Martin Monloubou

Myrthe A Bruning

Arnaud Saint- Jalmes

Benjamin Dollet

Isabelle Cantat

Blast wave attenuation in liquid foams: role of gas and evidence of an optimal bubble size †

Liquid foams are excellent systems to mitigate pressure waves such as acoustic or blast waves. The understanding of the underlying dissipation mechanisms however still remains an active matter of debate. In this paper, we investigate the attenuation of a weak blast wave by a liquid foam. The wave is produced with a shock tube and impacts a foam, with a cylindrical geometry. We measure the wave attenuation and velocity in the foam as a function of bubble size, liquid fraction, and the nature of the gas. We show that the attenuation depends on the nature of the gas and we experimentally evidence a maximum of dissipation for a given bubble size. All features are qualitatively captured by a model based on thermal dissipation in the gas.

Pressure wave propagation in liquid foams is very peculiar, with a low sound velocity, of the order of 50 m/s, and a high level of attenuation. This feature is of high interest in the context of soundproofing or of blast wave attenuation [1][2][3][4][5][6][7] . Recent progress has been made in the identification of the local processes controlling the propagation velocity and the attenuation length. In his seminal work, Wood 8 described the foam as an effective continuum, which average density and average compressibility are deduced from the bulk phase properties and from the volumetric liquid fraction φ l . The sound velocity predicted on the basis of these effective quantities, called Wood's velocity, is in good agreement with the experimental observations for bubble size below a frequency dependent critical value [START_REF] Mujica | [END_REF]10 . Above this bubble size, a resonant behaviour is observed: the energy absorption and the sound velocity reach maximal values 11,12 . In this regime, the liquid and gas phases, having very different inertia, do not follow the same trajectory and the simple continuum approximation made in Wood's model is not valid anymore. Finally in the large bubble size limit, most of the liquid phase stays at rest, as proposed by Kann's model 13 , and the velocity decreases toward its Kann's value, still larger than the Wood's prediction. These different models focus on the local deformations and motions induced by the pressure wave, but do not model the resulting energy dissipation. This dissipation is simply neglected in Wood's and Kann's models and taken into account by a single phenomenological internal time scale in the resonant model 12 .

Other studies focused in contrast on the fundamental origin of the energy dissipation. For a single bubble in an unbounded liquid, Prosperetti first modeled the high thermal dissipation induced by the large contact area between the gas phase, which temperature varies with the pressure, 0 a Institut de Physique de Rennes, UMR 6251 CNRS/Université de Rennes 1 -Rennes, France; E-mail: isabelle.cantat@univ-rennes1.fr and the liquid phase, which remains at constant temperature 14,15 . This model has been used for foams in [START_REF] Mujica | [END_REF]16 , assuming a continuum deformation, as in Wood's regime. The thermal dissipation model predicts a maximum of dissipation for a given bubble size, as the resonant model. In this paper, we present results on the propagation of a short overpressure, characteristic of a blast wave, in a liquid foam. We measure the pulse velocity and the maximal pressure attenuation, as a function of the bubble size R, the liquid fraction φ l , the initial pressure amplitude P 1 and the nature of the gas. This strongly broadens the parameter space already explored in our previous paper 16 , in which a single gas and a single liquid fraction had been used. It thus allows us to establish two important results: (i) the pressure attenuation depends on the nature of the gas, and that (ii) a maximum of attenuation is obtained for a critical bubble size, at a fixed liquid fraction. The first one is a very discriminant piece of information in order to build the right model of dissipation, whereas the second one may be useful to optimise the foam properties for practical use. Comparison with both resonant and thermal models allows us to conclude that the observed maximum of dissipation is due to the non monotonic variation of the thermal dissipation in the gas phase with the bubble size.

Experimental setup

In all experiments described in this paper, a foam is subject to a blast wave. In this Section, we describe the foaming solution and gas (Sec. 2.1), the foam production and characterisation (Sec. 2.2), the blast wave generation and the pressure measurements (Sec. 2.3) and the measurement of the wave velocity (Sec. 2.4).

Foaming solution and gas

The foam is produced with a solution of sodium dodecyl sulfate (SDS) at a concentration of 10 g/L, larger than the critical micellar concentration (cmc) which equals 2.8 g/L. The gas is either nitrogen (N 2 ) as in 16 , or hexafluoroethane (C 2 F 6 ). The thermal properties of these two gases are reported in Table 1.

Property N 2 C 2 F 6 Density ρ g (kg•m -3 ) 1.18 5.84 Thermal conductivity κ (W•m -1 •K -1 ) 0.024 0.0135 Specific heat c p (J•kg -1 •K -1 ) 1040 760 Molar mass M (g•mol -1 ) 28.0 138.0 Thermal diffusivity D T (10 -5 m 2 •s -1 )
1.95 0.304

Foam production and characterisation

The two experimental set-ups used in this paper are denoted by V and H (as vertical and horizontal) 16 .

In set-up H, the foam is produced by a turbulent mixing method 18 and injected between two horizontal plexiglass plates separated by a distance h. The liquid fraction is controlled by the balance between the gas and solution flow rates and pressures and can be varied over the range

A C C E P T E D M A N U S C R I P T φ l ∈ [0.02 -0.3].
For each experiment, it is measured by weighing a known volume of foam. The initial bubble radius is constant and close to R = 30 µm.

The foam coarsening leads to the bubble size increase and this parameter is thus tuned by changing the waiting time between the foam production and the blast. The setup allows to vary independently the control parameters R and φ l over a certain range, limited by several processes. The waiting time τ c required to double the bubble radius varies as R 2 /(D He), with He the Henry coefficient, which measures the gas solubility, and D the gas diffusivity in the liquid phase. This characteristic time is of the order of a few seconds for R = 30 µm and N 2 , and is much larger for C 2 F 6 , which is 30 times less soluble than N 2 19,[START_REF] Cantat | Foams. Structure and Dynamics[END_REF] . Moreover, τ c increases with liquid fraction. Another ageing process is gravitational drainage. The liquid phase begins to leave the foam and to leak on the bottom plate after a time τ dr scaling as 1/R 2 and decreasing with liquid fraction. The cell is thus turned upside down during the coarsening process at a frequency larger than 1/τ dr to maintain a homogeneous liquid fraction. Finally, this set-up can be used when the total coarsening time is smaller than 15 minutes and the draining time larger than 10 seconds. The resulting maximal bubble radius is typically 300 µm for φ l = 5 % and N 2 foams.

To extend the accessible range of parameters, the set-up V has been designed. In that case, the foam is produced by blowing gas in porous glass frits immersed in the foaming solution (see Fig. 1) and the bubble size is modified by changing the porosity of the frits. The foam then rises between two vertical plexiglass plates, and overflows at the top of the cell. Once a steady state is reached, a sample of known volume of the foam is weighed to measure the liquid fraction as with H. The latter is governed by the upward foam velocity, and can be tuned by changing the inlet pressure, which controls the global flux in the frits. Additionally, the individual flux in each frit is controlled by a valve (see Fig. 1). Indeed, as the bubble size slightly depends on the flux through the frit, it is important to get the same flux in each frit. When the foam looks homogeneous, and while it is still flowing, it is subject to the blast wave.

We checked in 16 for a given set of parameters that the pressure wave propagation is the same in set-up H and V. Results obtained with both set-ups can thus be directly compared.

For N 2 foams, the bubble size distribution of each foam sample is measured by spreading a few hundred bubbles on top of a thin layer of the foaming solution. The bubbles form a monolayer of spherical bubbles and their radius distribution can be obtained by image processing 10 . The size of a bubble in the foam is thus measured by the radius R s of the sphere of same volume.

For C 2 F 6 foams, the diffusion of the outside air into the bubbles is too fast to use this spreading method [START_REF] Maurdev | [END_REF] and an in situ measurement must be performed. We record an image of the layer of bubbles touching the transparent plexiglass plate and the contact area between the bubbles and the wall is determined by image processing. A calibration has been done using both techniques with N 2 foams to convert the contact area distribution into the R s distribution. The average value R = R s bubbles is the bubble radius used in the paper to characterise the bubble size in the foam. Each data point reported in this paper, at a target value of bubble radius, liquid fraction, blast strength, and gas, is the result of at least three independent experiments. All standard deviations and error bars come from the dispersion of the results of these individual experiments. The normalised standard deviation of the bubble distribution is close to 0.4 for all foams, which quantifies the foam polydispersity.

Blast wave generation and pressure measurements

A rigid PVC shock tube is connected at one end to the front plexiglass plate, in the middle of the plate (see Fig. 1). At the other end of the tube, a chamber is sealed by an aluminium foil, and connected to a gas bottle. To generate the pressure wave, we let the gas flow into this chamber; when the pressure in this chamber reaches a critical value, the foil breaks and a pressure wave propagates in the tube. A shock wave (i.e. a stiff pressure step followed by a finite interval of sustained high pressure) forms in the tube and evolves into a blast wave (i.e. a short overpressure followed by an underpressure) when exiting the tube. This blast wave then propagates in the foam. The pressure in the foam is recorded using four pressure sensors (PCB -piezotronics, model 113B28) installed at distances r i = [3.1, 5.1, 7.2, 9.2] cm (i = 1 to 4) from the symmetry axis of the experiment, corresponding to the center of the tube. They are connected to an oscilloscope (tektronics, model TDS 2004 B), which has a sampling time of 4 µS. Fig. 2 shows typical signals at sensors 1 and 4. At sensor 1, a rapid pressure increase is followed by a slower pressure decrease and an underpressure, also called rarefaction wave. This kind of profile is known as a Friedlander profile [START_REF] Friedlander | Proc. R. Soc. Lond. A[END_REF][START_REF] Dewey | Proc. R. Soc. Lond. A[END_REF] . The duration of the pressure increase is of the order of 0.1 ms, which is about 100 times larger than the nominal pressure sensor rise time. The duration of the overpressure is of the order of 2 ms. We observed that these two characteristic quantities do not depend on the foam, and that the shape of the pressure signals are quite reproducible (Fig. 2). Henceforth, for the sake of simplicity, we will thus consider that the pressure wave is entirely determined by its maximal value at sensor 1, denoted by P 1 in the following. It is varied by changing the number of aluminium foils used to close the high pressure chamber. The maximal value recorded at sensor 4 is P 4 and we define the pressure ratio as α = P 4 /P 1 .

