Nazihe Terfaya
email: terfaya@lma.cnrs-mrs.fr

Michel Raous
email: raous@lma.cnrs-mrs.fr

Abdelmadjid Berga
email: bergaabdelmadjid@yahoo.fr

Cohesive zone model and bipotential formulation: application to a pile/soil interface

Keywords: friction, adhesion, bipotential, soil/pile interface

The cohesive zone model developed by Raous, Cangémi, Cocou and Monerie includes strict unilateral contact to avoid interpenetration, initial adhesion progressively decreasing when the load increases, and Coulomb's friction which is progressively activated when adhesion decreases. Because of its implicit character, the Coulomb friction law is non-associated, and the notion of superpotential with normality rule cannot be used anymore. To overcome this difficulty, G. De Saxcé introduced the class of implicit standard materials based on the concept of bipotential. In the present work a specific bipotential adapted to the coupling of unilateral contact, friction and adhesion is constructed, and a numerical method is proposed and implemented. An application is presented, for modeling soil/pile interface behavior, on the simulation of pull-out experiments of a pile.

Introduction

In order to model the behaviour of complex interfaces, a model coupling adhesion, friction and unilateral contact was proposed in [START_REF] Raous | A consistent model coupling adhesion, friction and unilateral contact[END_REF] and extended in (Raous and Monerie, 2002) (the RCCM model). This model is based on the adhesion intensity variable, introduced by Fremond [START_REF] Frémond | Contact with adhesion[END_REF], which is a surface damage variable and takes its values between 0 and 1 (0 is no adhesion and 1 is total adhesion). This model gives a smooth transition from total adhesion to the usual Coulomb friction law with unilateral contact. Because of the non smooth character of the interface law set as multivalued applications (strict Signorini conditions and strict Coulomb law after collapse of the adhesion), the analysis regards non smooth mechanics. Because of its implicit character, the Coulomb friction law is non-associated, and the notion of a superpotential with normality rule, used in classical plasticity, cannot be used anymore. To overcome this non-associated character (loss of the normality rule) G. De [START_REF] Saxcé | The bipotential method : a constructive approach to design the complete contact law with friction and improved numerical algorithms[END_REF] introduced the class of implicit standard materials based on the concept of bipotential. A major result of the implicit standard material theory was the possibility of representing many non-standard dissipative laws by a suitable pseudo-potential depending on the dual variables, internal variable rates and associated variables. The contact law is then written as a differential inclusion and the choice of a convenient bipotential allows us to uncover a normality rule. The concept of bipotential is based on an extension of the Fenchel inequality, and is a generalization of the notion of pseudo-potential which enables us to write the evolution laws as implicit normality rules. In the present work, a specific bipotential adapted to the coupling of unilateral contact, friction and adhesion is constructed and a numerical method, based on a previous work of Terfaya and Berga in contact and plasticity [START_REF] Berga | Modélisation de la plasticité des sols et du phénomène de contact avec frottement par l'approche de matériaux standard implicite[END_REF] [START_REF] Terfaya | Modélisation des problèmes de contact avec frottement sec de Coulomb par la méthode du bipotentiel[END_REF], is proposed and implemented to solve the problem. As an application to Civil Engineering, the numerical simulation of a pull-out experiment of a pile is presented.

The RCCM model

The RCCM model has been successfully used for composite materials (matrix-fibre interfaces, ductile cracks) (Raous and Monerie, 2002), steel-concrete interfaces (pull out of reinforced concrete) (Raous and Karray, 2006) and interfaces in masonry construction [START_REF] Fouchal | Contribution to the modelling of interfaces in masonry construction[END_REF]. The model is based on an adhesion intensity variable, introduced first by Frémond [START_REF] Frémond | Contact with adhesion[END_REF], which is a surface damage variable that takes its value between zero (no adhesion) and one (perfect adhesion). The introduction of a damageable stiffness of the interface ensures a smooth transition between the two contact conditions (initial adhesion and final frictional sliding) during the competition between friction and adhesion. The contact of two deformable bodies is considered. Let u denote the gap between the two solids and R the contact force. They are separated into normal and tangential components as follows ( 1n is the normal vector to the contact interface):

t n u n u u + ⋅ = 1 (1) t n R n R R + ⋅ = 1 (2)
In the framework of continuum thermodynamics, the contact zone is assumed to be a material surface and the local constitutive laws are obtained by choosing the two specific forms (3) and (4) for the free energy and the dissipation potential associated to the surface. Details on the thermodynamic analysis can be found in [START_REF] Raous | A consistent model coupling adhesion, friction and unilateral contact[END_REF]. The relative displacements (u n , u t ), and the adhesion intensity β are chosen as state variables, and the contact force R and the decohesion force G are the associated thermodynamic forces.

