E Verron

G Chagnon

J.-B Le Cam

Hyperelasticity with volumetric damage

The present paper presents a simple framework to model continuous volumetric damage in elastomers. The formulation predicts phenomenologically the growth of microscopic cavities, and can be applied to both static and fatigue loading conditions. This first version of the approach cannot handle cavitation and is limited to small values of porosities. The derivation is based on the use of a simple scalar damage parameter, the irreversible volume change, and takes naturally into account the change in stiffness through the explicit dependence of the material parameters on the damage variable. The thermodynamic force which drives the volume change contains the hydrostatic stress and also a contribution due to stiffness evolution. As a first application, a damage compressible neo-Hookean constitutive equation is derived and a simple example is studied.

INTRODUCTION

Rubber-like materials are usually considered as incompressible. However, under multiaxial or fatigue loading conditions, cavitation and cavities growth take place, and lead to damage and finally to fracture [START_REF] Farris | The influence of vacuole formation on the response and failure of filled elastomers[END_REF][START_REF] Le Cam | Mechanism of fatigue crack growth in carbon black filled natural rubber[END_REF][START_REF] Gorju | Fatigue life of rubber components: 3d damage evolution from x-ray computed microtomography[END_REF]. Special experiments can be carried out to exhibit this behaviour as proposed by [START_REF] Gent | Forms of the stored (strain) energy function for vulcanized rubber[END_REF], [START_REF] Gent | Fracture mechanics and cavitation in rubber-like solids[END_REF] or [START_REF] Taylor | Fatigue initiation and propagation in natural and synthetic rubbers[END_REF]. For modelling, on the one hand the cavitation phenomenon under hydrostatic loading conditions is studied considering the stability conditions for the sudden growth of microscopic cavities in the incompressible bulk (see [START_REF] Ball | Discontinuous equilibrium solutions and cavitation in nonlinear elasticity[END_REF], [START_REF] Horgan | A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void[END_REF] for example). On the other hand, several phenomenological approaches have been proposed to predict the growth of pre-existing cavities; the corresponding models incorporate damage variables into compressible hyperelastic approaches (see [START_REF] Boyce | Constitutive models of elasticity: a review[END_REF] for a short review) to quantify the irreversible change of porosity [START_REF] Andrieux | Sur les solides hyperélastiques à compressibilité in-duite par 1'endommagement[END_REF][START_REF] Dorfmann | Shear, compressive and dilatational response of rubberlike solids subject to cavitation damage[END_REF][START_REF] Layouni | Compressibility induced by damage in carbon black reinforced natural rubber[END_REF][START_REF] Li | A constitutive model for cavitation and cavity growth in rubberlike materials under arbitrary tri-axial loading[END_REF]). These models can also be extended to cavitation by adapting the rate equation of the damage variable [START_REF] Dorfmann | Stress-softening of elastomers in hydrostatic tension[END_REF]. Nevertheless, they are limited to small values of the porosity.

In the present paper, similarly to [START_REF] Andrieux | Sur les solides hyperélastiques à compressibilité in-duite par 1'endommagement[END_REF], we propose a simple theoretical framework to model the compressibility induced by damage in hyperelastic materials. Our approach is phenomenological and is restricted to small values of porosity, such that the growing cavities do not interfere. The scalar damage variable is the irreversible volume change and its influence on the stiffness of the material is taken into account through the material parameters. The rate equation chosen here is not adapted to sudden volume change (cavitation) but only to continuous volume change (damage by continuous growth of cavities).

The derivation of the model is described in the next section, the emphasize being laid on the determination of the thermodynamic force which drives the volume change. Then, a very simple constitutive equation which generalizes the compressible neo-Hookean model is considered to illustrate the relevance of the method.

DERIVATION OF THE CONSTITUTIVE EQUATION

2.1 General formulation

Kinematics

It is considered here that under loading, the body exhibits an irreversible volume change due to what can be called a volumetric damage. In the general framework of large strain, the material is supposed homogeneous, isotropic and hyperelastic. Moreover, we assume that it initially contains small flaws that can be considered as holes. To simplify the derivation, the RVE is assumed to deform as sketched in Figure 1; the deformation gradient F is separated into two parts:

• an irreversible volume change between the initial configuration (C 0 ) and the intermediate configuration (C i ). It is defined by the deformation gradient J c 1 3

/ I, in which J c is the ratio of irreversible volume change between the two configurations. In the figure, this volume change is illustrated by the irreversible growth of small holes; nevertheless, we do not consider their microscopic evolution: interaction, coalescence, … • a classical elastic deformation, which gradient is denoted f, between the intermediate configuration (C i ) and the deformed configuration (C).

The well-established multiplicative decomposition of the deformation gradient is adopted:

F f I f = = . / / J J c c 1 3 1 3 (1)
Indeed, the deformation process can be described by the two following variables: the observ able strain (through F) and the internal variable J c which describes damage.

Constitutive equations

First, the reversible deformation between configurations (C i ) and (C) is considered hyperelastic: it exists a strain energy function w J c ( ) f per unit of volume in (C i ). It is highly important to note that this strain energy function depends on J c in two ways: through f, recalling Eq. (1), and through the material parameters which explicitly depend on the irreversible volume change, as notified by the subscript •J c . Second, the deformation between (C 0 ) and (C i ) being totally irreversible, no elastic strain energy is involved. Then, the total strain energy of the material is simply the strain energy w J c written per unit of undeformed volume, i.e. unit of volume in (C 0 ):

W J J w c c J c ( ) ( ) F f , = .
(2)

Once the strain energy function defined, one can easily derive the constitutive equations. Restricting the problem to a purely mechanical theory, i.e. ignoring thermal effects, the Clausius-Planck inequality is

D int = : -≥ , P F W 0 (3)
where D int is the internal dissipation and P is the first Piola-Kirchhoff stress tensor. Recalling that W depends on both the deformation gradient F and the internal variable J c ,

D int = : - : - ≥ . P F F ∂ ∂ ∂ ∂ W W J J J c c c F F 0 (4)
Following [START_REF] Coleman | The thermodynam ics of elastic materials with heat conduction and vis cosity[END_REF], F and J c ⋅ can be chosen arbitrarily and then the constitutive equation for F, i.e. the stress-strain relationship, is

P = ∂ ∂ W J c F , (5)
and the internal dissipation reduces to

D int with = ≥ =- G J G W J c c 0 ∂ ∂ F ( 6
)
where G is the thermodynamic force which drives the irreversible change in volume. Eq. ( 6) 2 is the constitutive equation for J c .

Stress-strain relationship. One can now derive Eq. ( 5):

P F f F f f F = = = : . ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ W J c J c J c c c J w J w ( ) (7)
Introducing

p f = ∂ /∂ w J c
the first Piola-Kirchhoff stress tensor with respect to the intermediate configuration (C i ), and after some algebraic manipulations, the nominal stress reduces to (see for example [START_REF] Holzapfel | Nonlinear Solid Mechanics. A continuum approach for engineering[END_REF] for such derivation)

P f f = . / J w c J c 2 3 ∂ ∂ ( ) (8)
Note that the derivation of p is straightforward: once the strain energy function w J c is chosen, the classical hyperelastic theory applies.

Thermodynamic force for change in volume. The thermodynamic force which drives the irreversible change in volume is

c 1/3 I f f 1 F (C) (C i ) (C 0 ) G J w J c J c c c = - = - - . ∂ ∂ ∂ ∂ W w J J c F F ( ) f (9)
The second right-hand side term can be calculated as follow

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ w w J J c c ( ) f f f J J w J c c J c c F = + : , expl (10)
where the subscript ⋅ expl denotes the explicit differentiation with respect to J c , which involves the differentiation of the material parameters. Considering again the intermediate engineering stress tensor p, we have

∂ ∂ ∂ ∂ w J J J c c c f f p F : = : - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ .
-/

1 3 4 3 (11)
So, invoking Eqs. ( 1) and ( 8), the thermodynamic force is

G J J w c c J c = - - - : ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . ∂ ∂ w J c expl 1 3 p f (12)
This equation can be transformed by introducing the Eshelby stress tensor Σ = WI -F T P:

G J J J c c c = - - : . - ∂ ∂ w J c expl 1 3 Σ I (13)
Moreover, considering the velocity gradient associated with the irreversible volume change L c , which is simply J J c c -/ 1 3 I here, and its symmetric part, the rate of deformation tensor D c (equal to L c in the present case), the internal dissipation can be written as

D int = - -: , J J J c c c c ∂ ∂ w J c expl Σ D (14)
and the thermodynamic force is then

G J J J c c c = - - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . - ∂ ∂ w J c expl 1 3 tr Σ (15)
Finally, this expression can be simplified by rewriting the Eshelby stress tensor as proposed by [START_REF] Verron | Definition of a new predictor for multiaxial fatigue crack nucleation in rubber[END_REF]: noting ( ) , , N i i=1 2 3 the principal strain directions in the reference configuration and recalling that the Eshelby stress tensor is symmetric for isotropic elastic materials, Σ becomes

Σ = - ⊗ = ∑ i i i i i W S 1 2 3 ( ) , λ N N (16)
where (λ i ) i=1,2,3 are the principal stretch ratio and (S i ) i=1,2,3 the eigenvalues of the second Piola-Kirchhoff stress tensor. Introducing the relationship between these principal stresses and the principal Cauchy stresses

σ i i i J S i = =,, -1 2 1 2 3 λ ( 17
)
where J = det F characterizes the total change in volume between (C 0 ) and C, i.e. reversible and irreversible, the Eshelby stress tensor can be written as

Σ = - ⊗ = ∑ i i i i W J 1 3 ( ) , σ N N (18)
and tr Σ 3

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = - W J s σ ( 19
)
where σ s is the spherical part of the Cauchy stress tensor, i.e. the hydrostatic stress. So, the thermodynamic force reduces to

G W J J J c c s = - + . ∂ ∂ expl σ (20)

Evolution equation

To close the general formulation of the model, we should precise the rate equation for the internal variable J c . Such equation can be written as:

J f G c = , ( ) F (21)
where the function f must be positive to ensure the positivity of the dissipation. In order to simplify the model, we simply choose a very simple "damagelike" evolution equation under the following form:

J k c = ⎧ ⎨ ⎩ G G G if < otherwise. max 0 (22)
In this equation, the parameter k is a positive real scalar value which depends on the material and G max stands for the maximum value of the thermodynamic force G previously endured by the material.