The first reflection of the wave on the external boundary of the foam reaches the sensor 4 at t ≈ 5 ms, leading to a second overpressure. We only consider events occurring before this first reflection.

To test the influence of the cell geometry, we compared pressure signals obtained from pairs of sensors placed at the same distance from the center, respectively on the top and the bottom plates, with a 3 cm gap, in the H geometry. At the position of the first sensor (r 3 cm), the bottom signals amplitudes are always greater than the top ones by a factor in the range [1 -1.5]. However both signals always superimpose at the position of sensor 4 (r 9 cm), with differences smaller than the experimental reproducibility. We also varied the H cell thickness from 1.7 cm to 8 cm. The pressure decrease from sensor 1 to sensor 4 (on the top plate) was not modified for gaps between 1.7 cm and 3 cm. For larger gaps, a faster decrease was observed, interpreted as a transition from a cylindrical (at small gap) to a spherical (at large gap) wave propagation. In Appendix A, we present comparisons between the pressure decreases obtained with the 3 cm and 8 cm gaps which confirm quantitatively this interpretation, thereby ensuring that geometrical effects have been correctly taken into account in the following data and interpretations.

In the following, we present experiments made in the cylindrical regime, with a 3 cm gap both in the H and V geometries.

Wave velocity measurements

The pressure wave deforms during its propagation, because of the non-linearities and of the viscous and dispersive nature of the medium (see Fig. 2). The wave velocity can thus be defined in different ways, leading to slightly different results. Experimentally, the maximal pressure P i is well defined on the pressure signal i, but its arrival time is difficult to determined accurately, as it turned out that the signal is quite flat around the maximum. In contrast, the arrival time of the pressure P i /2, denoted by t 1/2,i is a well defined quantity. We thus define the wave velocity as v = (r 4r 1 )/(t 1/2,4t 1/2,1 ).

In parallel to a velocity based on the pressure signals, optical measurements can also be performed. In that case, the foam sample is lit with an intense white light, used in transmission. The foam dynamics are recorded using a highspeed camera (Photron FastCam SA3), which is triggered by the same signal as the pressure sensors: both measurements are therefore synchronised. The dynamics remaining axisymmetric, we reduced the field of view to a narrow rectangle (640×32 pixel 2 ), allowing to record images at 57,000 frames per second. A spatio-temporal diagram is obtained from the images, on which different gray levels are clearly visible, produced by the spatial fluctuations of the transmission coefficient of the foam (see Fig. 14 in Appendix B). Before the pressure wave arrives, these fluctuations are static and vertical lines are thus visible on the diagram. The sudden inflection of these lines at a position r and time t im (r) is the signature of the pressure wave front reaching the position r. The wave velocity is deduced from the value t im (r) as detailed in Appendix B.

Both definitions of the wave velocity, based on the pressure signals or on light transmission, are equivalent, as evidenced in Fig. 14,Appendix B.

With the optical measure, we get the front velocity at any position during the propagation. For r < r 1 a high velocity transient is observed (see Fig. 14,Appendix B), but the velocity becomes independent on r for r > r 1 in all our experiments. This steady velocity increases with P 1 , as expected for a non-linear propagation 16 . However, for P 4 < 10 kPa, the wave velocity variation due to the pressure is smaller than the experimental dispersion. Therefore, the velocities discussed in the following are the average over all the experiments we performed which verifies P 4 < 10 kPa.

Results

Influence of the gas and of the bubble radius

The set-up V was used with glass frits of three different calibrated porosities 2, 1 and 0 to produce a C 2 F 6 foam with bubble radii in the range 200 to 800 µm. The pressure attenuation and the wave velocity are measured as a function of the bubble radius and of the pressure amplitude. To allow comparison with the previous study on N 2 foams 16 , the liquid fraction φ l was adjusted as close as possible to 4.8 % by tuning the gas flux through each frit. The bubble radius range of the previous study was also enlarged by adding a point at R = 840 µm for a N 2 foam.

Wave attenuation.

We report in Fig. 3 the pressure ratio α exp = P 4 /P 1 as a function of P 1 obtained for these experiments, and one data se- 16 , for sake of comparison. The pressure ratio increases with the pressure intensity. This behavior has been reproduced numerically by a slightly nonlinear numerical model in 16 . These simulations show an almost affine behavior, with a slope independent on the material attenuation. The extrapolation at P 1 = 0 is in contrast a signature of the attenuation properties, and the remaining part of the paper will thus focus on this extrapolated value. The affine fits at constant slope are indeed in good agreement with the data, even if a quantitative comparison is made difficult by the large experimental noise observed with some series. Only the N 2 foam with the largest bubble radius (R = 840 µm) is not compatible with such a fit as it exhibits a much faster increase of α exp with P 1 . This last data series corresponds to a different regime, as discussed in section 4 and is first discarded in the following discussion. 3) % (already published in 16 , Fig. 3); + R = 840 µm, φ l = (4.9 ± 0.2) %. Black dashed line: geometrical attenuation corresponding to a cylindrical propagation. Coloured dashed lines: affine fits with the imposed slope k 0 = 5.4 10 -3 kPa -1 .

A C C E P T E D M A N U S C R I P T ries from Fig. 3 in
An important result of this paper is that the pressure ratio α depends on the nature of the gas. A direct comparison can be made in Fig. 3 for R 210 µm: the pressure ratio is significantly larger with C 2 F 6 than with N 2 over the whole pressure range, which is the proof of a smaller attenuation with C 2 F 6 . This is a crucial hint to determine the fundamental nature of the dissipation processes leading to the wave attenuation. Indeed, as the bubble radius, the liquid fraction and the surfactants are the same for both series, the dissipation in the liquid phase and at the interfaces is not modified and thus cannot explain the attenuation variation between both series. This evidences that the dissipation in the gas itself contributes significantly to the total dissipation.

Moreover, for all P 1 values, the pressure ratio for C 2 F 6 foams first decreases with the bubble radius and then increases at larger bubble radius. This surprising nonmonotonous behaviour is discussed in the next paragraph. 3) (isothermal regime), with the prefactor λ T = 0.4 for both gases (red : N 2 , black: C 2 F 6 ). Dashed lines: predictions of eq. ( 4) (adiabatic regime) with λ a = 12, for both gases.

Attenuation length.

In the small amplitude regime, and in a non-dissipative, non-dispersive medium, the pressure attenuation only depends on the space dimension. A propagation with a cylindrical symmetry induces a pressure ratio α ref 2D (r) = r 1 /r between two points at the distance r and r 1 of the symmetry axis. Dissipation in the medium will induce an additional pressure decrease which can be characterised in the linear regime by an attenuation length a . The pressure ratio is in this case α(r) = α ref e -(r-r 1 )/ a . The attenuation length is the intrinsic property of the material, which must be measured to quantify attenuation.

To measure a , we extrapolated at P 1 = 0 the affine fits made in Fig. 3 to obtain the experimental value of α (for r = r 4 ) in the low pressure limit, i.e. in the linear regime. The low pressure data of each series (P 1 < 20 kPa), including those of Fig. 3 in 16 , were first fitted by the law α = k 1 P 1 + k 2 , with k 1 and k 2 fitting parameters which depends on the bubble size R and on the nature of the gas g. The obtained slopes k 1 were then averaged over all the series (excepted the N 2 series at R = 840 µm), leading to k 0 = k 1 R,g = 5.4 10 -3 kPa -1 . The different series were finally fitted by the one-parameter law α exp = k 0 P 1 + α 0 (R, g), as shown in Fig. 3. The experimental attenuation length is finally given, for each bubble radius and each gas, by

1 exp a = 1 r 4 -r 1 ln r 1 /r 4 α 0 (R, g) . (1)
As discussed in Appendix A, more data points have been obtained in spherical propagation geometry, and they were found to be consistent with the data obtained with a cylindrical propagation.

Fig. 4 shows 1/ a as a function of the bubble radius, for the two gases. As discussed in the previous section, the attenuation is larger in N 2 foams than in C 2 F 6 foams in the investigated bubble radius range. The non-monotonous behaviour of the C 2 F 6 foams appears more clearly in this representation: the attenuation in C 2 F 6 foams exhibits a max-

A C C E P T E D M A N U S C R I P T
imum for a bubble radius around 500 µm, as was conjectured in 16 for N 2 foams. This result is important for practical reasons, as the optimal pressure attenuation is sought in many applications. It is also a very discriminant experimental information to test different theoretical models of dissipation, as discussed in section 4.

Wave velocity.

