William M Campbell
email: william.campbell@uwa.edu.au

Michael E Tobar

Maxim Goryachev
email: maxim.goryachev@uwa.edu.au

Serge Galliou

Improved Constraints on the Minimum Length with a Macroscopic Low Loss Phonon Cavity

Many theories that attempt to formulate a quantum description of gravity suggest the existence of a fundamental minimum length scale. A popular method for incorporating this minimum length is through a modification of the Heisenberg uncertainty principle known as the generalised uncertainty principle (GUP). Experimental tests of the GUP applied to composite systems can be performed by searching for the induced frequency perturbations of the modes of mechanical resonators, thus constraining the degree of minimum length in certain scenarios. In this work previous constraints made with mechanical resonators are improved upon by three orders of magnitude, via the utilisation of a cryogenic quartz bulk acoustic wave resonator. As well as purely mechanical resonant modes; hybrid electromechanical anti-resonant modes are investigated, and shown to be sensitive to the same GUP induced effects.

I. INTRODUCTION

The existence of a fundamental minimum length scale, on the order of the Planck length l p = 1.62 × 10 -35 m, is a prevalent idea in many theoretical models that contain a quantum description of gravity [START_REF] Garay | Quantum gravity and minimum length[END_REF][START_REF] Hossenfelder | Minimal length scale scenarios for quantum gravity[END_REF]. Such a concept can be formulated in the context of non-relativistic quantum mechanics through modifications of the Heisenberg uncertainty relations. So called generalised uncertainty principles [START_REF] Maggiore | A generalized uncertainty principle in quantum gravity[END_REF][START_REF] Amati | Can spacetime be probed below the string size?[END_REF][START_REF] Jizba | Uncertainty relation on a world crystal and its applications to micro black holes[END_REF][START_REF] Scardigli | Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment[END_REF][START_REF] Amelino-Camelia | Gravity-wave interferometers as probes of a low-energy effective quantum gravity[END_REF][START_REF] Kempf | Hilbert space representation of the minimal length uncertainty relation[END_REF] provide a model independent way of introducing minimum length whilst recovering ordinary quantum mechanics on larger scales. GUP phenomenology has been subject to much theoretical examination [START_REF] Chang | Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations[END_REF][START_REF] Brau | Minimal length uncertainty relation and the hydrogen atom[END_REF][START_REF] Hossenfelder | A note on theories with a minimal length[END_REF][START_REF] Kempf | Minimal length uncertainty relation and ultraviolet regularization[END_REF][START_REF] Lewis | Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relation[END_REF][START_REF] Ching | Generalized coherent states under deformed quantum mechanics with maximum momentum[END_REF][START_REF] Pedram | Coherent states in gravitational quantum mechanics[END_REF][START_REF] Nozari | Gravitational induced uncertainty and dynamics of harmonic oscillator[END_REF][START_REF] Bosso | The minimal length is physical[END_REF], yet critical insights from experiment are highly limited due to the scales at which the introduced modifications induce an observable effect. The GUP may take the general form

∆x∆p ≥ 2 1 + β 0 l p ∆p 2 , (1)
where ∆x and ∆p denote the measurement uncertainties in the physical coordinate x and corresponding conjugate momentum p and β 0 is a dimensionless, model-dependent parameter that defines the minimum uncertainty implied by [START_REF] Garay | Quantum gravity and minimum length[END_REF]; ∆x min = √ β 0 l p . The parameter β 0 is usually assumed to be close to unity, which leads to corrections to measurement uncertainties that only become relevant at lengths l ∼ l p or energies E ∼ c lp = 1.2 × 10 9 GeV. Due to such extreme scales, direct tests of (1) are challenging. However, β 0 may be constrained by measuring the perturbations to the ground state energy of a harmonic oscillator implied by [START_REF] Garay | Quantum gravity and minimum length[END_REF]. Recently, resonant mass gravitational wave experiments, such as AURIGA exploited this effect to place upper bounds β 0 < 10 33 [START_REF] Marin | Gravitational bar detectors set limits to planck-scale physics on macroscopic variables[END_REF], however these bounds are still far from the expected Planck scale. Tighter bounds can be achieved by considering the application of this phenomenology to quantum mechanics. Extending this formalism; one can associate a deformed Heisenberg algebra to the GUP [START_REF] Garay | Quantum gravity and minimum length[END_REF] [x, p] = i 1 + β 0

l p p 2 , (2)
the introduction of which immediately has consequences to the energy spectrum for single particle quantum systems. Further constraint on β 0 has thus been made by looking for corrections to the Lamb shift, and Landau levels of scanning tunnelling microscopes [START_REF] Das | Universality of quantum gravity corrections[END_REF], implied by the reconstructed quantum mechanics of (2). However these constrains only manage an upper bound of β 0 < 10 20 .

A different approach to excluding minimum length can be found when considering the application of the deformed Heisenberg algebra (2) to the centre of mass modes of composite particle systems. Such systems typically exhibit larger momenta, increasing the magnitude of GUP induced effects. Interestingly, the exact implications of extending this formalism to the centre of mass mode of macroscopic systems is still unclear, with some arguments suggesting that the strength of minimum length induced effects may be suppressed by the total number of composite particles [START_REF] Amelino-Camelia | Relative locality and the soccer ball prob-lem[END_REF][START_REF] Amelino-Camelia | Challenge to macroscopic probes of quantum spacetime based on noncommutative geometry[END_REF][START_REF] Kumar | On quantum gravity tests with composite particles[END_REF]. However, it has been remarked [START_REF] Bawaj | Probing deformed commutators with macroscopic harmonic oscillators[END_REF] that the exact constituent level at which quantum gravitational effects intervene is completely unknown. Thus, further dedicated experimental investigation into these effects, under the assumption that (2) holds for the centre of mass mode of macroscopic systems, is necessary. Dedicated investigations into minimum length have recently been conducted that search for macroscopic GUP induced effects such as the broadening of molecular wave-packets [START_REF] Villalpando | Indirect probe of quantum gravity using molecular wave-packets[END_REF], various optomechanical effects [START_REF] Pikovski | Probing planck-scale physics with quantum optics[END_REF][START_REF] Kumar | Quantum-optical tests of planck-scale physics[END_REF][START_REF] Bosso | Amplified transduction of planck-scale effects using quantum optics[END_REF][START_REF] Girdhar | Testing generalised uncertainty principles through quantum noise[END_REF], and perturbations to the resonant modes of mechanical resonators [START_REF] Marin | Gravitational bar detectors set limits to planck-scale physics on macroscopic variables[END_REF][START_REF] Bawaj | Probing deformed commutators with macroscopic harmonic oscillators[END_REF][START_REF] Bushev | Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums[END_REF].

In this letter, previous tests utilising mechanical resonators are improved upon. The minimum length order parameter β 0 is constrained by measurement of the non-linear frequency perturbations to the resonant modes of a macro-mechanical quartz bulk acoustic wave (BAW) resonator. Such resonators exhibit extraordinarily high acoustic quality factors at cryogenic temperatures [START_REF] Galliou | Extremely low loss phonontrapping cryogenic acoustic cavities for future physical experiments[END_REF][START_REF] Lo | Acoustic tests of Lorentz symmetry using quartz oscillators[END_REF], and have long been used as timing standards with impressive short to mid-term frequency stability. This inherent frequency precision coupled with a gram-scale mass makes the quartz BAW resonator an ideal technology to probe minimum length induced effects, as well as performing other tests of fundamental physics [START_REF] Campbell | Searching for Scalar Dark Matter via Coupling to Fundamental Constants with Photonic[END_REF][START_REF] Goryachev | Rare Events Detected with a Bulk Acoustic Wave High Frequency Gravitational Wave Antenna[END_REF][START_REF] Goryachev | Next Generation of Phonon Tests of Lorentz Invariance Using Quartz BAW Resonators[END_REF]. The results presented in this work improve upon the upper bounds on β 0 placed by previous mechanical resonators experiments [START_REF] Bushev | Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums[END_REF] by up to three orders of magnitude.