A particular model

Once the general theory derived, particular models can be proposed by specifying the strain energy density w J c . We consider the case where the elastic deformation (from (C i ) to (C)) is compressible and we adopt one of the simplest strain energy densities for compressible hyperelastic materials: the generalization of the incompressible neo-Hookean model proposed by [START_REF] Simo | Remarks on rate constitutive equations for finite deformation problems: computational implications[END_REF] and adopted later by both Ehlers and Elipper (1998) and [START_REF] Bischoff | A new constitutive model for the compressibility of elastomers at finite deformations[END_REF]. So the complete strain energy density is

W J J C i i D i CJ J c c w c c J c ( ) [ ( ln ) (ln ) ] ( ) F f , = -- + = 1 3 3 2 3 -/ -+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , 2 3 1 2 3 2 4 I J J DJ J J c c C ln ln ( 23
)
where i 1 and i 3 are the invariants of the right Cauchy-Green elastic strain tensor f T f, i.e. its trace and determinant respectively. Note that for the deformation gradient f, the spherical-deviatoric split is not considered. In this equation the material parameter C is twice the shear modulus, and the material parameter D is proportional to the compressibility modulus. Two cases are now considered.

• The first one for which there is no stiffness decrease due to damage, i.e. material parameters are constant, C = C 0 and D = D 0 . In that case, the engineering stress tensor P and the thermodynamic force G are respectively

P F F = - + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ / - 2 2 8 0 1 3 0 0 C J C D J J J c c c T ln ( 24
)
and

G C J I J J C D D J J c c c = - - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + - - - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ -/ 0 2 3 1 0 0 0 3 1 2 8 4 ln ln .
(25)

• The second one for which the stiffness decreases, i.e. C and D depends on J c . To specify the dependence of these parameters on J c , we invoke the statistical theory of rubber elasticity which states that

C nkT = , 1 2 (26)
with n being the number of active polymer chains per unit of volume, k the Boltzmann constant and T the temperature. Considering the mass conservation equation between configurations (C 0 ) and (C i ),

n dV m n dV m i i 0 0 chain chain = ( 27
)
where n 0 and n i are the chain densities per unit of volume in (C 0 ) and (C i ), dV 0 and dV i the infinitesimal volumes in (C 0 ) and (C i ), and m chain is the mean mass of the chains, and recalling that dV i = J c dV 0 , one can state that

C J C J c c ( ) / . = 0 (28)
The second parameter D is not directly related to the chain density; nevertheless, we consider the same relationship with the initial compressibility modulus

D J D J c c ( ) / = 0 (29)
In that case, the engineering stress tensor P and the thermodynamic force G are respectively

P F F = 2 0 C J C D J J c c T - - - + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 0 0 2 8 / ln (30)
and

G J I J D J J J c c c c = - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ - - 2 3 1 8 0 5 3 1 0 C / ln . ( 31
)

FIRST RESULTS

In order to illustrate the previous theory, we consider a very simple problem: a sample is first subjected to cyclic triaxial loading conditions, then it is uniaxially stretched. More precisely, the loading conditions are

• Four equi-triaxial cycles under prescribed stretch ratio λ: from λ = 1 to λ = 1.25, then from λ = 1 to λ = 1.5, then from λ = 1 to λ = 1.75, and finally from λ = 1 to λ = 2. The deformation gradient and the engineering stress tensor are respectively

+ ⊗ λ 1 1 2 2 3 3 μ ( ) , (34)
P e e = 1 P ⊗ 1 . ( 35
)
The following values of the material parameters are adopted: C 0 = 1 MPa, D 0 = 10 MPa and k = 0.01.

First, for both models, we present the evolution of J c as a function of the loading time and the stress-strain response during the triaxial loading phase: in Figures 2 and3 for the first model without change in material parameters (Eqs. (24-25)), and in Figures 4 and5 for the second model with change in material parameters ). The comparison of the volumetric damage evolution in Figs. 2 and4 exhibits that ignoring the induced change in material parameters leads to a limit in the irreversible change in volume (see the two final constant steps in Fig. 2), whereas if this change is taken into account the volumetric damage evolves continuously during triaxial cyclic loading. This mechanical response can be also observed with the stress-strain responses in Figs. 3 and5: for the former model a unique curve is reached for about λ = 1.5 and unloading parts of the response become closer and closer as depicted in Fig. 3. For the latter model, Fig. 5 shows that the material stiffness evolves in a regular manner and the resulting cyclic response is similar to the one encountered with damage-like constitutive equations, see for example [START_REF] Chagnon | On the relevance of continuum damage mechanics for the Mullins effect in elas tomers[END_REF].

Finally, we examine the mechanical response obtained during the final uniaxial extension in Figures 6 and7. For both models, J c does not change under loading (the curves are not shown here). Nevertheless, the first model exhibits a "strange" behaviour: the stiffness of the material is revealed always greater than its initial stiffness (before damage), and the undamaged and damaged curves intersect which means that for stretch ratios greater than 1.75 the stress greater in the damaged material than in the undamaged one; this behaviour is physically irrelevant. For the second model, the stiffness of the damaged material is always lower than the initial stiffness and the curves never intersect.

CONCLUSION

In this paper, a new method has been proposed to consider irreversible growth of cavities in rubberlike materials. The general framework consists in introducing a scalar damage variable J c which represents the irreversible volume change at a material point; then, the constitutive equations are derived with in the framework of Thermodynamics or Irreversible Processes. The originality of the present approach consists in taking into account the stiffness change induced by damage through the explicit dependence of the material parameters on the damage variable. Moreover, considering the statistical rubber elasticity theory, the extension of the neo-Hookean model has been derived and the dependence of the shear modulus on J c has been easily established. Finally, a first proof of the model relevance has been proposed by considering a very simple loading history.

Further work will be carried out first to extend the model to cavitation through the choice of a more complex rate equation for J c and second to implement the model in the finite element context.

Figure 1 .

 1 Figure 1. Deformation of the RVE.

  J

F

Figure 2 .

 2 Figure 2. Model without change in material parameters: evolution of the volumetric damage during the four triaxial cycles.

Figure 3 .

 3 Figure 3. Model without change in material parameters: stress-strain response during the four triaxial cycles.

Figure 4 .Figure 5 .

 45 Figure 4. Model with change in material parameters: evolution of the volumetric damage during the four triaxial cycles.

Figure 6 .

 6 Figure 6. Model without change in material parameters: uniaxial stress-strain response.

Figure 7 .

 7 Figure 7. Model with change in material parameters: uniaxial stress-strain response.


T Guélon

E Toussaint

J.-B Le Cam

Mechanical characterization of rubber from one heterogeneous test

The present study deals with the mechanical characterization of elastomeric materials. Classically, three homogeneous tests are considered to identify constitutive parameters, namely Uniaxial Tensile (UT), Pure Shear (PS) and Equibiaxial Tensile (ET) tests. Here, a new method is proposed: it consists in inducing the three previous homogeneous tests from only one heterogeneous mechanical test. For this purpose, a conventional tensile machine is used and a new apparatus is designed to be adapted on the machine. The test-induced heterogeneity is discussed related to two criteria based on the existence of UT, PS and ET and on the distribution of the maximal principal elongation at each material points of the sample surface. Experimentally, kinematic fields are provided on the sample surface by an image correlation code suitable for large deformations. Finally, an inverse technique, so-called Virtual Field Method, is used to identify the material parameters in the framework of the Mooney hyperelasticity. Kinematic fields are provided by a Digital Image Correlation software suitable for large deformations: Correli LMT (Hild 2002). Constitutive parameters are identified using an inverse method called the virtual field method.

CHOICE OF THE SAMPLE GEOMETRY AND THE LOADING CONDITIONS

In this section, a heterogeneous test that combines UT, PS and ET from a conventional tensile machine is presented. A numerical approach is used to choose the sample geometry and the loading conditions in order to generate sufficient heterogeneity of the kinematic fields. Criteria used to estimate Test-Induced Heterogenity (TIH) are defined and the chosen sample geometry and loading conditions are presented.