The velocity v of the pressure wave front is plotted in Fig. 5 as a function of the bubble radius. The velocity in the C 2 F 6 foams (same experiments as in Fig. 4) is shown in the top graph and was measured from the pressure signal. One additional point at R = 45 µm has been obtained in cell H at large gap (spherical propagation). As the wave velocity is independent on the geometry, this data point can be compared to the other ones. The velocities for the N 2 foams have been measured on the images (same experiments as in 16 ), close to sensor 4. The point at R = 840 µm is a new experiment, with a velocity measured with the pressure signal. A classical prediction for the wave velocity in the linear regime is Wood's velocity [START_REF] Wood | A Textbook of Sound[END_REF] . This model assumes that the foam behaves as an effective medium of compressibility χ 0 and density ρ 0 , averaged over the volume fractions of the liquid and the gas phases. As the liquid phase is almost incompressible, χ 0 is based on the gas compressibility: χ g = 1/( κ p 0 ), leading to χ 0 = (1φ l )/( κ p 0 ), with φ l the liquid fraction measured at the atmospheric pressure p 0 , and κ the polytropic exponent 14 . The gas is modeled as an ideal gas with κ = γ or κ = 1 for an adiabatic or isothermal compression, respectively. Here, γ is the ratio of specific heats; it equals 1.4 for N 2 , and 1.085 for C 2 F 6 [START_REF] Hunt | Use of hexafluoroethane to simulate the inviscid real-gas effects on blunt entry vehicles[END_REF] . The foam density is given by ρ 0 = φ l ρ s + (1φ l )ρ g where ρ s and ρ g are respectively the solution and gas densities 8,25 . Wood's velocity is thus:

c 2 w = 1 χ 0 ρ 0 = κ p 0 1 -φ l 1 φ l ρ s + (1 -φ l )ρ g . (2)
The adiabatic and isothermal Wood's velocities c w,a and c w,T have been computed for each experiment, on the basis of the measured liquid fraction φ l . The obtained values are reported in Fig. 5. The experimental wave velocities remain close to Wood's velocities for small bubbles, and become larger than this prediction for bigger bubbles. The velocity is indeed 2.5 times larger than Wood's prediction for the largest bubble radius and the C 2 F 6 foam, and it reaches twice the predicted value for the N 2 foam with the largest bubble radius. This last sample is the one which also shows a specific behaviour in Fig. 3. A departure from Wood's velocity is a strong indication that a gas bubble and its contiguous liquid environment do not move with the same amplitude when the pressure wave arrives. The menisci, i.e. the liquid channels at the intersection of three films, having a much larger inertia that the thin films, have a different dynamics than the gas and the films. This effect is at the basis of the resonance observations and models established in 11,12 and becomes important above a critical bubble radius. Fig. 5 shows that for the bubble radius corresponding to the maximum of dissipation, i. e. R ≈ 450 µm, the wave velocity is still close to the Woods' velocity. This is a first indication that the maximum cannot be explained by the resonance.

Influence of the liquid fraction

The influence of the liquid fraction has been investigated for N 2 foams in the large bubble limit (R close to 900 µm) using the setup V and in the small bubble limit (R close to 95 µm) using the setup H. The shock was always produced with a single aluminum foil (low pressure range). For both series, the velocity is measured from the pressure signals.

Wave attenuation.

The pressure ratio for large bubbles is plotted in Fig. 6 and shows a decrease of α with the liquid fraction, i.e. an increase of attenuation. For a given frit porosity, the bubble radius fluctuates over a certain range and we kept only the experiments with bubble radius in the range 800 to 950 µm. The bubble radius is indicated for each data point by a colour code in Fig. 6. For the different liquid fractions, the largest value of α is obtained either for the smallest or for the largest bubble sizes. For bubbles larger than 900 µm, we expect that α increases with the bubble size (see Fig. 3), but the bubble size scatter in this figure is small enough for this dependency to be unobservable. The variation of α is thus solely due to the liquid fraction variation. Similarly, we only kept P 1 values in the range 4.8 to 6 kPa. From the value of α, we deduce the attenuation length of the foam using eq. ( 1), shown in Fig. 9. As the pressure intensity has not been varied, the pressure ratio cannot be extrapolated

A C C E P T E D M A N U S C

R I P T

to P 1 = 0, and the parameter α 0 in eq. ( 1) is identified with the raw data α exp .

The same study was performed for small bubble radius, as shown in Fig. 7. In that case, the pressure attenuation is equal to the geometric attenuation, within the experimental error bars. This means that the attenuation due to the medium itself is not detected. We can therefore not conclude about a potential dependency with the liquid fraction in this small bubble regime. This result also confirms that foams with small bubbles do not attenuate much sound waves, in agreement with Figs. 4 and13.

Wave velocity.

The wave velocity measured from the pressure signals is reported in Fig. 8 as a function of the liquid fraction for small bubble radii (R = 95 µm, same experiments as in Fig. 7) and for large bubble radii (R = 870 µm, same experiments as in Fig. 6, and additional points at φ l < 1 with R in the range 900 to 1100 µm). Wood's velocities for an adiabatic and an isothermal propagation (eq. 2) are plotted on the same graph, without fitting parameter. A very good agreement is obtained between the small bubble series and the isothermal Wood's velocity, whereas for the large bubble the experimental velocities are larger than Wood's velocity by a factor which reaches 2.5 for φ l close to 5 %. This confirms that the wave propagation is in the Wood regime at small bubble size, and departs from it at larger bubble size. The transition is difficult to quantify, but Fig. 5 indicates that it is around R = 800 µm. R = 95 µm (same experiments as in Fig. 7); ( ) Large bubble radius: R = (870 ± 44) µm for φ l > 1 % (same experiments as in Fig. 6) and R in [900 -1100] µm for φ l < 1 %. Wood's prediction is represented with a full line for the isothermal regime, and dashed line for adiabatic regime. The data at φ l = 4.8 % have already been shown in Fig. 5.

Models

Thermal dissipation model

The thermal model developed in [START_REF] Mujica | [END_REF]14,15 and already used in 16 assumes that the energy dissipation is mainly of thermal origin. The gas in the bubbles is compressed by the pressure wave, its temperature thus increases and dissipative heat transfers occur between the gas phase and the liquid phase, the latter acting as a thermostat. Such dissipation is strongly enhanced in foams because of the large contact area between the liquid phase and the gas phase, which temperatures differ strongly after the sharp pressure increase. At the end of the overpressure, the gas is thermalised by the liquid on a shell of thickness T = √ D T τ around each gas bubble, with D T the heat diffusivity given in Table 1 and τ the overpressure duration. For τ = 2 ms (Fig. 2), we thus compute T = 200 µm for N 2 and 80 µm for C 2 F 6 . A priori, if the bubble radius R is much smaller than T , the propagation is almost isothermal, and in the other limit, the process is almost adiabatic. The attenuation lengths in these two regimes are respectively 16 :

1 T a = λ T R 2 c w,T ρ 0 κT τ 2 , (3) 1
A a = λ A c w,A ρ 0 ρ g c p T R κ ρ g c p τ , (4)
with T the temperature, c w,A and c w,T the wave velocities for adiabatic and isothermal propagations given by eq. ( 2), ρ 0 the foam density and κ, c p and ρ g the gas properties given in Table 1. The parameters λ A and λ T are dimensionless fitting parameters, and all the other quantities are known for the two gases. A direct comparison with our experimental data can thus be made, to identify the origin of the observed dependencies with the liquid fraction, the bubble radius and the nature of the gas. )

Figure 9 Attenuation length as a function of the liquid fraction (same data as in Fig. 6). The red dashed line is the prediction of the thermal model in the adiabatic regime (eq. 4) with a prefactor λ A = 5.

The model predicts an attenuation 1/ a which increases with the liquid fraction φ l , both in the small and large bubble radius limits. Indeed, even if this model considers only a dissipation in the gas phase, 1/ a depends on the liquid fraction through Wood's velocity, scaling as φ -1/2 l , and through the foam density scaling as φ l . The attenuation lengths deduced from the pressure ratios shown in Fig. 6 (N 2 foam) are plotted in Fig. 9. As the bubble radius for these data is R = 870 µm, which is in the large bubble regime, they are compared with the prediction in the adiabatic regime. Fig. 9 shows a good agreement between the experimental data and the prediction of eq. ( 4) for a prefactor λ A = 5. This value is close to the value λ A = 7 used in 16 to reproduce the experimental data obtained with N 2 foams.

The dependency of 1/ a with the bubble radius predicted by the model is more complex. It predicts an increase of the attenuation as R 2 for small bubbles (eq. 3), followed by a decrease as 1/R for large bubbles (eq. 4). We show in 16 that the attenuation in N 2 foams is well fitted by this model for the couple of parameters (λ T = 0.4; λ A = 7). The prefactor λ A = 7 is too small to reproduce the data at large bubbles of Fig. 4, obtained with C 2 F 6 , that would be best fitted with λ A close to 35. The prediction of the model using the couple of parameters (λ T = 0.4; λ A = 12), chosen as a best compromise between the two gases, is shown in Fig. 4 together with the experimental data.

All important qualitative features are reproduced by this thermal model, despite the lack of quantitative agreement: a maximum of attenuation is observed for a bubble radius close to the thermal length; the attenuation is smaller for C 2 F 6 foams than for N 2 foams in the investigated parameter range; the order of magnitude of the predicted attenuation is in the right range; the attenuation increases with the liquid fraction at large bubble radius.

