A Review on Auxetic Structures and Polymeric Materials

Abstract.

Auxetic materials have been investigated since the 1990's, they are defined by a negative Poisson's ratio which gives them interesting capabilities in energy and shock absorption. They are usually lattice structures (2d or 3D array of unit cells), which gives them high stiffness to weight backdrop and their properties can exist tuned to adapt a given application. Several papers accept been published on new auxetic designs, their static equivalent mechanical properties, daze absorption properties and their manufacturing. Their dynamic behavior has not been studied to engagement and this is the main purpose of this paper. We will present a novel design that facilitates 3D printing with SLA printers and then examine the dynamic behavior of an auxetic structure between 0 and one kHz. In order to explain the unusual behavior of the structure, an initial theoretical model of not-linear bound has been adult with fractional success.

Keywords: auxetic, structures, dynamic, vibrations, frequency response, metamaterial, 3D printing, novel blueprint, non-linear leap.

1. Introduction

Auxetic materials are materials that take a negative Poisson's ratio (they contract or expand in any spatial direction when exposed to a stress), they well-nigh not be found in nature and their behavior emerges from their macro-structure. On a microscopic point of view, they are fabricated from a majority textile of positive Poisson's ratio.

They take been studied since the belatedly 1980s [1] for their expert properties in shock absorption, fracture toughness, indentation resistance and free energy dissipation. The auxetic properties often sally from acute angles in a lattice structure (re-entrant hexagon for example) or spiral geometry.

The geometry presented in this paper is the hexagonal re-entrant structure, i of the most normally studied [ii-vii] as it can be defined with very few parameters different other more complex structures. This geometry was kickoff invented in 2D only with the development of additive manufacturing was extended to 3D. Samples studied in the literature were usually produced using SLM [8, 9] or other powder bed fusion methods, this produced expert quality samples, but at loftier cost. The advantage of powder bed fusion methods is that the pulverization supports the structure equally it is printed.

The current literature focuses on prediction of the structure's stiffness, the energy absorption [10], fracture toughness [eleven] and quasi-static properties [eight]. To the author'southward knowledge, no study has been proposed that investigates their dynamic response.

This newspaper will therefore propose modifications on the lattice pattern to make it hands 3D printable with SLA printers (photo-sensible polymer resin that hardens with UV light). Enabling 3D press with SLA printers may accelerate research on those structures as samples can be obtained for a very depression cost (300 $ printers / 60 $ per liter of resin) and manufactured in whatever laboratory without specific investment. Their static properties accept been investigated and compared to analytical models to assess the generalizability of results. The showtime requirement to enable SLA based press methods is that the lattice is cocky-supporting during manufacture.

We will nowadays a dynamic study of this auxetic construction, studying the influence of two parameters on the lattice frequency response function (FRF): the strut diameter and the mass attached to the sample. Dynamic results are a novelty for auxetic structures so we will initially endeavour to use known analytical models to explain the construction's behavior.

2. Novel structure

This blueprint varies from the usual design used in previous studies as it has been developed to be 3D printed on an SLA printer. The vertical beams accept been modified and are at present not flat as shown in Fig. i.

Thus, they will not be printed in one layer, reducing the overhanging section, a recurrent trouble for SLA printers (a non-self-supportive press technique unlike SLS or SLM). The standard strut design was rectangular on most hexagonal re-entrant structures [12], this novel structure has cylindrical beams which reduces the number of parameters required to fully describe the structure. The vertical beam modification is washed via the add-on of another beam between the cell corner and a 16th of the angled axle.

Every bit a side issue of the horizontal border's modification, the beams junctions are at present stronger (the horizontal struts are thicker at their ends). The stress concentrations located at the beam junctions pointed out by Carneiro et al. [13] by an FEA analysis of a second hexagonal re-entrant structure tin can be reduced [4]. This may be an interesting feature to enhance the lattice stiffness and fatigue strength without adding much material and complexity. Information technology does also enable the production of the hexagonal re-aspirant structure past SLA printers which is a major improvement as it drastically reduces the cost of production and may aid further research on this topic.

Fig. 1. Novel lattice design for SLA printing: a) full sample with support, b) single prison cell

a)

b)

3. Methodology – experimental procedure

Several experimental procedures have been used, two different methods take been developed to measure the frequency response function (FRF) and make sure they were consistent, other tests have been conducted in a tensile/compression bench to evaluate the mechanical backdrop of the majority material and the lattice stiffness.

