Anisotropy in vascular bundle-inspired structures during in-plane compression

Bio-inspired metamaterials for mechanical applications offer exceptional structural capabilities, including high toughness, enhanced strength, toughness and energy absorption (Yin et al., 2021). Such metamaterials offer a wide range of applications in fields like soft robotics (shape-morphing and flexible electronics), biomedical implants (bone implants, vascular stents, biomimetic scaffolds, tissue engineering, etc.) and energy absorption/structural integrity (such as vibration damping and shock absorption (Zhao et al., 2023). Some of the bio-inspired structures used for metamaterials’ design include honeycomb, gyroid, spider web, bone, bamboo, vertex-offset, chiral, tissue, animals and plants, among others (Sharma et al., 2022). As demonstrated in references (Cetin, 2024; Atahan and Saglam, 2025; Ha and Lu, 2020; Karthikeyan, 2025) the bioinspiration from robust natural structures allows for the development of metamaterials with exceptional mechanical characteristics, primarily targeting lightweight design and enhanced energy absorption. A critical property offered by these designs is a significant improvement in crashworthiness and impact resistance compared to conventional engineering structures, making them vital for aerospace and transportation applications. These bio-inspired metamaterials often feature complex, multi-scale configurations, such as hierarchical patterns, porous foams and multi-cell reticulated shells, which are precisely fabricated using additive manufacturing techniques. Specific biological structures, including the beetle’s wing, nacre and pomelo peel, provide blueprints for creating designs optimised for high damage tolerance and structural integrity. Furthermore, lattice geometries inspired by the excellent load-carrying capacity of ants or thin-walled tubes featuring DNA-like helical ribs are employed to maximise critical performance metrics. Consequently, these innovative, bio-inspired approaches offer high specific stiffness, superior specific energy absorption (SEA) and enhanced crash force efficiency (CFE) for next-generation structural components. As such, extensive research on the relationship between the design parameters and mechanical performance of bio-inspired metamaterials has been undertaken.

Bamboo-inspired thin-walled structures, which were designed based on the individual vascular bundles of bamboo (specifically Type IV (double broken) vascular bundles), were investigated for the influence of the geometry of vascular bundles on the mechanical performance of the metamaterial (Palombini et al., 2020). The non-linear explicit finite element analysis (FEA) of axial, lateral and three-point bending tests was undertaken to evaluate mechanical behaviour such as peak force, mean force, crushing force efficiency and specific energy absorption. It was reported that the vascular bundle’s geometry is a significant factor in bamboo’s mechanical performance, showing higher compressive load per weight efficiency compared to conventional core shapes. A related paper (Li et al., 2022) proposed and investigated a bamboo-inspired porous lattice structure (BPLS) composed of prismatic microstructures and connecting plates. The axial compression characteristics, energy absorption and deformation modes of the BPLS were studied. The effects of five geometric parameters (cell height, radial dimension, short side length, thickness and cell arrangement) on the peak compression force and specific energy absorption were reported. The results showed that the BPLS exhibits a very stable deformation mode with excellent energy absorption capabilities and novel contraction/expansion patterns. The study demonstrated that geometric parameter optimisation significantly influences BPLS mechanical properties and compression stability. Mantis shrimp dactyl club-inspired multi-cell tubes were designed for crashworthiness (Huang and Xu, 2019). The work investigated the structure’s crashworthiness under axial and oblique loads using nonlinear finite element methods and then selected the best sectional configuration using a complex proportional assessment (COPRAS) method. The study revealed that the structure significantly improved energy absorption (EA) by 59.7% and crush force efficiency (CFE) by 29.4% on average compared to cylindrical tubes, although the maximum crush force (Fmax) increased by 23.9% (Huang and Xu, 2019). The structure exhibiting five cells was identified as the optimal design across multiple loading angles, with its optimal solutions for crashworthiness being dependent on specific loading angles and weighting factors. Energy absorption characteristics and crashworthiness of multi-cell thin-walled structures under impact loads were investigated (Wu et al., 2016). The results revealed that the mean crushing force and specific energy absorption increase with the number of cells, with the five-cell tube demonstrating the best energy absorption characteristics and excellent crashworthiness, making it a potential energy absorber. A 4D-printed metamaterial inspired by the arc-bearing structures of the human spine and the tortoiseshell was investigated by reference (Zhang et al., 2024). It was reported that the proposed metamaterial exhibits significantly tuneable mechanical properties, including stiffness, energy absorption and vibration damping, which can be regulated by varying structural parameters and temperature. For more bio-inspired metamaterials, readers are referred to the literature, such as butterfly-shaped auxetics (Sandak and Butina Ogorelec, 2023), bone-inspired (Song and Lee, 2017) and tissue-inspired (Vangelatos et al., 2020) materials, among others.