This allows to conclude that the thermal dissipation is non-negligible in the investigated parameter range, and that it is probably at the origin of the observed maximum of attenuation. The discrepancy between the model and the experiment may be explained by the fact that other dissipative and dispersive processes also contribute to the global pressure attenuation, with different scaling laws. For instance, viscous effects within the liquid phase, which are ubiquitous in foam rheology 26 , are also expected to play a role. The thermal model itself could be refined: in particular, the liquid films are so thin that the assumption that they act as thermostats is questionable, especially at large bubble size.

Furthermore, the thermal model implicitly assumes the validity of Wood's prediction. The velocity increase at large bubble radius is thus not predicted in this model, and is potentially related to the proximity of the resonant behaviour evidenced in 11,12 , as discussed in the next section.

Resonant film-meniscus model

Wood's model is based on the assumption that the foam behaves as a continuum and that, locally, the gas, the liquid in the films and the liquid in the menisci move at the same velocity v( r,t), which varies at the scale of the acoustic wavelength. However, at very large frequencies, the menisci do not move any more, and only the thin films, with a much smaller inertia, are displaced by the pressure wave 13 . Pierre et al. recently gave an extensive description, both theoretical and experimental, of the transition from one regime to another, at a critical frequency (for a given bubble radius) 12 or at a critical bubble radius (for a given frequency) 27 . This transition coincides with a maximum of dissipation and occurs for parameter values (bubble radius, time scale...) close to our experimental parameter range. More precisely, Fig. 3 in 12 shows that Wood's regime breaks down for a frequency f such that f (R/R 0 ) 1.5 = 10 5 Hz, with R 0 = 40 µm. Since the overpressure duration is in the ms range, it corresponds to typical frequencies in the kHz range. With f = 1 kHz, we thus obtain that Wood's regime breaks down at R = 0.9 mm, in agreement with our data on velocity (Fig. 5). It may thus a priori explain the maximum of dissipation which we observe, and is an alternative to the thermal model that must be carefully analysed. In contrast with the previous approach, this maximum of dissipation is not related to an especially efficient dissipative process at the bubble scale, but to an enhanced deformation of the film/meniscus structure. In this case, the dissipation is assumed to occur mainly in the connection between films and menisci and is simply modelled by a phenomenological time scale τ d , that does not depend on the bubble radius, nor on the frequency. The model developed in 12 predicts a specific dispersion relation for the plane wave propagation, in the linear regime. In order to compare our observations with the predictions of this model, we computed the propagation of a pressure pulse using this dispersion relation. The incoming signal, at the position z = 0, is assumed to be p 1 = sin(πt/τ) for t ∈ [0, τ] and 0 elsewhere, with τ = 1 ms the duration of the overpressure. This ansatz mimics well the shape of the overpressure signal at sensor 1 (Fig. 2), apart from its slight asymmetry, and neglects the rarefaction wave. With this simplified form, the Fourier transform of the signal has an analytical

The main Fourier components are in the range [0, 3π/τ], i.e. for ω smaller than 10 4 s -1 . The pressure at distance z can then be determined by the relation

p(t, z) = 1 π Re ∞ 0 p(ω)e -i[ωt-k(ω)z] dω . (7)
The dispersion relation which we used is based on eq. ( 2) in 12 :

k(ω) 2 = ω 2 ρ eff p 0 , (8)
with ρ eff = (1φ l )ρ g + φ ρ s the foam effective density, built on the complex number φ , playing the role of an effective liquid fraction. It equals the actual liquid fraction at low frequency and Wood's velocity is recovered in this limit. In the high frequency limit, the liquid phase contained in the menisci does not move any more, and φ is the volumetric fraction of the thin films. This is the Kann regime 13 . In the intermediate regime, a resonant behaviour is observed, with a maximal modulus of φ . The full expression for φ is 12 :

φ = φ l 1 + x 2 φ l φ f [1 -H (qa)] -iωτ d xH (qa) , (9)
with q = ω ρ s e/(2σ ), e ∼ 100 nm the thin film thickness, σ = 36 mN/m the surface tension, τ d = 10 -5 s, x the fraction of the bubble area covered by the thin films, which equals 0.38 for φ l = 4.8 % 28 , and a = √ xR the radius of the thin film. The function H (u) = 2J 1 (u)/[uJ 0 (u)] is built from the Bessel functions of order 0 and 1 and φ f = 3(1φ l )x 2 e/(4R) is the volume of liquid contained in the thin films per unit foam volume. Using the approximate value for H (u) for u 1, a simple expression is obtained for the resonant frequency: ω r = 12σ /(x 2 ρ s φ l R 3 ). For R = 500 µm, this yields ω r = 2 • 10 4 s -1 , which is thus just above the highest frequencies of the incoming signal. Fig. 10 shows the pressure signal at sensor 4 obtained with eq. ( 7) for bubble radius in the range 200 to 1100 µm, for N 2 . The signal is deformed due to the viscous and dispersive properties of the medium, but a well defined overpressure is still observable. The maximal pressure decreases with the bubble radius, as well as the slope of the initial pressure increase. However, the velocity deduced from t 1/2 is almost independent of the bubble radius and remains close to Wood's velocity.

The attenuation length is obtained from the P 4 /P 1 ratio, using the relation a = (r 4r 1 )/ ln(P 4 /P 1 ), because the simulated propagation is unidirectional, so there is no geometric attenuation. The experimental data of Fig. 3 are compared with the numerical attenuation length in Fig. 11 7), ( 8) and ( 9), for N 2 and for the bubble radius R = (200, 500, 800, 1100) µm (the attenuation increases with the bubble radius). are almost identical, as the density does not play a important role in the attenuation. As already visible in Fig. 10, the attenuation increases with the bubble radius. However it is impossible to explain the maximum of attenuation observed at R = 500 µm with the resonant model. We checked that until at least R = 3 mm (last computed value) the attenuation 1/ a keeps increasing with R. This strongly reinforces our conclusion that the thermal dissipation in the gas is at the origin of the observed maximum.

The resonant model nevertheless predicts the right order of magnitude of the attenuation length for bubble around 800 µm, and a transition from a thermal origin of the dissipation toward an origin based on the resonance model is possible. However, the wave velocity deduced from the numerical simulation varies by only 2 % on the bubble radius range 200 to 1100 µm, so the fast velocity observed for N 2 foams at R = 800 µm remains unexplained.

Finally, we made the same comparison with the experimental data of Fig. 6, with a bubble radius R = 870 µm. In that case, with a value of the adjustable dissipative time τ d ∼ 5 • 10 -5 (to be compared with τ d = 10 -5 obtained for another foam in 12 ), the agreement with the prediction of the model is fair (see Fig. 12). As the thermal and resonant models both predict the increase of 1/ a with φ l , these data does not bring additional information to discriminate between both models. However, the thermal model shown in Fig. 6 allows for a slightly better fit of the data.

Conclusions

In this paper, we have reported an extensive study of blast wave propagation and attenuation in liquid foams, mostly in the weak wave regime, where comparison with linear acoustics remains relevant. Thanks to our careful control of the foam production and parameters, we were able to study independently the influence of bubble size, liquid fraction, and gas. We showed for the first time that attenuation reaches a maximum for a certain bubble size, and depends on the gas. We also showed that attenuation increases at increasing liquid fraction. These measurements of attenuation were compared to a model accounting for thermal dissipation in the gas phase in contact with the liquid interfaces. It reproduces all the aforementioned qualitative experimental features, although it does not yield a quantitative agreement with the data. The propagation velocity of the blast wave agrees with Wood's velocity at small bubble size, but becomes larger for bubble sizes above 0.5 to 0.8 mm. This is in qualitative agreement with a resonance model recently proposed 12 . We also showed that such a model could not explain the attenuation maximum.

These unprecedented results pave the way toward a more rational optimisation of shock and blast wave mitigation by liquid foams. It also opens several perspectives. First, we did not model the nonlinear regime, nor the dependence of attenuation on the blast wave amplitude. Second, and related to the first point, the possible destruction of part of the foam by a strong enough blast, and its influence on propagation and attenuation of the blast wave, remains to be addressed. We obtained some recent data along this direction, which will be presented in a subsequent study. We thus hope that the current study is a first but crucial bridge between the most recent developments of linear acoustics of liquid foams and the regime of strong shocks and blasts. on this line. They have been reported on the pressure signals in Fig. 14 (bottom). At each sensor, the time t im i roughly coincides to t 1/2,i , defined as the time for which the pressure reaches half its maximal value at sensor i. This justifies that the velocity deduced from the image and t im i represents the same physical quantity as the velocity deduced from the pressure signal and t 1/2,i .

Figure 1

 1 Figure 1 Sketch of the vertical set-up. Pressure is monitored at the exit of the gas bottle and the flux in each frit is controlled by a valve (blue patches on the figure).

Figure 2

 2 Figure 2 Typical pressure signals in a C 2 F 6 foam measured at sensor 1 and sensor 4, for R = 422 µm and two aluminium foils. The time origin corresponds to the arrival of the blast wave on sensor 1.

Figure 3

 3 Figure 3 Pressure attenuation as a function of the shock amplitude. C 2 F 6 foams: • R = (218 ± 5) µm, φ l = (5.6 ± 0.5) %; R = (422 ± 13) µm, φ l = (4.4 ± 0.4) %; R = (794 ± 28) µm, φ l = (4.2 ± 0.3) %. N 2 foams: × × × R = 210 µm, φ l = (4.8 ± 0.3) % (already published in16 , Fig.3); + R = 840 µm, φ l = (4.9 ± 0.2) %. Black dashed line: geometrical attenuation corresponding to a cylindrical propagation. Coloured dashed lines: affine fits with the imposed slope k 0 = 5.4 10 -3 kPa -1 .

