Andre Chateau

Akue Asseko
email: aandre.akue.asseko@mines-douai.fr

Benoît Cosson
email: bcosson@mines-douai.fr

Fabrice Schmidt
email: cfabrice.schmidt@mines-albi.fr

Rémi Gilblas
email: drgilblas@mines-albi.fr

Yannick Le Maoult
email: eyannick.lemaoult@mines-albi.fr

Eric Lafranche
email: feric.lafranche@mines-douai.fr

André Chateau

Akué Asséko

Thermal modeling in

Keywords: Laser welding, light scattering, absorption, thermal modeling, thermal simulations, thermoplastics composites, continuous fiber composite

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The literature contains many researches on the modeling of laser welding process for calculating the temperature field in knowing the radiative energy interface (radiative source term) received at the interface. The phenomenon of absorption for semi-transparent materials (absorbent) is modeled using the Beer-Lambert law assuming cold and non-scattering media. The author [START_REF] Labeas | Optimization of Laser Transmission Welding Process for Thermoplastic Composite Parts using Thermo-Mechanical Simulation[END_REF] assumed a density of heat flux with Gaussian distribution to estimate the radiative energy at the interface and calculate the temperature field. In their contribution [START_REF] Shanmugam | A transient finite element simulation of the temperature and bead profiles of T-joint laser welds[END_REF] also considered the same assumptions. The authors in [START_REF] Grewell | Plastics and Composites: Welding Handbook[END_REF] use a simplified model for the diffused intensity into the material using the Beer-Lambert law with the laser beam. Most of the researchers as [START_REF] Coelho | Modelling the spot shape influence on high-speed transmission lap welding of thermoplastics films[END_REF], [START_REF] Ilie | Throughtransmission laser welding of polymers -temperature field modeling and infrared investigation[END_REF] and [START_REF] Ilie | Diode laser welding of ABS: Experiments and process modeling[END_REF] consider that the laser beam is absorbed at the interface (opaque medium) and in the second material (absorbent) is transmitted according to the Beer-Lambert. The author in [START_REF] Hou | General solutions for stationary/moving plane heat source problems in manufacturing and tribology[END_REF] and [START_REF] Suthar | Estimation of Temperature Distribution in Silicon During Micro Laser Assisted Machining[END_REF] proposed analytical general solutions (transient and steady state) to estimate the increase of the temperature field at any point of the weld interface. They performed that for fixed and mobile sources of heat considering several forms of surface source (elliptical, circular, rectangular or square).

Aims of the present work are the prediction of the heat source in the laser welding process thermal simulations, in composite materials and at the welding interface. This paper proposes to describe completely the laser welding process in composite materials (refraction and absorption coupling).

Laser welding process

Transmission Laser Welding technique [START_REF] Troughton | Chapter 13 -Laser Welding[END_REF] presents specific advantages for industrial applications over other conventional technologies: the method is accurate, flexible, small heat affected zone, easy to automate and control and non-contaminant, heat transfer with the ability to optimize the welding temperature (at the interface of the welding zone), absence of vibration during the welding process (contrary to the ultrasonic welding, friction welding), fast welding speed for welding plastic parts with an acceptable welding time. Transmission Laser Welding of composites involves two joining parts: one semi-transparent to the laser wavelength and the other part is absorbent in the same wavelength. The two parts are positioned together before the welding. The laser beam energy is transmitted through the semi-transparent material and is absorbed within the surface of the second materials (Fig. 1). The bonding between the two parts allows the heating of the semi-transparent part by thermal conduction. Thus, melting and fusion of the both materials interface occurs (the bonding between the two parts occurs when T>Tmelt in this area). The energy is deposited at the interface in a localized volume causing the formation of a weld zone. Comparing with welding traditional techniques, laser welding efficiency strongly depends on the optical properties of the material [START_REF] Churchill | Light-scattering by very dense monodispersions of latex particles[END_REF]: absorption, transmission, refraction, reflection, scattering and thermal properties: heat capacity, thermal conductivity and density.

Scattering and absorption phenomena coupling

The laser welding process requires the presence of two phenomena: refraction (causing optical scattering macroscopically semi-transparent in the composite) and the absorption that occurs in the absorbent composite. In order to model the optical behavior of thermoplastic composites, the coupling of these two phenomena must be undertaken. Unlike to thermoplastic welding, the two phenomena cannot be neglected. The absorption is taken into account in the analytical model in order to determine the heat generated in the welding interface and finally to perform a three-dimensional calculation of the temperature field.

The absorption in the second part leads to the exponentially decreasing attenuation along the propagation path of the incident laser beam [START_REF] Ozisik | Heat Transfer: A Basic Approach[END_REF]. To determine the generated heat source at Semi-transparent the welding interface, the profile of the distribution of the total transmitted power of the laser beam (the total power after penetrating into the semi-transparent composite and reaching at the welding interface) is used [START_REF] Akué Asséko | Analytical and numerical modeling of light scattering in composite transmission laser welding process[END_REF]. For a gaussian laser beam, the distribution of the total power of the laser inward beam in the UD semi-transparent composite is:

I x, y, z I e 2πσ σ z σ e for all x $ Ω & (1)
Where x ,y and z are the space variables, σ o is the spatial shape parameter of the initial laser beam (proportional to the beam diameter), σ x (scattering standard deviation) is a function of the thickness , refraction index of matrix and fiber volume fraction and microstructural arrangement and D is the optical scattering coefficient of the semi-transparent composite.

In the absorbent part, a high absorption of laser beam radiation is considered, according to Beer-Lambert's law, and a heat transfer by thermal conduction between the two parts. The heat source in the case of an assumed semi-transparent and absorbent composites coupling (Figure 2) is described by the following equation as in [START_REF] Hou | General solutions for stationary/moving plane heat source problems in manufacturing and tribology[END_REF][START_REF]The theory of moving sources of heat and its application to metal treatments[END_REF][START_REF] Mingliang | Gap Bridging in laser transmission welding of thermoplastics[END_REF][START_REF] Acherjee | Effect of carbon black on temperature field and weld profile during laser transmission welding of polymers: A FEM study[END_REF]:

z s is the distance from the surface with in the material. In our study, recall that the absorption in the strongly, scattering semi-transparent composite is neglected i.e. assuming ideal semitransparent material and we consider just a strong absorption at the surface of the absorbent composite (welding interface). The absorption coefficient K a is considered infinite. That is to model the heat source in surface flux (surface absorption).

Q x, y, z 1 -ρ & α 2 I x, y, z for all x $ Ω 8 (3)
With α 2 1 -ρ 2 (absorption factor of the absorbent composite), ρ & reflection coefficient of the transparent composite. Thus, for the Gaussian shape, eq.3 can finally be expressed as follows:

Q x, y, z 1 -ρ & α 2 I e = > 2πσ ? σ z σ ? e (4)
Where I o is the laser power (W).