These structures are typically sophisticated and exhibit anisotropic loading characteristics, which can be leveraged for the development of metamaterials with tailored mechanical responses, including programmable shape changes, tuneable stiffness, and enhanced crashworthiness, thereby providing versatile solutions for complex engineering challenges where loads are rarely axial. Such applications would include actuators, sensors, soft robots and shape-memory materials (Sandak and Butina Ogorelec, 2023). Anisotropic behaviour of metamaterials under mechanical loading has been reported in the literature (Sandak and Butina Ogorelec, 2023; Song and Lee, 2017; Vangelatos et al., 2020). Most of these studies, however, have focused on evaluating the anisotropic behaviour of bio-inspired metamaterials, and scant information is available on the anisotropic mechanical loading of the specific bio-inspired unit cells. This study, therefore, investigated the anisotropic axial loading behaviour of three distinct vascular bundle-inspired structures, envisioned as unit cells for the fabrication of high-performance bio-inspired metamaterials. The study involved both experimental and finite element simulations to comprehensively evaluate the deformation and failure mechanisms of the presented structures.

The structure design was inspired by vascular bundles in biological systems due to their potential for close packing. As such, there is a possibility of these structures being modular components that could interlock or connect with other similar structures to create larger assemblies. The outer contours provide a natural way for pieces to fit together for the fabrication of metamaterials. A similar approach was reported in an earlier research (Palombini et al., 2020). The structure appears like a multi-lumen extrusion with one central hole channel (or ring) and four larger circular channels arranged symmetrically around it. The variations in the design of the structure are shown in Figure 1. The structures were developed from a 20 mm × 20 mm × 20 mm cube. As shown, the three designs yielded different relative densities of 0.50 [Figure 1(a)], 0.34 [Figure 1(b)] and 0.29 [Figure 1(c)]. The samples, herein, shall be described using the relative densities.

The development involved a systematic workflow integrating computer-aided design, slicing, and additive manufacturing. A three-dimensional (3D) model replicating the hierarchical tubular architecture of vascular systems was designed in SolidWorks 2024 (Dassault Systèmes, France), emphasising channel geometry, branching patterns and spatial organisation to mimic biological analogues. The finalised model was exported as an STL file and processed in Ultimaker Cura 5.4.0 slicing software, where critical parameters were configured: a high-precision layer resolution of 0.06 mm, 0° print orientation (aligned to the build plate), 100% triangular infill (to ensure isotropic mechanical strength) and a print speed of 50 mm/s. Temperature settings were optimised at 195°C (nozzle) and 60°C (build plate) to ensure material flow stability and adhesion for polylactic acid (PLA) filament (BASF Ultrafuse PLA, Germany). The sliced design was fabricated on an Ultimaker S5 3D printer, calibrated for nozzle alignment and bed levelling to minimise dimensional inaccuracies. Post-printing, the structure was cooled to ambient temperature and carefully removed.

The fabricated structures were subjected to quasi-static compression testing using a 100 kN universal testing machine (Instron 3382 floor model, Boston, USA) in accordance with the ASTM D695-15 standard. Tests were conducted at a controlled crosshead displacement rate of 0.5 mm/min to ensure gradual load application. For each structure, compression was applied along two orthogonal directions (depicted in Figure 1) to evaluate anisotropic mechanical behaviour. However, structures with a relative density of 0.34 exhibited identical stress–strain responses in both directions due to their geometric symmetry; consequently, only one direction is presented in Figure 1(b) for this case. Before testing, samples were aligned centrally on the compression platen to minimise eccentric loading. Force–displacement data were recorded until structural failure, and pictures were taken at the end of the elastic region, the plateau region and the densification region. For each case, at least three tests were undertaken for repeatability and statistical accuracy (R et al., 2025).