Figure 4

 4 Figure 4Attenuation length for C 2 F 6 foams ( ) and for N 2 foams (•), as a function of the bubble size, for a liquid fraction φ l = 4.8 %. Solid lines: prediction of eq. (3) (isothermal regime), with the prefactor λ T = 0.4 for both gases (red : N 2 , black: C 2 F 6 ). Dashed lines: predictions of eq. (4) (adiabatic regime) with λ a = 12, for both gases.

Figure 5 (

 5 Figure 5 (Top) Wave front velocity for C 2 F 6 foams as a function of the bubble radius for φ l = 4.8 % ( ). Theoretical Wood's velocity given by eq. (2) for isothermal (• • •) and adiabatic ( ) regimes. (Bottom) Wave front velocity for N 2 foams and Wood's predictions, same symbols.

Figure 6

 6 Figure 6 Pressure ratio α exp = P 4 /P 1 as a function of the liquid fraction φ l for a N 2 foam in the V cell. Colours indicate the bubble radius in µm for each experiment. The average radius is R = 870 µm. The dashed line represents the pressure ratio α ref associated to the geometrical attenuation.

Figure 7

 7 Figure 7 Attenuation α = P 4 /P 1 for a bubble radius R = (95 ± 13) µm, obtained for a N 2 foam in the H cell. The dashed line is the geometrical attenuation.

Figure 8

 8 Figure 8 Wave front velocity for N 2 foams as a function of the liquid fraction, deduced from the pressure signals. (• • •) Small bubble radius:

π 2 -

 2 ω 2 τ 2 e iωτ/2 cos ωτ 2 .

6 timeFigure 10

 610 Figure 10Imposed pressure signal at the position z = 0 as a function of time (black line) and pressure signal at the position z = 6 cm determined numerically from eqs. (7), (8) and (9), for N 2 and for the bubble radius R = (200, 500, 800, 1100) µm (the attenuation increases with the bubble radius).

Figure 11

 11 Figure 11 Experimental attenuation length of Fig. 4 compared with the attenuation length obtained numerically for N 2 and τ d = 2 • 10 -5 s (solid line), 5 • 10 -5 s (dotted-dashed line) and 10 -4 s (dashed line).

Figure 12

 12 Figure 12 Experimental data of Fig.6 compared with the attenuation length obtained numerically for τ d = 2•10 -5 s (solid line), 5•10 -5 s (dotted-dashed line) and 10 -4 s (dashed line).

Figure 13

 13 Figure13 Attenuation length as a function of the bubble radius for N 2 foams in cylindrical geometry (•) (data from16 ) and in spherical geometry ( * ).

Table 1

 1 Gas properties of N 2 and C 2 F 6 17 .

Acknowledgements

The authors thank S. Kerampran for regular and insightful discussions and J.-C. Potier and P. Chasle for technical support. The shock tube has been designed by following the advice of G. Jourdan and L. Houas. M. Monloubou gratefully acknowledges financial support from the DGA.

Appendix A

In order to test our assumption of a cylindrical propagation in the 3 cm gap cell, we performed additional experiments in the H cell with a larger gap h = 8 cm, with φ l = 4.8 % and R in the range 50 to 300 µm. With this larger gap, it is clearly visible on the pressure signals that the maximal overpressure reaches sensor 4 before the first reflection on the bottom plate reaches the top plate. The signal measured by the sensor placed on the top plate is thus the same as a signal propagating in a half 3D space. As the foam was observed to slip at the top plate, we consider that the propagation is the same as in a full 3D space, and the reference attenuation α ref 3D (r) = r 1 /r can be used. For this test, we have not systematically varied the overpressure. We assumed that the attenuation is of the form, at sensor i:

This involves an attenuation length a , which is the intrinsic material property we want to determine, the geometrical attenuation r 1 /r i which corresponds to a linear propagation in a spherical geometry, and the non-linear correction K sph P 1 .

The slope K sph is assumed to be independent of the bubble radius and of the nature of the gas, as in the cylindrical geometry (see Sec. 3.1.1). Its value K sph = 7 10 -3 kPa -1 has been determined from a single data set obtained at different pressure values with C 2 F 6 , R = 45 µm, and r i = r 3 . This value is then used to compute the value of α 0 for all data obtained in the spherical geometry. The deduced attenuation lengths a , averaged over several data points for each bubble radius, is plotted in Fig. 13. The values are compatible with the ones obtained for the same gas N 2 and same liquid fraction φ = 4.8 % in the cylindrical geometry, thus confirming that the propagation between the two plates has a cylindrical symmetry, and that a does not depend on the cell geometry.

Appendix B

A spatio-temporal diagram of the foam is shown in Fig. 14 (top). The gray level fluctuations produce vertical lines when the foam is at rest, and oblique curves when it moves. The transition between both behaviours appears sharply on the diagram and corresponds to the front wave arrival time.

It has been determined by image processing and is represented by the black line on the diagram. The slope of this line is the front wave velocity. The times t im i are defined as the times corresponding to the positions of the four sensors The four squares represent the arrival time t im i of the wave front at each sensor position, as determined on the spatio-temporal image above.


Camila Freitas Salgueiredo
email: camila.freitassalgueiredo@ens.univ-evry.fr

Armand Hatchuel

Camila Freitas

Beyond analogy: A model of bio-inspiration for creative design

Keywords: Biologically inspired design, Concept-Knowledge Design theory, Innovation

Is biologically inspired design only an analogical transfer from biology to engineering? Actually, nature does not always bring "hands-on" solutions that can be analogically applied in classic engineering. Then, what are the different operations that are involved in the bioinspiration process and what are the conditions allowing this process to produce a bioinspired design? In this paper we model the whole design process in which bioinspiration is only one element. To build this model we use a general design theory, Concept-Knowledge (or C-K) theory, because it allows one to capture analogy as well as all other knowledge changes that lead to the design of a bioinspired solution. We ground this model on well described examples of biologically inspired designs available in the scientific literature. These examples include Flectofin ® , a hingeless flapping mechanism conceived for façade shading, and WhalePower technology, the introduction of bumps on the leading edge of airfoils to improve aerodynamic properties.

Our modelling disentangles the analogical aspects of the biologically inspired design process, and highlights the expansions occurring in both knowledge bases, scientific (non-biological) and biological, as well as the impact of these expansions in the generation of new concepts (concept partitioning). This model also shows that bioinspired design requires a special form of collaboration between engineers and biologists. Contrasting with the classic one-way transfer between biology and engineering that is assumed in the literature, the C-K framework shows that these collaborations must be "mutually inspirational", because both biological and engineering knowledge expansions are needed to reach a novel solution.

Introduction

Biologically inspired design (BID), also called biomimetic design or biomimicry refers to the use of biological systems as a source of inspiration for improving or developing new technical systems [START_REF] Vincent | Biomimetics: its practice and theory[END_REF], [START_REF] Shu | Biologically inspired design[END_REF].

Numerous examples of biomimetic design are found in different fields, such as materials, artificial intelligence, construction or transportation [START_REF] Benyus | Biomimetics: lessons from nature -an overview[END_REF], [START_REF] Bar-Cohen | Biomimetics: nature based innovations[END_REF]). According to the Velcro company website, George de Mestral, a Swiss engineer, developed Velcro ® based on the observation of burdock plants that attached to his dog fur. Genetic and ant colony optimization algorithms were inspired by natural selection and ant foraging, respectively. The Mercedes Bionic Car, a concept car presented in 2005, had the boxfish as a model for improving aerodynamic performance and the bio-inspired optimization methods such as Soft Kill Option [START_REF] Baumgartner | {SKO} (soft kill option): the biological way to find an optimum structure topology[END_REF]) for improving the vehicle structure [START_REF] Vincent | Biomimetics: its practice and theory[END_REF]). Besides, some researchers consider that natural systems are potential analogies for designing more sustainable systems, as they are submitted to natural selection and evolve in conditions respecting life [START_REF] Benyus | Biomimetics: lessons from nature -an overview[END_REF].

These examples of innovative designs and the need for more sustainable products motivate research and development to integrate the biologically inspired design approach to their new product development process, mainly in the ideas generation phase. This trend is reflected in the growing number of patents [START_REF] Bonser | Patented biologically-inspired technological innovations: A twenty year view[END_REF]) and articles [START_REF] Lepora | The state of the art in biomimetics[END_REF]) in this field. Understanding the advantages and limitations of adding biological knowledge into the design process, together with the organization of this process acquires a new significance: which interactions will be established between the different actors of the process (biologists, engineers, and researchers) and what will bring these dialogs between the different knowledge bases involved?

Literature has studied the biologically inspired design from the analogical transfer perspective. The biologically inspired design is interpreted as an analogical transfer between biology and engineering for problem solving [START_REF] Mak | Using descriptions of biological phenomena for idea generation[END_REF], [START_REF] Helms | Biologically inspired design: process and products[END_REF], [START_REF] Vattam | A content account of creative analogies in biologically inspired design[END_REF], [START_REF] Sartori | A methodology for supporting "transfer" in biomimetic design[END_REF]). [START_REF] Helms | Analogical problem evolution in biologically inspired design[END_REF]) recognize that these analogies can also provoke an evolution of the problem.

