Auxetic fixation devices can achieve superior pullout performances compared to standard fixation concepts

Despite bone screws being the most commonly inserted implant in orthopaedic surgery, 10% of fracture fixation failure is a result of screw migration or pullout. In this study, the effect of four auxetic structures on the pullout performance of a novel unthreaded bone fastener was investigated through experiments and numerical simulations. The auxetic fasteners included the re-entrant, rotating squares, missing rib, and tetrachiral structures. Parametric CAD models were developed for each, and polymer samples manufactured using a stereolithography process. Pullout testing using bone analogue material found the rotating squares fastener to achieve superior pullout resistance 2.5 times that of the non-auxetic control sample. With a pullout to push-in force ratio of 33.7, this fastener achieved high pullout resistance with a low insertion force improving ease of installation. The Poisson’s ratio of the structure was determined using image analysis to be −1.31, similar to the missing rib and re-entrant types. The low axial stiffness of 12.1 N mm−1 for the rotating squares fastener was the reason for superior performance, allowing axial and resulting transverse strain to be initiated at relatively low load. The effect of increased diametral interference was investigated, and the re-entrant structure found to be superior with pullout resistance improved by 342%. This work provides a foundation for further development of unthreaded auxetic bone fasteners, which have the potential to replace screws for some orthopaedic applications and significantly reduce the prevalence of pullout as a failure mode.

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Introduction
Osteosynthesis uses implants to aid the healing of fractures that cannot do so naturally.Screws are used to provide compression across fractured bone fragments, or to secure plates, rods or nails in location to achieve stabilisation for healing.As such, screws are a vital component to most fracture fixation procedures, making them the most commonly inserted implant [1].Despite this, over 10% of implant failures are a result of screw migration and pullout, a phenomenon where the screw loosens under axial loading [2].A leading cause of screw loosening is overtightening, which causes stripping of the bone associated with significant reductions in pullout resistance [3], over 80% [4].One in four non-locking screws are overtightened upon insertion [3], a result of the threaded connection of the screw, however no widespread alternatives are available.
Typical engineering materials used in orthopaedics include stainless steel (SS) and titanium alloys [5], both possessing Poisson's ratios approximately equal to 0.3 [6]: the Poisson's ratio, ν, of a material, is a measure of the ratio of transverse strain, ε 2 , to the axial strain in the direction of loading [7], ε 1 , shown by equation ( 1) A positive ratio implies that under axial extension, transverse contraction occurs, while conversely an auxetic material will undergo transverse expansion under axial expansion and is defined by a negative Poisson's ratio.The principle by which auxetics could benefit an unthreaded bone fastener is that under axial loading, the fastener diameter will increase, enhancing the frictional forces between itself and the bone to resist pullout.This is especially relevant to the application of suture anchor fixation, where the device is under constant tension.High resistance to pullout is a key criterion for anchors, which is currently achieved using threads, barbs or expanding devices by means of shape memory alloy (SMA) or suture balls [8].
The principle of expanding fasteners is commonly demonstrated by wall plugs; small devices that expand radially upon screw insertion to provide secure fixation in weak plasterboard walls.In bone, specifically osteoporotic bone, McKoy and An found unthreaded wall anchors to achieve a 74% greater pullout force than screws alone [9].However, this fixation method is limited by the strength of the bone's cortical layer, which is notably weakened in patients with severe osteoporosis [10].By instead implementing expansion of four barbed flaps at the distal end of a fastener, Oldakowski et al achieved pullout force improvements of 41% relative to conventional screws [11].While successful in improving pullout performance, both fasteners rely on multiple components which complicates insertion and introduces more points of failure.Smart materials have also been investigated to provide in-situ expansion of bone fasteners.Werner et al and Eshghinejad et al integrated SMA wires into screws to expand upon heating to body temperature [12,13].While both fasteners showed improved pullout resistance from synthetic bone material, Werner et al found these improvements to be negligible when tested in human bone, due to fragments and soft tissues blocking expansion of the wires.
The concept of using auxetics to develop fasteners with superior pullout resistance was first investigated by Choi and Lakes, in 1991 [6].The press-fit fastener, made from re-entrant copper foam with a Poisson's ratio equal to −0.8, withstood a tensile load over twice as large as the insertion force.This result confirms auxetic behaviour of the fastener, as the diameter shrinks upon insertion and expands upon removal.However, similar work conducted by Ren et al in the context of metallic nails found an average push-in force over 12 times the pullout force [14].Finite element analysis (FEA) confirmed that under tensile load, the diametral expansion of the nail was insufficient to prevent pullout due to the high stiffness of the structure.The nails used an auxetic structure consisting of elliptical perforations, a variation of the rotating squares unit cell with substantial material at the vertices [15].The same auxetic pattern was implemented into an inherently less stiff polymer auxetic dowel by Kus ¸kun et al [16], with an experimentally determined Poisson's ratio of −0.442, achieving a reduced mounting force of 100 N compared to a nonauxetic control sample of 290 N. Further work by Kasal et al found the auxetic dowels to typically achieve lower pullout forces than the control sample as a result of dowel failure [17], similar to the observations of Choi and Lakes [6].The dowels with superior auxetic performance correspondingly had minimum load-bearing material area, highlighting a trade-off between structural mechanical properties and auxetic performance.The pullout performance of threaded bone screws containing six unique auxetic patterns was investigated against a non-auxetic reference with equal surface area by Yao et al [2].The Poisson's ratio, strength and stiffness of each structure were tested experimentally using titanium samples, while the pullout performance was determined using FEA.
Despite improvements in pullout resistance of the auxetic screws relative to a control sample, the disadvantages of the bone screw threaded connection, such as over-tightening, are not mitigated by this work.Further work by Yao et al investigated the correlation between re-entrant structure stiffness and bone stiffness in auxetic pedicle screw pullout performance [18].A positive linear relationship was confirmed between the elastic modulus of screw and bone in those performing highest in pullout.The best improvement of 14.5% was achieved in low density bone representing the osteoporotic case, with the screw possessing the lowest stiffness.A drawback of the work of both Yao et al and Ren et al is the inclusion of solid material at each end of the tubular auxetic portion, which hinders the auxetic response by restricting the motion of the unit cell [18].
The literature presents a research gap for auxetic unthreaded bone fasteners; while threaded bone fasteners containing an auxetic structure have been investigated with success, they do not mitigate the negative effects of threaded connections.Unthreaded auxetic fasteners exist in literature for other applications including nails and furniture dowels, with partial evidence for improved pullout resistance.However, both studies used the same auxetic pattern of elliptical perforations, and other auxetic structures have not been investigated.The aim of this study was to investigate the pullout performance of different auxetic structures within an unthreaded bone fastener.

