Empirical Investigation of Energy Absorption for 3D-Printed Auxetic Tubular Structures

In light of the increasing imperative for sustainable and optimized design paradigms, 3D-printed structures have attained considerable prominence within engineering applications, particularly owing to their superior mechanical performance in energy absorption mechanisms. The objective of this investigation is to empirically characterize the fabricated cylindrical three-dimensional printed auxetic structures. The examined structure was produced utilizing fused deposition modeling (FDM) with polylactic acid (PLA) as the material. Following the empirical characterization of the energy absorption response of the structures subjected to quasi- static loading conditions, the critical parameters such as specific energy absorption (SEA) and crush force efficiency (CFE) were evaluated. 

Introduction

Auxetic materials display unique mechanical characteristics, such as lateral expansion under tensile stress and contraction under compressive stress. This unusual behavior can be ascribed to the negative Poisson’s ratio of these materials. Recent research has investigated the energy absorption properties of auxetic structures. Research has concentrated on various auxetic designs, including re-entrant honeycombs, chiral lattices, and modified stent structures [1]–[3]. Researchers have utilized various techniques, including experimental, numerical, and analytical methods, to evaluate the performance of these structures under compression and impact loads [4], [5]. The topology of the cells, material composition, and structural parameters are factors that affect energy absorption [6].

In comparison to conventional structures, some modified auxetic designs have exhibited a significant improvement in energy absorption [7], [6]. This discovery indicates that auxetic structures can be engineered to fulfill specific energy absorption requirements for diverse applications, including automotive, aerospace, and military sectors [4], [5]. Multi-material utilizing PLA, Nylon, and TPU enhanced energy absorption and mechanical characteristics [8], [9]. Foam-filling of auxetic structures has been proposed to enhance performance [10], [11]. Vyavahare and Kumar [12] developed numerical and finite element models to simulate and optimize the behavior of auxetic structures, indicating significant potential for the application of auxetic materials in the automotive, aerospace, sports, and protective equipment sectors. Different auxetic designs, including re-entrant, arrowhead, and honeycomb structures, have been investigated and produced through 3D printing [13], [14]. Empirical and simulation studies indicate that auxetic structures significantly enhance energy absorption and the crashworthiness of foam-filled tubes and crash boxes [15], [16]. These findings introduce novel applications for auxetic structures in engineering fields, including civil, mechanical, and aerospace engineering [17], [18].

A range of designs have been examined, including continuous fiber-reinforced composites [19], biaxially graded structures [20], and innovative struts in the forms of arcs and dumbbells [21]. Hybrid auxetic structures composed of re-entrant and standard honeycomb cells have all demonstrated enhanced compressive strength and energy absorption [22]. The advent of 4D printing technology has enabled the development of innovative dual-material meta-sandwich structures capable of programmable energy absorption [23]. Research has demonstrated that studies comparing various cellular structures with auxetic designs exhibit greater energy absorption properties, particularly the double arrowhead geometry [24].

In comparison to traditional designs, modified re-entrant auxetic cores in sandwich structures have demonstrated superior energy absorption capabilities [25]. A variety of auxetics, including re-entrant honeycombs, anti-quad configurations, and hourglass shapes, have been the subject of research in the context of thin-walled tubes and crash box applications [26], [27]. These structures exhibit enhanced energy transformation efficiency in comparison to conventional designs, concurrently manifesting negative Poisson’s ratios [28], [29]. Additive manufacturing processes have been utilized to create auxetic structures from materials such as thermoplastic polyurethane and polymer nanocomposites [30], [31] Auxetic behavior and energy absorption capacity have been the focus of research due to the influence of geometric parameters, relative density, and loading conditions [32], [33]. The extant research demonstrates that auxetic structures offer substantial advantages for applications necessitating lightweight materials with high energy absorption capabilities, such as crash protection and impact mitigation. Other works studied re-entrant diamond structures Logakannan [34], tetra-petal configurations Photiou [35], and even newer star-shaped auxetics. All of them show negative Poisson’s ratios, which further increase their mechanical properties as well as energy absorption capacity [36], [37]. The creation of complex auxetic geometries has been made possible with the aid of Additive manufacturing methods, primarily 3D printing [38], [39].