) ( ) ( ) ( 2 2 ) , , ( ] 1 , 0 [ 2 2 2 2 β β β β β I u I wh u C u C u u n t t n n t n s + + - + = Ψ + ℜ (3) ) ( ) ( 2 ) 1 ( ) , , , , ( 2 2 β β α β β µ β β - + + - - = Φ c t n n r n n n t I u u C R R u u ( 4
β t n u r n u u R R Ψ ∂ ∈ and ) , , ( β β β t n u u G Ψ ∂ ∈ - , (5) ) , , , , u ( ) , ( and t
) , u ( r t t β β φ β β n n r n n u R G R R R ∂ ∈ = . ( 6
)
The response of the interface is described by the following relations:

unilateral contact (Signorini conditions) with adhesion 0

). (R ; 0 ; 0 2 2 = - ≤ ≥ - n n n n n n n n u u C u u C R β β (7) -Coulomb friction with adhesion - ≤ - = = 2 r t t 2 R R ; β µ β n n n n r n t t r t u C R R R u C R (8) ( ) - - - = ≥ ∃ - = - = - < - r t t r t t t 2 r t t t 2 r t t R R R u , 0 R R 0 u R R R u C R u C R n n n n n n λ λ β µ β µ -evolution of the intensity of adhesion 1 si ) ) ( ) ( ( [0,1[ si ) ) ( ) ( ( 2 2 2 2 = + - ′ - = ∈ + - ′ - = - - β β β β α β β β β α t t n n t t n n u C u C h w u C u C h w (9)
The constitutive parameters of the model are:

-C n and C t are the initial stiffness of the interface, -w is the decohesion energy (as long as that threshold is not reached, adhesion stays to be complete and the behavior of the interface is elastic with the initial stiffness C n and C t ), -µ is the friction coefficient, -α is the viscosity associated to the evolution of the adhesion.

The bipotential model

In order to generalize the concept of pseudo-potential (introduced by J.-J. Moreau for multivalued constitutive laws) to non associated laws, G. De [START_REF] Saxcé | The bipotential method : a constructive approach to design the complete contact law with friction and improved numerical algorithms[END_REF] introduced the concept of bipotential and the one of Implicit Standard Materials (ISM). This generalizes the Fenchel's inequality and permits to recover a flow rule subnormality for non-standard behaviors (specially soils and unilateral contact with dry friction).

Concept of bipotential

The Coulomb's law is non-associated and the concept of pseudo-potential cannot be used. De Saxcé and Feng (De Sacxé et al., 19998) have generalized the normal dissipation rule by constructing a unique function depending on the velocities u and on the forces R, called bipotential and here denoted b :

[ ] ( )( ) R u b R u F V b , , : , : +∞ ∞ - ×
(10) b is biconvex, namely convex with respect to u , when R is fixed, and convex with respect to R, when u is fixed. Furthermore, the bipotential satisfies the fundamental inequality:

1 , , , β β µ β β β (18) ( ) ) ( ) ( , , , β β α β β α β β β β β β β β - - ∂ ∈ - ∂ + ∈ - ∂ ∈ C C I G I G G R u b G (19)

Local algorithm

The bipotential leads to a single displacement variational principle and a single inequality.

The unilateral contact with adhesion and the friction with adhesion are coupled. An augmented Lagrangian Method (Fortin et al.1982) is used. Its leads to the following implicit equation:

( ) µ τ K oj R , Pr = (20)
where τ is the modified augmented contact reaction defined by:

( ) [ ] n u u u R t n t . . µ ρ τ + + - = (21)
The parameter ρ is a positive number, chosen according to the eigenvalue of the contact flexibility matrix. The equality (20) means that R is the projection of τ onto the closed convex Coulomb cone µ K . The local problem is treated in a reduced system by means of reliable and efficient predictor-corrector algorithm. For solving the implicit equation ( 20), the Uzawa algorithm is used and it leads to an iterative process involving one predictorcorrector step:

-Predictor:

( ) [ ] .n u u u R i t i n i t i i + + - = +1 (22a) - ( ) µ ,K proj R i i 1 1 + + = (22b)