However, this analogical transfer does not allow to disentangle the roles of biological knowledge in the BID process: is it only used as an analogue model or has it effects on the creativity of the generated ideas? Where does this creativity come from? In this paper, Concept-Knowledge (C-K) theory is used to model the biologically inspired design process of selected biomimetic development examples. The aim of this modelling is to provide a framework for understanding not only the analogical processes taking place on biologically inspired design, but also the roles of biological knowledge and scientific knowledge in the process. In this paper, "scientific knowledge" refers to the non-biological knowledge used in the design process such as engineering knowledge. This better understanding of the process can contribute to a more systematic application of biologically inspired design in the ideas generation phase of new product development.

The remainder of the paper is organized as follows: Section 2 provides an overview of the biologically inspired design process theories and tools, identifying the points for which a general design theory could bring some new insights. Section 3 describes the methodology used to analyse biologically inspired design with a design theory. Section 4 presents the analysis of bio-inspired examples from literature using the C-K framework. Section 5 discusses the outcomes from this modelling process and Section 6 concludes the paper.

Theoretical background

By definition, biologically inspired design bridges two domains: biology and design. Two directions, or "high-level analogical processes for biologically inspired design" [START_REF] Goel | Biologically Inspired Design: Computational Methods and Tools[END_REF]) were identified for biologically inspired design. These directions refer to the motivations for seeking analogies between nature and design. If the analogical transfer is triggered by an existing design problem, the process is called top-down [START_REF] Speck | Process sequences in biomimetic research[END_REF]), problem-driven [START_REF] Helms | Biologically inspired design: process and products[END_REF], [START_REF] Nagel | Biologically inspired Design: Computational Methods and Tools[END_REF]) or challenge-to-biology (Biomimicry3.8 (2014)). On the other hand, if a biological phenomenon allows the identification of a design problem that could be solved using an analogy to this phenomenon, the process is named bottom-up [START_REF] Speck | Process sequences in biomimetic research[END_REF]), solutions-based [START_REF] Helms | Biologically inspired design: process and products[END_REF][START_REF] Nagel | Biologically inspired Design: Computational Methods and Tools[END_REF]) or biology-to-design (Biomimicry3.8 (2014)). [START_REF] Sartori | A methodology for supporting "transfer" in biomimetic design[END_REF]) and [START_REF] Badarnah | A methodology for the generation of biomimetic design concepts[END_REF]) compared studies discussing these approaches, identifying the main steps for each direction, summarized in Table 1.

Table 1: General steps for the biologically inspired design process [START_REF] Sartori | A methodology for supporting "transfer" in biomimetic design[END_REF] General process [START_REF] Badarnah | A methodology for the generation of biomimetic design concepts[END_REF])

General process [START_REF] Badarnah | A methodology for the generation of biomimetic design concepts[END_REF] Different studies about the BID process characterized this bridge as an "analogical transfer" or "cross-domain analogies" between biology and engineering [START_REF] Helms | Biologically inspired design: process and products[END_REF], [START_REF] Cheong | Using templates and mapping strategies to support analogical transfer in biomimetic design[END_REF]). The search for biological analogues and the transfer of attributes from the source to the target domain (in the case of BID from biology to design) constitute the two main foci of the research about the biologically inspired design process [START_REF] Shu | Biologically inspired design[END_REF]).

In the search for biological analogues, tools such as databases [START_REF] Deldin | Biologically inspired design: computational methods and tools[END_REF]), ontologies [START_REF] Vincent | Biologically inspired Design: Computational Methods and Tools[END_REF]), research on texts written in natural-language format [START_REF] Chiu | Biomimetic design through natural language analysis to facilitate crossdomain information retrieval[END_REF], [START_REF] Shu | A natural-language approach to biomimetic design[END_REF]), computational tools such as webcrawling [START_REF] Vandevenne | Webcrawling for a biological strategy corpus to support biologically-inspired design[END_REF]) and the functional basis (an organized search tool and engineering to biology thesaurus) [START_REF] Nagel | A computational approach to biologically inspired design[END_REF]) have been proposed. These tools depend on the contents of the databases and the designers ability in manipulating the data sources that will be used during the BID process.

For improving the analogical transfer outcome, representations of biological systems in a structure-behavior-function model (SBF) were developed using a platform that has algorithms for indexing and retrieving these models (DANE -Design by Analogy to Nature Engine, [START_REF] Wiltgen | Representation, indexing, and retrieval of biological cases for biologically inspired design[END_REF]). The SAPPhIRE model has also been studied for facilitating the analogical transfer in bioinspired design [START_REF] Sartori | A methodology for supporting "transfer" in biomimetic design[END_REF]). Other authors proposed functional decomposition to facilitate the identification of transferable attributes [START_REF] Nagel | Function-based, biologically inspired concept generation[END_REF][START_REF] Helfman Cohen | Biomimetics: Structure-Function Patterns Approach[END_REF]). Causal relation templates were proposed by [START_REF] Cheong | Using templates and mapping strategies to support analogical transfer in biomimetic design[END_REF]) in order to facilitate the analogical reasoning when designers use text descriptions of biological phenomena. The Bio-TRIZ approach [START_REF] Vincent | Biomimetics: its practice and theory[END_REF]) uses a modified TRIZ matrix of contradictions, made from the analysis of numerous biological phenomena to identify relevant principles in natural systems that could be used for solving a problem contradiction. Considering the whole BID process, [START_REF] Goel | The design study library: Collecting, analyzing and using case studies of biologically inspired design[END_REF]), elaborated a Design Study Library, containing case studies of BID from the initial idea until the conceptual design, which could be useful for helping novice designers learning about the BID process.

2.1 The role of "anomalies" in biologically inspired design [START_REF] Mak | Using descriptions of biological phenomena for idea generation[END_REF]) identified four similarity types between the source (a biological phenomenon) and the target (the design) in the biologically inspired design process: (i) analogy:

"implementation of the strategies found in the biological phenomenon without transferring biological forms"; (ii) literal implementation of biological forms and behaviours; (iii) biological transfer, that keeps the form, using it with a different purpose; (iv) anomaly, when there is no apparent similarity between the concept and the source biological phenomena. Apart from analogy, the other similarity types are considered "errors" [START_REF] Helms | Biologically inspired design: process and products[END_REF], [START_REF] Shu | Biologically inspired design[END_REF]). However, [START_REF] Wilson | The effects of biological examples in idea generation[END_REF]) observed that the novelty of designers' ideas was increased when they were exposed to biological examples and only half of these designs had an attribute transferred from the biological system. This may indicate that the biological example have provoked a cognitive stimulation on designers, which was useful to increase novelty. This represents an additional role of the biological knowledge in BID.

2.2 Problem evolution in biologically inspired design and the role of analogies [START_REF] Helms | Analogical problem evolution in biologically inspired design[END_REF]) proposed a model of analogical problem evolution that takes place during the BID process: a problem definition may evolve (extending or expanding) with the help of an analogy to an already existing solution. Analogies are then able to expand a problem, by adding other dimensions to the problem. Nevertheless, a biological phenomenon could provide the designer analogies that are not useful for his problem, but that could be valuable for solving another problem for which solutions were not specifically being searched for. This case can be found in companies covering large innovation fields, such as the automotive industry, for example.

Biologically inspired design goes beyond analogies?

Analogical transfer is indeed a part of the biologically inspired design process. Many of the bioinspired developments have clear analogies to the biological systems from which they drew inspiration from, namely when they are inspired by forms or "more visible" properties. However, as some studies in literature have pointed out, the biologically inspired design may go beyond analogies: explaining the increase in novelty provoked by the exposure to biological examples [START_REF] Wilson | The effects of biological examples in idea generation[END_REF]) or explaining how the misapplied analogies that involve identification of other knowledge bases, not originally related to the problem to be solved are also useful for stimulating creativity and innovation require the use of a more general design theory, capable of modelling creativity and cognitive stimulation processes.

Design theories have different levels of generativity, the ability to go beyond the search in a fixed set of solutions, and robustness capacity, the ability of producing robust designs (designs that have the expected performances). In order to model generativity and robustness, the tools used by formal design theories can change. To capture generativity it is important to model the expansion process, linked to searching outside a fixed set of solutions, while robustness is described by the ability of the design to resist to a disturbance (Hatchuel et al. (2011b)). The C-K theory [START_REF] Hatchuel | C-k design theory: an advanced formulation[END_REF]) allows a general interpretation of the design process, taking into account the generativeness and robustness capacities. This theory has already been used to disentangle properties from other design theories or methods, such as the creation of creative solutions using ASIT (advanced systematic inventive thinking) [START_REF] Reich | A theoretical analysis of creativity methods in engineering design: casting and improving asit within c?Äìk theory[END_REF]), the creativity process of Infused Design [START_REF] Shai | Creativity and scientific discovery with infused design and its analysis with c-k theory[END_REF]) or the innovative properties of Parameter Analysis (PA) [START_REF] Kroll | Steepest-first exploration with learning-based path evaluation: uncovering the design strategy of parameter analysis with c-k theory[END_REF]). In the creativity domain, it has been used to explain fixation effects (Hatchuel et al. (2011a), [START_REF] Agogué | The impact of type of examples on originality: Explaining fixation and stimulation effects[END_REF]).