The FEA was conducted in ANSYS software, beginning with the creation of the geometric model. Two rigid circular platens (40 mm diameter, 1 mm thickness) and the deformable sample geometry are designed using ANSYS SpaceClaim, ensuring proper alignment between the sample and the plates. The sample is assigned a flexible material model (elastic-plastic) with appropriate properties such as Young’s modulus and Poisson’s ratio, while the platens are defined as rigid bodies through ANSYS rigid body constraints to eliminate deformation. The elastic response of the material was represented using an isotropic elasticity model, while the plastic response was captured with a multi-linear isotropic hardening model. Material failure was described using a plastic strain-based criterion. Key mechanical properties of PLA, including Young’s modulus, Poisson’s ratio, density and the true stress–strain curve obtained from a uniaxial tensile test, were incorporated into the model. To enhance the reliability of the material representation, these properties were derived from experimental tensile test data reported by Otieno et al. (2026) and Mwema et al. (2025). Figure 2 compares the experimental results [Figure 2(a)] with the simulated response [Figure 2(b)] of the material model, which was defined using a density of 1244 kg/m³, a Young’s modulus of 1498 MPa and a Poisson’s ratio of 0.33.

Next, asymmetric frictionless contact interactions are established between the platens and the sample. The platens are designated as target bodies, and the sample surfaces are set as contact bodies using the augmented Lagrange formulation to minimise penetration. A frictionless interaction type is applied globally to the model body to prevent tangential forces. The deformable sample is discretised into a mesh with a size of 0.4 mm. The meshing is a combination of tetrahedral and hexahedral element types (Figure 3). Boundary conditions are applied as follows: the bottom plate is fully fixed in all translational degrees of freedom (Ux = Uy = Uz = 0), and the top platen is subjected to a prescribed displacement of 15 mm in either the y–y direction [Loading 1 as shown in Figure 3(a)] or x–x direction [Loading 2 as shown in Figure 3(b)]. The analysis is configured as a static structural simulation with large deflection enabled to account for geometric nonlinearity. Solver settings, such as the Newton–Raphson method with line search, are optimised for convergence. The assumption in the model is that the structures are perfect and does not consider the additive manufacturing inaccuracies or deficiencies. To validate the model, a comparison was made between FEM and experiments during compression tests [e.g. Figure 2(c), as an illustration for sample 0.50] and all the qualitative observations were compared with the experimental observations as it is demonstrated throughout the paper.

