domain: sde sdu

text:
Aurélie Verney-Carron

Mandana Saheb

Yassmine Ben Amor

Clarisse Balland

Bolou-Bi

Correlation between black crust formation on limestone monuments and air pollution

INTRODUCTION

Bibliographic references

METHODOLOGY RESULTS AND DISCUSSION

Impacts of air pollution on health and environment are a major issue. The addition of pollutants in the atmosphere is responsible for 4.2 million deaths every year and threatens 91% of the world's population (according to the World Health Organization). It also causes the reduction of agricultural yields, damage to ecosystems and degradation of historical monuments.

On monuments out of calcareous stones, air pollution leads to the formation of black crusts in sheltered zones (Fig. 1). Black crusts result from a sulfation reaction (SO 2 reacts with calcite to form gypsum (CaSO 4 •2H 2 O)) and the entrapment of particles (especially soot), causing its blackening [1][2][3].

In order to understand the first steps of this alteration and to investigate the influence of the environment, samples of limestone and marble were exposed in different sites for 1 year.

DEPOSIT CHARACTERIZATION: GLOBAL MEASUREMENTS:

The samples were characterized by weighing (mass of deposit), optical microscopy coupled with image processing (percentage of covered surface) and spectrophotometry (percentage of reflectance change).

The deposit was analyzed using SEM-EDS (Scanning Electron Microscopy -Energy Dispersive Spectrometer) and Raman spectroscopy. One piece of each sample (~10g) was cut and solubilized in an stirred vial of 20 mL of ultrapure water for 24h. Solutions were analyzed using ICP-AES (Indictively Coupled Plasma -Atomic Emission Spectrometry).

EXPOSURE CAMPAIGN AND ENVIRONMENT

An exposure campaign was performed in the framework of the International Co-operative Programme on Effects on Materials including Historic and Cultural Monuments (ICP-Materials, Chairs: J. Tidblad & P. Spezzano).

Samples of 'Saint Maximin roche fine' limestone and Carrara marble (6 x 6 x 1 cm) were exposed in sheltered conditions and in vertical position (Fig. 2) in 22 European sites and 1 site in USA (Fig. 3).

In parallel, environmental data were collected by each site: T°and RH for the climate and [SO 2 ], [NO 2 ], [PM 10 ] using passive samplers (from IVL Swedish Environmental Research Institute) for pollution.

CONCLUSION

 The percentage of the covered surface is the best parameter to assess the general soiling of stone materials.  The deposit is mainly composed of soot, metallic and terrigenous particles. After 1 year, gypsum is already formed on the samples (Raman, ICP-AES).  The study is in progress to complete the characterization of the deposit (Ion chromatography, C speciation) and to improve the link to air pollution.

MASS OF DEPOSIT

The mass of deposit was only measured for marble and glass (loss of material during handling and transport on the limestone samples). The available results are presented on Fig. 4a.

 The deposit on marble is generally slightly higher than on glass (Fig. 4b), which indicates a higher reactivity (with gases) and/or a higher roughness (to fix particles).

OPTICAL PROPERTIES AND COVERED SURFACE

The results of reflectance change are not relevant for the soiling measurement (Fig. 5) as the majority of the values indicates a gain of reflectance (due to a general lightning of the stone in contact with humidity), especially for limestone. However, particles are present on the surface (Fig. 6). The percentage of covered surface was calculated (Fig. 7).

The % of covered surface is higher for limestone than for marble as some dark zones initially present in the limestone were counted as particles. The values are low on glass due to its very low roughness.  The % of covered surface allows quantifying the soiling of stones.

CHARACTERIZATION OF DEPOSIT

Raman spectroscopy and SEM-EDS  The majority of the particles analyzed using Raman is composed of graphitic carbon typical of soot particles (Fig. 8).

Gypsum is detectable on some samples. Metallic and mineral particles can also be observed using SEM-EDS (Fig. 9).  Elements can be released by the dissolution of minerals (correlation between Ca, Mg, K for marble / Ca, B, Sr, Ba, K, Li, Mg, Na, P, Sr for limestone) or by air pollution (Cu, Ni, Zn) (Fig. 11).

The quantity of S on the stones (coming from gypsum) is slightly correlated to SO 2 concentration in air (annual average), which highlights that black crust are formed in polluted zones (Fig. 12.a). On the same way, the sum of all elements detected in ICP-AES is slightly correlated to PM 10 concentration for limestone, but not for marble (Fig.  The characterization of the deposit has to be pursued to improve the correlation of results with environmental data (not all available at this stage).

2.b).

Fig. 1 .

 1 Fig. 1. Images of black crust (a) on a gargoyle of Metz Cathedral and (b) on a tomb of the Père-Lachaise Cemetery (France)

Fig. 4 .

 4 Fig. 4. (a) Mass of the deposit (in µg/cm²) on glass and marble for each site. (b) Correlation between the mass of deposit on glass and on marble.

Fig. 5 .

 5 Fig. 5. Correlation between the percentage of reflectance change for limestone and for marble.

Fig. 6 .

 6 Fig. 6. Optical microscopy images of the surface of 2 limestone samples (Prague 01 and Berlin 41) and 2 marble samples (Stockholm 26 and Zagreb 61).

Fig. 7 .

 7 Fig. 7. % of covered surface for each site and for non exposed references (bands).

Fig. 8 .

 8 Fig. 8. Raman spectra of the marble from Split (Croatia) Fig. 9. SEM-EDS mappings of 3 particles found on the marble samples from Berlin 41, Split 60, Zagreb 61.