Preliminary thermal simulation

Knowing the generated heat source, the calculation of the temperature field at the welding interface and in the materials is performed numerically using a finite element model using the heat equation below [START_REF] Labeas | Optimization of Laser Transmission Welding Process for Thermoplastic Composite Parts using Thermo-Mechanical Simulation[END_REF][START_REF] Shanmugam | A transient finite element simulation of the temperature and bead profiles of T-joint laser welds[END_REF][START_REF] Grewell | Plastics and Composites: Welding Handbook[END_REF][START_REF] Coelho | Modelling the spot shape influence on high-speed transmission lap welding of thermoplastics films[END_REF], by taking a perfect contact between the two components subjected to welding. There are also studies that consider a thermal contact resistance between the joining components [START_REF] Ilie | Diode laser welding of ABS: Experiments and process modeling[END_REF] based on the roughness of the surface and the contact pressure. In the present study, a perfect contact assumption between the two components was considered due to a reduce thickness of the sample (2mm) and the good quality of the contact surface.

@ A BC BD @ E BC BF @ G BC BH I D, F, H J K L M C BC BN for all x $ Ω 8 (5)
Where k i is the thermal conductivity in x i direction, J K the density, and c p (T) the specific heat.

For the heat transfer between the two components and the surrounding medium, the boundary condition can be expressed as in [START_REF] Acherjee | Effect of carbon black on temperature field and weld profile during laser transmission welding of polymers: A FEM study[END_REF][START_REF] Acherjee | Modeling of laser transmission contour welding process using FEA and DoE[END_REF]:

-@O P PQ C. n PQ S T C U -C ? εV W C X Y -C ? Y for all x $ Ω Z &Z[\2] (6)
Where n PQ is the outward normal vector of the surface is the convection heat transfer coefficient (W/m 2 K), T s is the surface temperature (403.15 K), T ? is the ambient temperature (293.15 K), ε is material emissivity and b σ is the black body emission coefficient (5.67×10 -8 W/m²K 4 ). The convective heat transfer coefficient h c (10.81 W/m 2 K). We calculate this value as [START_REF] Andrieu | Étude expérimentale et numérique du chauffage infrarouge de plaques thermoplastiques pour le thermoformage[END_REF] in by relying on correlations of the exchange coefficients for a planar plate cooled on its top face. In order to determine the temperature field during the laser welding process, the thermal properties of the materials to be welded are required as input parameters to model the thermal process properly namely, thermal conductivity, specific heat and density, which are dependent on temperature. Thermal properties of the materials constituting the composite are different, a factor around 4 exists between resin and fibers conductivities for instance. Moreover, the final composite thermal properties also depend on the spatial organization of its constituents, resulting in anisotropic macroscopic effective properties. The thermal conductivity measurement in the composite is highly delicate, because of the anisotropy of the material on one hand and because of the reactivity of the resin on the other part. In this study, an experimental method is suggested in order to evaluate global conductivity tensor (k x , k y , k z ). A composite sample is submitted on its front face to a heat flux from an infrared lamp P21W (nominal power: 50W, 13.5V) (Fig. 3) focused by an ellipsoidal reflector, thus forming a beam (Gaussian shape). The temperature field on the back face is then measured using a FLIR infrared camera [7-13µm]. From its evolution, we will estimate by inverse method as in [START_REF] Bailleul | Thermal conductivity of unidirectional reinforced composite material. Experimental measurement as a function of state of cure[END_REF] using COMSOL software via its optimization tool, the conductivities of the material according to its anisotropy directions. The COMSOL optimization tool is particularly suitable for stationary and transient cases. The optimization solver used is the solver SNOPT [START_REF] Gill | User's guide for SNOPT version 7[END_REF]. The optimization criterions will use horizontal and/or vertical temperature profiles measured at the back surface of the composite. In our study we use a point (the center of the plaque).

The aim is therefore to minimize the squared difference between the experimental and simulated temperatures profiles. A least squares criterion J is then defined using in that case only the vertical temperature profile by the following relation [START_REF] Villière | Experimental determination and modeling of thermal conductivity tensor of carbon/epoxy composite[END_REF]:

` a b cC Z dZ[efZ\&2] center of plaque -C Uefj]2&Z= center of plaque k l ? (7)
The heat diffusion into the composite thickness is related to its transverse thermal conductivity (k z ) and along the perpendicular plane to the thickness of the composite to its longitudinal thermal conductivities (k x , k y ) which also have an influence on the shape of the laser beam. In our study, each reinforcement tissue is made of unidirectional glass fibers assembled in a quasi-isotropic draping as in [START_REF] Nakouzi | Modélisation du procédé de cuisson de composites infusés par chauffage infra rouge[END_REF]. The thermoplastic composite used in our study is a UD Polycarbonate (PC) Glass fiber reinforced. The initial parameters for the numerical simulation are referenced in the table below:

Table 1: Initial parameters for the numerical simulation Numerical simulation is compared to experimental data (Fig. 4a) versus time. The value of the error at convergence is less than 5% in the center of the plaque. We notice a significant difference between the two curves in the beginning (time < 6 seconds) due to the instability of the numerical model at the beginning of the simulation. Convergence occurs progressively as the time becomes important. As a preliminary validation, we compared our results to the analytical method developed in [START_REF] Villière | Experimental determination and modeling of thermal conductivity tensor of carbon/epoxy composite[END_REF] concerning modeling thermal conductivities in the unidirectional composites. This method used two steps. The first step consists in determining the transverse and longitudinal thermal conductivities of a single unidirectional lamina in which the fibers are parallel. In the second step, the different orientations of fibers in the stacking sequence are considered to obtain the laminate thermal conductivity tensor. The analytical method is based on an isotropic case due to ahomogeneous fibers distribution in the y and z plane. This leads to k y = k z . However the real microstructure of our composite is a quasi-unidirectional, which causes a variation of the conductivities in the y and z plane (Tab.2).

Table 2: Simulated thermal conductivities

We performed a simple thermal simulation of the laser welding process with our general analytical model. The laser power used is 30 W. We consider a surface heat source (eq.5) at the interface of two composite parts and we have observed the temperature distribution (Fig. 4b) after 5 second of simulation. This will enable us to make later a simplified optimization procedure. We observe a symmetrical distribution of the laser beam (same size and same shape) at the interface (due to surface absorption approach). The temperatures obtained are correct compared to those obtained during the laser welding process, the welding criterion being when T>T melt (glass transition temperature for the Polycarbonate [START_REF] Acherjee | Effect of carbon black on temperature field and weld profile during laser transmission welding of polymers: A FEM study[END_REF][START_REF] Acherjee | Modeling of laser transmission contour welding process using FEA and DoE[END_REF]).