The force–displacement curves for the uniaxial compression tests of the three samples are shown in Figure 4. Six samples were tested for each sample, and some of the curves are represented in both directions. For a sample with a relative density of 0.5 [Figure 4(a)], in load direction 1, a sharp linear increase in force with displacement (elastic region) is observed, followed by a distinct peak force at which the structure starts to yield or buckle significantly. After the peak region, there is a noticeable drop in force. This suggests that the structure undergoes a near-brittle collapse or buckling, although not an ideal brittle failure. In load direction 2, the curves have a lower stiffness and peak force than in load direction 1. After the peak force, the drop in force is more gradual and indicates a ductile crushing mechanism contrary to load direction 1. The structure can still carry some load as it deforms. The difference in response between load directions suggests that the internal architecture of the structure is anisotropic, meaning its mechanical properties are direction dependent. For samples with a relative density of 0.34 [Figure 4(b)], it initially displays a linear elastic response, with higher stiffness than in samples with a relative density of 0.5, before reaching the peak force. A plateau or slight force drop occurs at the peak force, indicating sustained crushing or buckling at a relatively constant load. Finally, a drastic drop in force signifies complete collapse, with consistent curves suggesting a uniform deformation mechanism involving elastic deformation, progressive plastic buckling/crushing and collapse. For a sample with a relative density of 0.29, under load direction 1 [Figure 4(c)], the structure exhibits an initial elastic region followed by a peak force, a plateau at peak force and a sharp decrease in force, indicating brittle failure. In contrast, load direction 2 shows a significantly lower stiffness, peak force and a sudden drop to near-zero forces at no displacement, suggesting a more brittle fracture. This anisotropic behaviour highlights that the deformation mode of the structure varies drastically depending on the loading direction, with direction 2 exhibiting lower stiffness and peak force. The symmetric sample (with relative density of 0.34) exhibits the highest peak force, followed by a sample with a relative density of 0.5 (load direction 1), as shown in Figure 5. The lower peak force is obtained on the sample with a relative density of 0.5 in the load direction 2. A one-way analysis of variance (ANOVA) test is performed to check if there is an overall significant difference between the values of peak forces of the individual five load directions (ANOVA table is shown as Table 1). From this, p-value of 0.0001 was obtained, which indicates that the peak forces of the cases are significantly different overall. In addition, a Tukey’s honestly significant difference (HSD) post hoc test is performed from which the results show that the sample with a relative of 0.34 is statistically and significantly different from the other samples (p-value less than 0.05, see Table 2. The peak forces of 0.5, direction 1, 0.5, direction 2, 0.29, direction 1, and 0.29, direction 2 are not significantly different from each other. The load–displacement curves presented herein can be related to those reported in an earlier work on compressive loading of Bambusa Vulgaris-Schrad in the transverse and longitudinal directions (Onche et al., 2020). It is an interesting observation that 0.34 structure exhibits higher load capacity than 0.5 structure despite having a lower relative density. This means that the internal architecture and load-path continuity contribute more significantly than mass density in defining compressive load-bearing capability in bioinspired unit cells. As demonstrated in an earlier study on open lattice structures by Dara et al. (2025) that structures with surrounding walls act like column members (e.g. in honeycombs) and provide better resistance to support larger loads, it implies that the four internal (symmetrical) struts in 0.34 sample act as extra supports and therefore provides better resistance to failure and can support larger loads than the 0.5 sample.

When subjected to compression, the sample with 0.50 relative density exhibits distinct deformation behaviours depending on the loading direction (Figure 6). At the beginning of displacement, both directions show the structure beginning to compress, with Loading Direction 1 exhibiting initial inward buckling of its “shape-eight” shape [Figure 6(a)], while Loading Direction 2 displays a more uniform flattening of the structure [Figure 6(b)]. As the displacement progresses, the differences become more pronounced: Loading Direction 1 experiences significant localised buckling and chaotic crumpling of the central structure, leading to a highly irregular and intertwined deformation with a significant loss of its original pattern [as shown in Figure 6(a)a1, b1 and c1]. In contrast, at higher displacements, Loading Direction 2 maintains a more organised and predictable compression, with its “shape-eight” segments flattening and stacking in a more orderly, corrugated fashion, thus retaining more of its overall structural integrity despite severe deformation [as shown in Figure 6(b)a2, b2 and c2]. In Loading Direction 1, the central hole fractures due to the severe buckling experienced by the central vertical strut at the progressively higher displacement [Figure 6(a)c1]. In Loading Direction 2, the central hole deforms but does not fracture, and the outer walls are in contact with the inner walls [Figure 6(b)c2]. These observations indicate that the structure exhibits near-brittle and ductile fracture behaviour in Loading Direction 1 and Loading Direction 2, respectively [as discussed in Figure 4(a)].

Under compression, the structure with a relative density of 0.34 transitions from an initial configuration of four distinct and symmetric circular cells [Figure 7(a)]. As displacements are applied initially, the internal struts begin to buckle inwards, causing the individual cells to deform and the overall structure to exhibit a slight lateral expansion [Figure 7(b)]. With the progressive displacement of the sample, the cells flatten significantly, and the internal connections fold extensively, leading to a substantial reduction in height while the structure continues to visibly widen. At high displacements, the original cellular architecture is largely lost, replaced by a densely crumpled and compacted mass of material [Figure 7(c)]. The central hole continuously closes, and eventually the walls fracture, as demonstrated in the experimental observations. These deformation observations corroborate the results of the load-displacement curves [Figure 4(b)].