Some researchers put their effort into auxetic composites by integrating lattice-type structures with soft materials to increase resistance to indentation and impact [37]. Both experimental and computational studies have been carried out to investigate the response of auxetic structures under various load conditions, including quasi-static compression and low-velocity impact [40]. These studies have demonstrated the potential of auxetic structures for applications that necessitate lightweight materials with high strength and exceptional energy absorption capabilities. It has been posited that these structures exhibit a negative Poisson ratio and augmented mechanical properties [41]. A substantial body of research has demonstrated that 3D-auxetic structures consistently exhibit superior energy absorption capabilities and impact resistance compared to 2D models [41], [42].

The fabrication of devices, including the more robust 3D models, has been achieved by leveraging thermoplastic polyurethane (TPU) and poly-lactic acid (PLA) as prospective materials for auxetic lattices [43], [44]. It has been posited that the optimization of the design, achieved through the alteration of performance-induced thickness gradients and hybrid structures, has undergone testing [45], [46]. As indicated by Tunay and Cetin [47], significant parameters such as strut thickness, angle, and orientation have been demonstrated to exert a direct influence on maximum energy commercial discharge. Auxetic structures have been shown to exhibit superior impact resistance and energy absorption capabilities when compared to non-auxetic configurations. This property renders them particularly well-suited for applications in protective armor, civil engineering, and crashworthiness [41], [44], [45].

Method and Materials

Geometry

For the present study, the R1A structure, illustrated in Fig. 2, was chosen as the specimen because preliminary simulations indicated superior energy-dissipation capacity. Its architecture consists of a uniform, axisymmetric tubular shell whose circumferential wall is formed by a single row of two-dimensional re-entrant auxetic cells, as seen in Fig. 1; these cells prove decisive for the structure’s load-bearing response. By stacking the unit cells across the wall’s thickness, the cylinder gains an enhanced capacity to cushion impacts when subjected to axial compression. The accompanying schematic of the auxetic cell delineates the precise geometric features that govern the negative Poisson’s ratio observed in laboratory trials. The structure is designed with an external diameter of 75 mm and a height of 50 mm.

Fig. 1. Two-dimensional re-entrant auxetic cells.

Fig. 2. Structure geometry illustrated from various perspectives.

Manufacturing

Auxetic tubes are manufactured by the fused filament fabrication (FDM) method, a subset of 3D printing, wherein thermoplastic filament is fed into a heated extruder. A continuous thermoplastic filament is introduced into a heated extruder, where it is melted and deposited in layers through a nozzle onto a build platform, following a layer-by-layer approach based on a digital model. The 3D objects are formed when the material cools and crystallizes, bonding with previously placed layers [48], [49]. A significant advantage of this technique is that it utilizes a single filament composed of a wide variety of materials, including composites and those that are filled with polymers and metals. This characteristic renders the technique popular due to its ease of use and cost-effectiveness. In this study, three cylindrical structures were created using a Polylactic Acid (PLA) filament on a Creality Ender-3 S1 Pro 3D printer. PLA is a thermoplastic that is derived from renewable materials such as corn starch or sugarcane, making it biocompatible. Kuznetsov, Czyżewski have demonstrated that it has a low melting point and is biodegradable, characteristics that render it a preferred material for 3D printing [50], [51]. Table I illustrates the optimized printing parameters. The utilization of a 0.4 mm nozzle optimizes the balance between resolution and printing speed Kuznetsov [50], Czyżewski [51], Kiński and Pietkowicz [52] despite the potential enhancement of mechanical properties associated with smaller diameters, albeit at the expense of prolonging printing times. Following the guidelines established by Giubilini and Minetola, [53], the thickness of each stratum was designated as 0.2 millimeters. The nozzle was maintained at a temperature of 200°C, which was found to be optimal for achieving smooth results without the risk of overheating. PLA is susceptible to degradation and overheating [54], [55]. Conversely, lower temperatures have been shown to enhance bonding and reduce porosity between layers [56].

Table I. Printing Standards

Printing parameter	Value
Layer height (mm)	0.2
Nozzle diameter (mm)	0.4
Infill density (%)	100
Top/Bottom thickness (mm)	0.8
Wall thickness (mm)	0.8
Plate temperature (°C)	60
Print temperature (°C)	200
Print speed (mm/s)	50
Cooling (%)	100