Conclusion

The aim of this study was to propose a numerical model in order to predict the heat source in the laser welding process thermal simulations, at the welding interface. This paper presents the complete description of the laser welding process in composite materials (refraction and absorption coupling). In order to determine the temperature field during the laser welding process, the thermal properties of the materials to be welded are required as input parameters to model the thermal transfer. An inverse design method based on experimental measurements performed using infrared camera was used in order to evaluate the global conductivity tensor. We performed subsequently a preliminary thermal simulation (considering a surface heat source) of the laser welding process with our general analytical model. In future prospect, experimental measurements will be performed in order to valid the numerical model.

Figure 1 :

 1 Figure 1: Laser welding system[START_REF] Troughton | Chapter 13 -Laser Welding[END_REF]

Figure 2 :

 2 Figure 2: Semi-transparent and absorbent composites coupling Q x, y, z ( 0 semi -transparent part for all x $ Ω & I x, y, z K 2 e 3 4 5 absorbing part for all x $ Ω 8 9(2)

Figure 3 :

 3 Figure 3: Experimental set-up

  Figure 4

Acknowledgement: This work could not be possible without the assistance of Jean Michel MOUYS (ICA-Albi) for helping with the implementation of the experimental set-up.


Eduard Marenić

Jurica Sorić

Zdenko Tonković

NANO-SUBMODELLING TECHNIQUE BASED ON OVERLAPPING DOMAIN DECOMPOSITION METHOD Summary

Keywords: molecular mechanics, multiscale, ghost forces, atomistic-to-continuum coupling, bridging domain method

A multiscale method that couples an atomistic with a continuum model is presented. Nano-submodelling (nSM) is an approach that enables the insertion of a nano-refined submodel (atomistic) in a global model (continuum). The applied concurrent atomistic-to-continuum coupling of the two models is based on the overlapping domain decomposition scheme called the bridging domain method or on a similar method, the Arlequin method. Different models overlap and the displacement compatibility is enforced via Lagrange multipliers. In this study, an analysis of the spurious effects that may arise in and near the coupling domain is performed. Some coupling options such as energy weighting, coupling zone geometry, and the Lagrange multiplier field interpolation are tested.

Introduction

Submodelling is a common engineering technique usually implemented in standard finite element (FE) method codes. It is used for detailed study of various geometrical discontinuities like grooves, welds, notches, and cracks. For example, in [START_REF] Marenic | On the calculation of stress intensity factors and J-integrals using the submodeling technique[END_REF], calculations of the stress intensity factor and the J-integral of cracked structural components are performed by using the submodelling technique. Therein, the results from the global model are interpolated on the boundary of the submodel. The global model has a relatively coarse mesh without a crack, and the submodel consists of a fine mesh with a crack.

Mesh refinement in the local-global strategy is selective and is carried out only on local subdomains without gradual transition. Only locally refined mesh considerably reduces the computational time. The s-method [START_REF] Fish | Multiscale Methods Bridging the Scales in Science and Engineering[END_REF] is an example of the local-global type multimodel framework that superimposes additional refined meshes to the existing global one. Another example is the Arlequin method [START_REF] Dhia | The Arlequin method as a flexible engineering design tool[END_REF][START_REF] Dhia | Multimodeling of multi-alterated structures in the Arlequin framework. solution with a domain-decomposition solver[END_REF][START_REF] Qiao | Implementation of the Arlequin method into ABAQUS: Basic formulations and applications[END_REF] where the FE models are not added but they partially overlap and are glued to each other. The fact that models are locally crossed with each other allows the coexistence of substantially different mechanical and mathematical models [START_REF] Dhia | The Arlequin method as a flexible engineering design tool[END_REF], and hence this approach can be used for multiscale modelling. Regarding coupling conditions between two scales, multiscale modelling approaches are divided into hierarchical (weak) and concurrent (strong) [START_REF] Ibrahimbegovic | Nonlinear Solid Mechanics[END_REF]. In the first approaches, the computations are performed on each scale separately and the scale coupling is often done by transferring the problem parameters leading to the classical problem of homogenization (e.g. see early work [START_REF] Sanchez-Palencia | Non-homogeneous media and vibration theory[END_REF]). In the concurrent coupling of scales, the behavior of each scale strongly depends on the other, and the appropriate model is solved on each scale simultaneously [START_REF] Ibrahimbegovic | Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures[END_REF]. Here, we focus on the concurrent, atomistic-tocontinuum multiscale modelling, strongly coupling atomistic and continuum scales. A good example of the application of this kind of multiscale modelling is material failure which is often initiated by nano-defects that exist on the atomistic scale and cannot be described on the continuum scale (fracture is intrinsically an interatomic bond breaking phenomenon). Fully E. Marenić, J.Sorić, Z. Tonković Nano-Submodeling Technique Based on Overlapping Domain Decomposition Method atomistic (or molecular) simulations, with current computer resources, still remain out of reach for engineering systems of practical interest. Hence, extensive work has been done in the development of atomistic-to-continuum multiscale modelling approaches starting with [START_REF] Mullins | Simulation of the (001) plane crack in alpha-iron employing a new boundary scheme[END_REF] and [START_REF] Kohlhoff | Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model[END_REF]. An overview is given in [START_REF] Curtin | Atomistic/continuum coupling in computational materials science[END_REF][START_REF] Park | An introduction and tutorial on multiple-scale analysis in solids[END_REF][START_REF] Liu | An introduction to computational nanomechanics and materials[END_REF] and a comparison of the methods performance in [START_REF] Miller | A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods[END_REF].

In this paper Belytschko's bridging domain (BD) [START_REF] Belytschko | Coupling methods for continuum model with molecular model[END_REF][START_REF] Xiao | A bridging domain method for coupling continua with molecular dynamics[END_REF][START_REF] Guidault | Bridging domain methods for coupled atomistic-continuum models with l 2 or h 1 couplings[END_REF] (or adopted the Arlequin method [START_REF] Bauman | On the application of the arlequin method to the coupling of particle and continuum models[END_REF][START_REF] Prudhomme | Computational analysis of modeling error for the coupling of particle and continuum models by the Arlequin method[END_REF]) is used for the atomistic-to-continuum coupling. These two methods are basically the same and are based on the overlapping domain decomposition (ODD) approach. Two different and partially overlapped domains are coupled, which enables the application of the nanosubmodelling (nSM) technique i.e. the insertion of the nano-submodel in the global FE model. The coupling based on the ODD method differs in the following options: 1. definition of coupling zone, 2. choice of linking parameters (e.g. continuity of displacement, displacement derivatives and forces), 3. way to impose linking (Lagrange method, augmented Lagrange method [START_REF] Belytschko | Coupling methods for continuum model with molecular model[END_REF]) and 4. energy distribution in the overlapping zone.

The paper is organized as follows. The governing equations and the coupling are described in section 2. In section 3, the performance of some of the mentioned coupling options is numerically tested. The numerical investigation is performed on a 1D academic example with the local and non-local interaction in the atomistic domain. Different options for energy weighting in the domain overlap and the Lagrange multiplier (LM) interpolation are studied together with its influence on coupling accuracy. Finally, some concluding remarks including further research plans are given in section 4.