Method
An overview of the testing evaluation scheme for the auxetic fasteners is shown in figure 1.

Design of auxetic structures
Auxetic structures from the re-entrant, rotating solids, chiral, and missing rib categories were selected for investigation, to allow the effect of different deformation modes on pullout performance to be analysed.Specifically, the re-entrant, rotating squares, tetrachiral and missing rib structures were chosen, as they are common within the auxetics literature.A notable similarity between these structures is that they can each possess a square unit cell.A hexagonal honeycomb structure was selected as a non-auxetic control.Matching the size and relative density of each lattice unit cell allowed the characteristics and performance of the four auxetic patterns to be compared.
Each unit cell was sized to be a 7.85 mm square, allowing four unit cells to be patterned around the circumference of a 10 mm diameter tube.The defining dimensions of each unit cell are shown in figure 2 and the parameters in table 1.The open area of each unit cell was set to 40% (resulting in a relative density of 60%); the minimum open area within a porous structure to encourage osseointegration [19].This value was selected to promote bone ingrowth while maintaining fastener strength.The wall thickness and axial length of each fastener sample were maintained the same across all designs at 1.5 mm and eight unit cells respectively.Two sample variants were designed; one possessing a tapered end to be used for bone pullout testing, and the other with solid material sections at both ends to be gripped by the tensile test machine.The pullout samples are shown in figure 3 with their respective unit cells.