Test-induced heterogenity

. This diagram allow us to determine the loading condition applied at

INTRODUCTION

The behaviour of rubber-like materials is generally modelled in the framework of hyperelasticity. Numerous constitutive relations are available in the literature [START_REF] Marckmann | Comparison of hyperelastic models for rubberlike materials[END_REF]. The identification of the material parameters that govern the constitutive equations is still a difficult task. Classically, three homogeneous tests are considered to identify constitutive parameters, namely uniaxial tensile (UT), pure shear (PS) and equibiaxial tensile (ET) [START_REF] Ward | An Introduction to the Mechanical Properties of Solid Polymers[END_REF], [START_REF] Sasso | Characterization of hyperelastic rubber-like materials by biaxial and uniaxial stretching tests based on optical methods[END_REF]) and a trade-off between the different sets of values obtained for each type of test must be found. These tests are based on the strong assumption of homogeneity of the kinematic fields induced by each test.

In the present work, a new approach based on a heterogeneous test that simultaneously generates the three types of strain states abovementioned is pro-posed. For that purpose, the authors propose to design and adapt a new apparatus on a conventional uniaxial tensile machine that generates a heterogeneous strain state. The sample geometry and the loading conditions are defined beforehand by numerical investigations. Two criteria are defined to discuss the heterogeneity induced by the test. In order to characterize TIH, we propose two criteria. Firstly, the test has to generate the three homogeneous loading conditions, namely UT, PS and ET. Secondly, for each loading condition, i.e. UT, PS and ET, a distribution of the maximal principal stretch ratio (defined as the ratio between actual and initial lengths) has to be observed and must be sufficiently large considering the chosen constitutive equations and its field of validity. In the following, these criteria are used to choose both sample geometry and loading conditions.

Sample geometry and loading conditions

In the present study, a uniaxial tensile machine is used. In order to generate heterogeneity in the kinelatic fields, a change in geometry (for instance by adding notches or holes) could be considered. However, because of large deformations, the strain state obtain in an elastomer tends to be homogeneous, i.e. TU, whatever the geometry. So, numerical investigations are carried out to choose both sample geometry and biaxial loading conditions that generate a high heterogeneity level. The sample geometry and the loading conditions applied are presented in Figure 2. The sample geometry, which corresponds to a three branch sample, is 2 mm thick, 60 mm high and the branches are 20 mm in width. The bottom branch is clamped, the loading is carried out by applying prescribed displacement following the axis of the two other perpendicular branches.

A Finite Element Analysis is performed using the Ansys 10.0 package to discuss the TIH. Element type plane 182 is used to ensure the incompressibility assumption. The mesh is composed of 640 elements. For the sake of simplicity, we have chosen the Mooney strain energy density [START_REF] Mooney | A theory of large elastic deformation[END_REF]:

W C I C I = -+ - 1 1 2 2 3 3 ( ) ( ) (1)
where C 1 and C 2 are the material parameters. Their values are chosen equal to 0.4 MPa and 0.04 MPa, respectively, which are typical values for carbon black filled natural rubber.

Figure 3 shows that in the (I 1 -I 2 ) diagram, the three homogeneous tests are satisfactorily represented and distributed. The corresponding maximum principal elongation (see Fig. 4) of each loading case at the sample surface is deduced from Figure 3.

Global stretch ratios (defined as the ratio between actual and initial sample lengths) of 1.71 and 1.42 are prescribed following the horizontal and the vertical axes, respectively. The equivalent maximum stretch ratio is higher for UT than for PS and ET. This is explained by the fact that PS and ET are obtained by the extension of the branches. From Figure 4, one can conclude that each loading case is satisfactorily represented in terms of distribution of the maximum principal stretch ratio.

EXPERIMENTAL SET-UP

The numerical configuration is now used to perform the test with a conventional uniaxial testing machine. To prescribe a biaxial loading condition, a new apparatus is designed. It is presented in the first part of this section. Then, the measurement of the displacement fields is precisely described.

New apparatus

The testing machine, referred to as MTS 858 Elastomer Test System, has a loading capacity of 15 kN. The loading cell is 1 kN capacity. A new tensile apparatus is designed to be adapted to the uniaxial tensile machine. It allows to generate ET at the sample centre. This apparatus is mounted on the left testing machine column (see Fig. 5).

Figure 6 presents a picture of the apparatus. It is composed of four parts:

• part A allows the apparatus to be fixed on one of the two columns; • part B is a grip in which the horizontal branch of the sample is fixed; • part C corresponds to a helical slide that ensures the horizontal branch of the sample to be stretched; • part D is a ruler used to measure the horizontal displacement of the grip.

In practise, the two vertical branches of the sample are fastened on the grips of the conventional testing machine, then the branch perpendicular to the previous ones is fastened on the apparatus grip.

Measurement of the kinematic fields

Digital Image Correlation (DIC) technique is used to measure the displacement fields at the sample surface. For that purpose, images are shot with a cooled 12-bit dynamic CCD camera with 1376 × 1040 squared pixels and CORRELI LMT software is used. The DIC technique is well-suited for measuring large strains and has already been used in the case of elastomeric materials [START_REF] Chevalier | Digital image correlation used to analyze the multiaxial behavior of rubber-like materials[END_REF]). To determine the displacement field of a given image with respect to a reference image, one considers a set of sub-images (i.e., a square region that contains N × N pixels). This set is referred to as Zone of Interest (ZOI). A suitable correlation function is used to calculate the displacement of the centre of a given ZOI in two images captured at different stages of an experiment. In the present work, special attention is paid to the choice of the size of the ZOIs that define the region of interest (ROI). Here, the size of the ZOI is equal to 16 pixels with a shift of 16 pixels. This parameter characterises the measurement grid. To improve the image contrast, white paint is sprayed on the sample surface before testing.

IDENTIFICATION OF THE MATERIAL PARAMETERS

In this section, we present the method used to identify the material parameters of a given model from one heterogeneous test. It must be emphasized that no closed-form solution generally exists for such a problem, thereby meaning that no simple relation between local measurements, load, specimen geometry and unknown parameters is available.

Extracting constitutive parameters in this case is a major issue which must be tackled using relevant tools. We propose here to use the so-called Virtual Fields Method (VFM) which has recently be used for characterizing elastomeric materials within the framework of large deformations [START_REF] Promma | Application of the virtual fields method to mechanical characterization of elastomeric materials[END_REF]).

The VFM relies on the Principle of Virtual Work (PVW) which can be regarded as the global equilibrium of the specimen under study or the weak form of the local equations of equilibrium [START_REF] Dym | Solid Mechanics A Variational Approach[END_REF]. In case of large deformations, assuming a plane state of stress and neglecting body forces such as weight, the PVW can be written as follows:

Π Π : * ( . .). * * . . S S U X dS n U dl U K A 0 0 0 0 ∫ ∫ ∂ ∂ + =∀ ∂ ( 2
)
where Π is the first Piola-Kirchhoff stress tensor (PK1), the X s are the Lagrangian coordinates, U* is a kinematically admissible virtual field, S 0 is the surface of the specimen in the initial configuration the boundary of S 0 and n the unit vector perpendicular to this boundary. Considering the Mooney strain energy density, one can express the stress components as functions of the actual strain components. This leads to the following equation:

C A U X dS C B U X dS n U dl U K A S S S 1 0 2 0 0 0 0 0 : * : * ( . .). * * . . ∂ ∂ + ∂ ∂ = =∀ ∫ ∫ ∫ ∂ Π (3)
where A and B are quantities depending on the components of the deformation gradient tensor F. This equation is linear with respect to the unknown parameters C 1 and C 2 . In the integrands above, there are two types of terms: A and B on one hand, ∂ ∂ U X * on the other hand. A and B are directly deduced from the full-field measurements whereas ∂ ∂ U X * is derived from the virtual field U*. At this stage, the idea consists in writing the PVW with two different virtual fields U*. Since actual strain fields are heterogeneous, this leads to two independent equations where C 1 and C 2 are unknown. They are obtained by inverting the linear system.

Choosing at best the two virtual fields is a key issue in the method. In the present case of large deformations and hyperelasticity, it has been decided to use the following heuristic method. A large number of virtual fields is first randomly generated. Then, the set of two virtual fields that leads to the best conditioning of the linear system (assessed with the condition number) is finally chosen for identification purposes. Those two virtual fields provide parameters less sensitive to noisy data. In the present work, virtual fields have been defined piecewise [START_REF] Toussaint | The virtual fields method with piecewise virtual fields[END_REF]) because of the particular shape of the specimen. Four subregions are used to mesh the specimen because of its particular shape. The virtual displacement defined in each of the four sub-regions is described by polynomial shape functions multiplied by the virtual displacement of the nodes defining the corners of the sub-regions. These shape functions are similar to those employed in the finite element method. Figure 7 presents an example of optimized initial fields used for identifying C 1 and C 2 for the maximum global stretch ratios.