II. MODEL

In order to describe the perturbations to a mechanical oscillator due to the existence of a minimum length; it is commonly assumed [START_REF] Bawaj | Probing deformed commutators with macroscopic harmonic oscillators[END_REF][START_REF] Pikovski | Probing planck-scale physics with quantum optics[END_REF][START_REF] Bushev | Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums[END_REF] that any two conjugate position and momentum observables, attributed to the centre of mass mode of a composite system, obey the new deformed commutator of equation [START_REF] Hossenfelder | Minimal length scale scenarios for quantum gravity[END_REF]. Whilst the form of the resonator's classical Hamiltonian remains unchanged. It then follows that a suitable representation for the physical position and momentum operators x, p in terms of some general operators X, P can be chosen such that the canonical commutation relation [ X, P ] = i , is recovered. One such representation can be described by the transformation [START_REF] Quesne | Composite system in deformed space with minimal length[END_REF] x, p → X, P (1

+ β 0 1 3(m p c) 2 P 2 ), (3)
where m p denotes the planck mass. It has been shown [START_REF] Bawaj | Probing deformed commutators with macroscopic harmonic oscillators[END_REF] that this representation leads to a non-linear correction to the resonator's Hamiltonian, resulting in a dependence of the device's perturbed resonant frequency ωr on it's oscillation amplitude x 0 :

ωr = ω r 1 + β 0 2 m eff ω r x 0 m p c 2 . (4)
Equation ( 4) immediately applies to the frequencies of the purely mechanical resonant modes of the quartz BAW resonator. However, the piezoelectric nature of the device allows for the further study of anti-resonant modes in which an electrical half degree of freedom is contributed by the capacitive electrodes and the external circuit. Application of the same GUP formalism to these antiresonant modes can be understood by first considering the Butterworth-VanDyke model of the resonator in the vicinity of a resonant mode λ, in which the piezoelectric crystal is described by a lumped element electrical circuit consisting of a series RLC branch shunted by the capacitance of the external electrodes. An example can be seen in the shaded blue region of figure 1. Assuming a linear coupling; the charge form equation

q = κ λ x (5)
can be introduced to relate the electrical charge variable q back to the mechanical displacement x of the crystal plate. This allows for full interchangeability between lumped element electrical and mechanical descriptions of the resonator.

In order to explore the GUP phenomenology of the antiresonant modes of this system; it is helpful to employ a Hamiltonian formalism to the electrical circuit description. This allows for GUP induced effects to be introduced in a fully consistent manner. Following the circuit quantisation procedure by Vool et al [START_REF] Vool | Introduction to quantum electromagnetic circuits[END_REF] the Hamiltonian of the lossless resonator can be written as

H = q2 m 2C λ + q2 e 2C 0 + φm -φe 2 2L λ . ( 6
)
Where φi is the magnetic flux node operator corresponding to the circuit node i with conjugate node charge qi . These node operators are defined such that for a lumped element E connected to nodes i and j the magnetic flux of that element φ E = φiφj , with a similar definition for the conjugate charge q E . The motional node denoted by m is the intersection of the motional inductance L λ and capacitance C λ , whereas the electrical node e connects L λ and the electrode capacitance C 0 . With the motional resistance R λ made to vanish; the remaining circuit node is then identified as the electrical ground g such that φg , qg = 0. The Hamiltonian of equation ( 6) generates two sets of coupled equations of motion, one for each pair of variables φi , qi . Substituting these coupled equations into each other gives

φL + ω 2 a φL = 0, ω a = C λ + C 0 C λ C 0 1 L λ , (7)
where φL = φmφe . This differential equation clearly generates oscillatory solutions with frequency ω a corresponding to the anti-resonant modes of the circuit. Conversely, by shorting the electrical node to ground such that φe , qe = 0, we recover the equations that lead to the resonant solutions;

φL + ω 2 r φL = 0, ω r = 1 L λ C λ . ( 8
)
This distinction can be understood as coupling an ideal port to the electrical node of the circuit with either infinite output impedance giving anti-resonant solutions, or zero output impedance shorting the electrical node to ground and giving resonant solutions. In a physical dissipative system, both modes can be coupled to.

Employing the deformed Heisenberg algebra of equation ( 2) and utilising the charge form [START_REF] Jizba | Uncertainty relation on a world crystal and its applications to micro black holes[END_REF]; it follows that a suitable representation for the physical charge and flux operators q, φ in terms of some general operators Q, Φ, that recovers the canonical commutation relation Q,

Φ = i is q, φ → Q, Φ(1 + β 0 1 3(m p c) 2 κ λ Φ2 ). (9)
This representation thus perturbs the structural form of the Hamiltonian of equation ( 6), to leading order in β 0 , by the addition of the term ( Φm -Φe ) 4 /L λ , where = β 0 κ λ /3(m p c) 2 . The Heisenberg equations of motion in this new representation can be found to be

Qm = Φm -Φe L λ 1 + 4 Φm -Φe 2 (10a) Φm = Qm C λ (10b) Qe = Φm -Φe L λ 1 + 4 Φm -Φe 2 (10c) Φe = Qe C 0 (10d) (10e)
where the hat to denote an operator has been omitted for ease of notation. Solving these coupled differential equations for Φ L (t) = Φm (t) -Φe (t) via Poincare's method [START_REF] Strogatz | Nonlinear Dynamics and Chaos[END_REF], and assuming 1, gives a solution with a perturbed anti-resonant frequency

ωa = ω a 1 + β 0 2 κ λ Φ L m p c 2 . (11)
This relation can be converted back into the mechanical description by the substitution of κ λ Φ = P = m eff ωX.

It is thus follows that under the GUP formalism the electromechanical anti-resonant modes of the quartz resonator experience the exact same degree of amplitude dependent frequency perturbation as that of the purely mechanical resonant modes described by equation ( 4).

For completeness; if the electrical node is grounded such that Φe , Qe = 0, then the solution to the resulting differential equation from [START_REF] Brau | Minimal length uncertainty relation and the hydrogen atom[END_REF] is that of the resonant mode with frequency given by equation ( 4). Non-linear effects in quartz associated with high order elasticity terms in the crystal Hamiltonian induce a similar amplitude dependent frequency shift. This is known as the the amplitude-frequency effect and has been extensively studied in quartz resonators [START_REF] Tiersten | An analysis of nonlinear resonance in contoured-quartz crystal resonators[END_REF]. As a result an observed frequency perturbation due to the deformed commutator cannot be decoupled from that due to the crystal's inherent non-linearity. However, an upper bound on β 0 can be determined up to some confidence limit by fitting equation (4) to any observed change in ω 0 as a function of oscillation amplitude. Minimum length can thus be constrained by measuring the frequency response of a cryogenic quartz resonator's modes under increasing oscillation amplitude.

III. METHODOLOGY

The resonator used in this work consists of a 30 mm diameter by 1 mm thick quartz crystal plate cut along the SC axis [39], with a plano-convex surface in order mitigate phonon losses due to anchoring. Copper electrodes are held close to the crystal's surface by an additional supporting quartz structure, with the entire assembly placed into a surrounding copper enclosure and sealed under vacuum pressure. A copper mount attaches the resonator assembly onto the cold plate of a dilution refrigerator, where it is subject to a stable cryogenic environment of temperature T = 4 K. Sub-Kelvin dilution temperatures are also attainable with this set-up, however the non-linear effects in the quartz crystal become far more dominant at these extremely low temperatures [START_REF] Goryachev | Observation of Rayleigh phonon scattering through excitation of extremely high overtones in low-loss cryogenic acoustic cavities for hybrid quantum systems[END_REF]. Additionally, the quality factors of certain modes may degrade for T<1 K. This occurs as the crystal's loss mechanisms undergo a phase transition into a regime which is dominated by two level systems associated with crystalline disorder. Thus, a higher temperature of 4 K is preferred for this work.

Two separate electrical topologies were employed in measuring frequency perturbations of the resonator in order to investigate both resonant and anti-resonant modes. Schematics of both topologies are presented in figure 1. The motivation behind these configurations is understood by once again considering the electric circuit description of the crystal resonator. In this model; resonant or series modes of the crystal can be described by the minimum impedance modes of the series RLC branch parametrised by the crystal's motional degrees of freedom. In addition, the anti-resonant or so called parallel modes can be described as the minimum admittance modes of the motional RLC branch in parallel with the physical electrode capacitance. Thus, different electrical configurations can be employed to optimise coupling to either the purely mechanical resonant modes, or hybrid electromechanical anti-resonant modes of the resonator. The first topology utilised was a series configuration where each of the resonator's electrodes were coupled to individual co-axial signal lines that feed out of the cryogenic system. This allows for an electrical drive signal supplied by external source to excite resonant acoustic modes in the crystal bulk, due to the piezoelectric coupling of the quartz lattice to a charge density on the electrodes. The same mechanism can be exploited in order to read out the mechanical motion of the crystal plate. In the parallel configuration, the quartz electrodes were directly soldered to a capacitive T-network. This circuit was then coupled to the signal lines in such a way as to shunt the quartz resonator to the electrical ground plane. This configuration allowed for direct coupling to the antiresonant modes of the resonator, through the same piezoelectric mechanism. Studying the anti-resonant modes of this system is of interest as the resonance shares a half degree of freedom with the purely electrical component in the parallel capacitance, perhaps modifying the degree of frequency perturbation due to inherent crystal non-linearities. Impedance measurements of the resonant network were made utilising a vector network analyser (VNA) locked to a stable frequency reference supplied by Hydrogen Maser, with signals supplied by co-axial cables that feed into the cryogenic system and connect to the resonators electrodes. The measurement set up is displayed diagrammaticality in figure 2. Measurements of the system transmission parameter S 21 as a function of frequency ω in the vicinity of a mode, show maximum transmission at the frequency corresponding to the resonant mode ω r for the case of the series configuration, or the anti-resonant mode ω a for the case of the parallel T-network configuration. The exact frequency of each mode, as well as their quality factors Q λ , can thus be determined by fitting a Lorentzian to the corresponding S 21 (ω) measurements. The VNA supplies a power P VNA to the system through the signal input line that drives the acoustic modes. The transfer function that relates P VNA to the crystal displacement x in the vicinity of resonance can be written as:

x(ω) = 1 ωκ λ (1 -Γ)(P VNA + α) Re [Z input (iω)] Z gen (iω) ( 12
)
Where Γ is the reflection coefficient of the resonant mode, α is the loss in the input line, Z input (iω) is the complex input impedance of the resonator network and Z gen (iω) is a generalised transfer function that relates the current input to the resonator network to the motional branch current I λ . The coupling constant κ λ is experimentally determined and has units of C/m. The term under the square root sign in equation ( 12) represents the total current at the input of the resonator network as I 2 input = P input /Z input where P input = (1 -Γ)(P VNA + α). The term Z gen (iω) is then the ratio of impedances that form the current divider equation which determines I λ from the input current. This ratio is different for each resonator network topology. The values of Γ and α for each resonant mode are experimentally determined from S 11 (ω) measurements of the resonator network in reflection.