Manufacturing fastener samples
Additive manufacturing (AM) was selected to produce the fastener samples, as it is well suited to complex geometries and parts can be produced on demand.A Stereolithography (SLA) printer (Form 2, Formlabs, USA) was used, as it possesses a higher resolution than other AM techniques, such as fused deposition modelling (FDM).The samples were printed such that the flat surfaces were at an angle of 20 • to the build platform as recommended by Formlabs [20].This is not expected to affect fastener properties; SLA prints using Formlabs printers have been found experimentally to possess limited anisotropy unlike FDM prints, likely a result of the post-build curing process [21,22].Typical materials of bone screws include biocompatible metals such as SS and titanium alloys [5].While the fastener samples used in this study were manufactured from polymer, the resin was selected to best replicate the failure mode of conventional screw materials in the context of resins available with the prototyping process, i.e. it has a relatively high stiffness and toughness.A direct comparison however with realistic-typically metallicbone fastener materials would not be useful.Tough V5 resin (FLTOTL05, Formlabs, USA), with the material properties in table 2, was chosen due to its high strength and toughness, to prevent fastener failure before pullout.Following printing, the samples were washed in isopropyl alcohol and subsequently cured with UV light and heat using the Form Cure system (Formlabs, USA) to achieve optimal material mechanical properties.The system was set to a temperature of 60 • C for 60 min, as recommended by the manufacturer for Tough V5 resin [23].

Tensile testing
The samples of each structure with solid material at either end were tensile tested using a 5 kN Instron (Instron 5965 Tabletop testing system, USA) at a rate of 1 mm min −1 based on previous literature [2].The sample deformation was measured with a static axial clip-on extensometer of 50 mm gauge length (Instron 2630-111, USA) centred along the length of the sample, and removed following the elastic deformation test region.A digital camera (Sony a6100) with a 35 mm lens (SIGMA F1.4,Japan) was positioned with line of sight perpendicular to the sample centre and each test recorded.
Frames of each test at 0% and 1% axial strain were extracted from the video footage using MATLAB (version R2022a, MathWorks, USA), and the change in sample gauge length and diameter measured in ImageJ photo analysis software (version 1.53k, USA).A line was drawn between the same pixels on both images, and the difference in the line length recorded.To reduce variability in the line position, the ends were placed at the intersection between four pixels.The experimental strain across the diameter and gauge length of each structure was calculated by the ratio of the extension (line length from 1% strain image-line length from 0% strain image) to the original length (at 0% strain).The experimental Poisson's ratio was then calculated from these strains using equation (1), where ε 1 = ε axial and ε 2 = ε diameter .The stiffness of each structure was determined from the force-extension relationship, and this was calculated in the strain interval 0.0005 ⩽ ε ⩽ 0.0025 from the extensometer, based on ISO 527.Using the same strain interval range, the elastic modulus of three solid samples was calculated to characterise the material properties without structural effects.

FEA
The FEA was performed using ABAQUS (Version 2022, Dassault Systèmes Simulia Corp., Johnston, RI, USA) with an implicit linear static solver.Linear elastic material properties were assigned (E = 1.11GPa, ν = 0.35).For the tensile tests  one end of the geometry was fixed in all degrees of freedom on the flat axial faces where the grips held the sample, and a tensile load was applied to the opposite side of the sample to generate up to 16% strain.The flat faces on the loaded end of the sample were constrained for axial rotation.A mesh convergence study was performed, and a converged global element size of 0.7 mm was selected for the models using quadratic tetrahedral elements (C3D10).An element set of just the central 50 mm gauge region was used for the strain analysis; and the same process was used to analyse the FEA models as was used for the video footage analysis described above.

Push-in and pullout testing
Blocks of bone simulant material (30 PCF Solid Foam Block, Sawbones, Sweden) with height 45 mm and width and depth of 40 mm were used for push-in and pullout testing.Preliminary results found the minimum nominal hole size for insertion of all fastener types without failure to be 9.9 mm, and that the rotating squares, re-entrant and missing rib samples could be inserted into a nominal 9.5 mm hole without failure.The latter was the smallest size of hole tested.These combinations of fastener type and hole size were tested in push-in and pullout from bone simulant material.
Push-in testing of the fasteners was conducted using the same 5 kN Instron to an extension of 40 mm.A test speed of 5 mm min −1 was selected in line with the standard for medical bone screw testing (ASTM F543-17) [26].The forceextension data for one repeat of each sample type was recorded.The remaining samples were installed into the bone simulant blocks using a mechanical Arbor press for time efficiency.Pullout testing was conducted using the Instron machine fitted with mechanical wedge action tensile grips (Instron 2716-015, USA).The test was run for three repeats of each pattern type until the fastener failed or pullout occurred.