RESULTS

Experimental kinematic fields

The biaxial tensile test is carried out by prescribing a 25 mm displacement along both the x-and y-directions shown in Figure 2. The corresponding global stretch ratios are 1.71 and 1.42 along the x-and y-directions, respectively. In order to avoid the well-known stress accommodation [START_REF] Mullins | Effect of stretching on the properties of rubber[END_REF][START_REF] Meunier | Mechanical experimental characterization and numerical modelling of an unfilled silicone rubber[END_REF][START_REF] Godin | Acoustic emission potentialities for characterization of mullins effect in natural rubber materials filled with carbon black[END_REF] Figure 7. Example of one optimized virtual fields (dotted line) used for identifying C 1 and C 2 . The maximum global stretch ratios is superimposed with the virtual mesh (solid line).

over the first mechanical cycles, three cycles are first carried out with the same maximum stretch ratio, thereby partially stabilizing the mechanical response of the specimen. Images are stored for every 1 mm of prescribed displacement in both directions. The material considered here is a carbon black filled natural rubber. Its composition and some of its mechanical properties are given in Table 1.

A typical view of the horizontal and vertical displacement fields obtained for the maximum prescribed global stretch ratios is shown in Figure 8.

Differentiating these fields provides the four components of the displacement gradient. They are obtained by using a centered finite differences scheme. In the present case, 740 experimental points are processed. This number is close to the number of elements used for numerical simulations and makes it easier to compare numerical and experimental TIH visualizations (see Figs. 3 and9).

The states of strain of the three homogeneous tests (UT, PS and ET) are represented and the equivalent maximum stretch ratio is higher for UT than for PS and ET. Figure 10 presents the maximum principal stretch ratio for each ZOI.

Here, each loading case is satisfactorily represented in terms of maximum stretch ratio distribution. With regard to the previous analysis and numerical prediction, the TIH satisfies the two criteria used to define a sufficient level of TIH. Some points are encircled in Figures 9 and10. They correspond to points taht are located at the border of the sample i.e. to zones for which there are insufficient measurement points to obtain an efficient image correlation. To summarize, these experimental results are in good agreement with the numerical prediction in terms of TIH.

Identification results

Identification results are presented in Figure 11 for each step of the loading. This step is equal to 1 mm along both the x-and y-directions. C 1 and C 2 are calculated at each loading step. The global stretch ratio reported along the horizontal axis in Figure 11 is that prescribed along the y-direction.

Apart from stretch ratios inferior to 1.05, both C 1 and C 2 remain approximately constant while the global stretch ratio increases, thereby showing that the Mooney strain energy density presently used correctly describes the actual mechanical response of the specimen. The mean values of C 1 and C 2 deduced from these curves are 0.509 MPa and 0.012 MPa, respectively. These values are acceptable with regard to thermodynamic considerations [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF]. It should be noted that only the loading steps corresponding to a global stretch ratio along the y-direction ranging between 1.05 and 1.42 are considered here. In fact, identifying C 1 and C 2 in the domain of low stretch ratio leads to unreliable results. This is explained by the fact that the material parameters are not sufficiently activated in this domain. It should be noted that this observation is valid whatever the law considered.

CONCLUSION

The aim of the present paper is to propose an alternative to the classical method of identifying constitutive parameters of rubber. For that purpose, only one heterogeneous test is performed. Sample geometry and loading conditions are chosen using numerical simulations in order to involve UT, PS and ET at the sample surface. The test-induced heterogeneity is discussed related to two criteria.

To perform the heterogeneous test, a new apparatus is designed and is adapted on a conventional tensile machine. Displacement fields are measured on the specimen surface using the DIC technique. Unknown constitutive parameters are then deduced from these fields using a suitable identification procedure: the Virtual Fields Method extended to hyperelasticity. Results obtained are in agreement with theoretical and numerical expectations, thus confirming the feasibility of the present approach. An interesting perspective would be to identify parameters governing more complicated constitutive models. For instance, it could be interesting to take into account the significant change in volume observed under extension in such a material [START_REF] Le Cam | Volume variation in stretched natural rubber: competition between cavitation and stress-induced crystallization[END_REF].

  each material point of the kinematic fields. In this diagram (see Fig. 1), the three curves UT, PS and ET are plotted. The localization of any point in this diagram indicates the corresponding loading condition: it can be only one of the three previous loadings or a combination of two of them.

Figure 1 .

 1 Figure 1. Three basic loading cases.

Figure 2 .

 2 Figure 2. Sample geometry and loading conditions.

Figure 3 .

 3 Figure 3. Loading cases in the (I 1 -I 2 ) plane.

Figure 4 .

 4 Figure 4. Numerical TIH visualization: maximum value of the principal stretch ratio.

Figure 5 .

 5 Figure 5. Experimental set-up.

Figure 6 .

 6 Figure 6. Proposed apparatus.

Figure 8 .

 8 Figure 8. Displacement fields.

Figure 9 .

 9 Figure 9. Loading cases in the (I 1 -I 2 ) plane.

Figure 10 .

 10 Figure 10. Experimental TIH visualization: maximum value of the principal stretch ratio.

Table 1 .

 1 Material formulation (parts per hundred rubber) and mechanical properties.

	Components	NR
	Rubber	100
	Zinc oxide	9.85
	Oil	3
	Carbon black	34
	Sulfur	3
	Stearic acid	3
	Antioxidant	2
	Accelerators	4
	Density	1.13
	Shore A hardness	58
	Stress at break (MPa)	22.9
	Elongation at failure	635


Patrice Guyot

Julien Pinquier

Soundscape visualization: a new approach based on automatic annotation and Samocharts

The visualization of sounds facilitates their identification and classification. However, in the case of audio recording websites, the access to a sound is usually based on the metadata of the sounds, i.e. sources and recording conditions. As sonic environments, or soundscapes, are mostly composed of multiples sources, their compact description is an issue that makes difficult the choice of an item in a sound corpus. The time-component matrix chart, which is abbreviated as TM-chart, has been proposed recently as a tool to describe and compare sonic environments. However their process of creation is based on a subjective annotation that makes their creation time-consuming. In this paper, we present a new method for urban soundscape corpus visualization. In the context of the CIESS project, we propose Samochart: an extension of the TM-chart that is based on sound detection algorithms. We describe three original algorithms that allow the detection of alarms, footsteps, and motors. Samocharts can be computed from the results of these algorithms. This process is applied to a concrete case study: 20 urban recordings of 5 minutes each, from different situations (places and time). An application case shows that Samocharts allow an identification of different situations. Finally, the whole method provides a low-cost tool for soundscape visualization that can easily be applied to the management and use of a sound corpus.

Introduction

The visualization of sounds has always been a source of questioning. If the composer Olivier Messiaen, known as synesthetes, was able to perceive colors when he heard certain musical chords, the translation of sound to image is generally not easy to formalize. In the case of ethnomusicological studies for instance, transcriptions have traditionally been distinguished between descriptive and prescriptive notations according to their aims: the former are intended for analysis purpose, and the latter to reproduce the music [START_REF] Seeger | Prescriptive and descriptive musicwriting[END_REF]. This illustrates the difficulty to find an representation adapted to the needs of the users.

In the framework of environmental recordings, or soundscapes as introduced by Murray Schaffer, the issue of representation has been addressed since the 1970s [START_REF] Schafer | The tuning of the world[END_REF]. If some works have been done in the focus of designing soundscapes (see [START_REF]COST Action TD0804 -Soundscape of European Cities and Landscapes[END_REF] for interesting articles on this topic), one objective of soundscape representations is to analyze and compare them.

When looking at the websites that propose browsing and downloading of field recordings, the display-(c) European Acoustics Association ing of the sounds is usually based on a time/energy representation which can be slightly informative [START_REF] Guyot | Browsing soundscapes[END_REF]. It seems indeed difficult to choose a sound from its waveform without hearing it. On websites such as freesound 1 , textual metadata add a relevant information but remains generally insufficient to get an idea of what sounds compose the soundscape.

Through the CIESS project 2 , we investigate the field of soundscape analysis and representation. Perception studies lead us to identify some sound events that seem characteristic of modern urban soundscapes [START_REF] Tardieu | A method to collect representative samples of urban soundscapes[END_REF]. We assume that a representation based on this sound events can be efficient to represent a sound corpus.

Beside temporal representation, we decided to focus on illustrations that could globally display the presence or absence of these sound events. The TM-chart representation provides us an efficient tool to represent these recordings [START_REF] Hiramatsu | The physcial expression of soundscape: An investigation by means of time component matrix chart[END_REF]. However, the costly human annotation makes it difficult to use in the context of a big corpus. In the regard of the growing improvement of automatic sound detection, we proposed a new framework of representation, inspired by TM-charts, that we call Samocharts [START_REF] Guyot | Browsing soundscapes[END_REF]. TM-charts, Samocharts can be based on automatic detection algorithms, that makes their use possible without human annotation.

In this paper, we present a new approach for soundscape visualization based on automatic event detection algorithms. Section 2 introduces some relative works in this domain. Section 3, 4, and 5 describe our algorithms of sound detection to obtain confidence values, respectively on alarms, footsteps and motors. Section 6 presents an application case through the urban recordings of the CIESS project.

Relative works

Soundscape visualization

Soundscapes are usually materialized as quite long recordings, as its relative concepts focus significantly on the relationship with time. For instance, the duration of our recordings from the CIESS project are usually about 5 minutes. Longer observation recordings have also been made in the context of acoustical ecology. For example, a recent study reports the visualization of temporal change in soundscape power over a 4-year period [START_REF] Gage | Visualization of temporal change in soundscape power of a Michigan lake habitat over a 4-year period[END_REF], and revealed discrete patterns such as changes in the time of the occurrence of dawn and dusk choruses.

