180° (Fig. 2b) and then mirrored along the xy plane at the center of the bounding box (Fig. 2c). Finally, a series of arcs are drawn perpendicular to the center of each of the curves shown in Fig. 2c following a cosine function (Fig. 2d). This is the base eighth unit of the gyroid has an x, y, and z length of \(\pi\) and is composed of 6 NURBS surfaces. Here, two of the fundamental patches described previously would equate to 1 NURBS surface. The eighth unit of the gyroid (Fig. 2e) can be rotated and reoriented to complete the full unit of the gyroid (Fig. 2i). The eighth unit of the gyroid was copied in the x direction at a distance of \(\pi\) and rotated 180° around the xy plane at its center (Fig. 2f). The resulting two-eighth units were copied at a distance of \(\pi\) in the y direction and rotated 180° around the xz plane at its center (Fig. 2g). Finally, the now bottom half of the full gyroid unit was copied at a distance of \(\pi\) in the z direction and reflected along the yz plane at its center (Fig. 2h), thus completing the gyroid full unit (Fig. 2i). The resulting gyroid has an x, y, and z length of 2 \(\pi\), and this geometry can be arrayed in all axes seamlessly joining with adjacent units. Through NURBS modeling, the single gyroid unit is composed of 48 surfaces and any meshes derived from these NURBS surfaces can have exclusively quadrilateral mesh faces.

3.2 Geometric density studies

The cross section of each layer of the extruded 3DPC was determined to be 28 mm wide by 7 mm thick based on the extruding nozzle diameter. With these layer dimensions, a series of geometric density studies were conducted within a 500 mm by 500 mm by 500 mm bounding box. Figure 3a shows one gyroid unit (2 \(\pi\)) within the specified bounding box. Figure 3b, c shows two and three gyroid units, respectively, within the same bounding box. The two-unit gyroid structure was selected for further investigation.

Fig. 3
figure 3

Gyroid geometric density studies within a 500 mm by 500 mm by 500 mm bounding box: a one gyroid unit—2 \(\pi\); b two gyroid units—4 \(\pi\); c three gyroid units—6 \(\pi\)

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3.3 Toolpath design

For AM, concrete extruders do not often have built-in stop/start function between the pump and the end-effector, because of factors related to residual pressure overflow within hoses, changes in curing times, or the auger’s relation to the accelerator injection. Due to these challenges, especially as they relate to multiple short line segments, a continuous tool path between print layers was developed for gyroid-based prints with multiples of two sine curves similar to that of previous research by the authors (Conrad et al. 2023). The continuous tool path was developed by adding additional vertical NURBS surfaces (magenta and pink in Fig. 4) to the NURBS-based gyroid (blue in Fig. 4). Figure 4 demonstrates the continuous tool path from horizontal contouring of the digital model through three critical sections within the proposed gyroid prototype. Layers 5 and 6 show a change in the tool path direction and similarly, so do layers 12 and 13. Layers 17 and 18 demonstrate a potential issue in the printing process where a 28 mm cantilever was added to the tool path in Layer 18 to make the path continuous. The added surfaces to make the toolpath continuous made up 45% of the total toolpath length for each overhang prototype. The curves along the tool path were divided into line segments no shorter than 20 mm apart. The end points of each segment were used to define the robot targets in the RAPID code.

Fig. 4
figure 4

Toolpath design for gyroid with 45° overhang (2 \(\pi\)) geometry

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3.4 Materials and equipment

A six-axis robotic arm, ABB Robot Model IRB 6650S (3.9 m reach, 90 kg handling capacity) on a 3.7 m linear external track, IRBT 6004 Track, and a XtreeE mortar extrusion head was used for printing the proposed gyroids, as shown in Fig. 5. A pumpable mixture with a mixture composition as shown in Table 1 and a flow diameter of 17 cm (± 1.5 cm) was prepared in a Mortarman Mixer. The mortar mixture was then pumped through XtreeE’s XFeed to achieve constant pressure at the top of the printing head of 4 bar (± 0.5 bar). The mortar was then pumped through the printing nozzle with an extrusion rate of 1.7 L/min (± 0.1 L/min). An accelerator was injected at the extrusion head at a rate of 8 mL/L of mortar and sheared prior to extruding the layer through a 20 mm nozzle. Each print had a tool path length of 325 meters and consumed 64 L of 3DPC mortar mix. A total of six batches of material were prepared for the overhang tests. Each overhang test was printed on a plywood base covered with a thick plastic sheet (Fig. 5).