Testing

The energy absorption capacities of the unique cylindrical auxetic structure are evaluated using a quasi-static compression test. The objective of the experiments was to ascertain the specific energy absorption (SEA) and the crush force efficiency for the structure. The experimental apparatus consists of a universal testing machine (MTS CMT5305), as illustrated in Fig. 3. The apparatus is equipped with load cells and displacement sensors, both of which are characterized by their high degree of accuracy. The experimental apparatus comprised a compression testing machine, which had been calibrated for the purpose of applying loads at the specified velocities. Testing was performed at a velocity of 10 mm per minute. The total specified displacement is set at 25 mm, constituting 50% of the height of the cylindrical structures. Before the initiation of the specimen examinations, a lubricating agent was applied to its contact surfaces in order to minimize frictional resistance. The experiment began by compressing the specimen by 25 mm to achieve maximum compression. During the test, force and displacement data were recorded. Also, the specimen mass was documented before and after the evaluation. Subsequent to the conclusion of the tests, the specimens were meticulously stored in bags that were distinctly labeled to prevent any potential misidentification. Subsequent to the completion of the experiment, the analytical software of the testing apparatus was utilized to evaluate the experimental outcomes. This evaluation yielded a material response curve that illustrated the relationship between force and displacement during the compression phase.

Fig. 3. Universal testing apparatus (MTS CMT5305).

Processing

Quasi-static crushing of the auxetic tube was undertaken as explained in Section 2.3. For each experiment, the time variation of the force $F\left(t\right)$ and displacement $\delta \left(t\right)$ were observed. A typical force-displacement curve is illustrated in Fig. 4. Three distinct zones of deformation may be observed a starting with an elastic zone, then a progressive crushing zone, and completed by a densification zone.

Fig. 4. Force-displacement curves R1A structure for three samples.

By finding the force-displacement curve, it is possible to extract the specific absorbed energy E as follows:

(1) $$\begin{array}{r}E\left(\delta \right)=\int F\left(\delta \right)d\delta \end{array}$$

It refers to the area underneath the force-displacement graph. The energy is written in terms of time:

(2) $$\begin{array}{r}E\left(t\right)=\int F\left(t\right)\delta \left(t\right)dt\end{array}$$

You can use any of the Eqs. (1) or (2) to find the energy if you know the force and displacement.

Also, figure out the specific absorbed energy (SEA) to see how well the auxetic tubes can absorb energy. To find it, you divide the total absorbed energy $(E_a)$ by the tube’s mass ($m)$:

(3) $$\begin{array}{r}SEA=\frac{E_a}{m}\end{array}$$

It is also interesting to investigate the crush force efficiency (CFE). It is determined by an average-to-maximum force ratio throughout the progressive crushing section. So,

(4) $$\begin{array}{r}CFE=\frac{F_{avg}}{F_{max}}\end{array}$$

where $F_{avg}$ is the average and $F_{max}$ orce within the progressive crushing region.

The crushing force efficiency $(CFE)$ It is investigated to analyze the fluctuation of force in the progressive crushing region. Ideally, the $(CFE$) is equal to (1), if the structure is crushed under a perfectly constant force. In that case, the maximum force is equal to the average force. For real cases, $(CFE)$ is lower than 1. The closer $(CFE)$ is to unity the smoother and more uniform the progressive crushing.

Results and Discussion

Force Curves

The present part examines the characteristics of the force-displacement curves. Fig. 4 illustrates the relationship between force and displacement at a velocity of 10 mm/min. The force-displacement curves display four stages. The initial phase pertains to the elastic response of the structure, during which the cells undergo uniform deformation [26]. The force escalates linearly with displacement until it attains the yield point. Yielding signifies the commencement of plastic deformation or the establishment of plastic hinges, predominantly occurring at the cell corners of the auxetic structure, which impede the rotation of the cell walls. The second stage, after yielding, is defined by the ongoing folding of the cell walls and the development of supplementary plastic hinges [57].

Upon complete folding of the auxetic cells, the third phase, referred to as the densification stage, commences. At this stage, the cells experience compaction due to the imposed compressive load, resulting in a substantial rise in force with displacement until a maximum value is attained. At this phase of densification, the structure begins to function as a constituent material [26]. This maximum force usually signifies the initiation of internal failure or the collapse of the auxetic cells, ultimately resulting in structural failure. A detailed analysis of these stages provides valuable insight into the deformation mechanisms governing energy absorption in auxetic structures and highlights opportunities for optimizing their design for enhanced crashworthiness.