With long recordings, the traditional tools of audio representation, based on the temporal evolution of the signal, are slightly informative [START_REF] Guyot | Browsing soundscapes[END_REF]. For example, urban recordings are too noisy to make appear semantic information from a waveform or a spectrogram. However, these recordings are usually composed of the same kind of sound events, for instance vehicles, voices or birds. This fact provides an interesting entry point to alternative representations of sounds.

A new paradigm has been introduced in 2008 by Hiramatsu and al [START_REF] Hiramatsu | The physcial expression of soundscape: An investigation by means of time component matrix chart[END_REF]. The time-component matrix chart, which is abbreviated as TM-chart, allows describing and comparing sonic environments. It is a matrix that can be easily visualized as a figure to analyse and rapidly compare sonic environments. From a set of predetermined sound events, TM-charts provide a SoundSource×Soundlevel information for each target sound. It shows the percentage of time of the sound source audibility and the percentage of time of its level range. However, their production process rely on the identification of the predominant sound at each time, which can be very costly in time. Thus, the TM-charts seem quite difficult to use on a big data set.

In a previous paper, we introduced Samocharts, a new approach inspired by TM-charts [START_REF] Guyot | Browsing soundscapes[END_REF]. As the TMcharts, the Samocharts are matrices that represent the percentage time of sound sources. Moreover, they can be computed from confidence values of sound detection algorithms.

Audio event detection

Automatic annotation of audio recordings made significant progress these last years. These developments can be observed in the field of human voice with automatic transcription, and emerge in the field of music with some applications like Shazam3 . From these observations, we shall assume that the detection of sound events in field recordings will become reliable in coming years.

Yet, various methods have been proposed on Audio Event Detection (AED) from audio files that have been recorded in real life. These works can be divided in two categories. The first one aims at detecting a large set of sounds in various contexts. These approaches generally use machine learning techniques to automatically model a set of sounds from the evolution of acoustic features [START_REF] Giannoulis | Detection and classification of acoustic scenes and events: an IEEE AASP challenge[END_REF]. For example, the detection of 61 sound events, such as bus door, footsteps or applause, has been reported in [START_REF] Mesaros | Acoustic event detection in real life recordings[END_REF]. In this work the author model each sound class by an Hidden Markov Model (HMM) with 3 states, and Mel-Frequency Cepstral Coefficients (MFCC) features.

A second category of methods aims at detecting specific sound events. These works are built up privileging accuracy over the number of sounds that can be detected. They generally rely on a specific modeling of the target sound that is based on acoustic observations. For example, some studies propose to detect gunshots [START_REF] Clavel | Events detection for an audio-based surveillance system[END_REF], water sounds [START_REF] Guyot | Water sound recognition based on physical models[END_REF], or alarms sounds [START_REF] Lutfi | Automated detection of alarm sounds[END_REF][START_REF] Ellis | Detecting alarm sounds[END_REF].

In this last reference, the authors compare the results of alarm sounds detection with the two kinds of method mentioned. They show that a specific and simple approach can be more efficient that blind machine learning techniques.

Moreover, due to numerous overlapping sound events, the urban sound recordings of our project are very noisy. In this context, the machine learning approach seems currently inefficient. Thus, we choose to focus on particular sounds that slightly emerge from the noisy background and try to formalize the reasons of this emergence. Therefore, we built up ad hoc algorithms to detect these sounds. In the following parts, we will present our methods to detect alarms (such as car-horns), footsteps and motors.

Alarm detection

Overview

Alarms are emblematic of urban soundscapes. They produce a sound event that convey urgent information or prevent efficiently a danger. In this regard, alarms such as car horns, sirens, and bicycle bell are usually built up to be easily audible and clearly identifiable. humans seem able to identify a particular sound as an alarm even when they have never heard it before. However, the distinctive characteristics of alarm sounds are not formally defined. We suppose in the following work these sounds include acoustic invariants.

Method

In the CIESS corpus, we focused on three kinds of sound alarms: car horn, bus horn, and bicycle bell. Based on the observation of these sounds, we notice different acoustical properties that can be considered as invariants of alarm sounds. Our detection of these sounds is based on the following properties:

• duration longer than half a second, • frequential stability over time (spectral lines),

• medium and high frequencies preferred.

Moreover, the different kinds of alarms seen in the CIESS project show different distribution on spectral lines. Car horns are generally highly harmonic sounds. Bicycle bells in our corpus show one spectral line around 4000 Hz. Buses of the city of Toulouse show a particular horn with a spectral line around 1200 Hz. Figure 1 Our method for alarm sound detection is based on the following steps: high-pass filtering, computation of a spectrogram, selection of spectral lines and identification or reject of the founded candidates.

3.2.1. High-pass filtering As alarm sounds are generally higher than 1000 Hz, we use a high-pass filter to suppress low frequencies and avoid potential false alarms. We use a high-pass digital Butterworth filter at order 10 with a cut frequency of 1500 Hz.

Computation of a spectrogram

We compute a spectrogram on the sound signal with a Fast Fourier Transform algorithm. As we look for a good frequency resolution, we compute each spectrum on windows of 120 milliseconds (2048 samples) with an overlapping of 75%.

Selection of spectral lines

The next step is to extract spectral lines from the spectrogram. We focus on small time/frequency rectangles of 0.4 second width that are extracted from the spectrogram. We assume this duration is the minimal duration of an alarm sound.

On each rectangle, we compute temporal mean for each frequency. Each mean is afterwards compared with its neighbors on a frequency band of 300 Hz. If the mean if 3 times higher than its neighborhood, we consider that the spectral line is detected. Each rectangle of the signal which contains spectral lines is then considered as a potential candidate.

Identification or reject of candidates

We use a simple algorithm based on our observations mentioned above to validate the candidate. This algorithm relies on conditions on the number of spectral lines and their frequencies. This validation step allows to reject a false candidate or to identify a rectangle as car-horn, bus alarm, or bicycle bell sound.

This identification is then converted into values between 0 and 1.

Footstep detection

Overview

As walking remains one of the main way of moving, lots of people still walk in our city. Depending on their movement, their body and their shoes, these persons produce various sounds.

The footstep sounds are characterized by regular impacts. In the following method, we analyze the signal to detect regular impacts.

Method

Computation of a spectrogram

We compute a spectrogram on the sound signal with a Fast Fourier Transform algorithm. As we look for a good temporal resolution, we compute each spectrum on windows of 30 milliseconds (512 samples) without overlapping. From this spectrogram, we focus on frequency bins ranging between 300 and 4000 Hz.

Peak extraction

We extract peaks frames whose energy content is 3 times superior to the mean of its temporal neighborhood (on 100 frames around). We obtain a list of attack times.

Rhythm spectrum

To compute a rhythm spectrum, we focus on windows of 5 seconds and consider the attacks detected. We compute the Fourier transform of this attack times, as shown in the following equation:

RS(f req) = N bAttacks i=1 e -2 * j * π * f req * time(i) (1)
Where RS is the rhythm spectrum, N bAttacks the number of attacks, and time(i) the time of the i th attack. This equation is applied to frequencies varying between 0.2 and 10 Hz.

Thus, on a particular frequency, a significant value in the rhythm spectrum reflects recursive attacks at this frequency.

Tempogram computation

By computing a rhythm spectrum at each sliding window of the signal and concatenating them, we obtain a matrix that we call tempogram [START_REF] Coz | Spectre de rythme et sources multiples: au coeur des contenus ethnomusicologiques et sonores[END_REF]. Each frame of the tempogram is a rhythm spectrum.

We extract the values of the tempogram that are superior to a masking threshold. We can see on figure 2 the orignal tempogram (2a) and tempogram superior to a masking threshold (2b).

Then for each frame of the tempogram, we compute its sum divided by our decision threshold. We use 0.6 for the masking threshold and 250 for the decision threshold. With setting values between 0 and 1, we obtain a confidence of repeated impacts at each time (2c).

Motor detection

Overview

Motors of vehicles such as cars, mopeds, scooters and trucks, produce important noises that are characteristic of urban sound environment. In this step, we aim at detecting passages of motorized vehicles that emerge from the background sound.

Method

Computation of a spectrogram

We compute a spectrogram on the sound signal with a Fast Fourier Transform algorithm. We compute each spectrum on windows of 120 milliseconds (2048 samples) with an overlap of 50%.

Spectral spread of low frequencies

We suppose that the motors sounds can be retrieved from their low frequencies. Thus, we extract from the spectrogram the frequency bins of the spectrogram ranging between 0 and 250 Hz. For each frame, we compute the spectral spread [START_REF] Peeters | A large set of audio features for sound description (similarity and classification) in the CUIDADO project[END_REF] on this frequency band.

Normalized local minimum

We compute the mean of these values on the whole signal. Local minima are then computed on group of frames of 6 second total duration, and divided by this mean. With setting values between 0 and 1, we obtain a probability of motors at each time. The duration of each file is about 5 minutes. The recordings have been made with a Soundfiled SPS 200 microphone and a Tascam DR-680 recorder.