Fastener properties
The four structures tested were found experimentally to possess negative Poisson's ratios, confirming their auxeticity.A  4. This is consistent with the FEA results, which are pictured in figure 5.The deformation with increasing compressive strain is shown in figure 6.The FEA data followed the same trends in Poisson's ratio as observed in the experiments with the exception of the Tetrachiral structure which was predicted to be much more auxetic by the numerical analysis compared with the near zero Poisson's ratio observed experimentally (figure 7).The stiffness of the structures was found to vary across a wide range.The stiffness of the rotating squares (12.1 N mm −1 ) and missing rib (29.5 N mm −1 ) types was low compared to the re-entrant (187.1 N mm −1 ) and tetrachiral (200.0N mm −1 ) auxetics, and hexagonal control sample (173.5 N mm −1 ).The FEA models over-estimated the stiffness of all samples but showed equivalent trends to the experiment data (figure 8).The modulus used in the models was found experimentally and it was much lower than both the manufacturer estimation and published independent results from the Tough V5 material (table 2).

Pullout force
All four of the auxetic fasteners were found to achieve a peak force greater than the control sample during pullout from a 9.9 mm hole (figure 9).The rotating squares type achieved the highest mean force of 99.5 N, at a relatively high mean extension of 2.90 mm (5.8 × 10 4 µε).The re-entrant fastener achieved the lowest mean force of the auxetics, of 44.0 N at 0.23 mm mean extension (4.6 × 10 3 µε).A non-parametric Kruskal-Wallis test using SPSS (Version 28, IBM, USA) found the results for all fastener types to be significantly different.This includes the peak force (p = 0.025) and extension results (p = 0.024), as well as the Poisson's ratio (p = 0.026) and stiffness (p = 0.028), although the low sample size n = 3 should be noted.
The pullout curves for each fastener from a 9.9 mm hole are presented in figures 10-14.The rotating squares fasteners (figure 10) were observed to consistently fail at the peak load without pulling out, indicated by several sudden reductions in load with minimal change in extension.The other auxetic fasteners and control pulled out intact, characterised by a sharp increase in load to a peak followed by a gradual decrease with increasing extension, as the fastener slid from the hole.
The pullout force for some auxetic fasteners was found to increase when installed in a smaller hole (9.5 mm diameter).The effect of increased diametral interference on mean peak pullout force and corresponding extension is shown in figure 15.The re-entrant fastener showed the best improvement, increasing by 342% to 194 N. Conversely, the rotating squares type withstood less load before failure than in the 9.9 mm hole case, reducing from 99.5 N to 64.1 N. At this load, the structure was observed to fail.The missing rib structure also failed under the peak load, despite pulling out intact from the 9.9 mm hole.The re-entrant fasteners mostly pulled out intact, with one of the three repeats failing instead.

Pullout to push-in ratio
As a result of the negative Poisson's ratio, auxetic fasteners are characterised by a pullout force which exceeds the pushin force.This effect, shown in figure 16, was observed for installation and removal of the auxetic fasteners from a 9.9 mm hole, except for the tetrachiral type.The rotating squares fastener achieved the highest ratio of 33.7, made possible by the lowest installation force (2.95 N) and greatest removal force.Table 2. Post-cured properties of Formlabs Tough resin material [24].

Material property
Manufacturer data Independent data [25] Experiment data from solid sample