4-Simulation of the pile-soil interface during a pull-out experiment

The simulation of pull-out experiments of a pile conducted by R. Franck and P. Mestat on the sites of Plancoët [START_REF] Frank | Aspects expérimentaux et numériques du frottement unilatéral des pieux[END_REF] is presented. The RCCM model is used to describe the pile/soil interface. The pile has a diameter D = 0.27m and a length L=13m. Linear elasticity is considered both for the pile (E=3.E06 kN/m 2 , ν=0.3) and for the soil (E=3.E04 kN/m 2 and ν=0.33) according to [START_REF] Frank | Aspects expérimentaux et numériques du frottement unilatéral des pieux[END_REF]. The finite element mesh and the boundary conditions are given on figures 1 and 2. Axisymmetric three-node elements have been used in the computations carried on using the SYMEF finite element code developed at the FIMAS in Béchar. For the in-situ soil condition prior installation, the soil is considered to be normally consolidated and the initial lateral stress coefficient K 0 is taken to be 0.43. The pull-out is simulated by applying displacement increments on the pile head. The initial normal stresses due to the pile installation are taken into account by applying a prescribed displacement on the pile shaft.

On figures 3 and 4, results of the numerical simulations are presented and compared to the experiment results given in (Frank et al. 200).

Results, as the ones presented on figures 3 and 4, have shown the strong influence of the characteristics of the soil/pile interface on the behavior of the pile, and the necessity of

Conclusion

In this work, a bipotential formulation for the RCCM model coupling contact, friction and adhesion has been presented. It has shown that the contact law on the interface can be derived from a unique bipotential. A numerical method is deduced. The convenient agreement observed between simulation and experimental results for a pull-out test demonstrates the ability of the model to simulate the behavior of pile-soil interfaces.

I

  impose the unilateral conditions and the condition β ∈ [0, 1], respectively. In (4), the indicator function that adhesion cannot be regenerated).The state laws and the complementary laws are then written in the sense of differential inclusions in order to obtain the contact behaviour laws

Figure 1 :

 1 Figure 1:Pull-out of a pile Figure 2: Finite element mesh

Figure 3 :

 3 Figure 3: Results with classical Coulomb's law Figure 4: Results with Interface model

(11) This generalization of the Fenchel inequality is the cornerstone of the formulation. The couples ( ) R u, related by the dissipative law are said extremal when equality is reached:

) Then any extremal pair satisfies:

This is equivalent to the differential inclusions characterizing an implicit subnormality law:

The relations ( 14) define a multivalued constitutive law and its inverse.

Bipotential coupling contact, friction and adhesion

Let µ K be the Coulomb's cone, which defines the set of admissible forces satisfying:

where:

, and ) 1

The relative velocity u is not normal to the Coulomb's cone, because the normal relative velocity n u is equal to zero. Normality does not apply. A change of variable for the sliding velocity is introduced and the following bipotential ( 16) is constructed. It is composed of two parts, one controlling the interface law and the other one controlling the adhesion evolution:

where u is the relative velocity, the intensity of adhesion, R the contact force and β G the thermodynamic force associated to the state variable . The parameter α is the adhesion viscosity and µ is the friction coefficient. I s denotes the indicator function of the specified sets S. It has been shown that this bipotential verifies the suitable properties of biconvexity and satisfies (11) ( 12) and ( 13). Then the contact law and the equation for the evolution of , which are explicitly given by expression (7) ( 8) and ( 9), are deduced from the state and complementary laws expressed in term of differential inclusions:

( )


Peter Yun

Tenghui Yang

Rodolphe Boudot

Emeric De Clercq

Constructive, destructive and differential detection of coherent population trapping resonance

Keywords: coherent population trapping (CPT), electromagnetically induced transparency (EIT), electromagnetically induced absorption (EIA), differential detection

come

I. INTRODUCTION

Coherent population trapping (CPT) [START_REF] Arimondo | Coherent population trapping in laser spectroscopy[END_REF] is a quantum interference process that consists of making atoms interact with a dual-frequency field that connects two ground states of the atomic ensemble to a common excited state. At null Raman detuning, atoms are pumped in a so-called dark state in which electromagnetically-induced transparency (EIT) of the atomic medium is obtained.

The opposite behavior, increased absorption, can be obtained with electromagnetically-induced absorption (EIA) [START_REF] Affolderbach | Electromagnetically induced transparency and absorption in a standing wave[END_REF]. The latter can be observed in 3-level Λ-schemes with properly-tuned counterpropagating bi-chromatic fields such that a dark state created by a dual-frequency field is a bright state for the second field.