Sound event detection

The recordings are transformed in mono files with 16 kHz sampling rate. Our three detection algorithms are tuned on a recording of the corpus. Afterwards, we ran them on the full corpus and obtain time confidences for car horns, footsteps and motors for each file.

For each recording, our extraction methods give us temporal values of confidence on the presence of carhorns, footsteps, and motors. These values are interesting but cannot directly be interpreted as global and semantic information.

Visualization

As defined in [START_REF] Guyot | Browsing soundscapes[END_REF], we use the Samochart visualization to represent our corpus. The figure 4 represents all the Samocharts we plotted from our corpus. In these figures, we can see important duration of the signal (in white) where none of our target sounds were detected.

A quick viewing shows differences in the recordings that are mainly relative to the presence of motors and footsteps. For instance, we notice that the last three recordings (in the direction of reading) exhibit a relative larger part of footsteps. These recordings have indeed been made in pedestrian streets. Moreover, the other recordings, that have been recorded in different kind of streets, show a presence of motors more or less important. This could reflect different intensity of traffic according to different streets.

A zoom on the Samochart from the recordings lets appear the alarms (horns) sounds. Indeed, as horns sounds are usually very short events, the cumulative duration of these sounds can be very brief in comparison of the duration of the whole signal. We can see on figure 5 two Samocharts, the first one with no car horn detected and the second one with car horn sounds.

Conclusions and perspectives

In this paper, we presented methods to detect automatically particular sound events in sound recordings and to plot them with a new paradigm, the Samocharts.

Sound detection algorithms

The sound detection algorithms provide time values of confidence for the following sound events: car horns, footsteps and motors. Overall the algorithms seem to produce interesting results. However, they could be significantly improved with more annotated data. Indeed, the methods have been set up on a single annotated file, and could be tuned to a larger set on heterogeneous files. Moreover, a greater number of manually annotated recordings could make possible to obtain a test data set and objective results of our algorithms. In this regard, our algorithms could be compared to state-of-the-art methods in order to choose the best approach. Some improvements could also be considered, based for instance on the use of the Doppler effect in the motors sounds or the onset of the alarm sounds.

Furthermore, other methods and target sound events, such as voice, could also be added to our package. After obtaining a stable version of our whole method, we should also consider saving computation time by pooling common processing, as the Fourier transform.

Browsing a sound corpus

The Samocharts based on automatic annotation algorithms provide an efficient way to browse a corpus of soundscapes. The whole method allows a fast comparison of recordings. It seems to make possible the identification of particular situations, such as pedes-trian streets. However, some experiments should conducted to validate this concept.

Beside the zoom function, some improvements could be considered in our online displaying to facilitate the browsing. For example, we could add research and sort functions based on sound events.

Elsewhere, it shall be mentioned that our approach describes a recording by the sum of its sound events, but does not take into consideration a more global perception of the Soundscapes (for example, we could consider a globally frightening recording). The concepts and methods needed to recognize such global feeling and display them should give us some works for coming years.

  represent a spectrogram on which we can see two examples of these sounds.

Figure 1 .

 1 Figure 1. Spectrogram of an urban soundscape extract from the CIESS project. Spectral lines from alarm sounds appear around 2 seconds at 1200 Hz, and at 6 seconds at 3800 Hz. The first sound is produced by a bus of the city of Toulouse and the second one by a bicycle.

Figure 2 .

 2 Figure 2. Tempograms and confidence values of footstep sound events on a 5 minutes urban soundscape extract from the CIESS project. The highlight parts correspond to the segment where we distinctly heard footsteps.

Figure 3 .

 3 Figure 3. Confidence values of motor sound events on a 5 minutes urban soundscape extract from the CIESS project. The highlight parts correspond to the segment where we distinctly heard the passage of a motor vehicle.

Application 6 .

 6 1. CorpusWe use audio files from the CIESS project. From the city of Toulouse, at different place and time, we have recorded sound environment. There are three different schedules and situations: • Non pedestrian street at 8 am,• Pedestrian street at noon, • Non pedestrian street at 9 pm.

Figure 4 .

 4 Figure 4. Samocharts of recordings of the CIESS project. Each recording is represented by a square showing the percentage of time of presence of sound events. On each sound event, the confidence of the detection algorithm is shown by shades of colors.

Figure 5 .

 5 Figure 5. Zoom on Samocharts of two recordings of the CIESS project.

https://www.freesound.org/

http://www.shazam.com/

Acknowledgement

This work is supported by a grant from Agence Nationale de la Recherche with reference ANR-12-CORP-0013, within the CIESS project.


Nicolas Verstaevel
email: verstaev@irit.fr

Christine Régis

Marie-Pierre Gleizes

Fabrice Robert

Principles and experimentations of self-organizing embedded agents allowing learning from demonstration in ambient robotic

Keywords: Adaptive agents, Cooperative multi-agents systems, Robotic in ambients systems, Machine Learning, Context-Awareness

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Once confined in a science of control in structured environments, researches on robotic are now considering the integration of intelligent systems in the real world for applications where tasks are multiple, complex and evolutive [START_REF] Kaminka | Autonomous Agents Research in Robotics: A Report from the Trenches[END_REF] . Service robotic differs from its industrial version by the interest in providing services to humans. Consequently, researchers now consider the use of robotic components in ambient applications [START_REF] Chung | Ambient robotics for meaningful life of the elderly[END_REF] .

Ambient systems are characterized by their high dynamic and their complexity. Many heterogeneous devices can appear and disappear along the system lifecycle and interact opportunistically together. According to the definition of Russell and Norvig, the environments of such systems are [START_REF] Russel | Artificial Intelligence: A modern approach[END_REF] :

-Inaccessible: each device composing the system has a partial observation of the environment.

-Continuous: considering applications in the real world, the number of observations and actions is not discrete.

-Non-deterministic: consequences of performed actions in the real world could not be determined in advance with certainty. -Dynamic: system's actions, user activity, appearance and disappearance of devices may change the environment. Consequently, designing an ad hoc controller of an ambient system is a complex task that requires lot of knowledge. This complexity is increased if we take into account that users have multiple, specific and often changing needs. Bringing the ability to learn and adapt to users' needs is then a particularly challenging task [START_REF] Hagras | Creating an ambient-intelligence environment using embedded agents[END_REF] . To be as natural as possible, such learning ability needs to rest on a process that does not require any kind of knowledge for users (i). Furthermore, it needs both genericity, to be applicable on any kind of devices with any kind of users, and openness properties to deal with the appearance and disappearance of devices. The genericity and openness properties require then using agnostic learning technics that makes as few assumptions as possible [START_REF] Kearns | Toward efficient agnostic learning[END_REF] (ii). To deal with (i) and (ii), we propose to use Learning from Demonstration, a paradigm to dynamically learn new behaviours, as the engine of self-organization in ambient applications.

The paper is organized as follows: first, we present the problematic and challenges of LfD in ambient systems. Then, we present ALEX our solution to handle this challenge. An experiment illustrating ALEX behaviour on a collecting task is then provided. At last, conclusion will discuss perspectives and future works.

Learning from demonstration

General principle

Learning from Demonstration, also named "imitation learning", is a paradigm mainly studied in the robotic field that allows systems to self-discover new behaviours [START_REF] Argall | A survey of robot learning from demonstration[END_REF] . The main idea is that an appropriate controller can be learnt from the observation of the performance of another entity (virtual or human) named as the tutor. A tutor can interact with the system to explicit the desired behavior. This interaction is done through the natural process of demonstrations, which consists in the performance of an action in a particular context. The system then learns a mapping function correlating observations of the environment to tutor's action. The main advantage of such technique is that it needs no explicit programming or knowledge on the system. It only observes tutor's action and current system context to learn a control policy. Recent surveys [START_REF] Argall | A survey of robot learning from demonstration[END_REF][START_REF] Billard | Robot programming by demonstration[END_REF] proposed an overview of the LfD field. Billing and Hellström 8 propose a complete LfD formalisation.

LfD is a problem of imitation as an entity tries to produce a behaviour similar to another entity. A tutor evolving in a world can perform a set of actions ( could be empty). The tutor follows a policy (1) that can associate to any world state a particular action. It is supposed that (1) is the optimal policy to satisfy the user.

(

An imitator (named as the learner) disposes of a set of observations (named observation space) on the states space (named the world) and follows another policy (2) in order to produce a behaviour similar to the observed one. such as (2) In many cases, the tutor and the system have a different observation of the world. It is particularly true in real world problems with human tutors where the world is observed by the system through sensors whereas the human observes it through its own body. It results in a problem of perceptual equivalence [START_REF] Dautenhahn | The correspondence problem[END_REF] . The tutor demonstrating a particular behavior can observe modifications of the world that the learner cannot perceive. However, equivalences of perception can be found by the system. For example, a user is cold and turns the heating on. Observing through sensors that a user is cold is complex, but observing the temperature, wind and humidity levels is easily feasible. A learner can make correlation between the current situation and the action of turning the heating on and learns that it is necessary to turn the heating on.

The learner has to find correlations between its own observations and the performance of an action by the tutor. This raises the question of the possible lack of equivalence. A tutor can perform a demonstration dependent of a phenomenon that the learner cannot observe. In this paper, we consider that the learner disposes of sufficient observations to perform the task. Nevertheless, some clues to handle this problem are proposed on section 5 as perspectives.