Discussion
This study found that auxetic fixation devices can achieve superior pullout performances relative to a non-auxetic equivalent.The rotating squares type withstood a mean peak removal force 2.5 times the control sample during pullout from a 9.9 mm hole.The structure was found to have a Poisson's ratio of −1.31, confirming its auxeticity.From figure 7, it can be seen that this does not differ significantly from the Poisson's ratio of the re-entrant and missing rib fasteners.However, the tensile stiffness of the rotating squares fastener of 12.1 N mm −1 was observed to be substantially lower than the other types.This implies that, relative to stiffer structures, the rotating squares fastener requires less load to undergo an equal axial extension.This is a possible explanation for the superior frictional force obtained during pullout; for a given load, the axial extension (and corresponding radial extension due to the negative Poisson's ratio) exceeds that of the other auxetic structures.This produces higher contact pressure between the fastener and the bone simulant material, and thus more friction to resist pullout.As the fastener failed before pullout occurred, the frictional force at the fastener-bone interface exceeded the ultimate tensile strength (UTS) of the structure.Localised yielding of the hinge points, implied by the load increasing at a decreasing rate to a peak (figure 10), led the smallest cross section of the fastener to reach a stress level sufficient for failure.With optimisation of the auxetic structure, the UTS could be tailored to exceed the frictional force such that the fastener would pull out before breaking.The tetrachiral fastener obtained the second highest peak load, 197% that of the control, of the auxetic types, pulling out intact from the 9.9 mm hole.The fastener diameter remained approximately constant under strain due to its near-zero Poisson's ratio, allowing the contact pressure at the fastener-bone interface to remain constant.Interestingly, the FEA predicted a tetrachiral structure with high auxeticity (ν = −0.6)which deviated significantly from the near-zero Poisson's ratio observed in the experiment.There is debate within literature about the tetrachiral structure, and whether its Poisson's ratio is positive or negative.An analytical solution studied by Mousanezhad et al suggests a zero Poisson's ratio independent of unit cell geometry, while FEA showed a tetrachiral structure, with a similar r/R ratio but comparatively low unit cell relative density, to possess a positive value (ν = 0.2) [27].It appears that tetrachiral structures exhibit behaviour which is not being captured by standard FEA analyses.The FEA was able to predict the overall trends in all of the other structures but overestimated the stiffness values and underpredicted the auxeticity (with the exception of tetrachiral).The error is likely to be an effect of the SLA manufacture not perfectly replicating the CAD geometry, variability in the material properties resulting from curing parameters [24], and measurement error from the experimental method.Another cause could be the high unit cell relative density     (60%), used to maintain consistency to the other lattice structures while taking into consideration manufacturing capabilities and strength requirements.This is significantly greater than the critical value of 29% recommended by Gibson and Ashby to allow for ligament bending, which is the dominant deformation mode for honeycomb structures [28].In that context, the through thickness dimension is much greater, but the in-plane behaviour is comparable.
A day-to-day drawback of the screws and anchors that are currently used is that the greater the axial load they are subject to, the less axial resistance there is.The control sample, with a positive Poisson's ratio of 1.41, was observed to contract significantly across its diameter under loading.This effect, visible in figure 4, reduces the contact pressure and subsequently the frictional resistance to pullout.
The pullout curves for the auxetic re-entrant and missing rib fasteners, shown in figures 11 and 12 respectively, typically show a greater peak force relative to the remainder of the curve than the non-auxetic control.This implies that the auxetic structure allows a higher pullout force to be achieved, but when the limiting friction is overcome and the fastener begins to slide, the fastener diameter contracts as the tensile 13.Pullout force of the tetrachiral fastener from a 9.9 mm diameter hole.axial strain is released.This reduces the contact pressure at the fastener-bone interface, resulting in decreased frictional resistance to pullout.It is speculated that this contraction of the diameter is represented by the decreasing rate of load reduction from the peak, as the diameter expands to its relaxed size.This is followed by a linear reduction in load as the fastener is removed from the hole.This effect is more clearly seen in the missing rib type than the re-entrant fastener, due to the difference in stiffness.The missing rib fastener has a relatively low axial stiffness of 29.5 N mm −1 in comparison to the re-entrant stiffness of 187.1 N mm −1 (figure 8).This combined with the similar Poisson's ratio of the fasteners, of −1.23 and −1.31 for the missing rib and re-entrant types respectively, implies that per unit load, the expansion of the missing rib fastener diameter exceeds that of the re-entrant fastener.This enhanced auxetic effect before pullout results in a more significant reduction in contact pressure and friction when the load is removed.This is supported by the large mean fastener extension at peak load for the missing rib fastener of 0.93 mm (1.9 × 10 4 µε) relative to the re-entrant value of 0.23 mm  (4.6 × 10 3 µε) (figure 9).Despite this, the mean pullout force of the two was similar, at 44.0 N and 46.3 N for the re-entrant and missing rib types respectively.
An additional benefit of some of the tested designs is that the force required for insertion is minimal due to the auxetic effect, indicated by a ratio of pullout force to insertion force greater than one.The rotating squares fastener obtained the highest pullout to push-in force ratio of 33.7, aided by the low structural stiffness.The low insertion force of this fastener is a desirable operative feature, as it avoids large loads being exerted on the bone and improves ease of installation for the surgeon.By pre-compressing the auxetic fastener prior to insertion such that the hole diameter exceeds the fastener diameter, insertion force could be eliminated.Insertion of the missing rib fastener into the hole was made difficult by twistaxial coupling under compressive loading, although the same effect may be less significant in a metallic fastener with higher stiffness.The pullout to push-in ratio of the tetrachiral fastener was 0.53, which typically implies non-auxetic behaviour.With a near-zero Poisson's ratio, a pullout to push-in ratio of one is expected, and the reason for discrepancy in this case is not clear.From figure 9, the tetrachiral fastener can be seen to pull at a low mean extension of 0.25 mm (4.9 × 10 3 µε) relative to some other auxetic fasteners.This implies that pullout occurred shortly after the test began with little deformation of the structure.The shape of the pullout curve, given in figure 13, tends to agree with this, showing a sudden peak in load as the resistive friction is overcome, followed by a reasonably linear load decrease.Providing the contact pressure remains constant, this linear relationship is expected as the fastener slides out of the bone.This effect is a result of the decreasing fastener length within the hole reducing the frictional surface area.Similarly, an explanation for the significant reduction in load from the peak to the linear region is that the dynamic coefficient of friction is typically less than that of static friction [29], with the abrupt drop in load representing the transition phase between the two.A similar trend is shown by the pullout curve for the hexagonal control in figure 14.The pullout to push-in force ratio of the re-entrant and missing rib fasteners was 2.00 and 1.63 respectively, with the missing rib fastener showing a 5.29% improvement in pullout force for a 29% higher insertion force than the reentrant fastener.This aligns with the auxeticity of the structures, with the re-entrant structure having a negative Poisson's ratio of marginally greater magnitude than the missing rib type.
Increasing diametral interference was found to increase fastener pullout force, providing the resulting heightened stresses in the structure do not exceed the material strength.The increased initial friction provided by the interference fit allows the fastener to achieve a higher axial strain before pullout occurs, and in the case of auxetic fasteners, greater radial expansion.This was investigated by installing the fasteners into a hole of nominal diameter 9.5 mm, and all fastener types achieved peak load at a higher extension during pullout from the smaller hole.The rotating squares fastener, withstanding the highest peak load during pullout from the 9.9 mm hole, broke at a 35.6% (mean) lower force during this test, shown in figure 15.This was attributed to heightened stresses in the fastener due to the increased diametral interference.The resulting radial stresses in the fastener act in conjunction with the tensile stress, exceeding the material strength at lower applied tensile load and subsequently leading to failure.This was confirmed by the observed failure mode of these samples, which failed near to the embedded portion of the fastener due to the increased stress in this location.A similar trend was observed with the missing rib samples, failing parallel to the surface of the Sawbones block at the internal corner subject to angular expansion.The mean failure load of the missing rib fastener increased by 197% from the 9.9 mm case to 137 N, occurring when the fastener broke.During tensile testing, the missing rib unit cell was seen (figure 4) to rotate anti-clockwise in addition to the opening of the 90 • angle between the ligaments.It is suspected that increased friction prevented this rotation, introducing shear stress within the ligaments at the surface of the bone where the structure is free to rotate.In fact, the reentrant fasteners were the only type to pull out intact from the 9.5 mm hole, with the mean pullout force increasing by 342% from the 9.9 mm hole case.The pullout force of 194 N was the highest achieved in this study, with the best performing structure matching the findings of Yao et al; the group found that the highest pullout force from high density (3.5 × 10 −4 g mm −3 ) bone was achieved by the auxetic screw containing a reentrant structure [2].The density of the bone simulant material used in this work exceeds that of the highest density cancellous bone [2] investigated, making the result the fairest comparison.
There are some limitations with this study.Future work should focus on optimising the dimensions and reinforcement of the rotating squares structure to achieve improved strength while maintaining a negative Poisson's ratio and low axial stiffness.Similar optimisation of the re-entrant structure is recommended due to the pullout performance improvements with increased diametral interference.Additional optimisation through simulation and physical studies would enable stresses within the part to be considered so that the maximum interference fit can be achieved without material failure.The failure mode of metallic samples in natural bone could be more closely replicated using polymer samples by using bone simulant material with reduced stiffness, to maintain the ratio of bone-to-implant stiffness.Following this, metallic samples should be manufactured and tested in natural bone to validate fastener performance.The fasteners could be further improved by adding texture to their surface so that the friction with the bone surface and resistance to pullout is increased.Computerised tomography scans could be implemented at this stage to improve understanding of the interactions at the fastener-bone interface.Alternative insertion methods and failure mechanisms for auxetic fasteners that align with clinical practice could also be explored, although the failure model tested here (pullout) is by far the most clinically relevant, given the modality in which such anchoring devices would be used [30].For metallic samples, manufacturing quality and defects at the necessary scale will be a greater challenge than for polymers and distinct causes of failure, e.g.fatigue, will need to be considered.In a clinical setting, an applicator for the fastener would likely need to be developed, which could introduce additional risks.Separate studies would need to assess the options for such an applicator.Future tests should aim to reduce the sources of error identified in this study, by developing a specialised jig to ensure accurate alignment of samples during pullout, and use of a camera with higher resolution to determine the Poisson's ratio.