Fig. 5
figure 5

Photo of robot printing concrete with XtreeE end-effector

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Table 1 3DPC mixture design
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3.5 Overhang tests

It is known that the early age mechanical properties of the mortar mixture along with the accelerator injection, and toolpath velocity will affect the limits of how much the deposited material can overhang. Additionally, the shape of the geometry also plays a role in the degree of overhang. Section 3.3 describes how additional surfaces can be added to the 500 mm by 500 mm by 500 mm gyroid TPMS which has a base sine curve wavelength of 4 \(\pi\) in x, y, and z directions (2 gyroid units in all cartesian directions). Based on this geometry, three overhang tests were designed by scaling the toolpath geometry in the z direction while maintaining a 500 mm by 500 mm bounding box. Figure 6a illustrates the no scaling in the z direction; thus, a 45° overhang angle is exhibited through the 3DPC gyroid prototype. Figure 6b, c demonstrates a 34° and 27° overhang angle relative to the vertical, respectively, by scaling the geometry in the z direction by multiples of 1.5 and 2.0. Each overhang test had 71 print layers for a total toolpath length of about 325 m and was printed at travel velocity of 153 mm/s with an average accelerator injection of 1.7 L/m. The extruding nozzle was perpendicular to the plywood base throughout the entirety of each print. The total print time per overhang test was estimated to be about 35 min.

Fig. 6
figure 6

Photos of overhang print tests: a gyroid uniformly scaled in x, y, and z directions resulting in a 45° overhang; b gyroid scaled by 1.5 in the z direction resulting in a 34° overhang; c gyroid scaled by 2.0 in the z direction resulting in a 27° overhang

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Figure 6a demonstrates that uniform scaling of the gyroid in all Cartesian directions will have a 45° slope, and when scaled in the vertical direction by 1.5 and 2, the gyroid will have a general slope of 34° and 27°, respectively. However, the general slope does not describe the overhang angle of all the mesh faces within the gyroid geometry. Draft angle analysis can be used to describe each angle of the mesh face relative to the vertical through a color gradient. If 0° is totally vertical and 90° is horizontal, then one can articulate locations of significant overhang between each of the gyroid prototypes through a color gradient (Fig. 7). In Fig. 7, 0°—totally vertical, is all the blue mesh faces and those mesh faces that are most horizontal are colored in red from the perspective of the worm’s-eye isometric—an isometric looking up as opposed to down. Each gyroid has a different percentage of mesh faces that are vertical verses horizontal. The angle of each mesh face was rounded to the nearest whole number. Those mesh faces that were totally vertical 0° were culled from the data set to examine only the overhang, 1°–90° and were tallied per each overhang angle (Fig. 8). The graph on Fig. 8 demonstrates that the uniformly scaled prototype had the most significant overhang in which 26% of the mesh faces had more than a 45° overhang. While only 13% of the mesh faces in the gyroid scaled by 1.5 had an overhang beyond 45° and 8% of the mesh faces in the gyroid scaled by 2.0 had an overhang beyond 45°.

Fig. 7
figure 7

Worms eye isometric of each gyroid prototype: a gyroid uniformly scaled in x, y, and z directions; b gyroid scaled by 1.5 in the z direction; c gyroid scaled by 2.0 in the z direction

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Fig. 8
figure 8

Draft angle analysis per gyroid prototype: describes the percentage of mesh faces per each overhang angle between 1° and 90° for each prototype

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4 Results

The research presented demonstrates an NURBS-based modeling method to digitally model gyroid TPMSs. The advantage of this method, as opposed to isosurface-based modeling methods, is that the mesh(s) derived from the NURBS-based modeling method are more uniformly constructed making it a more exact geometry than previously demonstrated by others and can easily be joined with other digital geometry or transformed through various parametric modeling methods. This is not so easy to accomplish with present isosurface modeling methods which can create non-planar naked edges, making it difficult to join with other geometries which also affects the horizontal slicing of the model. Additionally, isosurface modeling methods can create triangulated meshes making it more difficult to orient the first and second directions of orthotropic materials for FEA. In the case of the research presented in this paper, the NURBS-based modeling method was implemented to help construct a continuous toolpath for the gyroid TPMS prototypes.

The 3DPC, used for both the overhang tests and the prototypes for compression testing, maintains its pumping characteristics for at least 20 min without increasing pumping pressures. The in-house developed mixture can be categorized as a high-performance concrete with a compressive strength of 75 MPa at 7 days, 90 MPa at 28 days. With this 3DCP mixture and the extrusion head described in Sect. 3.4, the cantilever tests did demonstrate that a 45° cantilever gyroid TPMS could be printed, however two issues arose during the printing process and immediately following. As mentioned in Sect. 3.3, there were two locations where a 28 mm cantilever was required to keep the tool path continuous, which caused part of the printed prototype to collapse (Fig. 9a). Yet, the equivalent cantilever location on the 34° overhanging prototype only showed mild slumping (Fig. 9b). Additionally, the 45° overhanging gyroid prototype exhibited cracking in the concrete within a few hours after it had been printed (Fig. 9c). It is plausible that the cracking is mostly like the result of the steep overhang within that area. Because the 34° overhanging gyroid showed only mild slumping in the cantilever area and no visible cracking during the curing process, it was selected for compression testing. It should also be noted as it is illustrated in Fig. 6 that the 27° overhang test did not finish printing, because there was a miscalculation in the number of batches required to print all three overhang tests. This miscalculation was a product of the numerous extrusion tests done prior to printing the first overhang test.