Governing equations and coupling in nSM technique

The scheme of the coupled model domain Ω consisting of three subdomains is shown in Fig. 1. The atomistic domain Ω a is treated with molecular mechanics (MM), whereas the continuum domain Ω c is discretized by FEs. The role of the continuum model is to replace the molecular model with a coarser, and thus cheaper, model in Ω c ⊂ Ω away from the region of interest, e.g. crack or dislocation. Linear elasticity is selected for the continuum model assuming small deformations. The atomistic and continuum domains overlap on Ω b = Ω a ∩ Ω c which is called the bridging domain. This overlapping domain is also called the handshake or coupling domain. Traction t and displacement boundary conditions ū are given on the part of the surface ∂ t Ω and the complementary part of the surface ∂ u Ω, respectively. Since ∂ t Ω, ∂ u Ω ⊂ Ω c , the treatment of the boundary conditions is standard as in the continuum FEM.

Total potential energy of the system may be written as

E tot,w = E c w (u) + E a w (d) -E ext (u, d), (1)
where u i d are displacement vectors in the continuum and atomistic domain, respectively. Furthermore, E ext is the work of external forces, while E c w and E a w are weighted continuum and atomistic energies, defined as

E c w (u) = Ω c w c (X)W (F) dΩ c , E a w (d) = 1 2 i,j∈Ω a w a V ij . (2)
In the above equations, W (F) stands for the continuum strain energy density as a function of the deformation gradient F, V ij describes the atomistic interaction (bonding energies) where subscripts i and j indicate atom pairs in the two-body potential. In the bridging domain the atomistic and continuum models overlap, and the weighting functions w c and w a in (2) partition the energy. The weighting function serves to blend the behavior from the continuum model (w c ) and the atomistic model (w a ) and to avoid the double counting of the energy in the bridging domain. Furthermore, the use of an overlapping subdomain obviates the need for the FE nodes of the continuum model to correspond to the atomic positions. Thus, the finite element mesh does not need to be refined to the lattice dimensions around the atomistic subdomain, which is consistent with the nSM concept. The weighting functions w c and w a define a partition of unity of the energy in the bridging domain as follows:

w c (X) = 1 for X ∈ Ω c \ Ω b , w a (X) = 1 for X ∈ Ω a \ Ω b , (3)
w c (X) + w a (X) = 1 for X ∈ Ω b .
The coupling of the atomistic and continuum models is achieved by forcing displacement compatibility in the bridging domain as u(X = X i ) = d i , ∀i ∈ Ω b . The Lagrange multiplier method is used to convert the problem of constrained minimization into finding the minimum of the larger, unconstrained problem. Hence, LMs, λ, are used to enforce the compatibility between the discrete atomistic displacement and the continuum displacement field, which gives the following Lagrangian

W L = E tot,w + G = E tot,w + i∈Ω b Ω b λ(X) • [u(X) -d i ] δ(X -X i )dΩ, (4)
where G is the constraint in terms of the energy, and δ(X -X i ) is the Dirac delta function.

Regarding the numerical implementation, the displacement field in Ω c and LM fields in Ω b are approximated by the shape functions N i (X) and N λ k (X) as

u(X) = i∈S N i (X)u i , λ(X) = k∈S λ N λ k (X) λk , (5)
with u i and λk as nodal values. The set of all FE nodes is represented with S, and S λ is the set of all λ-nodes. The equilibrium configuration corresponds to the stationary point of the Lagrangian. The discrete equilibrium equations are obtained from the stationarity of the Lagrangian 4 with respect to u i , d i and λi :

∂W L ∂u i = ∂E c w ∂u i - ∂E ext w ∂u i + ∂G ∂u i = 0 ∀i ∈ S, (6)
∂W L ∂d i = ∂E a w ∂d i - ∂E ext w ∂d i + ∂G ∂d i = 0 ∀i ∈ M, (7)
∂W L ∂ λi = ∂G ∂ λi = 0 ∀i ∈ S λ . (8)
E. Marenić, J.Sorić, Z. Tonković

Nano-Submodeling Technique Based on Overlapping Domain Decomposition Method In [START_REF] Sanchez-Palencia | Non-homogeneous media and vibration theory[END_REF] M denotes the set of all atoms. The derivatives of the above equations give the forces, where uppercase F is used for continuum and lowercase f stands for the atomistic forces (indices int and ext refer to internal and external, respectively) as follows

∂E c w ∂u i = F int i , ∂E a w ∂d i = f int i , (9)
∂E ext w ∂u i = F ext i , ∂E ext w ∂d i = f ext i . ( 10
)
3. Numerical investigation of one-dimensional coupling

Model description, nomenclature and symmetry boundary condition

According to the general scheme from Fig. 1, a chain-like one-dimensional (1D) model is used as a numerical example, see Fig. 2 a). The lengths l a , l c and l b are simplified 1D counterparts of Ω a , Ω c and Ω b respectively. The given parameters for the numerical examples considered in the following sections are: l 0 = 1, l b = 16l 0 , l a = 80l 0 , l c = 64l 0 . The size of the FE is denoted as l e , and all the lengths are taken as the integer multiple of the lattice size l 0 . Parameter R c defines the cut-off radius which governs the atomistic potential interaction. In order to represent the real material adequately, the potential that governs the atomistic interaction should be extended beyond the nearest neighbour atoms. The potentials are thus non-local, in theory extending over the whole space but in practical implementations extending over a range R c on the order of the first few neighbour distances. In this paper, the harmonic potential [START_REF] Allen | Computer simulation of liquids[END_REF][START_REF] Griebel | Numerical Simulation in Molecular Dynamics[END_REF][START_REF] Elizondo | Horizontal Coupling in Continuum Atomistics[END_REF][START_REF] Ercolessi | A molecular dynamics primer[END_REF][START_REF] Buehler | Atomistic Modeling of Materials Failure[END_REF] is considered only, with the nearest and the second nearest neighbor interaction represented by the springs k 1 and k 2 , respectively. Due to the practical implementation, in classical atomistic potentials, the total atomistic energy is partitioned into energies on a per-atom basis, even though the quantum energy cannot be treated in this manner [START_REF] Curtin | Atomistic/continuum coupling in computational materials science[END_REF]. The i-th atom scaled energy in the 1D case is

E a w,i = 1 2 w a i,i-1 k 1 2 (d i -d i-1 ) 2 + w a i,i+1 k 1 2 (d i+1 -d i ) 2 + w a i,i-2 k 2 2 (d i -d i-2 ) 2 + (11)
w a i,i+2 k 2 2 (d i+2 -d i ) 2 .
In the above equation the half in front of the bracket is to avoid double counting when summing up the terms to obtain the total atomistic energy E a = i E a i , ∀i ∈ Ω a . The leftmost atom is ... fixed and the rightmost node is loaded with F = 1. The symmetry boundary condition (BC) is set (see Fig. 2 a) on the left), which corresponds to the nSM inserted in the continuum chain as presented in Fig. 2 b). The symmetry is modelled by including the energy of the pad atoms, i.e. the energy of the dashed springs k 2 connecting the pad atoms 1p and 2p with the atoms 1 and 2. The displacements of the pad atoms 1p and 2p are mirrored displacements of the atoms 2 and 3, respectively, so the energy is doubled. This remedy corrects the boundary effect on the left edge of the atomistic domain with the non-local interaction in the atomistic domain, see Fig. 3. When the pad atoms are not included in the total atomistic energy, the atoms 1 and 2 do not have non-local neighbours to the left. This causes errors i.e. oscillations in the strain field (Fig. 3). However, it should be noted that this effect does not occur when the interaction in the atomistic domain is local.