IV. RESULTS

To utilise equation [START_REF] Kempf | Minimal length uncertainty relation and ultraviolet regularization[END_REF] the values of the motional resistance R λ , capacitance C λ , inductance L λ and electrode capacitance C 0 for each mode must be known in order to construct Z input as well as Z gen . Accurate measurements of R λ for multiple modes have already been conducted in previous works [START_REF] Goryachev | Observation of the fundamental Nyquist noise limit in an ultra-high Q-factor cryogenic bulk acoustic wave cavity[END_REF]. All other parameters can be determined by solving the analytical expressions that minimise

a) d) b) c) e) f)
FIG. 3. Plots a)-c) show the experimental results of S21 measurements as a function of input power for the third order 5.5 MHz resonant mode, as well as the fifth order 8.4 MHz and 9.2 MHz resonant modes. The center frequency found by Lorentzian fitting has also been marked. Plots d)-f) show the net frequency perturbations ∆ω as a function of crystal displacement amplitude, as well as the fitted limits on β0, for the same corresponding modes. I. Results from the investigations of resonant and anti-resonant quartz BAW modes. Each mode is denoted by λ = Xn,m,p where X is the polarisation (A for longitudinal, B and C for fast and slow shear polarisations), and integers n, m, p denote the longitudinal and x-y plane wave numbers respectively. β Z input for resonant modes or 1/Z input for anti-resonant modes, given the experimentally determined values for ω r , ω a and Q λ . In order to vary the excitation amplitude; P VNA was increased in increments to the maximum value attainable before the excessive power input causes the cryogenic environment to increase in temperature. At each stage the S-parameter measurements were made with a narrow resolution bandwidth in order to avoid any ringdown effects. With the modal frequencies determined at each step, equation ( 4) can be fit to the resulting data with β 0 set as a free parameter. The errors in the center frequencies ω r,a associated with the Lorentzian fitting are accounted for by running monte-carlos simulations in which β 0 is fit for multiple Gaussian distributed trajectories. Confidence limits to 3σ on the fitted values of β 0 are then obtained from the aggregate statistics. Examples of such fitting of multiple resonant modes are presented in figure 3, with the full list of all results and measured parameters given in table I, and final exclusion limits plotted in figure 4. In this work only fundamental longitudinal overtone modes where investigated as they typically exhibit higher quality factors, and larger effective mode mass. In general the resonant or series modes provided better exclusions on the minimum length order parameter β 0 than the anti-resonant modes. This is potentially due to the external circuit capacitors of the T-network contributing extra losses and non-linear behaviours when coupled to the resonator. The strongest limit established to 3σ confidence was β 0 < 62.4, this was achieved with the 3rd order fast shear mode B 3,0,0 at 5.505 MHz. When compared to previous experiments that exclude minimum length with mechanical resonators, the previous most stringent limits where set by a similar quartz resonator system at room temperature [START_REF] Bushev | Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums[END_REF]. We improve upon these bounds by three orders of magnitude thanks to the improved accuracy of the analysis in which equation ( 4) is fit to experimental data, as well as the operation of the resonator in a cryogenic environment. 4. Exclusion limits from this work are plotted as well as the previous best limits generated by the sapphire mechanical resonator of ref. [START_REF] Bushev | Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums[END_REF]. Also shown is a previously projected estimate for the sensitivity attainable with an LD-cut quartz resonator.

λ = Xn,m,p ωr/2π (Hz) ωa/2π (Hz) Q λ (10 6 ) R λ (Ω) m eff (mg) κ λ (10 -4 C/m) β (r) 0,3σ β (a) 0,3σ B3,0

V. DISCUSSION

It is acknowledged that tight bounds of β 0 1 have been proposed [START_REF] Kumar | On quantum gravity tests with composite particles[END_REF] by reusing results from dated experiments measuring the period of heavy pendulums [START_REF] Smith | Precision measurement of period vs amplitude for a pendulum[END_REF][START_REF] Fulcher | Theoretical and experimental study of the motion of the simple pendulum[END_REF]. However, the lack of metrological treatment in these experiments limits the quantitative significance of these results [START_REF] Bushev | Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums[END_REF]. For genuine validation of these bounds such experiments would need to be repeated in a controlled context; where system frequency stability, external environmental influences and spurious mode induced non-linearities can be measured. Multiple discussions around the physical conclusions that can be drawn from experimental tests of minimum length remain highly active and seemingly problematic. As mentioned above; the application of the GUP to center of mass coordinates is uncertain. In the case that GUP induced effects in macroscopic bodies are to be suppressed by either particle number or the number of elementary interactions, the results presented in this work would be scalable and remain of large significance as the most stringent metrological bounds to date. It has also been noted [START_REF] Bawaj | Probing deformed commutators with macroscopic harmonic oscillators[END_REF] that the phenomenological model which leads to (4) suggests large deviations from the classical trajectories of astronomical objects, possibly leading to conclusions such as β 0 displaying some mass dependence. In this case it becomes well motivated to explore minimum length over a varying range of mass. The quartz BAW resonator thus presents an attractive platform for such explorations thanks to a large density of overtone modes giving access to the effective mass regime of a few milligrams and below. An additional complication arises as exclusion of β 0 , based on the model of equation ( 3), approaches bounds of O(1). The chosen representation of the physical position and momentum operators is a perturbative approach that recovers minimum length for β 0 1. A more exact nonperturbative treatment is therefore necessary for smaller length scales [START_REF] Bosso | The minimal length is physical[END_REF]. As the results presented in this work investigate β 0 at an order of magnitude larger than β 0 = 1; this perturbative approach is sufficient. Further to this point, the conclusions that should be drawn by these results are an exclusion bound on the potential scale of minimum length intervention in macroscopic systems, and not any sort of confirmation of the GUP approach or phenomenology. These arguments motivate the need for further experimental investigation into the behaviours of center of mass modes of macroscopic bodies close to the quantum regime. Exploring the classical to quantum transition in mechanical resonators with minimal decoherence to the external environment could reveal further insights into GUP implications of composite systems as well as other quantum phenomena such as spontaneous wave function collapse [START_REF] Diósi | Testing spontaneous wave-function collapse models on classical mechanical oscillators[END_REF][START_REF] Vinante | Narrowing the parameter space of collapse models with ultracold layered force sensors[END_REF][START_REF] Tobar | Testing spontaneous wavefunction collapse with quantum electromechanics[END_REF]. Previous work has been undertaken in bringing low noise nano-mechanical resonators into the quantum regime through various cooling procedures [START_REF] O'connell | Quantum ground state and single-phonon control of a mechanical resonator[END_REF][START_REF] Teufel | Sideband cooling of micromechanical motion to the quantum ground state[END_REF][START_REF] Chan | Laser cooling of a nanomechanical oscillator into its quantum ground state[END_REF][START_REF] Cattiaux | A macroscopic object passively cooled into its quantum ground state of motion beyond single-mode cooling[END_REF][START_REF] Naseem | Ground-state cooling of mechanical resonators by quantum reservoir engineering[END_REF][START_REF] Seis | Ground state cooling of an ultracoherent electromechanical system[END_REF], setting the stage for the confirmation of purely quantum phenomena in composite particle systems [START_REF] Safavi-Naeini | Observation of quantum motion of a nanomechanical resonator[END_REF]. Further development of quantum limited mechanical devices of even larger mass is ongoing [START_REF] Georgescu | A 10 kg mechanical oscillator close to its motional ground state[END_REF]. Thus, pushing the quartz BAW platform to single phonon occupancy, which would be attainable for ω > 200 MHz in a 10 mK thermal environment, is an ideal next step towards the demarcation of the quantum-classical limit.