Lfd and ambient systems

Ambient systems are rich of interaction possibilities for users. We propose to exploit the inherent interactivity of ambient systems in order to learn and adapt from users activity. LfD then appears to be a good paradigm to guide system adaptation. Therefore, users can be seen as system's tutors. Each user's action on a device is then seen as a demonstration of the desired behaviour. The idea is that if a user has to act on a particular device, the reason is that the service provided by this device is not satisfying anymore. The device can then use this information to self-adapt. However, ambient systems have some particular properties that are challenging for LfD.

An ambient system is open, which means that entities can appear and disappear during system activity. This openness property is challenging for LfD, as it does not allow doing assumptions on system's composition. With respect to the openness property, it must be considered that the set of observations is a priori unknown. Moreover, to provide a generic learning technic, it has also to be considered that the set of actions is a priori unknown. Then, a learner on an ambient system must adapts its policy to integrate new observations and actions. This adaptation must be dynamic and transparent to users to provide a good quality of service. Thus, approaches separating learning phase from exploitation (supervised learning) appear to be not pertinent.

Users are also a source of complexity as they have specific and often changing needs. In ambient systems, many different users have day-to-day interactions with the system. Capturing users' needs in such a system cannot be an ad hoc process and has to be made along system lifecycle. As users can often change their needs and contradict themselves. The system dynamically adapts its policy to integrate any change in the user's need. An approach based on user profiles appears to be not relevant. LfD in ambient systems needs for adaptivity to the task to perform (what to do), to the reasons to do it (when to do) and to the users (whom to imitate).

An extremely sensitive system

Nevertheless, what and when are not the only questions to bring adaptivity in ambient systems. The classical approach considers the system as a whole, which means the system is seen as an omniscient process that exercises a direct control over all its parts. However, this centralized approach shows some limitation for highly dynamic and open systems where the observability is local. Works among the community now consider decentralized approach where there is no supervisor of the activity [START_REF] Kaminka | Autonomous Agents Research in Robotics: A Report from the Trenches[END_REF][START_REF] Hagras | Creating an ambient-intelligence environment using embedded agents[END_REF][START_REF] Guivarch | Self-Adaptation of a Learnt Behaviour by Detecting and by Managing User's Implicit Contradictions[END_REF] . This shift of paradigm raises the question of who learns. In literature concerning robotic, the answer is often "the robot", as the entity composed of sensors and effectors. The more there are sensors and effectors, the more complex the robot is. The same reasoning can be applied to ambient systems. We argue that this vision is restrictive and propose what we call "Extreme Sensitive Robotic" [START_REF] Verstaevel | Extreme Sensitive Robotic: A Context-Aware Ubiquitous Learning[END_REF] .

The eXtreme Sensitive Robotic (XS Robotic) vision considers each functionality as an autonomous entity. A robot is then composed of many eXtreme Sensitive Functions (XS Functions), globally one for each sensor and effector. An XS Function is sensitive (such as a temperature or ultrasound sensor) or effective (a motor, a led). Each XS Functionality has to be designed as nonfinal, which means that the functionality can self-evolve. Self-observation capacities are the key for sensing variation in the environment and adapt in response. With this approach, the production of a complex behavior by a robot emerges from the interaction between the different XS Functionalities and the environment. There is thus no difference of design between a single robot, a multi-robot application or a complex ambient system. All those systems are composed of independent functionalities that need to cooperate to perform a global function that is greater than the sum of its parts.

The next section presents ALEX, a multi-agent system for learning by demonstration in ambient robotic, built upon the eXtreme Sensitive Robotic vision. Its conception is based on the Adaptive Multi-Agent System (AMAS) approach and uses recent results on context-aware learning [START_REF] Guivarch | Self-Adaptation of a Learnt Behaviour by Detecting and by Managing User's Implicit Contradictions[END_REF] .

ALEX: Adaptive Learner by EXperiments

The Adaptive Learner by Experiments (ALEX) is a multi-agent system designed to learn from demonstration to control a functionality (as described in section 2.2). It has been built upon the Adaptive Multi-Agent System approach (AMAS) [START_REF] Gleizes | Self-adaptive complex systems[END_REF] , which addresses the problematic of complex systems with a bottom-up approach where the concept of cooperation acts as the core of self-organization. The theorem of functional adequacy [START_REF] Capera | The AMAS theory for complex problem solving based on self-organizing cooperative agents[END_REF] states: "for all functionally adequate systems, there is at least one system with an internal cooperative state that realizes the same function in the same environment". The role of an AMAS is then to automatically detect and repair non-cooperative situations by self-organizing to reach a functionally adequate state. ALEX has been developed according to the ADELFE 14 methodology that guides the design of an adaptive system.

Context-Learning of an XS functionality

Section 2 illustrates that making an exhaustive list of all situations that an ambient system may be faced to is impossible. It is then necessary for the system to dynamically adapt to contexts. The term context refers in this paper to all information external to the activity of an entity that affects its activity. This set of information describes the environment [START_REF] Guivarch | Self-Adaptation of a Learnt Behaviour by Detecting and by Managing User's Implicit Contradictions[END_REF] . A system capable of exploiting context information is called "context-aware".

ALEX is designed to continually interact with its environment and dynamically learn all the different contexts that can occur. It associates to each context the adequate action to perform. ALEX uses self-observation capacities to dynamically build correlations between the performance of an action and effects of this action on the environment. For thus, an ALEX is composed of a non-finite set of context agents that makes a collective control over an XS functionality. The set of context agents is empty and ALEX dynamically and autonomously creates those context agents. In this section, we describe the structure and behaviors of context agents. The next section will provide an experiment to illustrate how contexts make an efficient control of XS functionalities on a concrete application. Context agents are described by decomposing their behaviors in two kinds. The nominal behavior is the normal behavior that the agent performs when the system is in a functionally adequate state. The cooperative behavior is a subsumption of the nominal behavior that occurs when the agent needs to self-organize to bring the system toward a functionally adequate state.

Context agents

Context agent nominal behavior

A Context Agent associates a low-level context description (see section 3.2.2) to a unique action. An action corresponds to maintaining or adjusting the XS functionality. This action is given at the agent creation and never changes (see section 3.2.3.2). It can be a high-level command (such as "go Forward", "go Left") or a low-level command ("speed at 42%", "x=2.1",) depending on the functionality to control. A Context Agent receives signals from the environment and uses them to characterize current context. When the observed environment corresponds to its low-level context description , it decides to perform its associated action. Otherwise, it does nothing. When a context agent performs an action, the agent is said selected.

Low-level context description

Value ranges (named validity ranges) manage the low-level context description. A Context Agent receives a set of observations from the environment. Each is a continuous value such as . A validity range is associated to each observation such as corresponds to the validity range associated to the observation . The set of all validity ranges is named the validity domain.

A [START_REF] Lemouzy | Principles and properties of a mas learning algorithm: A comparison with standard learning algorithms applied to implicit feedback assessment[END_REF] , a tool that can dynamically find a value from feedbacks greater, lower and good.

Context agent cooperative behavior

At each time step, only one action can be performed over the controlled functionality. Thus, each context agent has to determine if its action is the most adequate. To be in a functionally adequate state, the system then needs that only one context agent proposes an action. However, situations where more than one context agent proposes an action can occur. Those situations are non-cooperative situations (NCS). Three NCS can occur. The first one (a) occurs when two (or more) agents propose to perform an action (regardless of the nature of the action). The second one (b) occurs if the context agent proposes an action that is not in adequation with the tutor's one. The last one (c) occurs when no context agent proposes an action. To solve these situations, context agents follow a cooperative behavior to dynamically self-organize.

Confidence

A Context Agent determines a confidence value that represents the relevance of the action to perform. This confidence value allows context agents to compare their actions. If a context agent wants to perform an action with a confidence value lower than another agent, it has to self-organize its context description to exclude the current situation. Thus, the next time the same context occurs, the context agent will not try performing its action.

The confidence value is ruled by a lambda function (3).

(

) 3
is the confidence value at the step . is a feedback value and is a given parameter that models the importance of the feedback. Each time a context agent makes a correct proposal, . Otherwise, . In our model, is fixed at 0.8. The more the context agent makes correct proposals, the more its confidence value increase. At creation, . The value is then used in the decision process of the context agent. Each context agent can observe the current best confidence value to decide if its action is the most adequate. Confidence allows solving the ambiguity made by the non-cooperative situation (a).

A low confidence means that the context agent is often mistaken. A context agent with a low confidence level will decide to destroy himself to not disturb the system. The minimal confidence value is set at 0.1, meaning that any context agent with a confidence lower than 0.1 will self-destroy.

Adequation with the tutor's action

A tutor can interact at any time with the system by demonstrating the desired behavior. The demonstration consists in the performance of a particular action . Thus, at any time step, there is a value corresponding to the tutor's action at this time step. Each context agent can perceive this value. If , the tutor has performed no action. Thus, it is context agent's duty to find the most adequate action to perform. However, if , context agents can observe tutor's action. If a context agent finds himself valid but proposes a different action than the tutor's one, it has to self-organize to exclude the current situation. Each valid validity range manages its bounds to exclude the current situation. If the set of observations contains values that are not associated to a validity range (for example, when a new sensor is added on the system), the context agent adds a new validity range to its validity domain corresponding to the new observation. However, if a context agent proposes an adequate action, it reinforces its. This allows solving the ambiguity made by the non-cooperative situation (b).