Apart from preventing boundary effects, the purpose of the symmetry is to obtain the configuration equivalent to the nSM inserted into the continuum chain (Fig. 2 b). The symmetry BC is used to avoid modelling of two bridging zones, one on the left and one on the right end of the nano-submodel. This remedy gives simple model which is needed to properly analyze the coupling and the errors that occur in and near the bridging zone.

with sym BC without sym BC Fig. 3 Strain distribution plot for non-local interaction. The symmetry BC corrects the boundary effect and gives the constant strain field.

On the Lagrange multipliers and energy weighting

Two limiting cases regarding the LM field (see eq. ( 5)) are taken into consideration: the strict (also called non-interpolated) coupling where the LMs coincide with atoms i.e. N λ k (X i ) = δ ki , and the interpolated coupling where the λ-nodes are coincident with FE nodes and the LM shape functions N λ k correspond to the FE shape functions N k . The distribution of the λ-nodes for the two cases is shown in Fig. 4.

The FE size is taken as an integer multiple of the lattice size l 0 as mentioned, and the uniform meshes are considered only. The idea of the nSM technique is to avoid mesh refinement and to have a relatively coarse mesh, that is one or at most two elements in the bridging zone. However for the sake of analysis of atomistic-to-continuum coupling accuracy, the size of the FE is in the range from the lattice dimension to the size of the bridging zone, l e = l 0 . . . l b . This is encompassed by the parameter ES = 0 . . . 1 which defines the relative size of the FE element with respect to the size of the bridging zone:

ES(l e ) = l e -l 0 l b -l 0 = 1
for l e = l b , 0

for l e = l 0 .

E. Marenić, J.Sorić, Z. Tonković

Nano-Submodeling Technique Based on Overlapping Domain Decomposition Method

The FE size which is less than the lattice dimension is inconvenient in the sense of the nSM technique and is not studied here.

The energy weighting functions are varied and are taken to be constant (value 0.5), linear (ramp) and cubic functions of x in Ω b as depicted in Fig. 5. The distribution of the three atomistic weighting functions w a is shown, and due to simplicity, only the linear continuum weighting function (dashed line in Fig. 5) is shown.

Strict coupling

The strict coupling is an expensive option which generates a new unknown for every atomistic DOF in Ω b , as shown in Fig. 4. In 1D examples the number of additional unknowns, namely LMs, n λ = n ab , where n ab is the number of atoms in the bridging zone. However, it will be shown that the strict coupling is more accurate. In fact, it exactly reproduces results from full molecular simulation for examples studied herein concerning the local and non-local atomistic interaction.

As the first example, the simplest molecular system with the local atomistic interaction is considered. The local or 1 st neighbour interaction implies R c = l 0 or k 1 = 1 and k 2 = 0 in [START_REF] Curtin | Atomistic/continuum coupling in computational materials science[END_REF]. The displacement and strain distribution are depicted in Fig. 6. For this simple molecular system the coupling is consistent and the model passes the patch test, i.e. it is able to reproduce a constant strain field exactly. For the local interaction in Ω a and strict coupling, the constant strain field is obtained for any FE size ES or weighting function. Although it is not shown in this study, the exact strain field is obtained for any bridging size l b , too. However, the LM field depends on the weighting function which is depicted in Fig. 7. For this very simple molecular model by hand calculation of the LM is carried out in [START_REF] Fish | Multiscale Methods Bridging the Scales in Science and Engineering[END_REF][START_REF] Belytschko | A continuum-to-atomistic bridging domain method for composite lattices[END_REF]. It is easy to show that λ i ∝ (w a i,i-1w a i,i+1 ). Since l 0 = 1 the LM distribution for the local interaction corresponds to the first derivative of the weighting functions, e.g. the LM for the constant weighting function is zero in Ω b except on the boundary, see Fig. 7.

In the next example, the non-local interaction is modelled in Ω a by setting R c = 2l 0 , i.e. k 1 = k 2 = 1. For the non-local interaction case the accuracy depends on the strategy of calculating the atomistic weighting function w a i,j from eq. ( 11). In Zhang et al. [START_REF] Zhang | Mechanics of defects in carbon nanotubes: Atomistic and multiscale simulations[END_REF], the bonds are weighted by using the linear interpolation between an atom pair, which will be labeled here as strategy A

w a i,j = w a (X i ) + w a (X j ) 2 . (13)
In the original BD method [START_REF] Xiao | A bridging domain method for coupling continua with molecular dynamics[END_REF], and e.g. in [START_REF] Zhang | A bridging domain and strain computation method for coupled atomistic-continuum modelling of solids[END_REF], the weighting function is calculated exactly at the half distance, denoted as strategy B The weighting using strategy A (13) yields some errors, as it can be seen in Fig. 8. Oscillations in the strain and small deviation in the deformed shape (not shown here) is due to ghost forces (GFs). GFs is a common name for the unphysical, spurious effect that occurs because of the locally affected symmetry of the global stiffness matrix [START_REF] Curtin | Atomistic/continuum coupling in computational materials science[END_REF][START_REF] Miller | The quasicontinuum method: Overview, applications and current directions[END_REF]. Fig. 8 shows that strain error highly depends on weighting strategy. For the constant weighting function the strain error is unacceptable, more than 50%, whereas it drops below 2.5% and 0.5% if the linear and cubic functions are used. On the other hand, the strict coupling and the weighting strategy B reproduce a constant strain field and yield no error for any weighting function or FE size (ES).

w a i,j = w a X i + X j 2 . (14)

Interpolated coupling

The interpolated coupling, where λ-nodes are coincident with FE nodes is computationally cheaper than the strict coupling, but consequently less accurate. In 1D examples n λ = n cb , where n cb is the number of FE nodes in bridging zone, (see Fig. 4). The main idea of the nSM technique is to replace the molecular model with a much coarser continuum model, i.e. to achieve that n cb n ab . This is not so obvious in the 1D case but it becomes so in the 2D and especially in the 3D case. The exact distribution of the LM field for the strict, local atomistic interaction, shown in Fig. 7, does not change with the number of elements in the bridging zone. It is clear that for the interpolated coupling case, where N λ = N , the LM distribution changes and considerably varies with the FE size (ES), as shown in Fig. 9. The limiting case ES = 0, clearly, gives the exact LM distribution, see Figs. 7 and9, and is equivalent to the strict coupling. Inability to reproduce an exact LM field for the case of the interpolated coupling (even for the local interaction) causes discrepancy from the constant strain field, see Fig. 10.