VI. CONCLUSIONS

In summary; minimum length has been tested by exploring the effect of the GUP on the center of mass mode of a macroscopic quartz BAW resonator. Constraints on the minimum length order parameter β 0 have been placed to 3σ confidence by fitting to the observed frequency perturbations of the resonant and anti-resonant modes of several crystal resonances. The constraints derived in this work improve upon the previous most stringent limits obtained with dedicated metrological systems by at least three orders of magnitude, thanks to an improved analysis technique as well as a low noise cryogenic operating environment. As further bounds on β 0 by macroscopic resonators approach the theoretically predicted regime of β 0 = 1, stringent approaches and analysis will become necessary in order to draw physical conclusions. Further exploration of the quantum regime in macroscopic acoustic systems, such as single-phonon mechanical resonators, will hopefully provide the necessary developments in order to make conclusive statements on the GUP phenomenology, as well as other testable predictions of quantum theories.

FIG. 1 .

 1 FIG. 1. Schematics of the resonator network topologies utilised. The quartz resonator in the vicinity of a resonant mode is described as a single RLC branch in parallel with the electrode capacitance as shown. The shaded blue box distinguishes this lumped element resonator model from external electrical components. a) Series network for coupling to the resonant modes ωr. b) Parallel T-network for coupling to the anti-resonant modes ωa.

FIG. 2 .

 2 FIG.2. Diagram of the experimental set up for the case of the resonator connected in the series configuration. The loss in the input line is denoted by α. The output signal is amplified at room temperature in order to improve the signal-to-noise of the measured acoustic modes.

  are the excluded values of β0 to 5σ confidence derived from fitting to the resonant and anti-resonant modes respectively.

  FIG.[START_REF] Amati | Can spacetime be probed below the string size?[END_REF]. Exclusion limits from this work are plotted as well as the previous best limits generated by the sapphire mechanical resonator of ref.[START_REF] Bushev | Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums[END_REF]. Also shown is a previously projected estimate for the sensitivity attainable with an LD-cut quartz resonator.


Jason Govilas

Violaine Guicheret

Cédric Clevy

Vincent Placet

Fabien Amiot

Violaine Guicheret-Retel

Cédric Clévy

Introducing compression platen misalignment in single fiber transverse compression analytical models

Keywords: compression test, fiber mechanical characterization, parallelism error, extended Jawad-Ward model

Introduction

Fibers are a widely used structural and functional material in a wide variety of applications ranging from textiles and composites to energy harvesting, temperature regulation, biomedical use and more [START_REF] Chang | Structural and Functional Fibers[END_REF]. Fibers can be natural (plant, animal, mineral) or man-made (carbon, polymer, aramide...) providing different properties and sustainability benefits depending on their source [START_REF] Chang | Structural and Functional Fibers[END_REF][START_REF] Khan | The challenges of natural fiber in manufacturing, material selection, and technology application: A review[END_REF][START_REF] Bourmaud | Towards the design of high-performance plant fibre composites[END_REF]. Knowledge of the transverse mechanical properties of these fibers is key in modeling and predicting the behavior of the structures that incorporate them or in cases where fibers are subject to important transverse loads such as ballistic impacts [START_REF] Abtew | Ballistic impact mechanisms -A review on textiles and fibre-reinforced composites impact responses[END_REF], composite material compression molding [START_REF] Comas-Cardona | Unidirectional compression of fibre reinforcements. Part 1: A nonlinear elastic-plastic behaviour[END_REF] or plant fiber extraction [START_REF] Bourmaud | Towards the design of high-performance plant fibre composites[END_REF].

To characterize fiber transverse mechanical properties, Single Fiber Transverse Compression Tests (SFTCTs) represent the established direct testing method. The test consists of compressing a unitary fiber between two compression platens, one of them being fixed and the other mobile. Analytical models predict, through various formulations, the fiber's behavior under this transverse compression. The origins of all SFTCT models can be traced to Hertz's contact theory [START_REF] Hertz | On the contact of rigid elastic solids and on hardness[END_REF] and more specifically McEwen's [START_REF] 'ewen | XLI. Stresses in elastic cylinders in contact along a generatrix (including the effect of tangential friction)[END_REF] formulation of contact stresses in parallel elastic cylinders under transverse compression. Based on these works Hadley et al. [START_REF] Hadley | The transverse compression of anisotropic fibre monofilaments[END_REF] proposed a formulation to calculate the contact half-width b, in the context of single fiber transverse compression. Morris [START_REF] Morris | 39-The Determination of the Lateral-Compression Modulus of Fibres[END_REF], Phoenix et al. [START_REF] Phoenix | Transverse Compressive Moduli and Yield Behavior of Some Orthotropic, High-Modulus Filaments[END_REF], Jawad et al. [START_REF] Jawad | The transverse compression of oriented nylon and polyethylene extrudates[END_REF] and Cheng et al. [START_REF] Cheng | Experimental investigation of the transverse mechanical properties of a single Kevlar® KM2 fiber[END_REF] incorporated Hadley's formulation of the contact half-width and iterated on models that offer the Introducing compression platen misalignment in single fiber transverse compression analytic fiber's radial displacement, as a function of the applied compressive force and the fiber's geometric and material properties.

In all these aforementioned analytical models, the fiber's geometry is approximated to a right circular cylinder. Since a fiber's length is significantly larger than its diameter, the problem is formulated in plane strain conditions, omitting the fiber's longitudinal strain. An infinitesimal strain approach is used. While fibers can exhibit complex inelastic behavior, isotropic or transversely isotropic elastic behavior hypotheses are used in these models. With the use of these models, an apparent transverse elastic modulus E T of the fiber can be identified by inverse method by performing SFTCTs.

Analytical models of SFTCTs also make hypotheses that are not inherent to fiber nature. Since the compression platens are made from materials (typically sapphire, glass) that are significantly stiffer than the fiber, they are considered as rigid. Furthermore, the two compression platens are considered as perfectly parallel. In practice however, assuring this parallelism experimentally can prove challenging. Failing to find any mention of parallelism control in the SFTCT literature is not a rare occurrence. Our team has demonstrated that platen misalignment can have a major influence on the fiber's identified apparent transverse elastic modulus [START_REF] Govilas | Platen parallelism significance and control in single fiber transverse compression tests[END_REF]. The study was focused on the rotation of the upper compression platen around the fiber's pitch axis, as seen in Fig 1 . The angle associated to this rotation around the x axis was termed "tilt angle" and noted ϕ. The tilt angle produces the largest variation of the contact surface between the platen and the fiber, resulting in a large error in the identification of E T , with an angle of 1°leading to an identification error of around 90%. An experimental protocol was proposed to minimize the tilt angle under 0.1°, however a custom force-displacement sensor and precise actuation were required to achieve it. Other methods of minimizing platen misalignment Introducing compression platen misalignment in single fiber transverse compression an in SFTCTs have been proposed in the literature such as manual adjustments [START_REF] Hadley | The transverse compression of anisotropic fibre monofilaments[END_REF], passive alignment methods [START_REF] Jones | The lateral deformation of cross-linkable PPXTA fibres[END_REF][START_REF] Guo | Transverse compression of two high-performance ballistic fibers[END_REF], platen preloading [START_REF] Sockalingam | Transverse compression behavior of Kevlar KM2 single fiber[END_REF][START_REF] Naito | Transverse compressive properties of polyacrylonitrile (PAN)-based and pitch-based single carbon fibers[END_REF] and active motors [START_REF] Wollbrett-Blitz | Multiaxial mechanical behavior of aramid fibers and identification of skin/core structure from single fiber transverse compression testing[END_REF], however no quantification of the platen parallelism accuracy was provided in these studies. b) cross section view with the fiber with a contact width of 2b, c) side view of SFTCT with a tilt angle. The origin of the reference system is placed, along the z axis, on the center of zone of the fiber that is in contact with the upper platen.

An alternative-complementary approach to an active correction of the tilt angle before SFTCTs lies in introducing it directly in the analytical model of the test. This would allow a better identification of E T when a small tilt angle persists after parallelism setting procedures, or even potentially eliminate the need for such procedures and the related equipment altogether. Resources and time constraints related to platen parallelism control could thus be significantly reduced. In this paper, such a SFTCT model is developed by extending the model developed by Jawad and Ward [START_REF] Jawad | The transverse compression of oriented nylon and polyethylene extrudates[END_REF], to account for the presence of a tilt angle (i.e., a rotation of the upper platen around the x axis). To facilitate comparisons the term "tilt model" is used to refer to the newly developed model while "Jawad-Ward model" is used for the original formulation. A Finite Element Model (FEM) of a SFTCT performed with a tilt angle is used to validate the predictions of the tilt model and compare its identification error with the Jawad-Ward model.

SFTCT analytical model tilt angle extension

In this section the development of the analytical model that accounts for the presence of a tilt angle in SFTCTs is presented. The widely used model developed by Jawad and Ward [START_REF] Jawad | The transverse compression of oriented nylon and polyethylene extrudates[END_REF] serves as basis in this development and is extented to incorporate the tilt angle. Its original formulation provides the radial displacement of a fiber, represented as a right circular cylinder, under compression between two parallel and rigid compression platens with:

U = 4F L π α 1 sinh -1 (R/b) + ln(2) - 1 2 α 1 + α 2 + α 2 R b 1 + (R/b) 2 -R/b (1)
with: F L the compressive force per unit of length, U the fiber's radial displacement along the y axis and R the fiber's radius (see Fig. 1). The term b corresponds to the half contact width given by:

b = 4F L R π α 1 (2)
The model represents the fiber as transversely isotropic with the fiber's logitudinal axis (z axis), being the axis of symmetry. The terms α 1 and α 2 relate Introducing compression platen misalignment in single fiber transverse compression an the fiber's material properties to the rest of the model through:

α 1 = 1 E T - ν 2 LT E L (3)
α 2 = - ν T T E T - ν 2 LT E L (4)
with: E T the fiber's transverse elastic modulus, E L the longitudinal modulus and ν LT and ν T T the Poisson ratios in the longitudinal and transverse plane respectively.