Conclusion
All the auxetic structures investigated showed improved pullout resistance relative to a non-auxetic control sample.The rotating squares structure showed the maximum pullout force of 9 9.5 N, 2.5 times that achieved by the control.It also showed the best pullout to push-in ratio of 33.7, requiring low force for insertion and improving ease of installation.The structure had a mean Poisson's ratio of −1.31, and a comparatively low axial stiffness of 12.1 N mm −1 which was identified as the reason for its superior pullout performance.
Increasing the interference between the fastener and hole improved the pullout resistance of the re-entrant and missing rib types, with the premature failure of the rotating squares fastener likely due to the resulting additional radial stresses.The overall trends of Poisson's ratio and stiffness found by experiment were as predicted by FEA, except for the tetrachiral structure which showed lower auxeticity than expected.This was attributed to efforts to maintain a constant density for all samples which reduced the influence of bending deformation.At the same time, discrepancies between FEA and experiments highlight gaps in our understanding of the underlying material properties and manufacturing repeatability.The findings in this study demonstrate the potential of auxetic devices in bone fixation, not only from their ease of insertion, but their improved failure performance.Future work using metallic materials will enable further exploration of the potential impact of these fixation designs in advancing orthopaedic surgery.Realization of these structures in a clinical setting requires additional work on manufacturability, scaling and durability.