Key Parameters

Stiffness and Yielding

Important features of the first elastic stage of deformation are highlighted in this section. It specifically looks at the three auxetic structures’ performance in terms of yield strain (yield displacement relative to original length), specific stiffness (stiffness per unit mass), and specific yield force (yield force per unit mass). The measured values, average, and standard deviation for each of the three specimens are shown in Table II. Results show good repeatability of the results and good performance of the auxetic structure. The manufacturing process and the dependability of the experimental setup. The findings also show that the auxetic designs perform promisingly in maintaining high stiffness and yield force despite their lightweight nature, which is crucial for applications that need effective energy absorption with minimal mass.

Table II. Key Parameters for Stiffness and Yielding

Key parameters	Test 1	Test 2	Test 3	Average	Standard deviation
Specific stiffness (N/mm/g)	217.4	216.5	241.0	225.0	13.9
Specific force at yield (KN/g)	414.8	391.0	395.8	400.5	12.6
Strain at yield (%)	6.90	5.30	4.58	5.59	1.19

Crushing Stage

This section concentrates on the second stage, the crushing phase, which is the most pivotal phase. Table III reports the crushing force efficiency (CFE) and specific energy absorption (SEA) for the axisymmetric auxetic structure. The obtained results show good repeatability. The average SEA of the axisymmetric auxetic tubular structure is equal to 4.29 J/g. The crushing force efficiency (CFE) is about 0.887.

Table III. Key Parameters for Stiffness and Yielding

Key parameters	Test 1	Test 2	Test 3	Average	Standard deviation
SEA (J/g)	4.08	4.35	4.68	4.29	0.19
CFE	0.898	0.870	0.894	0.887	0.015

The findings of this study are consistent with those documented in the extant literature concerning tubular auxetic structures, where the specific energy absorption (SEA) ranges from 0.38 J/g to 8.13 J/g, and the compressive energy failure (CFE) fluctuates between 33% and 88%. It is noteworthy that the attainment of our (SEA) values was accomplished through the utilization of a significantly more pliable material, namely PLA, in contrast to the materials commonly employed in prior research, such as steel, aluminum, and ABS, which are renowned for their rigidity and durability. In addition, the CFE values are among the highest documented. The axisymmetric shape facilitates an effective crushing phase, as the crushing force remains nearly constant throughout deformation. Despite the employment of comparably malleable materials such as PLA, the encouraging outcomes observed in SEA and CFE, particularly for axisymmetric tubes, underscore the considerable potential of this auxetic geometry for energy absorption applications. This research will provide a foundation for subsequent numerical and experimental investigations aimed at enhancing these skills, to attain superior (SEA) values while preserving elevated (CFE).

Limitations

Neglect of Strain-Rate and Temperature Effects. The test series held strain rates and ambient temperature steady, yet literature shows both factors shape polymer stiffness and fracture in quasi-static and impact regimes required for engineering applications. Single Material Investigated. Because the study focused exclusively on polylactic acid, its mechanical observations and any design recommendations are provisional until similar investigations are performed on other semi-crystalline and amorphous filament grades used in large-scale additive production. Idealized Boundary Conditions. Finite-element models simulated perfectly clamped or pinned edges, a simplification that ignores fixture compliance, thermal expansion, and tangential slip characteristic of real printing rigs and test frames. 3D-Printing Imperfections Not Modeled. Most additively manufactured parts contain voids, weak inter-layer bonds, and anisotropic bead alignment, but these commonplace defects were removed from both lab experiments and failure-influenced simulations, leaving their contribution unknown. Quasi-Static Loading Only. Dynamic and cyclic loading regimes were omitted, so questions about impact energy absorption and long-term fatigue behavior-relevant to protective and structural components, still open until future tests introduce high rates and repeated cycles. Narrow Geometric Scope. Only three auxetic unit-cell topologies were printed and tested; a much wider universe of cell density, wall thickness, multi-material, and hybrid lattice concepts remains outside current findings, yet likely exhibit different deformation pathways. Small Specimen Count. The analysis includes fewer than five replicates per condition.

Conclusion

This study examined the energy absorption characteristics a 3D-printed cylindrical auxetic structures subjected to quasi-static loads. The samples were fabricated using fused deposition modeling (FDM) with polylactic acid (PLA). The results indicated that the average SEA value is about 4.29 J/g, while the CFE average value is about 0.89, consistent with the values documented in the literature for tubular auxetic structures. The axisymmetric structure demonstrated (SEA) values at the upper limit of this range and attained some of the highest (CFE) values. This exceptional performance is ascribed to their effective crushing behavior, characterized by a practically constant crushing force throughout deformation.

Conflict of Interest

The authors declare that they do not have any conflict of interest.