The first step in development of an extended model that accounts for the tilt angle consists in the nondimensionalization of the initial equation :

Ũ = b2 sinh -1 b-1 + ln 2 - 1 2 1 + α 2 α 1 + α 2 α 1 b-1 1 + b-2 -b-1 (5)
where:

Ũ = U R (6) b = b R (7)
Assuming that b is small enough, Eq. 5 may be approached by a Taylor expansion:

Ũ = -b2 ln b + ∞ k=1 (-1) k+1 (2k)! b2(k+1) 2 2k (k!) 2 2k + 2 ln 2 - 1 2 1 + α 2 α 1 b2 + α 2 2α 1 b2 + ∞ k=2 (-1) k+1 k-2 p=0 (2p + 1) 2 k-1 k! b2k (8)
Springer Nature 2021 L A T E X template

Introducing compression platen misalignment in single fiber transverse compression analytic One assumes that the plates are much stiffer than the tested fiber, so that the tilt angle ϕ results in a linear compression of the fiber. Considering that the origin of the reference frame is placed, along the z axis, at the center of the contact zone between the fiber and the upper platen, as seen in Fig 1, the following stands:

Ũ (z) = U (z = 0) -z tan ϕ R = Ũ0 - z R tan ϕ (9)
where:

Ũ0 = U (z = 0) R (10)
The contact width field b(z) resulting from the linear compression field defined by Eq. ( 9) is now sought. Let us denote b0 the value such that:

Ũ0 = b0 2 sinh -1 b0 -1 + ln 2 - 1 2 1 + α 2 α 1 + α 2 α 1 b0 -1 1 + b0 -2 -b0 -1 (11)
The field b(z) is sought as:

b(z) = ∞ k=0 bk z k = b0 1 + ∞ k=1 bk b0 z k = b0 1 + ∞ k=1 bk z k (12)
where:

bk = bk b0 (13)
The terms involved in Eq. ( 8) thus read:

ln b = ln b0 + ∞ k=1 c k z k (14) Introducing compression platen misalignment in single fiber transverse compression an bn = ∞ k=0 d k z k (15)
where c k are obtained using (0.315) in [START_REF] Gradshteyn | Table of integrals, series and products[END_REF]:

c 1 = b1 c 2 = b2 - b1 2 2 c 3 = b3 -b1 b2 + b1 3 3 etc... ( 16
)
and the d k are obtained recursively using (0.314) in [START_REF] Gradshteyn | Table of integrals, series and products[END_REF]:

d 0 = b0 n d m = 1 m b0 m k=1 (kn -m + k) bk d m-k for m ≥ 1 (17)
Ones assumes that Eq. ( 5), which has been derived for plane strain, holds locally. Keeping the Taylor expansion (8) up to the second order (which is a very good approximation up to b R ≃ 0.4 for any values of the material parameters), the normalized compression field therefore reads:

Ũ (z) ≃ Ũ0 +z -b0 2 c 1 -2 b0 b1 ln b0 + 2 2 ln 2 - 1 2 b0 b1 +z 2 -b0 2 c 2 -2 b0 b1 c 1 -b1 2 + 2 b2 b0 ln b0 + 2 ln 2 - 1 2 b1 2 + 2 b2 b0 +z 3 -b0 2 c 3 -2 b0 b1 c 2 -b1 2 + 2 b2 b0 c 1 -2 b1 b2 + b0 b3 ln b0 +2 2 ln 2 - 1 2 b1 b2 + b0 b3 Springer Nature 2021 L A T E X template
Introducing compression platen misalignment in single fiber transverse compression analytic

+ . . . (18)
Comparing Eqs ( 9) and ( 18) directly yields the linear system to be solved:

- tan ϕ R = -b0 2 c 1 -2 b0 b1 ln b0 + 2 2 ln 2 - 1 2 b0 b1 0 = -b0 2 c 2 -2 b0 b1 c 1 -b1 2 + 2 b2 b0 ln b0 + 2 ln 2 - 1 2 b1 2 + 2 b2 b0 0 = -b0 2 c 3 -2 b0 b1 c 2 -b1 2 + 2 b2 b0 c 1 -2 b1 b2 + b0 b3 ln b0 +2 2 ln 2 - 1 2 b1 b2 + b0 b3 . . . ( 19
)
The system is easily solved sequentially with b0 being found through Eq.( 11):

the first equation of the system yields b1 , the second subsequently yields b2 and so on. The proposed sequence is therefore easily coded to yield all the desired bk and thus, assuming convergence, the function b(z). It is worth noting that the proposed correction does not depend on the fiber's elastic properties. From Eq. ( 2), the force per unit of length as a function of z can also be expressed:

F L (z) = πb 2 (z) 4α 1 R (20)
The force per unit of length may be integrated over the contact length L to yield the net force F applied to the fiber:

F = L F L (z)dz (21)
In conclusion, the net force applied to the fiber F , can be calculated, for any tilt angle, with the fiber's displacement at z = 0 (U (z = 0)) and the rest Introducing compression platen misalignment in single fiber transverse compression a of its geometric and material properties as an input. Using an inverse method, the fiber's transverse elastic modulus E T can be identified if force-displacement data along with the tilt angle and the fiber's properties are provided.

3 Materials and Methods

Finite element model

In order to validate the newly developed tilt model, its predictions are compared to those of a FEM. A model developed by our team in COMSOL Multiphysics ® and first presented in [START_REF] Govilas | Platen parallelism significance and control in single fiber transverse compression tests[END_REF] is used for this purpose. A brief description of the model is given below. Half of the fiber and the platens are modeled by applying a symmetry condition along the yz plane. The lower platen is fixed while the upper platen is mobile with an imposed displacement of 1 µm along the y axis and rotated with tilt angles varying from 0°t o 1°. A linear strain formulation is used to better match the infinitesimal strain approach used in the analytical model. The contact between the platen and the fiber is modeled through a Lagrangian formulation. A polyamide 11 (PA11) fiber is modeled as a transversely isotropic right circular cylinder. Its geometric and material properties are given in Table 1. 1 PA11 fiber geometric and material properties. Sources and measurements of these parameters are given in [START_REF] Govilas | Platen parallelism significance and control in single fiber transverse compression tests[END_REF].

Parameter

The mesh of the model consists of prisms with triangular cross sections for the fiber and rectangular ones for the platens. Second-order elements are Springer Nature 2021 L A T E X template Introducing compression platen misalignment in single fiber transverse compression analytic used. The mesh is slightly adjusted compared to the model presented in [START_REF] Govilas | Platen parallelism significance and control in single fiber transverse compression tests[END_REF].

A potential contact zone between the fiber and the upper and lower platen is defined along an arc of 30 • , with 14 elements placed along its length. An adjustment of the mesh along the fiber's length is also made, with a ratio of |1 -500 • sin ϕ| between the length of the first element, situated at the edge of the fiber that is in contact with the upper platen, and the last element on the opposite end of the fiber, which is potentially not in contact with the platen, depending on the tilt angle. In this way, a uniform mesh along the fiber length is obtained when the platens are parallel (ϕ = 0 • ) while an active refinement along the contact zone is performed as the tilt increases. The

Transverse elastic modulus identification

In order to validate the developed analytical model, its ability to identify the fiber's transverse elastic modulus E T by inverse method at different tilt angles is evaluated. To do so, force and displacement data issued from the Finite Element Analysis (FEA) are used. The displacement of the upper platen is taken, which is equivalent to the fiber's vertical displacement at the point where the first contact occurs. The applied compressive force F , is evaluated by integrating the y axis component of the contact pressure on the top surface of the fiber where it is in contact with the upper compression platen.

The inverse identification is performed through a trust-region algorithm by providing the compressive force F , the upper platen displacement U , the tilt angle ϕ and the fiber's geometric and material properties (R, E L , ν LT , ν T T ).

Eq. ( 12) is truncated at orders k ≤ 5 in the formulation of the tilt model. To evaluate the model's identification ability at different tilt angles, the identified transverse elastic modulus E T id is compared with the one identified for a compression with parallel platens E T0 by calculating their relative difference ∆E T :

∆E T = E T id -E T0 E T0 (22)
The influence of the truncation order in the expression of the field b(z) in Eq. ( 12) is also studied. Specifically, its influence on the identification of E T is assessed by evaluating ∆E T as a function of a truncation order, varied from 1 to 5. A convergence in the value of E T that is identified by the tilt model can thus be verified.