As for the strict coupling, the model with the non-local, harmonic interaction in the atomistic domain is studied for the interpolated case, too. The weighting strategy B is used here. This model causes even more significant oscillations in the strain field. This is expected because the interpolated coupling for the local interaction yielded errors, and adding non-locality only degrades the accuracy. However, no significant deviation from the linear displacement field can be noted, Fig. 11. The strain error caused by the coupling and due to GFs is a localized effect which is influencing the strain neither in the atomistic nor in the continuum domain.

Conclusion

Nano-submodelling is a multiscale method that combines the atomistic with the continuum model and seizes the benefits of the submodelling as a common engineering technique for inserting the refined submodel in the global model. In the nSM, the atomistic submodel is inserted in the global FE model. These two models are mathematically different and thus special attention has to be paid to the coupling. In this study, different atomistic-to-continuum coupling options are numerically tested on a 1D coupled model. The performance of the following options was investigated: the strict and interpolated LMs, the local and non-local atomistic interaction, the weighting strategy and the weighting function.

The strict coupling, the more expensive option, gives exact results for the local atomistic interaction and shows neither dependence of the choice on the weighting function nor the size of the elements in the bridging zone. The strict coupling for the non-local interaction considerably E. Marenić, J.Sorić, Z. Tonković Nano-Submodeling Technique Based on Overlapping Domain Decomposition Method varies with the choice of the weighting function in the case of the weighting strategy A. On the other hand, for the weighting strategy B, the exact constant strain field is obtained for all weighting functions.

The interpolated coupling reduces the number of degrees of freedom by adding the interpolated LM field. This coupling yields errors for both local and non-local atomistic interaction. It significantly depends on the number of FEs in the bridging zone. It is thus recommended to use strict coupling at least in 1D and 2D cases due to accuracy. However, for 3D problems with many DOFs, the interpolated coupling should be considered to optimize the accuracy and the computational time.

It should be noted that for both the strict and the interpolated coupling the discrepancy from the constant strain field caused by GFs is a localized effect which does not influence the atomistic or continuum domain. Hence, it is possible to use the nSM technique as an approach to study nano-defects, but bearing in mind that the area of interest has to be far enough from the coupling domain.

In the future study, the weak form of displacement compatibility (often called L 2 and H 1 ) will be considered. This approach averages the coupling error, due to the integral form of the coupling constraint, but requires an additional approximation of the atomistic displacement. Furthermore, the force based coupling should be considered as in [START_REF] Fish | Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force[END_REF]. The displacement coupling presented herein and the force coupling represent upper (Voigt) and lower (Reuss) stiffness bounds, respectively [START_REF] Markovic | On micro-macro interface conditions for micro scale based FEM for inelastic behavior of heterogeneous materials[END_REF]. The influence of the displacement and the force coupling on the stiffness will be studied in future. The research plan also includes two-dimensional (2D) and three-dimensional (3D) generalization of the proposed algorithm in order to be able to include the crack-like defects in the nano-submodel. In order to describe the material more realistic, the complex potentials, like the Brenner and Dreiding potential for carbon structures should be used. Furthermore, the use of the nSM technique is planned to be studied on nano-defects in carbon structures e.g. carbon nanosheets (2D), and carbon nanotubes (3D).

Fig. 1

 1 Fig. 1 Scheme of the coupled model and the nSM technique.

3 Fig. 2

 32 Fig. 2 1D coupling model scheme with the symmetry BC on the left end of the atomistic domain a). The utilization of the symmetry BC yields the configuration equivalent to the nSM inserted into the continuum chain b).

Fig. 4

 4 Fig.4Scheme of the distribution of the LM nodes for strict and interpolated coupling.

Fig. 5

 5 Fig. 5 Energy weighting function distribution in the bridging zone.

Fig. 6 Fig. 7

 67 Fig. 6 Displacement and strain plots versus position for R c = l 0 , k 1 = 1 and ES = 1.

Fig. 8

 8 Fig. 8 Strain distribution for R c = 2l 0 , ES = 1, strict coupling, weighting strategy type A, with the weighting function varied.

Fig. 9

 9 Fig. 9 Values of LMs for local interaction, interpolated coupling and constant weighting with different FE sizes (ES).

Fig. 10

 10 Fig. 10 Displacement and strain plots versus position for the local, interpolated coupling, with 2 elements per Ω b (ES = 1/15).

Fig. 11

 11 Fig. 11 Displacement and strain plots versus position for the non-local, interpolated coupling, with 2 elements per Ω b (ES = 1/15).

Acknowledgments

This study was supported by the Ministry of Science, Education and Sports of the Republic of Croatia within the framework of the research projects No 120-1201910-1812 and 120-1201910-1906.

REFERENCE


Claire Dupin

come

Not to be seen. Sordid aspects of prostitutions in illustrations.

Claire DUPIN DE BEYSSAT

NOT TO BE SEEN

Sordid aspects of prostitutions in illustrations

"Prostitutes are as inevitable in a great urban centre as are sewers, roads and rubbish dumps;

[it is thus necessary] to avoid by all possible means the inconveniences which are inherent to them, and, to that end, to hide them, to relegate them to the darkest corners, in short, to make their presence go as unnoticed as possible." This sentence is from Alexandre Parent-Duchâtelet, first to study prostitution as a phenomenon both social and political. More than his surveys, his recommendations form the basis of the regulationist system, instituted in France in the 1840s and then adopted all around Europe. Its main goal was to control prostitution: at this time, even if necessary, prostitution was not to be seen.

However, in the second half of the 19th century, prostitution appears in artistic or literary representations. Publications started to describe the realities of prostitution by diverse means: songs, novels and short stories, testimonies and scientific papers. In art, prostitution became a motif used to modernize traditional genres or to add some excitement to more and more pompous compositions. Yet, it seems that contemporary prostitution is in itself was an audacious enough subject that artists were not willing to paint its dirtiest aspects. The major works of art would lead one to think that loneliness and boredom are the two most unpleasant situations a prostitute can endure. To find representations of the crudest aspects of prostitution, such as misery, exploitation, repression, disease and violence, one has to seek through illustrations, as those kind of depiction do exist there.

Most of these pictures date from the end of the 19 th century, it is to say from the 1880s and 90s. The explanation is simple: in those decades, illustrations became easier to make and publish, and therefore soon appeared in most of the medias. After the liberalization of the press in 1881, 2 / 8 newspapers were allowed to put every opinion into texts but also to publish new kinds of images.

In parallel, editors continue to publish books with more frequent illustrations and of higher qualities. Actually, most of the illustrations newly produced used these improvements, be it technical or be it judiciary, to present comic or erotic pictures. This is particularly true with prostitution as a subject. But some supports distinguish themselves by representing prostitution also through its most despicable aspects. Those images, being rare, are quite unknown: it is these that I would like to study with you today.