Fig. 9
figure 9

Photos from 3DPC overhang tests: a 28 mm cantilever caused partial collapse of the test; b 28 mm cantilever caused minor slumping; c 45° overhang test exhibited cracking within hours of fabrication—fracture line is notated in red

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Based on the overhanging test, the 34° overhang prototype was selected to investigate the mechanical performance of gyroid cubes under compressive loading. Three identical 34° overhang prototype cubes were printed following the methods described in Sects. 3.4 and 3.5 for compression testing. The printed cubes had an average tool path length of 456 cm per layer and was 28 mm wide, and thus, the average cross section between layers was 1277 cm.2. The theoretical load capacity of the cubes (\({P}_{\text{th}}\)) was estimated considering the cube’s cross-sectional area (\({A}_{\text{c}}\)), and the material’s compressive strength (\({f}_{\text{c}}^{\prime}\)), as shown in crushing Eq. (3):

$${P}_{\text{th}}={A}_{\text{c}}{f}_{\text{c}}^{\prime}.$$
(3)

Using Eq. 3, the cubes were found to have a theoretical compressive load capacity of 9475 kN considering the material compressive strength at 7 days of 74.2 MPa. The cubes were moist cured and tested at age of 7 days using a 1780 kN Baldwin Universal Testing Machine with a loading rate of 0.5 mm/min (Fig. 10). The load–displacement curves for the tested gyroid cubes are shown in Fig. 11. The cubes were able to carry a maximum compressive load of 1050 kN, 1275 kN, and 1235 kN with a displacement at maximum load of 5.7 mm, 3.2 mm, and 4.5 mm, respectively. This indicates that the cubes were able to support 11–13% of their theoretical compressive load capacity. However, the theoretical compression capacity does not consider the stress distribution due to gyroid’s geometry. As shown in Fig. 10, the cracks run perpendicular to the direction of the direction of the applied load, thereby suggesting that the gyroid prototypes failed due to tensile stress or shear stress.

Fig. 10
figure 10

Photos of compression testing

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Fig. 11
figure 11

Results of compression testing

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When comparing the yield stress of the gyroid under compressive and tensile loads, the results are “quasi-identical” (Timercan et al. 2023) or near similar depending on the cellular infill density (Araya et al. 2024). However, in the case of the gyroid prototypes of this paper, 3DPC is the material, which has a high compression strength relative to its tension and shear strengths. According to the Poisson effect, concrete tends to expand laterally when subjected to axial compression loading. The three specimens subjected to compression loading in this paper contract between 3.2 and 5.7 mm. Unfortunately, no strain gauges were used to measure the lateral displacement of the specimens, so it is difficult to hypothesize on the exact relationship between axial and transverse strain for each of the gyroid prototypes tested. Furthermore, the interior of the gyroid geometry is quite complex, as illustrated in Fig. 12, making it rather complex to properly apply enough strain gauges to fully and empirically define locations of the high strain. Nonetheless, it is probable that the varying distribution of local tensile stresses throughout the gyroid prototypes are integral to the mostly vertical cracks in each of the specimens.

Fig. 12
figure 12

Section through isometric of 34° gyroid model for 3D printing

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Given the loading direction and the percentage of the vertical or near vertical surfaces in the gyroid cubes, it is unlikely that the cubes failed due to pure shear. However, it is possible that local shear failures contributed to the cracking and splitting of the 3DPC in each of the gyroid prototypes. To specify the types of local failures that lead to cracking and splitting in the 3DPC, future FEA is needed to study the stress distribution in 3DPC gyroid structures. The FEA investigation results can also be used for toolpath optimization.

5 Conclusion

The research presented in this paper describes a method to accurately model the gyroid TPMS, such that resulting geometry can easily be joined with NURBS-based geometries or meshes. Due to the fact that the XTreeE mortar extrusion end-effector does not have a start/stop function to pause the flow of the concrete being extruded, a continuous tool path was developed. However, the toolpath design did exhibit several areas in the overhang tests that had a 28 mm cantilever. In the 45° overhang sample, the cantilever existed in two locations, and in the 34° and 27° overhang samples, this condition presented itself twice, once in each sample. Due to the fact that the 27° overhang sample did not finish printing, the cantilevering condition was only exhibited in the digital geometry. Though the compression tests demonstrated that 34° could support up to 1280 times its own self weight, a significant variability was observed between the theoretical and measured values possibly due to internal cracking within each test.

Finite-element analysis will be used in future investigations to structurally simulate and optimize the gyroid geometry for larger 3DPC wall assemblies. Additionally, the toolpath design will be further refined to reduce or eliminate the issue of the 28 mm cantilever exhibited in Fig. 7a, b. Part of this future investigation will also look to develop a continuous toolpath for gyroid-based structures that does not reverse back on itself between layers. The current toolpath travels in opposite directions between odd and even layers. Potentially, layers will have more time to cure if the toolpath is designed to move in the same direction throughout the entirety of the print. Finally, other TPMS surface typologies will be investigated based on different loading scenarios and integration of rebar systems into the printed geometry. Further studies will also examine the probability of cracking during the curing period given various overhang angles to determine a more exact range of allowable overhang angles for doubly curved 3DPC structures.