Results and discussions

The results of the identification procedure using the Jawad-Ward and tilt model with an expansion order of 5, is shown in Fig. 3. Regarding the Introducing compression platen misalignment in single fiber transverse compression analytic force-displacement results, the typical non-linearity obtained for purely elastic materials under SFTCT can be seen in Fig. 3.a. The tilt model closely matches the FEA results for all tilt angles. On the other hand, the Jawad-Ward model, that does not take into account the tilt angle, only fits the FEA results well for a tilt angle of 0°with important differences appearing for non-zero tilt angles.

These differences are reflected in the values of the identified transverse elastic modulii. As seen in Fig. 3.b, the Jawad-Ward model significantly underestimates the fiber's transverse elastic modulus, as attested by the negative values of ∆E T , which can reach up to -90% for tilt angles of only 1 It can be noted that for a tilt angle of 0 • a relative difference of -0.6% and -0.7% between the identified transverse elastic modulus and the one imposed in the simulation exists, for the Jawad-Ward and the tilt model respectively.

This small difference can be explained by the presence of longitudinal strains in the 3D SFTCT finite element model, as opposed to the 2D, plane strain analytical models used in the identification, as explained in [START_REF] Govilas | Platen parallelism significance and control in single fiber transverse compression tests[END_REF]. The non-uniform loading along the z direction for a tilt angle ϕ ̸ = 0 • may also be expected to be at the origin of an additional deviation from the plane strain assumption.

The quality of this assumption may be further assessed by analyzing the effect of G LT on the identification error. Varying G LT over almost 10 decades, the identification error is shown to vary by a few percent, thereby proving the negligible sensitivity of the reported results to G LT . This further validates the use of plane strain assumption. The impact on ∆E T of the truncation order used in Eq. ( 12) can be seen in Fig. 4. A decrease in ∆E T is observed with the increase in expansion order.

This decrease is significant when switching from order 1 to 2. For higher orders however, ∆E T converges towards values of 1.3%. Compression platen misalignment represents a significant source of error in the identification of a fiber's transverse elastic modulus through SFTCts. In this paper, an analytical model which accounts for the presence of a tilt angle is developed by extending the model proposed by Jawad and Ward [START_REF] Jawad | The transverse compression of oriented nylon and polyethylene extrudates[END_REF]. Using data from a finite element model of a SFTCT with a tilt angle, the proposed "tilt model" is shown to identify the fiber's transverse elastic modulus with an error under 1.5% for all tested tilt angles. The identification of a fiber's transverse elastic modulus E T can thus be made with very low errors compared to those obtained with the standard model, regardless of the tilt angle. The predictions of the tilt model also closely match the results of the FEA. Modeling SFTCTs with a tilt angle can thus be achieved with the tilt model instead of a 3D FEM, saving development and computation time [START_REF]SITARIST : Single fIber TrAnsverse compRessIon teSt with Tilt[END_REF].

Furthermore, the tilt model contributes significantly in reducing experimental constraints on compression platen parallelism in SFTCTs. As long as the tilt angle between compression platens can be measured, with optical means for example, the uncertainty on the identified value of E T that is related to platen parallelism is significantly reduced. The need for complex or timeconsuming parallelism setting procedures and their related equipment could thus be reduced or eliminated.

Further work can be performed to account for platen rotations around the z axis as well. Additionally, the parallel identification of both the transverse elastic modulus and the tilt angle could be considered, eliminating the need for misalignment angle measurements. This parallel identification however requires further data in order to provide reliable results, such as the displacement fields at the fiber's edge. Finally, it can be noted that while the Introducing compression platen misalignment in single fiber transverse compression a proposed model is developed in the case of the SFTCT, the underlying principle of its development could be used for the compression of other objects at any scale, providing an alternative to parallelism setting procedures across various mechanical tests and scientific disciplines.

Fig. 1 a

 1 Fig.1 a) Representation of a single fiber transverse compression test with a tilt angle ϕ, b) cross section view with the fiber with a contact width of 2b, c) side view of SFTCT with a tilt angle. The origin of the reference system is placed, along the z axis, on the center of zone of the fiber that is in contact with the upper platen.

  geometry, boundary conditions and mesh of the model are shown in Fig 2.

ig. 2

 2 Finite element model of SFTCT with an upper platen tilt angle misalignment: a) boundary conditions, b) cross section view of mesh with defined potential contact zones, c) side view of mesh with tilt angle-adaptive mesh. Introducing compression platen misalignment in single fiber transverse compression a

Fig. 3

 3 Fig.3Transverse elastic modulus identification results with the use of the Jawad-Ward and tilt model: a) force-displacement FEM results for different tilt angles (markers) with fitted analytical models (dotted lines), b) evolution of ∆E T as function of tilt angle for both analytical models.

OrderFig. 4

 4 Fig.4Evolution of ∆E T as a function of tilt angle for different expansion orders in Eq. (12) of the tilt model.

  • . Conversely, the modulii values identified with the tilt model are close to the one identified with parallel platens. Only, a slight increase in ∆E T takes place with the increase in angle, that reaches 1.3% for a tilt angle of 1 • . Despite this small increase, the model's overall ability to account for the presence of a tilt angle is demonstrated.

Acknowledgements

This work has been supported by the EIPHI Graduate School under grant "ANR-17-EURE-0002".

Data and code availability

An Octave/Matlab implementation of the approach proposed herein is available [START_REF]SITARIST : Single fIber TrAnsverse compRessIon teSt with Tilt[END_REF].

Conflicts of interest or competing interests

The authors declare no competing interest.

Supplementary information

Not applicable.

Ethical approval

Not applicable.


Clémentine Havrez

* Sophie Lepreux

Yoann Lebrun

Sylvain Haudegond

Pierrette Ethuin
email: pierrette.ethuin@univ-littoral.fr

Christophe Kolski

A Design Model for Tangible Interaction: Case Study in Waste Sorting

Keywords: TUI Design Model, Tabletop, Tangible object, Serious Game, RFID, Waste sorting 

In this paper, we propose a design model for applications with Tangible User Interfaces (TUI). This model aims to support designers in organizing the different elements and their relationships. We then implement the model through the design of a Serious Game for an RFID tabletop that uses tangible objects: this Serious Game simulates microbiological waste sorting in a practical educational setting. In this case study, a scenario is described, highlighting the use of the design model. The conclusions and suggestions for future research are then presented.

1. INTRODUCTION Shneiderman (1983) says the following about video games: "perhaps the most exciting, well-engineered -certainly, the most successful -application of direct manipulation is in the world of video games". With improvements in technology, inputs and outputs are getting closer, enhancing the feeling of direct manipulation (e.g. touch, stylus, movements). One of the emerging technologies that fosters this enhanced closeness involves Tangible User Interfaces (TUIs) [START_REF] Ishii | Tangible bits: Towards seamless interfaces between people, bits and atoms[END_REF]. TUIs allow interaction to take place with a virtual application through real objects that can represent application data (see the definition of [START_REF] Hornecker | A design theme for tangible interaction: Embodied facilitation[END_REF]). And finally, to become closer, inputs and outputs have merged [START_REF] Ullmer | Emerging frameworks for tangible user interfaces[END_REF].

In the video games field, we can refer to Activision Skylanders released in 2011. In this game, one or more real figurines are selected and placed on a base. A playable avatar of one or more figurines appears in the virtual world via the TV screen. We can also mention the STARS platform [START_REF] Magerkurth | Towards the next genera-tion of tabletop gaming experiences[END_REF] to generate games for tangible and tactile tabletops. This platform aims to emulate board games, enhancing them by adding a virtual layer (sound, a screenbased game board equivalent, a vertical screen for general data and Personal Digital Assistants for private data). In this case, a player directly interacts with a figurine and moves it.

The question that subsequently arises is: how to design a TUI application taking into account every physical or digital element? In order to clarify this point, we propose in this paper a design model which takes into account the different elements we can find with TUI as well as their relationships.

After identifying what kind of video toys considered as TUIs exist on the market, we will present several HCI models in Section 2. From these models, we propose a design model for tangible interaction described in Section 3. A case study showing the development of a Serious Game on a TUI is then presented in Section 4. The implementation of the serious game is presented in Section 5 in a scenario illustrating the model. The article ends with a conclusion and suggestions for future research.

STATE OF THE ART

To begin our study, we first considered video toys currently available on the video game market; they are for us a source of ideas for designing new types of industrial or educative applications. We then studied several HCI models that could be compared with each other to create a model for designing TUI applications.

Video Toys as Tangible Objects

By video toy we mean one or more physical elements, such as figurines, linked to at least one digital application. This link augments the figurine, giving it a life, capacities, and so on.