3.i. Dynamic experiments

The measurements are performed nether dynamic loading, the sample is fastened by one stop to a shaker and a mass ($g=$0.124 kg) is fixed to the other end. The output is measured by a light amplification by stimulated emission of radiation vibrometer, the Polytech PDV100, that measures velocity with high accurateness. Both apparatuses are controlled via MATLAB and a NI-4431 acquisition card, the input is generated by MATLAB and converted past the conquering carte into a variable voltage. The output variable is the free end velocity of the sample, there is no need to integrate this variable in order to get the displacement as this would but result in a 90° phase in the Fourier domain. The sampling rate is set at 48000 Hz, the Fourier spectrum is however only exploitable up to 4 kHz at best due to noise in the measurement.

The transfer function of the shaker is evaluated in a carve up experiment and used to normalize the measured FRF and account for organization functioning.

The offset method uses a white dissonance as input, the normalized output's Fourier transform is then compared to the input: the ratio of the 2 spectrum is the FRF of the sample. This method is the most widely used in this paper equally it gives a very clean spectrum between 0 and 1 kHz, the noise starts to boss at loftier frequencies.

The 2d method uses a sinusoid input and goes through the desired frequency range step by step. The amplitude of the output is then filtered and normalized past the same procedure as above (a reference measurement was conducted without sample) and plotted. This measurement method takes longer than the "noise measurement" as each frequency is tested one past one only enables an arbitrarily high frequency resolution.

3.ii. Quasi static tests

Pinch tests have been performed on the lattice to measure out their stiffness, they are performed using an INSTRON 3366, the testing speed is 1mm/min. All samples from a strut diameter of 0.5 mm to 1.2 mm accept been tested three times in order to quantify the experimental dubiety. Only the linear region of the stress-strain curve is considered, the tests are not destructive: they are stopped before the deformation becomes plastic (load lower than fifty N). As the deformation is small-scale enough and the stiffness is calculated in a linear regime, we can presume the tensile modulus is similar to the compressive modulus.

four. Results

four.ane. Static tests

We compared the experimental results to two theoretical models developed past Yang et al. [14] and Shokri et al. [xv] that predict the equivalent tensile modulus of an hexagonal re-aspirant lattice. The first assumes only two modes of deformation are significant: it uses a Timoshenko beam model to describe the angled beam deformation and a classic compression model for the vertical axle. The second author uses Castigliano'southward theorem to calculate the tensile backdrop, the change in strain free energy caused by a force divided past the force itself is equal to the displacement. Note that both methods need to be adapted for cylindrical beams (dissimilar cross department area, moment of inertia and Timoshenko beam factor). Although the two approaches are radically different, they requite most similar results as seen in Fig. 2. The novel hexagonal re-aspirant structure is comparable to previously studied designs: its mechanical properties can be predicted from the analytical models adult for previous designs.

Fig. ii. Compressive tests results comparison to ii analytical models

iv.2. Dynamic experience – strut diameter influence

From the range of samples produced (7 samples from 0.5 mm strut diameter to 1.ii mm), we observed the FRF and compared them, the results are shown Fig. 3. The frequency response of an auxetic sample has a clear resonance frequency that can be observed in both the amplitude and phase spectrum. The peak width varies linearly with the superlative frequency and their width at –3 dB is on average 32 % of the meridian frequency.

Fig. 3. Strut bore influence on the FRF

We can see that each sample has a distinct pinnacle corresponding to a resonance phenomenon in the structure, the stiffer the structure, the higher the resonance frequency. Nosotros can plot the resonance frequency VS strut diameter equally shown in Fig. 4 to run into quantitively how the strut diameter affects the dynamic behavior of the structure. We can see a clear linear tendency between the resonance frequency and the strut bore.

Fig. four. Strut diameter influence on the resonance frequency

We can first try to explain that trend using a usual linear jump-mass model. As the resonance frequency is $f_{re\mathrm{due\; south}}=\sqrt{\mathrm{grand}/\mathrm{grand}}$ if $k\sim t^2$, we find $f_{re\mathrm{south}}\sim t$. The 2d equation tin be hands justified as the strut cross sectional surface area is proportional to $t^2$. This suggests the structure works more in axle pinch than in angle (moment of inertia of a beam is proportional to$t^{\mathrm{iv}}$). This consideration is nonetheless incorrect as Yang et.al analytical model suggests the beam compression is simply responsible for 5 to 25 % of the lattice deformation (5 % for $d=$0.5 mm, 25 % for $d=$i.2 mm).