3 The C-K theory [START_REF] Hatchuel | A new approach of innovative design: an introduction to c-k theory[END_REF]) defines design as an "interplay between two interdependent spaces", the space of concepts (C) and the space of knowledge (K). The space K contains the available knowledge of designers. The space C contains concepts, propositions that are neither true nor false considering the available knowledge in K. Design proceeds by the expansion of the initial concept into other concepts (concept partitioning) and / or into new knowledge, until a partitioned concept becomes a true proposition in K. These two spaces can expand and have different structures. In C space only partitioning (which means adding or subtracting properties) is allowed, giving it a tree-like structure, in which each node represents a partition into several sub-concepts. The K space grows like an "archipelago", with new propositions being added without following an order or a direct connection [START_REF] Hatchuel | C-k design theory: an advanced formulation[END_REF]).

The transformations between and inside spaces are called operators in C-K. These operators have different functions [START_REF] Hatchuel | C-k design theory: an advanced formulation[END_REF]). The C→K operator searches attributes in K that can be used to partition concepts in C or that contribute to the generation of new propositions in K. The modification of an initial concept by a new attribute must be followed by a verification determining whether the new proposition is still a concept or has become knowledge. The K→C operator generates new concepts by assigning new attributes to the existing concepts. The C→C operator allows the analysis of design paths and the K→K operator represents the classic types of reasoning.

The key notion of C-K theory for the analysis of the BID process is the notion of expanding partitions. These partitions expand the definition of an object, by introducing properties that are not known properties of the object. In addition to the revision of the identity of the object, expanding partitions guide the expansion of knowledge in new directions. Using the example of conceiving new car tires [START_REF] Hatchuel | Towards an ontology of design: lessons from c-k design theory and forcing[END_REF]), the partition "tires without rubber" is considered expansive. If the partition was already known in K, "tires with white rubber" it would be considered a restricting partition, as it only selects knowledge without simultaneously allowing knowledge expansion. The expanding partitions capture usual processes of creativity, such as inspiration, imagination, analogies and metaphors.

4 Results: A C-K model for biological inspiration

C-K modelling of biologically inspired design examples

The construction of the C-K framework begins with the formulation of concepts linked to the technical issues identified in each case (Aspect 1). Using the information about the biological properties observed, we identified the general biological bases mobilized during the design process (Aspect 2) and the properties from these biological knowledge bases that triggered concept partitioning with the K→C operator (Aspect 3). We observed that in each case this partitioning had an expanding effect: the concept partitioned using biological knowledge stimulated a revision of the scientific knowledge bases and of the biological knowledge bases.

Flectofin ®

First concept formulation (K→C): Deployable structures with simpler technical actuators are an issue to architects and engineers [START_REF] Knippers | Design and construction principles in nature and architecture[END_REF]). This is a concept in C-K theory: "design of a deployable system with simpler technical actuators", as it cannot be defined only using knowledge available to architects and engineers.

Concepts partitioning (K→C):

The usual actuators for deployability use hinges and rollers that require constant maintenance [START_REF] Lienhard | Flectofin: a hingeless flapping mechanism inspired by nature[END_REF]). One partition for the first concept, formulated using this knowledge about hinges and rollers would be "improving hinges and rollers to require less maintenance".

Biological knowledge activation (C→K) and concept partitioning: Previous research of this group had

shown that plants have some deployability mechanisms that do not require hinges, and are based on elastic deformations [START_REF] Matini | Application of "abstract formal patterns" for translating natural principles into the design of new deployable structures in architecture[END_REF], [START_REF] Lienhard | Abstraction of plant movements for deployable structures in architecture[END_REF]). The activation of this knowledge base partitions the first concept with "deployable systems in architecture without hinges and rollers", and "using elastic deformations", as it adds new properties to the first concept.

Exploration of biological and scientific knowledge (K→K) and concept partitioning:

The research group tried to identify deployability phenomena in plants that could be useful for improving deployability in architecture. They explored the biological knowledge on plants deformation without hinges. In parallel, scientific knowledge was also explored: they built prototypes that helped explaining and understanding the physical phenomena behind these deformations [START_REF] Matini | Application of "abstract formal patterns" for translating natural principles into the design of new deployable structures in architecture[END_REF], [START_REF] Lienhard | Abstraction of plant movements for deployable structures in architecture[END_REF]). They identified that elastic deformations could be useful for these deployability properties. One of these deployability phenomena, the bird-of-paradise flower pollination mechanism, was identified as particularly interesting to these researchers as it was "actuated externally", making use of "a special form of lateral torsional buckling … not unfamiliar to engineers but mainly perceived as an undesirable failure mode to be avoided when planning architectural constructions" [START_REF] Lienhard | Flectofin: a hingeless flapping mechanism inspired by nature[END_REF]).

According to [START_REF] Knippers | Design and construction principles in nature and architecture[END_REF]) the external actuation in the bird-of-paradise flower refers to the opening of the two adnate petals that form a perch in which birds can land when searching for the plants' nectar. The mechanical pressure exerted on the perch by the bird causes the bending of the two adnate petals, which exposes the flower pollen. The pollen is attached to the bird's feet, helping the pollination mechanism. When the birds fly away, the perch returns to the closed position. Scientific knowledge revision (K→K) and final design path (C→K): Researchers revised their knowledge about this phenomenon, in order to understand how this failure mode could turn useful for deployable structures. They identified that "attaching a thin shell element to a rib" allowed obtaining this special form of lateral torsional buckling (Figure 1) and studied materials most appropriate for the technical realization, Flectofin ® , a façade shading system without hinges, has vertical backbones and laminae attached to each backbone (Figure 2). Moving the lower support of the system up and down causes the bending of the backbones and the opening / closing of the laminae [START_REF] Lienhard | Flectofin: a hingeless flapping mechanism inspired by nature[END_REF], [START_REF] Knippers | Design and construction principles in nature and architecture[END_REF]). In this case, the two expanding partitions "without using hinges" and "using lateral torsional buckling" came clearly from the biological knowledge about plants movements and the bird-of-paradise pollination mechanisms.

The C-K framework for this case is summarized in figure 3.

WhalePower

First concept formulation (K→C): Humpback whales superior manoeuvrability properties compared to other whales seemed to be linked to their flippers hydrodynamic performance. These flippers have protuberances, called tubercles, on their leading edge, contrasting with the "well-streamlined, engineered hydro and air foils" [START_REF] Bushnell | Drag reduction in nature[END_REF], [START_REF] Fish | Hydrodynamic design of the humpback whale flipper[END_REF]). We hypothesize, based on the information presented on the article of [START_REF] Fish | Hydrodynamic design of the humpback whale flipper[END_REF], that this contrast between engineering and biology was the starting point for this case. The first concept could then be formulated as the "design of hydro-and airfoils with improved hydro-/aerodynamics properties".

Concepts partitioning (K→C): the contrast between engineered and biological systems partitions this first concept: "by improving well-streamlined leading edges" and "without using well-streamlined leading edges".

Exploration of biological and scientific knowledge (K→K) and concept partitioning: In order to better understand the whale's flippers hydrodynamic properties, researchers activated knowledge about fluid mechanics and hydrodynamics of foils. The concept "without using well-streamlined leading edges" is an expanding partition that allows scientific knowledge about fluid dynamics expansion and revision.

Using this newly activated knowledge base, researchers understood the reasons for the improved performance of the tubercles in the leading edges of the whales flipper [START_REF] Miklosovic | Leading-edge tubercles delay stall on humpback whale (megaptera novaeangliae) flippers[END_REF][START_REF] Johari | Effects of leading-edge protuberances on airfoil performance[END_REF]) and linked tubercles to passive control around a wing like structure [START_REF] Fish | The tubercles on humpback whales' flippers: Application of bio-inspired technology[END_REF]).

The presence of tubercles was claimed to "delay stall and both increase lift and reduce drag at the same time" [START_REF] Fish | The tubercles on humpback whales' flippers: Application of bio-inspired technology[END_REF]). This was an unexpected property of these structures, that could be applied in the design of hydro and air foils, partitioning the concept of "without well-streamlined leading edges"

into "introducing tubercles to the leading edge", not to increase manoeuvrability (as in whales) but to improve hydro and aerodynamics of fan blades, and air turbines. Frank Fish © WhalePower Corporation.

Scientific knowledge revision (K→K) and final design path (C→K):

The introduction of tubercles in blades of ceiling fans as those currently commercialized by Envira-North (Altra-air fans) represents the final design path. There also current possibilities for applying the bumps in turbines (Figure 4) or in aviation [START_REF] Fish | The tubercles on humpback whales' flippers: Application of bio-inspired technology[END_REF]).

The C-K framework for this case is summarized in figure 5. The previous two examples modelled with the C-K framework and operators show some similarities. Firstly, the identification of interesting properties in a biological knowledge base allows the partitioning of concepts of the existing design paths. Secondly, these bio-inspired concepts are expansive concepts: They provoke a revision of the existing knowledge about an object or a phenomena (as in the buckling properties of materials or the aerodynamic properties of foils) and guide the expansion of the knowledge in new directions, activating knowledge bases that would not otherwise be activated.

A general model for BID using the C-K framework should then be composed of the two spaces, and the knowledge bases. These knowledge bases are the scientific knowledge bases, i.e. the initial knowledge bases available to designers and the biological knowledge bases, which can exist initially (in the case of WhalePower, biologists were studying the hydrodynamic properties of whales), or not (in the Flectofin ® case, architects did not have the biological knowledge at the beginning, but established a partnership with plants biomechanics experts to get access to this knowledge).

The activation of biological knowledge during the design process guides the knowledge expansion phenomena and partitions concepts:

-Scientific knowledge bases are activated to explain or better understand the properties observed in biological phenomena;

-The properties of biological knowledge that activate scientific knowledge bases not spontaneously activated, guide the knowledge revision and expansion and can be conceptualized in the C-space, forming expanding partitions.