Activision launched the first successful toy of this kind in 2011. It is Skylanders: Spyro's Adventure game. The digital character you play corresponds to the physical figurine placed on the base and linked to the game console. Similarly, in 2013 Pokemon Rumble U by Ambrella uses the NFC component of the Wii U game console to incorporate figurines. Placing a Pokemon figurine on the GamePad allows the player to use this Pokemon in the game. The Pokemon and Skylanders figurine characteristics, which evolve throughout the game, are stored in the figurines. A similar game released by Avalanche Software in 2013 is Disney Infinity. It uses trophy and disc objects to enhance the current character's abilities. These discs can be superimposed and the character figurine can be placed on top. More recently, we can cite Nintendo Amiibo figurines which are available since late 2014. All these physical toys are physical avatars, behaving like an ID card. Once a figurine is placed on the base, users no longer need to interact with them and can play using pads and screens.

Prodigy by Hanakai Studio is a game developed in 2015. It works with figurines, a base and cards. The base is linked to a computer. It is a tactical game. Contrary to previous games, in this one, players interact with physical elements. The users can strategically place their figurines in different squares on the base. Putting a card on the base allows a figurine to perform a specific action that will be digitally represented (through the screen). Here, the digital layer augments the real situation, adding a fantasy layer to the game and making the characters seem alive. On tablet (iPad), the version of Cars 2 developed by Avalanche Software in 2011 allows controlling a physical small car called appMATES, while the screen of the iPad displays the landscape for the race. The car is placed directly on the iPad and allows the user to control the direction taken. Thus, the object is directly immersed in the virtual world. Finally, the ePawn company (2010) was able to develop video toys using its Arena surface (2012) and tracking technology. As for the Cars 2 iPad application, physical elements are detected and immersed. Several elements can be detected at the same time, and mobile elements can even be linked to the surface.

By analyzing these games, we can observe three ways of using a digital layer with physical objects: (1) A physical object enables an avatar with associated stored data to be selected. Then, all happens digitally. (2) The physical object is used to play. But to endow it with a life, movement, abilities and things that cannot be provided physically (or in a very complex way), a digital layer is used. This layer augments the object, or the image we have of it. (3) The physical object is immersed in a digital environment. One would not function without the other. The digital layer brings meaning to the game (for example, the road and obstacles), and the physical elements must respect this meaning. More than augmenting the object with effects, the digital part forms the structure. This notion of structure can be applied to a set of objects as in Disney Infinity application. The discs, alone, don't have any sense. But with a figurine, it increases one of its ability. The figure is a part of structure for the disc. This section deals with objects that represent the targeted element with which the user wants to interact. But sometimes, objects in augmented reality are tools, intermediaries between the user and the targeted object. This is what is explained in the next section.

Domain Object and Interaction Instrument

Beaudouin-Lafon (2000) says that "Our interaction with the physical world is governed by our use of tools". He defines then what he calls domain objects and interaction instruments. The domain objects are directly linked to the aim of the task; they are "objects of interest", whereas interaction instruments are intermediaries used to perform the task and to interact with the objects of interest.

The AppMATES car from the Cars 2 application is a domain object, as it is the object of interest in this racing game. In contrast, the application introduced by [START_REF] Fitzmaurice | Bricks: Laying the foundations for graspable user interfaces[END_REF] uses tangible bricks to manipulate digital forms. In this case, the bricks are interaction instruments. If we refer to Ishii and Ullmer works (2012), we can tell that physical domain objects are linked to digital information and physical interaction instruments are linked to computational function.

Beaudouin-Lafon stipulates that depending on the task to be performed and the user's focus, an interaction instrument can turn into a domain object. He gives the example of a pencil, which is actually an instrument for writing text. The pencil becomes an object of interest when the user needs to sharpen it; the pencil sharpener is the interaction instrument.

To go further in the analysis of the relation between the physical and the digital layers, we refer to the notion of feedback: how a user can glean information from his/her experience through digital and physical interactions.

Tangible User Interfaces and Feedback

Ishii and Ullmer (2012) explain that three feedback loops exist in this field. The first loop involves immediate tactile feedback received by the user when he/she manipulates an object. It is a passive feedback felt by the user, for example, when moving the object, changing its direction or putting it on the table. This feedback allows the user to note the new state of the object through his/her haptic perception. The second feedback loop pertains to what the user perceives from the digital device. The system's state is updated according to the new state of the object(s); this update is directly communicated to the user through digital information (the feedback could be visual or audio). It concerns all nontactile feedback. The third kind of feedback loop involves an updating of the object without user intervention. In the model presented in [START_REF] Ishii | Tangible user interfaces[END_REF], the computer actuates the object (the object can then modify its appearance or move by itself). For example, we observe that, in this era of smart objects, the object can update itself with contextual changes. Thus, there are three feedback loops to take into account when designing a TUI application: (1) sensory feedback linked to the object, (2) digital feedback linked to the application and (3) actuation feedback of the object by itself or by the application. The two first loops are the loops experienced most frequently when using TUIs. They are even considered as essential In Ishii's and Ullmer's opinion in Tangible Interaction.

The Execution and the Evaluation Phases

We can refer to the Norman's action cycle [START_REF] Norman | A. The Design of Everyday Things[END_REF] to understand when the previously studied feedback loops can be exploited. There are four elements to consider in Norman's model: (1) the goal of the user, (2) the execution phase (i.e., what the user does to achieve his/her goal), (3) the world and (4) the evaluation phase (i.e., when the user perceives the world and compares it to his/her expectations). The execution phase is divided into three stages: (1) the intention (by what means does the user want to achieve his/her goal), (2) the sequence of actions to perform to fulfill the intention and (3) the execution stage, when actions are performed, changing the state of the world. Then, the evaluation phase is divided into three stages. During the first one, the user perceives the world. Then, he/she interprets the perceived world according to his/her expectations and finally, the user compares this with his/her initial goal as well as with the intention chosen to achieve the goal. Has the goal been achieved? Was it achieved as desired?

Going back to the augmented reality field, [START_REF] Dubois | Classification space for augmented surgery, an augmented reality case study[END_REF] refer to the Action Theory of Norman to show that two types of augmentation exist: during the execution phase and during the evaluation phase. Let us use the work by [START_REF] Pedersen | Tangible bots: Interaction with active tangibles in tabletop interfaces[END_REF] to illustrate this theory via the less well-known object actuation feedback loop. Augmentation during the execution phase means that a digital layer is added to accompany the user throughout this phase. This corresponds to the interaction commands terms used by [START_REF] Pedersen | Tangible bots: Interaction with active tangibles in tabletop interfaces[END_REF]; the imitation is proposed as an example. The user records movement for some objects, one by one. These movements can then be replayed simultaneously thanks to a command. This is of interest when the number of objects to manipulate simultaneously is superior to the number of available hands. Augmentation in the evaluation phase helps a user execute his/her evaluation phase by adding some digital information to the real world. In the work of [START_REF] Pedersen | Tangible bots: Interaction with active tangibles in tabletop interfaces[END_REF] the used words are interaction feedback. One of the proposed examples is to create resistance when turning an object used to modify a value on a scale when the user approaches one of the end points of the scale (like when increasing sound volume).

We saw that three feedback loops can be used during the execution and evaluation phases: sensory feedback, augmented sensory feedback and augmented digital feedback. These feedback loops are part of the user experience, and it is necessary to take them into account when designing an interactive application. Whether real world or virtual world, users act on tokens and the observable results of these actions allow them to evolve and appreciate the experience. Let us see how the various feedback loops and their functioning can be taken into account for the design of applications incorporating virtuality and tangibility.

PROPOSAL: A DESIGN MODEL FOR TANGIBLE INTERACTION

This section presents our model proposal. Its purpose is to show how to consider every element together when designing a digitally augmented application. We took into account the different data we collected through previous models and video toys analysis. The model is represented by the Fig. 1.

We share previously studied elements into two categories: physical and digital. A physical object represents at least one digital element: a computational function, digital information and/or a scenery. Function of what it represents, this physical object can be specified as being an Avatar Object, a Domain object or an Interaction Instrument. But this specification does not always apply. Indeed, we want to consider all devices of the system including interactive surfaces for example. Each object, whether physical or digital can structure other objects, giving them constraints, indicating ways of using them [START_REF] Ullmer | Tangible query interfaces: Physically con-strained tokens for manipulating database queries[END_REF].

Existing elements must have related to the user(s)' goal(s).

The best we known the different goal of the user, the best we can give him or her feedbacks to help in the evaluation or execution phase as defined in Norman's action cycle. We integrate then the three feedback loops described by Ishii's and Ullmer's. The two feedback loops concerning the haptic field are gathered with the Haptic Feedback class. The last feedback loop concerning the other fields (e.g. audio, visual) is represented by the Digital Feedback class. These classes are association classes. Indeed, in this situation a feedback cannot exist without an action of the user on an element of the system.

How to use the model

First of all, goals have to be defined. Then, each physical object and digital element which can be manipulated has to be connected to a user goal. When these elements are manipulated, the user will receive one or several feedbacks in relation with his/her action, improving interactivity. The feedback can help him/her in the execution phase (manipulating the object), the evaluation phase (interpreting the results of his/her action) or even both phases. Each physical object must be linked to at least one digital element it represents. It can be whether a digital information (a data from the model) or a computational function (which controls a digital information). For example, the "Zoom object" of the road traffic simulation application presented in [START_REF] Lebrun | A model for managing interactions between tangible and virtual agents on an RFID interactive tabletop: Case study in traffic simulation[END_REF] represents the control of the display scale. The model and the object are illustrated in Fig. 2.