For a sample with a relative density of 0.29, the deformation occurs starting with outer “shape-eight” loops and a central diamond-like cell during compression loading in both loading directions (Figure 8). At initial displacements, Direction 1 shows initial inward bowing of vertical connections and flattening of loops [Figure 8(a)a1-b1], while Direction 2 exhibits more uniform flattening and a visible lateral expansion. As the displacement progresses, Direction 1 continues to buckle, leading to a localised, chaotic crumpling and entanglement of struts in its central region and fracture of the central walls [Figure 8(b)a2-b2]. Conversely, at increasingly more displacements, Direction 2 demonstrates progressive horizontal folding of its internal elements, resulting in a more organised, layered compaction across its width [Figure 8(b)b2]. Direction 1 collapses into a tangled mass [Figure 8(a)c1], whereas Direction 2 maintains a discernible, albeit highly densified, layered structure [Figure 8(b)c2]. In Load Direction 1, the fracture occurs on the walls of the central hole, whereas in Direction 2, the fracture occurs on the outer walls of the “shape-eight” structure as per the experimental (pictorial) observations. In corroboration with Figure 4(c), the stability of the structure in Loading Direction 2 is responsible for larger deformation and ductile behaviour of the load–displacement curves.

The damage mechanisms of the structures in different directions have been analysed from the von Mises stress contours of the finite element model, with emphasis on the stress concentration on the cells and walls. A similar approach was adopted by Li et al. (2025) to demonstrate the damage mechanisms in high-energy absorbing cellular structures. These results are used to explain the physical/pictorial observations of failure of the structures in Figures 9–11 for samples with relative densities of 0.50, 0.34 and 0.29, respectively. As shown in Figure 9, for a sample with a relative density of 0.50, in Load Direction 1, as compression progresses, the critical vertical struts experience significant buckling, which directly contributes to the severe distortion and eventual collapse of the central void. FEA reveals high stress concentrations along the walls of these vertically oriented struts (indicated as “S” in Figure 9), which are the primary drivers for the observed cracks, fracture and subsequent complete failure of the central hole (as indicated by the pictorial observations). Conversely, in Load Direction 2, the failure mechanism shifts to the outer vascular walls of the structure. These walls undergo progressive collapse, eventually touching the walls of the central hole. The FEA stress contour map indicates pronounced stress concentrations occurring at the corners of these outer walls and, critically, at the interface where the collapsing walls make contact (indicated as “A” in Figure 9). These localised stress concentrations are the precursors to the observed cracking and fracture at these specific points (pictorial observations). Notably, in this loading scenario, the central hole itself maintains its overall shape and does not collapse at the point of structural failure; instead, failure initiates and propagates from the outer vascular walls. The failure mechanisms for the two directions can be reconciled by incorporating struts into the load path in direction 2 analogous to proposition that infill geometry and topology influence the mechanical strength and energy absorption of nature-inspired 2.5D structures (Ashok et al., 2023).

The symmetric structure (sample with a relative density of 0.34) under compression loading undergoes failure as indicated in Figure 10. As per the finite element model, the maximum stresses are concentrated along the primary load path, specifically identified as “S” in Figure 10, which corresponds to the vertical strut connected to the central hole. This intense stress concentration around the central hole is the direct cause of macroscopic cracking and ultimate fracture of its internal walls (shown by the pictorial observations). Furthermore, significant stresses are also observed at the corners of the outer (vascular) walls, leading to the initiation of microscopic cracks in these regions (as observed via the zoomed camera pictures). In contrast, the mid-section of the horizontal struts experiences the lowest stress concentration, which explains their resilience; they neither undergo fracture nor exhibit significant buckling or bending even under high compressive loads. The behaviour explains the ability of the structure to stay stable under longer displacement as per force–displacement curves in Figure 4(b). However, small perturbations in the symmetry of such a structure due to geometrical errors or manufacturing discontinuities may interfere with the stress concentration, maybe off the primary load path and shift to the outer struts of the structure, leading to asymmetric stress distribution as reported by Dara et al. (2022).