Furthermore, if we calculate $F=\sqrt{\mathrm{grand}/m}$ with $g$ being the measured static stiffness, we find the right trend just in that location is a factor ~2 between the calculated resonance frequency and the dynamic experimental results.

4.3. Mass variation

Equally we saw previously, the linear spring mass model cannot explain the diameter-resonance frequency relationship. We varied the stiffness and observed a discrepancy in the results, in this part, we varied the mass and observed the variation in resonance frequency. Results are presented in Fig. five along with the predictions of a non-linear model which nosotros will discuss in the next section. The experiments were conducted for various masses between 124 yard (fixation weight) and well-nigh 500 g (maximum allowed weight for the shaker).

Fig. 5. Mass variation results and non-linear leap model predictions

v. Discussion

The results obtained in department 3.3 cannot be explained by a linear mass-jump model: the variation in resonance frequency does not stand for to any possible prediction of this model if the stiffness is kept constant (for the aforementioned sample). If we calculate a "virtual stiffness" for each experiment using $\mathrm{thousand}=f^{2}m$ in that location is a clear linear relation between this fictional parameter and the mass (with 0 intercept). There is nevertheless a model that explains those unusual results almost perfectly as seen in Fig. 5, it is based on a non-linear spring with governing equation $F=\mathrm{southward}\mathrm{i}\mathrm{one\; thousand}\mathrm{northward}\left(\mathrm{\Delta }\right)\mathrm{thou}\text{'}{\mathrm{\Delta }}^{\mathrm{iv}}$. The $\mathrm{south}\mathrm{i}\mathrm{g}\mathrm{due\; north}\left(\right)$ function here is used to forcefulness $F\left(\mathrm{\Delta }\right)$ to be odd, this equation is therefore not defined the same mode for positive and negative deflections. This is one of the reasons why information technology is simply not possible to solve the governing differential equation of the mass-spring arrangement, the only way to discover the resonance frequency is by solving it numerically. We used the ode45() function on MATLAB to simulate the beliefs of this non-linear system with 0-initial deflection and dispatch and one-initial velocity. Information technology is worth noticing that such a organisation has an unusual behavior and its resonance frequency varies with the initial velocity, the calculated stiffness thou' has therefore no physical sense here as it has been arbitrarily set by choosing an initial velocity of 1 (setting one will force the other parameter during the optimization process to fit the results). To our knowledge in that location is no theoretical justification that the lattice cloth behaves like a not-linear fabric, especially for very modest amplitude vibrations. As we saw, this model has still a bang-up predictive ability, this unusual dynamic beliefs may be due to dissipative furnishings in the construction'southward bulk fabric that could be non-linear (a viscoelastic model could perhaps be used to describe the resin dynamic beliefs).

vi. Conclusions

A novel structure has been adult with success, it enables 3D press with SLA printers and has mechanical backdrop comparable to previously adult structures. It tin can exist described by very few parameters: strut bore, cell size (the cell fills a cubic book here, and so a single parameter is needed), vertical length ratio and additional strut location parameter (1/sixteen here) are sufficient to fully depict our novel design. It has been experimentally verified that analytical models developed for previous structures also predict the mechanical properties of our design with good accuracy.

2 dynamic methods accept been used to determine FRF of auxetic samples: they requite like amplitude spectrum for unlike samples which validates them. The results obtained with those method cannot exist explained with a simple mass-spring organisation (sample acting equally a linear leap): fifty-fifty though the global trend of the linear system prediction is similar to the of the experimental results, there is an important multiplicative factor betwixt both methods. Furthermore, when we vary the mass attached to the complimentary end of the sample, the modify in resonance frequency is not explained by a linear spring model. Even though there is no theoretical justification to the use of a polynomial spring model, it happens that a non-linear spring with governing equation $F=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{northward}\left(\mathrm{\Delta }\right)\mathrm{1000}{\mathrm{\Delta }}^4$ explains virtually perfectly the experimental results. Dissipative furnishings inside the resin that are not considered in the theoretical models may play a significant role in the dynamic behavior of the sample and be absolutely negligible in quasi-static behavior. This would accept to be verified in further studies, which would be highly facilitated by the use of SLA printed samples, which are much easier to produce in significant quantities in a relatively brusque time and at a moderate toll.

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