A physical object can be constrained by one or several other physical and/or digital elements. These elements create a structure for the physical object to be manipulated. The model encourages defining this relation.

To test this proposal, we implemented it in the context of a Serious Game that aims to teach microbiology students to properly sort their waste during practical exercises.

IMPLEMENTATION: SERIOUS GAME ABOUT WASTE SORTING

In this section, we present a Serious Game about waste sorting on the interactive tabletop TangiSense. It is implemented using the proposed design model for tangible interaction. Game concept story, equipment used and gameplay are successively described.

Game Concept Story

University of Littoral Côte d'Opale (ULCO), located in the North of France, incorporates microbiology into almost all courses of study from first-year university level to the Masters degree level in the Department of Life Sciences and Nature. For many years, the university has been welcoming non-french speaking students for whom the domain-specific vocabulary is difficult to master as they learn the language. Furthermore, for health reasons (e.g., disability, serious illness, pregnancy), it is sometimes difficult or impossible for some students to attend practical sessions (use of dangerous products). In another context, people who wish to receive professionalized training need to boost their general microbiology knowledge. Several issues emerge: problems of accessibility for foreign students due to limited vocabulary, problems of improving for beginners and problems of practical facility accessibility.

In order to develop necessary best practice in microbiology knowledge, the university decided to integrate Serious Games that are accessible outside of the practical setting into different learning modules. These games are likely to be more attractive [START_REF] Abt | Serious Games[END_REF] than a traditional educational medium (e.g., textbook, e-learning). Physical handling is essential in this area, which is why maintaining this physical appearance seemed important. We decided to use the tangibility of a tabletop, which, in the future, may represent the laboratory bench top for other applications. We also took into account the laterality of the user: indeed, during manipulation in a laboratory, the arms of the different users should never cross, or accidents could occur.

The first module on which we focused is the sorting of waste and materials in a practical work session. In addition to addressing the aforementioned problems, this module caters to more specific issues identified during the practical sessions: students do not distinguish between types of waste (household, biological, glass), they confuse contaminated waste with uncontaminated waste, they do not dare ask for help from an assistant or an educator, they do not take the time to sort waste at the end of the session and they do not necessarily separate liquid media from solid media, which has repercussions when sterilizing containers in an autoclave.

In an autoclave, solid media will melt during sterilization, becoming liquid, but will harden again during cooling.

People in charge of disposing of the contents of each container may confuse media.

Fig. 3. Collection containers, photo taken during a practical session

After the needs analysis and practical session observations (Fig. 3 shows how the collection containers are placed in practical sessions; note that we only present the collection containers to be used by students), we organized elements into three categories: container type, contents type and associated contamination risk. These are the three categories that let you know which collection container will be appropriate. If a category is not specified, it means that it is not necessary for sorting the waste in question. We should specify that for pipettes, it is necessary to soak the dirtied part in bleach for at least seven minutes before placing it in the collection container.

Equipment

The game is developed for the TangiSense tabletop. This interactive tabletop is equipped with RFID technology for manipulating tangible and traceable objects. It is designed by the RFIdees company. These objects are equipped with one or more RFID tags that are automatically identified and located by the drive array comprising the interactive tabletop.

The table is equipped with a Full HD 47-inch screen display to immerse the user in a universe both real through tangible objects and virtual through the screen. The architecture used is the one proposed by [START_REF] Kubicki | New human-computer interactions using tangible objects: Application on a digital tabletop with RFID technology[END_REF]. It uses several layers: (1) the application layer involving a multi-agent system that links each tangible object to a software agent (see [START_REF] Kubicki | Simulation in contexts involving an interactive table and tangible objects[END_REF] and (ISO, 2010) for more descriptions),

(2) the traceability layer for detecting the position of objects and launching the associated events, and (3) the capture and user interface layer, capturing the presence or absence of objects and creating virtual agents.

Gameplay

The goal of this serious game is to successfully combine waste with its appropriate waste collection container. The play area consists of a grid on which collection containers appear randomly one by one. The area also contains a zone with the score and the representation of bleach in which the contaminated pipettes need to be soaked, and an information zone in the event that the user needs help. The position of this zone adapts to the lateral dominance of the user defined by his/her avatar object. Once the objects are associated with a collection container, it is impossible to reuse them for the current game; they are considered discarded. The collection container display and the score are then updated according to the sorting. Each user has an avatar object which permits to know his/her laterality and his/her best score.

We opted for positive scoring: only correct actions earn points. We integrated the combo concept to benefit consecutive correct choices made without using the combo increases by one. The score increases by the value of the combo. The combo is reset when a bad choice is made, when incorrect sorting has taken place or when the information zone is used. The score display changes color with the value of the combo.

The information zone, which moves functions of player's laterality (stored with the avatar object), allows the player to obtain clues about an object that has not yet been discarded.

The object must therefore be put on the zone representing a microscope so that the representation of its screen provides one of the following clues, depending on the object: (1) it is biologically contaminated or (2) it is considered to be sharp or pointed or (3) it has a chemical risk or (4) it is recyclable or (5) it is safe.

The collection containers defined above randomly appear for T time, T depending on the game level, in a 3x4 grid. If no waste has been placed in the current collection container, it fades after t seconds. If waste is placed in it, the collection container remains apparent and two other elements appear to assist the user during the evaluation phase; an icon, in the bottom right corner of the square, lets the user immediately know whether or not the sort is correct or not. In addition, in the event of error, a visual element placed above the collection container lets the user know why the sorting has been incorrectly performed: chemical contamination, bacterial contamination, or inadequate sorting (which will imply issues in organization or loss of money). A collection container can receive only one waste item. A given collection container can appear more than once.

The presence of bleach is situated above the information zone to simulate the bleach's position on the lab bench. A pipette that is not dipped in bleach before being discarded bacteriologically contaminates the selected collection container, even if it is the pipette collection container. On this zone a pot is placed, the same as the one we can find during practical work sessions to make users understand what this zone is for. After seven virtual minutes in the bleach zone, the object in the zone adopts the bleached state, thereby increasing the score if the action is useful. Otherwise the value of the combo is reset to zero.

SCENARIO ILLUSTRATING THE PROPOSED MODEL

As mentioned before, a contaminated pipette has to be soaked in bleach before being put in the collection container. This is this part of the game we will use in this section. First, the goal of the user is to place the pipette in the pot representing bleach (Fig. 4). The pot was previously placed on the "bleach zone"; a scenery representing the bleach. The pot gives a structure to the pipette when placed into it and indicates to the user how to place the pipette. It helps the user in the execution phase. When the pipette is placed in the pot the bleach scenery highlights. The model of this action is illustrated in Fig. 4. Then the user has to wait for seven "virtual" minutes (faster than in reality). After what our project is to make the pot vibrate as an alarm-clock (cf. centre of Fig. 5). The goal for the user is to know that the pipette is ready to be discarded.

To finish, when the correct collection container appears, the user discards the pipette putting it on the display of the container (cf. Figure 5 -right). The container appearance changes functions of the sorting result. We have chosen to illustrate a part of scenario with the three types of feedbacks; more illustrations and models are available but cannot be integrated in this section because of lack of space.

The process followed during the design of this game is shown on Fig 6. Fig. 6. Actual design process used to establish the model.

CONCLUSION

The concept of feedback is very important when interacting in digital mode or in the real world (Ishii & Ulmmer, 2012). Feedback allows a user to understand, learn and evolve from his/her actions. This is why the proposed Design Model for Tangible Interaction associates each component to a user's goal through feedbacks. Feedbacks can happen during evaluation and/or execution phases. The relationship between objects is also present: in the physical world, every physical element is delimited by other physical elements and this fact must be taken into account. It represents the space. We put this model into practice through a Serious Game dedicated to learning how to sort waste during a practical microbiology work session (evaluation from the design point of view). We illustrated this application in relation to the proposed model with a scenario. We are currently organizing an evaluation with users (evaluation from the learning point of view). For future research, we should consider applications with several users collaborating around the same tangible and/or collaborating remotely [START_REF] Pauchet | Tabletops: Worthwhile experiences of collocated and remote collaboration[END_REF][START_REF] Bouabid | Context-sensitive and collaborative application for distributed user interfaces on tabletops[END_REF] in order to validate the design model in several contexts.

Fig. 1 .

 1 Fig. 1. The proposed Design Model for Tangible Interaction

Fig. 2 .

 2 Fig. 2. Traffic simulation application objects (above); our model proposal applied to the Zoom object (below).

Fig. 4 .

 4 Fig. 4. "Soak the pipette" action (above); "Soak the pipette" model (below).

Fig. 5 .

 5 Fig. 5. Left and center: "The pipette is ready to be discarded". Right: "Discard the pipette".

13th IFAC/IFIP/IFORS/IEA Symposium on Analysis, Design, and Evaluation of Human-Machine Systems Aug.30 -Sept. 2, 2016. Kyoto, Japan