Figure 1 .
Figure 1.Schematic overview of the evaluation process for the auxetic fasteners.

Figure 4 .
Figure 4.The deformation of the five physical samples under increasing axial extension.

Figure 5 .
Figure 5. Predicted maximum principal strain and shape change from FEA simulation.Images show deformation between 0% and 16% tensile axial strain in evenly spaced intervals.

Figure 6 .
Figure 6.Predicted maximum principal strain and shape change from FEA simulation.Images show deformation between 0% and 16% compressive axial strain in evenly spaced intervals.

Figure 7 .
Figure 7. Experimentally measured Poisson's ratios (n = 3) compared with FEA predictions for the different fastener types.

Figure 8 .
Figure 8. Experimentally measured stiffness values (n = 3) compared with FEA predictions for the different fastener types.

Figure 9 .
Figure 9. Peak removal force from a 9.9 mm hole and corresponding extension comparison between fasteners.

Figure 10 .
Figure 10.Pullout force of a rotating squares fastener from a 9.9 mm diameter hole.

Figure 11 .
Figure 11.force of the re-entrant fastener from a 9.9 mm diameter hole.

Figure 12 .
Figure 12.Pullout force of the missing rib fastener from a 9.9 mm diameter hole.

Figure 14 .
Figure 14.Pullout force of the hexagonal (control) fastener from a 9.9 mm diameter hole.

Figure 15 .
Figure 15.Effect auxetic fastener pullout performance when pilot hole size decreased from 9.9 mm to 9.5 mm.

Figure 16 .
Figure 16.Push-in and pullout force for the fasteners in a 9.9 mm hole.

Table 1 .
Parameters for the structures under evaluation.The hexagonal control sample was found to have a Poisson's ratio of 1.41.The effect of Poisson's ratio on the deformation of each structure with increasing axial strain is pictured in figure •θ2 90• ζ 90 • φ 45Poisson's ratio of −1.31 was identified for both the rotating squares and re-entrant types.The missing rib structure possessed a Poisson's ratio of −1.23, and the tetrachiral near-zero.