These elements can be summarized in the following four general steps, as shown in Figure 6: This identification corresponds to the conceptualization of a design issue that will trigger the activation of scientific knowledge bases for clarifying this issue.

Activation of biological knowledge related to the first concept

The choice of the biological knowledge base can be guided by the existing methods for retrieving biological phenomena, such as functional decomposition [START_REF] Nagel | Function-based, biologically inspired concept generation[END_REF][START_REF] Helfman Cohen | Biomimetics: Structure-Function Patterns Approach[END_REF]), SBF modelling [START_REF] Wiltgen | Representation, indexing, and retrieval of biological cases for biologically inspired design[END_REF]), the SAPPhIRE model [START_REF] Sartori | A methodology for supporting "transfer" in biomimetic design[END_REF] or search in natural-language [START_REF] Shu | A natural-language approach to biomimetic design[END_REF]). Identifying interesting properties partitions the first concept and guides a deeper exploration of the biological knowledge base.

If the observation of an interesting property in a biological system is the starting point of the design process, then the first step becomes the identification of a design path that could benefit from this property.

Expansions in both biological and scientific knowledge bases

These expansions represent the activation of biological knowledge (screening process of different plant movements in the case of Flectofin ® ) and of scientific knowledge related to these interesting biological properties being explored (the modelling and simulations of flippers hydrodynamic properties in the WhalePower case). With these expansions in both knowledge bases, interesting properties allowing concepts partitioning are identified.

Return to the scientific knowledge bases

Once a design path becomes knowledge, its development does not depend on the biological knowledge anymore: the pollination mechanism of the bird of paradise flower was useful for understanding how to apply the buckling phenomena, but once researchers understood how to reproduce this phenomena with existing materials and techniques, the development of the product only used scientific knowledge. Similarly, in the WhalePower case, the identification of flippers properties allowed the bumps to be installed in fans, independently from research on whales' movements.

Discussion

The modelling of the bio-inspired design process with the C-K theory aimed at disentangling the reasons for using biological knowledge during the design process.

The two directions pointed out in literature for the BID process, top-down and bottom-up, are retrieved in the C-K framework: the two examples, Flectofin ® and WhalePower belong to each one of these directions. The C-K framework gives a more general interpretation to these two directions, not only encompassing the operations in the K space (search for analogues, transfer, and emulation) but also relating these operations to the concepts space, showing that biological knowledge stimulates concepts partitioning and that these partitions are expanding partitions: they guide the exploration and revision of knowledge. These expansions and revisions of both knowledge bases are easily visualized in the C-K framework, facilitating the exploration of the design paths shown in the C space and the associated knowledge bases.

The creative power of biologically inspired design comes from these expanding partitions it generates, as they stimulate the activation of scientific knowledge bases that would not otherwise be activated. The "interesting" character of biological knowledge is a consequence of this ability to generate expansive concept partitioning. This aspect was not easily visualized with the analogical transfer interpretation of the BID process, as it does not show how the analogous properties found in biological systems interact with the existing knowledge and if this interaction stimulates the search of new knowledge bases or if it leads to remaining in the same knowledge bases.

The C-K model for BID also provides some insights about the organization of this process in practice. Firstly, the C-K model indicates that the use of biological knowledge in the design process requires more than a superficial knowledge about biological systems. For example, when designing a novel multi-energy system for vehicles, it is not sufficient to know that nature also has multi-energy systems. Engineers must also understand details about the design of these natural systems, and reinterpret this biological knowledge using their own knowledge bases, and possibly new ones that they will have to acquire. This collaboration between engineering and biology differs from a simple knowledge sharing or transfer process: biology specialists must work together with engineers, also revising their knowledge with the revisions engineers are making, while engineers should be able to accurately identify the knowledge bases that biological knowledge is revising and the new ones being activated.

This C-K framework offers a representation of the design process, including the design paths and the use of knowledge bases. It provides designers with a representation of the design paths and how the knowledge bases being activated can interact. Designers can easily see, using this framework, that the biologically inspired design process involves not only finding an interesting property in biology. It implies relating these properties to existing knowledge bases, which can lead to revisions and expansions in these scientific knowledge bases.

One limitation of this C-K model is that it does not indicate how the suitable biological knowledge bases should be found. The methods and tools proposed in literature for the BID analogical transfer could represent a solution to this aspect. For example, functional decomposition [START_REF] Nagel | Biologically inspired Design: Computational Methods and Tools[END_REF])

can be used to expand and revise both biological and non-biological knowledge bases, helping in the identification of non-spontaneously activated knowledge bases while natural-language (Shu ( 2010)) can help in the identification of other biological bases to explore during the design process. However, it is important to notice that while classic analogical transfer methods consider misapplied analogies as errors in analogical transfer, the C-K framework shows that the most important role of biological knowledge is concept partitioning, independently of these properties being used or not in the final design path.

Conclusions and perspectives

The interpretation of the biologically inspired design process with the C-K theory framework presented in this article highlights two main aspects of this process. The first one refers to the knowledge base construction that takes place during the biologically inspired design process. Two knowledge bases, biological and scientific (non-biological) are expanded and revised during the process. The identification of interesting properties in the knowledge space allowing concept partitioning is made possible only with the expansions in both knowledge bases. The second one is related to the organizational aspects of the BID process. Expanding and revising these two knowledge bases requires a collaboration between biology and traditional engineering, this collaboration is "mutually inspirational" because both knowledge bases are expanded using the interesting properties. Therefore, these interesting properties can only be identified by making connections between the two knowledge bases.

This initial modelling of the biologically inspired design process using the C-K theory framework and operators is based on a limited number of case-examples. This model constitutes a first step towards a comprehension and systematic inclusion of other process besides analogy into the design process.

Future work will elaborate on these findings and explore practical conditions for applying these two findings in a real industrial context.

  the interpretation of BID as an analogical transfer process and the search for biological analogues and transfer are steps found in both directions.

  Research methodology: Analysis of biologically inspired design examples with C-K theory Our research methodology involves the search for examples of biologically inspired design developments (products or processes) in literature. The descriptive accounts of the biologically inspired design process were used to model the bioinspiration process using the C-K theory framework and operators. This modelling of the case-examples with the C-K theory required a full description of the design process, including the concept paths not followed and the usual design paths and knowledge bases accessed. The examples were identified using a systematic research in scientific databases such as Scopus ® and journals and books dedicated to biological inspiration such as Bioinspiration and Biomimetics (IOP Publishing), (Bar-Cohen (2012)) book and Design and Nature Conference Proceedings (WIT Press). The search queries were bioinspir*, biologically inspired, biomim*. Among the numerous examples of biologically inspired developments, we chose five examples, in which the bioinspiration process was clear and fully explained.

From

  our initial screening of examples in literature on biologically inspired design, we selected five examples to interpret using the C-K framework, operators and the expanding partitioning notion. These five examples are: Flectofin ® , developed by the Institute of Building Structures and Structural Design (IKTE) of the University of Stuttgart and the Plants Biomechanic Group of the University of Freiburg, in Germany; WhalePower technology, conceived by Dr. Frank Fish and Dr. Philip Watts from West Chester University in the United States; the textiles inspired by the Pine-Cone effect developed by Julian Vincent of the University of Bath and Veronika Kapsali from the London College of fashion, theGecko effect, that allowed the development of different bio-inspired designs, studied in many laboratories around the world and the Lotus-Effect, first characterized by[START_REF] Barthlott | Purity of the sacred lotus, or escape from contamination in biological surfaces[END_REF]) and used in self-cleaning coatings and surfaces.4.1 Cases overviewModelling each case using a C-K framework required the identification of the following aspects of the design process: Aspect 1: The technical issue addressed by the biomimetic development (first concept or design path);  Aspect 2: The biological property observed that could bring some insight for this issue (the biological inspiration);  Aspect 3: The reasons for using inspiration from nature (how the biological inspiration was used during the design process).Table2summarizes these three aspects for each case. These aspects were used in the construction of the C-K representation of each design process. In this article we detail two of these representations, the WhalePower case and the Flectofin ® case. This last C-K framework, along with the framework of the Gecko and Lotus case were presented in a previous study[START_REF] Freitas Salgueiredo | Modeling biologically inspired design with the c-k theory[END_REF]), and is now revised and expanded with a clear identification of the expanding partitions and the construction of the knowledge base. With respect to the two directions for biological inspiration, WhalePower represents a case of bottom-up bio-inspiration process and Flectofin® a case of top-down process. Table 2: Main aspects identified in each of the case examples of biologically inspired

Figure 1 :

 1 Figure 1: The deformation principle of the bird-of-paradise flower realized with a physical model. The

Figure 3 :

 3 Figure 3: C-K framework for the Flectofin ® case

Figure 4 :

 4 Figure 4: WhalePower technology applied to a turbine blade. Reproduced with permission from Dr.

Figure 5 :

 5 Figure 5: C-K framework for the WhalePower case

Figure 6 :

 6 Figure 6: General C-K framework for the biologically inspired design process

Acknowledgements

The authors thank the researchers of the Design Theory and Methods for Innovation Research and Teaching Chair; Philippe Doublet from Renault; Drs. Sébastien Glaser, Olivier Orfila, and Guillaume Saint Pierre from the LIVIC-IFSTTAR laboratory; Professor Stéphane Doncieux from ISIR-UPMC; and Professor Véronique Billat from UBIAE for their support and feedback throughout this work.