Best CPT clocks exhibit short-term fractional frequency stability levels in the range of a few 10 -13 at 1 s [START_REF] Yun | High-performance coherent population trapping clock with polarization modulation[END_REF][START_REF] Hafiz | Symmetric auto-balanced Ramsey interrogation for high-performance coherent-population-trapping vapor-cell atomic clock[END_REF][START_REF] Yun | High-performance coherent population trapping atomic clock with direct-modulation distributed Bragg reflector laser[END_REF]. In [START_REF] Brazhnikov | Electromagnetically induced absorption scheme for vapor-cell atomic clock[END_REF], an EIA clock was demonstrated with a short-term stability of 4x10 -12 at 1 s.

While the short-term stability of CPT clocks is comparable to the one of best microwave vapor cell clocks [START_REF] Micalizio | Metrological characterization of the pulsed Rb clock with optical detection[END_REF], it remains about a factor 10 higher than the ultimate photon shot noise limit. When ultra-low noise local oscillators are used [START_REF] Francois | Simple-design ultra-low phase noise microwave frequency synthesizers for high-performing Cs and Rb vapor cell atomic clocks[END_REF] such that the Dick effect contribution is well rejected, the short-term stability of high-performance CPT clocks is in general limited by the laser noise, especially the laser amplitude noise.

In this study, we propose and demonstrate a method for probing coherent population trapping resonances in an atomic vapor based on the simultaneous detection of mirror-images of transmission (dark state-EIT) and absorption resonances (bright state-EIA) for the clock transition, and subsequent use of a differential detection stage at the cell output [START_REF] Yun | Coherent population trapping with high common-mode noise rejection using differential detection of simultaneous dark and bright resonances[END_REF][START_REF] Yun | Double-modulation CPT differential detection method and system[END_REF]. This technique leads to the detection of an atomic resonance with enhanced amplitude and suppressed common noise.

II. EXPERIMENTAL RESULTS

Figure 1 shows a simplified scheme of the experimental setup implemented for demonstration of the proposed method. The laser source is a distributed Bragg-resonator (DBR) laser, tuned on the Rb D 1 line, directly-modulated at 3.417 GHz for producing optical sidebands frequency-split at 6.8 GHz for CPT interaction. The laser frequency is stabilized using the dual-frequency sub-Doppler spectroscopy [START_REF] Hafiz | Doppler-free spectroscopy on the Cs D 1 line with a dual-frequency laser[END_REF] on an evacuated vapor cell. Initially prepared into a CPT state using the pushpull optical pumping scheme [START_REF] Liu | Coherent-population-trapping resonances in buffer-gas-filled Cs vapor cells with push-pull optical pumping[END_REF] created with a Michelson-like system, atoms are then probed with a phase-delayed linearly polarized bichromatic probe beam, such that the initial CPT state is a dark state or bright state for respective orthogonal circularly polarized components of the probe field. EIT and EIA resonances are then simultaneously generated due to the fact that one of the electric dipole moments of the double-Λ system is of opposite sign to the three other ones. The atomic resonances are created in a buffer-gas filled Rb vapor cell, temperature-stabilized, surrounded by a static magnetic field and protected by a magnetic shield. At the cell output, opposite polarizations are separated and balanced photodiodes are used to obtain the output diff-CPT signal. Figure 2 shows an example of EIT, EIA and diff-CPT signals detected in the vapor cell. The three peaks correspond to transitions of the Zeeman spectrum, the central one being the 0-0 clock transition. The diff-CPT signal exhibits an amplitude twice higher that those of respective EIT and EIA signals, and is detected with a dc background level that is almost nulled. The linewidth of the diff-CPT signal is between those of EIT and EIA resonances. Also, we have observed common-mode noise rejection onto the diff-CPT signal.

III. CONCLUSIONS

We have proposed and demonstrated a scheme that allows the simultaneous detection of dark and bright resonances in a vapor cell that leads, through the use of a differential detection stage, to a resonance signal with doubled amplitude and reduced common-mode noise. These ingredients might be the basis for the demonstration of a CPT clock with improved short-term fractional frequency stability.

Fig. 1 .

 1 Fig. 1. Experimental setup.

Figure 1 :

 1 Figure 1: Experimental setup.

Figure 2 :

 2 Figure 2: EIT, EIA and differential-CPT signal detected in the buffer-gas filled Rb vapor cell.