First of all, these illustrations come from different types of publications. They are not the doing of a sole editor, writer or artist, even if they are mostly signed by few authors known for their preliminary engagement. In fact, most of them are from Théophile Steinlen, whose political and social commitment is today well known and studied. Indignant to the strong injustices of his time, he has denounced them through his art and represented numerous types of misery, among them the sordid realities of prostitution. Steinlen's images constitute the majority of this corpus, which comes as no surprise, as it is the result of the convergence of his concerns with the social and political tendencies of the publication he illustrated.

Most of the images I present were published in press publications belonging to boulevardière culture: Le Gil Blas illustré and Le Courrier français. Born in 1891, Le Gil Blas illustré was a derivative of Le Gil Blas, and was given to the readers every Sunday. It was directed by René Maizeroy, and the line, even if more entertaining, was close to the one of the original newspaper: a moderate republicanism, sensitive to the social questions. This commitment is well visible when one studies the authors and illustrators published in their pages. Most of the short stories are signed by author close to the naturalistic movement: Jean Richepin, Gustave Coquiot, Oscar Méténier or Camille Lémonnier, among others. Le Courrier français, also a weekly paper, was directed by Jules Roques and was more bound to the culture of Montmartre, where prostitution soon become part of social and political claims. Regarding illustrations, Théophile Steinlen appears to be the dedicated illustrator of Le Gil Blas illustré, in the company of Balluriau and Albert Guillaume, whereas Le Courrier Français is illustrated by Adolphe Willette, Henri Rivière, Ferdinand Lunel and others. But those are not the only press publication to be interested in prostitution. In fact, La Petite Rosse is a weekly review published by and for prostitutes. Its director is Gaby la Bordelaise and it presented itself as "Gazette féministe et demi-mondaine". Unknown today, as are their writers and illustrators, the publication proposes yet an original view on prostitution, not abolitionist but quite virulent against procurers and clients. Coherently, illustrations which appear on the cover and inside the pages often denounce the work and life conditions of prostitutes.

The second major source for those representations are songs illustrations. As social song becomes more and more appreciated at the end of the century, with its dedicated places and the growing of its audience, it will soon be subjected to illustrated publishing, within newspapers or as a booklet. Among them, the ones by Aristide Bruant are the most virulent against prostitution and are usually illustrated by Steinlen. The two of them share closed convictions regarding social injustices and, among them, exploitations inherent to prostitution. Other artists depicted the phenomenon as well, including women describing their life as a prostitute, through songs and the accompanying illustrations.

The last main source is novels, which become more and more frequently illustrated at the end of the century. Therefore, as a great number of them deal with prostitution, numerous images are produced, sometimes representing its crudest aspects. The illustrations by Jean-Louis Forain for Marthe histoire d'une fille, from Huysmans, are well known, but many novels by Zolanot only Nanaand other prostitution novels, such as La Traite des Blanches from Dubut de Laforest or Nathalie Madoré by Abel Hermant were illustrated too. In those cases, illustrations could be used as an advertising poster or as the cover, and the artist should then tempt the audience without altering the story told. Within the book, illustrations were placed by choice of the editor, and they must depict what is written in the immediately previous or following pages. Yet, the illustrators are not entirely passive, as they had to compose an original picture at the same time appealing for the readers and faithful to the text. However, some artists choose to stay close to the crudest situations described in the novels, and represented the particularly harsh living and working conditions of the prostitutes, heroines of those novels.

As those images are bound physically to texts, being nearby or following, it is necessary to ask whether the illustrators remain faithfully to the situations described in the original text. Most of them do: sometimes, the link is even reinforced by a quote, which acts as a caption. It is the case of the illustrations of L'Assommoir and Nana from Zola. With the text this close, artists have to keep most of the elements originally present in it and, more important, need to put into image a one line-caption. Regarding the songs, be it published in a booklet or in a press publication, illustrations appear next to the verses. They are not, properly speaking, a caption but are easy and quick enough for the reader to recognize the accuracy of the image to the text. For instance, for "A Saint Lazare", Steinlen multiply the signs that help identify the hospital-prison: bar on windows, religious outfits, line of inmates. Indeed, most of the time, illustrators try to respect the place and time of the narration and to include all the characters. It is also the case when the image is not immediately nearby the text and there is no caption, as for the covers of Le Gil Blas illustré.

Even when they stay faithful to the texts they illustrate, artists need to make choices so that the transcription is both coherent and harmonious. In fact, these short stories and songs often stage more than one situation, forcing the illustrators to choose which one they are going to put into image. It is then interesting to see that artists often favour the saddest passages. It is the case in the illustration of the song "Fille" from Ivanof, appearing in Le Gil Blas illustré from February the 11th, 1894, which is much more pessimistic than the text. The artist only represents a street walker, roaming the cold night, choosing to ignore the passage where the prostitute joins a customer in a warmed room and is truly loved by him. Sometimes, the interpretation of the illustrator, without being unfaithful to the text, lets a criticism shine through: while the song "Les Municipaux" details several situations where the absurd stubbornness of the gendarmes is made obvious, Balluriau rather show the violent arrest of a prostitute only. Much in the same way, "Monsieur Bart" short story is organized around the figure of the eponymous police captain in the vice unit, in charge of questioning and restoring the girls taken during the police roundups and raids to their families.

Despite that, the illustration, by Jean Pauwels, very clearly favours the representation of the arrested prostitutes waiting in front of his office.

There are also certain images which have little or no link with the text they illustrate. The artist goes considerably off the original story to propose his own representation of prostitution, even if the phenomenon has not be mentioned in the text. It is in particular the case of the illustration of "Le regard vers l'amour inconnu" from Guy de Téramond. Whereas the short story tells the gallant meeting between a bourgeois and a miserable wife, Steinlen chooses to represent a woman tempted by prostitution, symbolized by the house with a large number and by the men waiting in front of it. The example is all the more striking in "La Conversion d'Angèle": Claude Berton writes about a high-society dinner party, whereas the artist represents, without any link with the text, a quarrel between cocottes. It is thus the illustrators' choice to show prostitution even if not asked: this can be interpreted as the sign that prostitution is omnipresent in popular culture, and is latent in every narration dealing with seduction or misery.

In fact, prostitution through its harshest aspect is also represented in pictures not really illustrative. Those compositions can be found inside the newspaper pages, such as Le Courrier français, where illustrations appear alone and have no link to the published texts. However, those representations are bound to the editorial line adopted by the publication, and consequently by its writers and illustrators. As said before, Le Courrier français along with Le Gil Blas illustré treat prostitution as a phenomenon both entertaining and deplorable. It is then no surprise to seen that sometimes, their illustrators need no text to deal with the unpleasant face of prostitution. Those images are easily read, as they use most of the times a caption to highlight their intention. They use quite schematic composition, opposing in stark contrast the poverty of the prostitutes to the comfort of the bourgeoisie.