The adaptation of a validity range can result in a situation where . If such situation occurs, the context agent has a non-coherent structure making it useless. To allow the system to stay in a cooperative internal state, the agent will selfdestroy.

Validability and context agent creation

The system functional state requires that at least one context agent is valid at any step (see section 3.2.3). If at the end of a decision cycle, no context agent proposes to perform an action, the system is in a situation of incompetence. Two mechanisms can solve this situation: extending an already existing context agent or creating a new context agent to represent the situation.

To anticipate a situation of incompetence, the concept of validable is added. A validity range is validable if and only if and . are two values (one for each bound) dynamically managed by the AVTs to control the evolution of each bound. can be interpreted as the next increment of the bound. With the concept validable, context agents can propose their action in situation that are not so different to their validity domain. If the context agent proposition is selected, it has to adapt its bound by updating the nearest bound to include the current situation. This mechanism allows context agents to dynamically increase their validity ranges.

The second mechanism occurs when no context agent is valid or validable. The previously selected context agent (and not valid anymore) can create a new context agent associated with the tutor's action. The created context agent then initializes its validity range around the current position in order to represent the system's context. If , two mechanisms can occur depending on the desired level of autonomy. With a collaborative approach, the system can ask the tutor what is the action to perform. On contrary, with a more autonomous approach, system will maintain the last known action, considering that the inaction of the tutor correspond to the maintaining of the current action.

Those two mechanisms, the validable concept and context creation, allow solving the third non-cooperative situation (c). On the next section, we propose to illustrate ALEX on a concrete application.

Experiment

We want to highlight that ALEX presents interesting properties for ambient applications that differ from the traditional approaches:

-Genericity: the learning process is independent of the task to perform and uses no semantic on signals.

-Openness: new signals can be dynamically integrated in the decision process.

-Distributed: self-observation allows each functionality to be autonomous while remaining cooperative.

-Real-time: self-organization is performed without any system downtime.

To illustrate these properties, the following experiment has been performed.

Description

A two-wheeled rover with no sensors is immerged in a 2mx2m arena composed of a blue area and a green block. An intelligent camera located perpendicularly to the arena at 2m of its center can analyses pixels to capture the position of artefacts (the blue area and the green block) relatively to the rover orientation (determined by two red markers on the rover) (Figure 1). Thus at each time step, the camera can produce four observations on the scene (Table 1). To show ALEX openness properties, the camera sends observations only if an artefact is in front of the rover. At each time step, the camera can then produce either zero, two or four observations. This means that some observation can appear and disappear. A human user performs a direct control over the rover through a 2-joystick gamepad. Each joystick controls the speed of one wheel (left joystick for left wheel and reciprocally). Speed value belongs to and correspond to the percentage of the maximum speed to be applied and the sign of the rotation.

The aim of the experiment is to show ALEX capacity to learn complex behaviors from the observation of the user activity. For thus, the user can perform a range of activities in the arena and it is ALEX duty to exploit this interaction to imitate the user performance.

The experiment is performed both on real and virtual world. To show both genericity and distributed properties of ALEX, an ALEX is associated to each wheel allowing each wheel to act autonomously. Each wheel is then seen as an autonomous device. The role of an ALEX instance is then to control the wheel speed by correlating the observations from the camera to the actions performed by the user. An ALEX disposes at each step of a set of observations from the camera and, if relevant, the action made by the user. One-step of decision occurs every 250ms. The exact same implementation of ALEX is used on both experiments; only network protocol will differ depending on ALEX controls either a simulated rover or a real rover. OpenCV is used on both experiments for image analysis. ALEX uses the collaborative approach, which means that an ALEX instance will ask to the user the action to perform when no context agent is proposing an action. The virtual experiment has been developed on Webots™ simulator using the given Boe-bot model. The real world implementation has been made with a Boe-bot rover and uses an Xbee communication protocol. One of the activity the user can perform on the arena is a collecting task. In order to do so, solid whiskers are added to the rover to allow the rover to capture boxes. Whiskers are not movable and there is no sensor on them. They are just a physical part of the robot. Whenever the boxes are inside the blue area, boxes are moved randomly inside the arena.

Results

Context agent creation

Figure 2 illustrates the creation of context agents in the two ALEX instances. It shows the trajectory performed by the rover during the first demonstration of the collecting task. In this demonstration, the user takes control over the rover and drives it to collect and transport the box to the blue area.

We can observe that ALEX instances create new context agents when the user changes direction. On contrary, when the user maintains the direction of the rover, no context agents are created. Each user action is observed and each context agent determines if the current situation belongs to its own context description. If there is no such context agent, a new one is created. This illustrates ALEX capacity to detect new contexts and dynamically self-organize to integrate these new contexts in its decision process.

Self-adaptation of context agents

Figure 3 shows the structure of a particular context agent at its creation and at the end of a demonstration. Each line corresponds to the structure of a validity range associated to an observation. The yellow range corresponds to the valid range whereas the green area corresponds to the validable range. White boxes correspond to the current value of the signal. We observe that each validity range has its own evolution. This evolution is the result of the self-organization process. More precisely, validity ranges associated to the perception of the green block (GreenA and GreenD) are smaller than the one associated to the perception of the blue area (BlueA and BlueD). This particular context agent is valid when the block is close to the front of the rover and the rover is in front of the blue area. It is involved in the part of the activity where the rover brings back the block to the blue area.

Performances

To observe the capacity of the system to imitate the user performance, the user realized a 5 minutes demonstration in which 12 boxes were collected. The number of collected box by the user serves as a metric for performance comparison. The system is then let in autonomy and each 5 minutes the score is computed. During the autonomy phase, context agents have to find the most adequate action. Contrary to the collaborative demonstration phase where the user interacts with the system to teach previously unknown situations, the user never acts on the system. The figure 4 shows typical results we obtained. In the worst case, the system performs the task as well as the user does: 12 boxes are collected. However, the number of collected boxes is often better than the user's ones. Two factors influence this result. The first one comes with the randomness of the box movements. In some case, boxes are moved farther away from the blue area and it takes more time to reach and bring back the block. The other one, more interesting, lies in the fact that the user needs more time to take a decision than ALEX does. Moreover, the user can contradict itself. This phenomenon is observable in the figure 2. At midpoint between the start position and the box position, we can observe a change in the trajectory. This change is in fact a user tele-operation mistake. Context agents corresponding to this situation will never be reelected as they correspond to a non-desired action and will then self-destroy. The learnt behavior is then "filtered" of user mistakes allowing it to perform the task more efficiently.

Conclusion and future work

This paper deals with the challenges of LfD in ambient systems. It presents ALEX, a multi-agent system to learn from demonstration in ambient applications. Experiments tend to show the capacity of ALEX to learn from the interaction with its user. This means concretely that a simple demonstration of the task to perform allows each multi-agent associated to all the effectors to understand autonomously what the relevant data are in order to mimic collectively what the tutor does, without any central control. Each ALEX creates and self-organizes its context agents to produce collectively a behavior that is user satisfying. The experiment has shown that two ALEX instances can cooperate without direct interaction. Moreover, it illustrates that the system is able to perform a task more efficiently than the user.

This paper is a proof of concept that multi-agent context learning can deal with the complexity of ambient systems. However, many works remain to be made. We want to consider the use of ALEX technology in more complex problems coming from industrial needs. We have a particular interest for collaborative robotic applications where workers and robots have to work collaboratively. Factories of Future (FoF) are a good illustration of such applications [START_REF] Sicilliano | EuRoC-The Challenge Initiative for European Robotics[END_REF] . Evolutions of ALEX algorithm have to include the capacities to filter useless data and to discover that data are missing. Such processes could be done by adding percept agents associated to each signals who will have the responsibility to learn their utility for context agents and to help them. At last, we want both to formalize our approach and to compare its performances with other well-known learning techniques.

Figure 1 :

 1 Figure 1 : The experiment in Webots™. On the left, the rover inside the arena. On the right, the camera detection.

Figure 2 :

 2 Figure 2: Rover's trajectory and context creation during a demonstration

Figure 4 :

 4 Figure 4: Number of collected boxes each 5 minutes. The step 0 corresponds to the reference score.

Figure 3 :

 3 Figure 3: On the left, the validity domain structure at the creation of a context agent. On the right, the same context agent at the end of the demonstration.

  validity range is composed of two values, such as . A validity range is said valid if and only if as . Then, a context agent is said valid if and only if is valid. Validity ranges allow the context agent to determine if is similar to . Adaptive Value Range Trackers (AVRT) manage validity ranges. AVRT is an extension of Adaptive Value Tracker (AVT)

Table 1 .

 1 Camera observations

	Camera observations	Symbol	Domain
	Distance in pixel to the center of the green block		[-
			680;680]
	Angle between the rover front and the green block in radian		[-2 , ]
	Distance in pixel to the center of the blue area		[-
			680;680]
	Angle between the rover front and the blue area in radian		[-, ]