The failure mechanisms of the sample with a relative density of 0.29 are shown in Figure 11. In Load Direction 1, the analysis consistently shows that the maximum stresses are critically concentrated along the vertical struts forming the internal wall of the structure (indicated as “S” in the FEM model in Figure 10). This intense stress localisation directly instigates the fracture of these internal walls, a failure mode corroborated by macroscopic and microscopic (experimental) observations. In addition, significant stress concentrations are also identified at the four outermost corners of the structure (marked as “S” in Figure 11), which contribute to the cracking and eventual fracture of the outer vascular walls. Conversely, in Load Direction 2, the primary locations for high stress concentrations are found at the critical connection points between the outer and internal strut walls (indicated by “S” in the FEM model). These localised stresses are directly responsible for the initiation of cracks that propagate predominantly perpendicular to the applied load direction, ultimately leading to the fracture of the outer wall. The localised stresses are indications of critical stress regions and can further be determined via critical stress approach as earlier illustrated in a 3D open lattice structure (Ashok et al., 2022). This distinct failure mechanism contrasts sharply with that observed in Load Direction 1.

Table 3 summarises the key findings from compression loading experiments and finite element modelling, outlining the characteristic deformation and damage mechanisms. It distinctly illustrates how the geometry, relative density of the structure and the applied loading direction significantly influence its mechanical response, including stiffness, deformation patterns and specific points of failure.

The polylactic acid (PLA) structures discussed herein exhibit properties that make them suitable for design of metamaterials with a wide array of applications, primarily focused on shock absorption and lightweight design. In the areas of Packaging and Logistics, these structures serve as crucial protective elements, including shock-absorbing inserts for fragile electronics, medical equipment, and goods, reusable cushioning foams, lattices, and dunnage trays to mitigate shipping vibrations. Their utility extends to Automotive (non-critical parts) where they are used for interior trim panels with energy-absorbing lattices, prototypes for crash boxes in low-speed impact studies, and padding or seat inserts requiring moderate impact dampening, as well as lightweight crash protection for unmanned aerial vehicles (UAVs).

Beyond industrial uses, PLA structures are valuable in Sports and Personal Safety Equipment, forming helmet liners for light industrial or recreational helmets (like bicycling and skating), bio-based foams for elbow pads, shin guards and body protectors, and protective casings for sports gear. In Medical applications, their biodegradable nature is advantageous for orthopaedic uses, such as shock-absorbing splints or braces, cushioning devices for patient support and single-use packaging with energy absorption for fragile instruments. Finally, the material is applied to various Consumer Products, including protective cases for electronics (laptops, phones and VR headsets), impact-dampening feet or cushioned joints in furniture components, and the manufacturing of children’s toys with safer impact properties. However, these applications shall be fully evaluated for when the metamaterial is developed based on the presented unit cells.

The study reveals that the compression response and failure mechanisms of the vascular bundle-inspired structures are distinctly influenced by the geometry, relative density and loading direction. As per the compression test, pictorial observations, and finite element modelling, the following conclusions can be drawn from the study:

• For the sample with the relative density of 0.5, Load Direction 1 exhibits brittle behaviour, characterised by non-uniform deformation, buckling of vertical struts and fracture of the central hole walls due to maximum stresses along these struts. In contrast, Load Direction 2 displays ductile behaviour with uniform deformation, where the structure folds consistently, leading to fracture at the four outer corners while the central hole remains stable.

• The sample with the relative density of 0.34 shows isotropic behaviour across loading directions due to the symmetric geometry, undergoing stable and uniform deformation where the central hole closes like a “human eye”, with maximum stresses concentrated along the vertical strut.

• Finally, for the sample with the relative density of 0.29, Load Direction 1 results in an abrupt fracture after the peak force, driven by vertical strut buckling and severe stress concentration, causing the central hole to fracture and collapse. Conversely, Load Direction 2 experiences a more gradual fracture after the peak force, where deformation is uniform and the outer walls collapse and fracture due to maximum stresses at their corners, notably leaving the central hole stable.

Overall, the failure mechanisms are dictated by how the geometry and loading direction influences stress concentration: either along internal vertical struts leading to central hole collapse (brittle for 0.5 Direction 1, 0.29 Direction 1), or at outer corners/connecting points causing outer wall fracture while preserving the central hole (0.5 Direction 2, 0.29 Direction 2), with the 0.34 density sample demonstrating a more uniform, less catastrophically localised stress pattern. The best structure, 0.34, shall be assembled into a porous structure following existing design strategies such as unit cell tailoring, hierarchical structuring and functional grading (Ashok and Bahubalendruni, 2023) in the future to develop multifunctional materials for mechanical applications.