In order to be easily understood by the reader, most of the artist don't hesitate to create iconographic types to represent some characters. This results from a visual tradition, from the caricatures to the illustrated physiologies, but also from the constitution of types in literature. Even if progressive, artists, both literary and visual, tend to contain the prostitution phenomenon by applying to its actors a representation grid. Illustrators form no exceptions, and even used it to denounce the miserable aspects of prostitution.

Besides the prostitute, clients and procurers both take parts in prostitution as well. To represent them, illustrators rapidly adopt a proper look for each and make them easily recognizable by their attributes. As such, the madam is the character whose treatment is the most caricatural.

She is depicted as a corpulent middle-aged woman, taller than the young womeneven girlsshe is talking to. Her dominance is emphasized by the richness of her toilet and finery. Often associated to the maternal figure, she can also be likened to an octopus, whose tentacles symbolize the pressure which she exercises on the girls, and given an openly sinister face -if, that is, they did not paint a beard on her. For his part, the customer is most of the time represented as a bourgeois, recognizable by his top hat -an omnipresent item in the painted images of prostitution too. By favouring bourgeois over working-class clients, illustrators insist on both sexual and economic domination.

In fact, compositions often oppose a prostitute represented alone, young and obviously poor (as shown by her clothes) and several men, acting like predators, dressed in sophisticated clothes, monocles and walking sticks. By contrast, the procurer appears essentially as a man originating from the lower classes. He wears the beret, a short coat -probably from standardized stores -and example of their versatility. The illustration of the song's booklet underlines the loneliness of the street walkers. Far from investing all the space of the page, the artist confines his illustration in a medallion where the street walker seems alone, roaming at night. The representation insists also on the absurdity of their condition, by missing to show potential clients. In the version appeared in Le Gil Blas illustré, the treatment is similar. Steinlen contain the prostitute in a long and vertical space, that stress her thinness and isolate her from the rest of the page. He also shows the extent and the gravity of the prostitutionnal phenomenon, by depicting numerous silhouette in black in the background of the image. The same effect is used in the illustration published in Le Mirliton, but much more advanced. Here, the image invests all of the page, living just the minimum space to the text. The image is extremely synthesized -the girls are no more than silhouettes wandering on the pavements -and composed around lighting and colours effects. Steinlen refuse to depict a proper narration but succeed in creating on ambiance, perfectly coherent to the illustrated text. The artist proposes thus a variation around Bruant's song that invests the iconography of the street walker at night, often used to depict misery. Indeed, numerous contemporary testimonies and historic studies insists on the rarity of jobs for women and on the poor incomes, which urge numerous women to resort to prostitution to fulfil their needs. Therefore, this "heure du gaz", admirably represented by Steinlen, was the time where prostitutes could roam the pavements to accost future clients, which they did in huge number.

Black is also used to picture uniforms and to signify the omnipresence of the police. In fact, the regulationnist system was particularly repressive to the prostitutes, who are treated more like criminals than victims. To depict the harassment of which the girls were victims, artists do not hesitate to stage in a rather caricatural way a row of gendarmes watching the prostitute's every move, who seems not to be the most threatening individual. But the most spectacular aspect of the repressive system is the roundup, of which the illustrators represent the violence by resorting to heavy and dynamic compositions. They also underline the injustice and the humiliation felt by the prostitutes, by reducing them to silhouettes surrounded by black uniforms.

Colour itself appears rarely in these illustrations, as the technique or the economy of the publication don't permit it: the internal pages of newspapers and of books were hardly ever in colour. However, some covers of Le Gil Blas illustré make a strong use of colours, especially red: when used, those colours most of the rime signify violence. First, violence is aimed to the prostitutes: some illustrations represent them facing prospect or real violence of their customers, 8 / 8 thus revealing their vulnerability. There, red serves to show the inherent danger of their conditions.

But the prostitutes themselves are not exempt from violent behaviour, as shown by some illustrations. Here, colours and patterns from the prostitutes' clothes reveal their vanity, as their job is to seduce. But they are also used to highlight the dynamism of certain situations. In both these pictures, Steinlen depict a fight between two prostitutes, even though of different social classes. This points out the despair inherent to prostitutes' business: whatever their status, they cannot allow themselves to lose a single customer. But mainly, it shows how colours is used to represent the two faces of prostitution: the entertaining one and the threatening one.

The corpus which I have just analysed is naturally not exhaustive. It certainly shows that representations of the harshest aspects of prostitution exist, but it also shows the rarity of this kind of depictions. The stark contrast between those images and the ones we are used to see and study is thus illuminating. It underlines the fact that popular culture and high art had favoured a more sensual and exciting view of the prostitution, that is yet contradictory to the realities of the phenomenon largely described in testimonies and in essays.

cheap shoes. Everything thus indicates that he arises from the same class as the prostitute who he "protects", and he actually plays a role more akin to the one of a lover. But artist insist on his ambivalence and deceit: the face of the pimp is often emaciated and shows no emotion, when it is not simply hidden. Indeed, artists do not ignore the underlying exploitation, and stage, via the composition or the legend, the interest which he pulls out of this relation. Thanks to those relatively fixed iconographic types, each character plays a well-defined role, in both the visual composition and the narration illustrated.

On the other hand, the treatment of the prostitute figure in illustrations has little to do with caricature. In the same way as illustrators use the types defined in literature, this is also due to the authors who, most of the time, don't characterise the prostitute they write or sing about. Both texts and images of prostitutes refer to no particular persons: from then on, the suffering they endure is not individual and isolated, but affects the whole prostitutionnal population. But, whereas a writer can give no depiction and name to his character, illustrators have to assign a body. Therefore, even if they do personify prostitutes, artists often refuse to draw them a precise face and their expressions appear strangely neutral. However, prostitutes are not just a symbol, and they are represented in wide diversity of morphology, hair colour or situations. Diversity favours the recognition of the girls' individuality, which goes against the wide-spread opinion at the time of a prostitutionnal essence which would be shared by a homogeneous population. The prostitutes are also humanized through the harshness of the shown situations, which makes the reader / spectator feel an instinctive pity towards the girls. In fact, thanks to the absence of a real iconographic type, artists multiply subtle back-and-forces between individualization and iconisation of the prostitute. On this point, it is necessary to notice that the prostitute, even if personalized, is, in the great majority of the images, isolated and embodies the prostitutionnal figure by herself. This way, illustrators can join in a sole picture the denunciation of the work and living condition of the prostitutes with the claim that those women are not to be relegated into the background: they are rather humans and victims of a system of sexual and economic domination directed toward them.

Apart from this, artists also use many tools particular to the illustrative lithography. First among these is the integration of the image to the page, which forces artists to consider its various elements (titles, texts, dedicated space) and to favour synthetic compositions and colours. The various illustrations by Steinlen for the song "Les Marcheuses" from Aristide Bruant offer a perfect