Fig. 12 .

 12 Fig. 12. Correlation between: (a) [SO2] in air and S in the deposit, (b) [PM10] in air and the sum of all elements analyzed using ICP-AES.

Fig. 11 .

 11 Fig. 11. Major and minor elements (in g/g of rock) on marble and limestone.

Acknowledgements -The authors would like to thank all the partners involved in the ICP-Materials program and the Ile de France Region (DIM MAP) for its financial support.


N Triantafyllidis

Y M Leroy

Stability of a frictional, cohesive layer on a viscous substratum: Validity of asymptotic solution and influence of material properties

Stability of a frictional, cohesive layer on a viscous substratum: validity of asymptotic solution and influence of material properties

N. Triantafillydis, Y. M.

Introduction

The model considered here and by Le roy and Tri antafyllidis [1996] consists of a cohesive, frictional over burden overlaying a viscous substratum of finite thick ness and lower density and a rigid basement. The choice of an elastoplasticity model for the mechanical response of the overburden material renders possible the concur rent analysis of two sets of instabilities. The first set consists of diffuse modes, such as folding in compression and necking in extension, detected fr om the solution of an eigenvalue analysis for the whole stratified system. Shear band mode of instability constitutes the second set and marks the onset of localized faulting which is detected by the local condition of loss of ellipticity of the governing equations.

The motivation for solving the buckling problem of elastoplastic layers has several origins, discussed at length by [START_REF] Leroy | Stability of a frictional, cohesive layer on a substratum: Va riational fo rmulation , and asymptotic solution[END_REF], and only two points of interest are briefly mentioned here. First, the role of the density contrast, the tectonic fo rces and the over burden stiffness in driving salt diapirism in sedimentary basins have been studied traditionally in the laboratory [START_REF] Nettleton | Geologic models made from granular materials[END_REF] and more recently by nu merical means [START_REF] Poliakov | Initia tion of salt diapirs with frictional overburdens: Nu merical experiments[END_REF]Daudre and Cloet ingh, 1994]. Our stability analysis complements these earlier works by quantifying the influence on stability of the nonlinear work hardening properties of the fric tional and cohesive overburden material which is either difficult to scale for granular materials in the nonlin-ear range of deformation or disregarded fo r numerical reasons. This line of work should be completed with the introduction of a rheological model for the base ment and could then contribute to the elucidation of the role played by tectonics forces and density contrast in the formation of salt structure during thin-or thick skinned compression in sedimentary basins [ Vendeville and Jackson, 1992;St ewart and Coward, 1996]. Sec ond, studies of the oceanic lithosphere recognize that the upper layers yield and thus do not contribute to the bending strength in an elastic manner [Mc Adoo and Sandwell, 1985]. Such an idea was explored by [START_REF] Wallace | Buckling of a pervasively fa ulted lithosphere[END_REF], who studied numerically the buck ling of a pervasively faulted lithosphere with the model of an elastic plate in which faults were accommodat ing slip with no friction. An alternative approach is to smear those faults over the top part of the lithosphere and to make use of an elastoplasticity model such as the one considered here to introduce dissipation. It should be stressed that initiation of buckling for elastoplas tic solids occurs in general under increasing load and Overburden is thus initially stable in a force-control set up [Sh an le y, 1947;Hi ll, 1958]. The buckling of geological struc tures modeled with the theory of plasticity is unlikely to lead to the catastrophic failure discussed by Wallace and Melosh and thus deserves further attention.

Le roy and Triantafylli dis [1996] (hereinafter referred to as paper 1) presented a variational fo rmulation of the stability problem to detect diffuse modes of instability. It is applicable to a general class of strain rate indepen dent plasticity models for the cohesive, frictional over burden material. The timescale relevant to the struc ture's stability is associated with the substratum vis cosity. However, the details of the substratum rheol ogy are not important since the passive compression necessary to detect nonlinear viscosity effects [Sherwin and Ch app le , 1968;[START_REF] Fletcher | Wavelength selection in the folding of a sin gle layer with power-law rheology[END_REF] is disregarded and time independent equilibria assumed. An asymptotic solution to that variational formulation was presented for an overburden thickness which is small compared to the perturbation wavelength. This solution differs from previously published asymptotic solutions [e .g. , Sm olu- ch owski, 1909] by the nonlinear response of the overbur den, the presence of a stress gradient with depth, and the erosion and redistribution condition [Bi ot and Ode, 1965] at the top surface. Results fo r the onset of fo ld ing mode of instability have been presented for the flow theory version of the plasticity model of [START_REF] Rudnicki | Co nditions for the local ization of the deformation in pressure-sensitive dilatant materials[END_REF] . It was fo und that there is a range of lateral compressive stress, which includes the domain of elastic response, for which the system is stable to perturba tions of any wavelength despite the density contrast. The conditions fo r neutral stability are not affected by the precise nature of the boundary conditions between substratum and basement. However, in the unstable range of stress, these boundary conditions are found to be important if the perturbation wavenumber times the substratum thickness is smaller than 4. In that instance, the thinner the substratum, the slowest the initial rate of growth of the instability. A condition of perfe ct bond with the basement reduces that rate com pared to a condition of slip with no friction.

This paper has two objectives. First, it is necessary to define the range of validity of the asymptotic so lution discussed above. For that purpose, a finite ele ment solution of the variational fo rmulation is proposed which approximates with the appropriate accuracy the exact solution which could not be obtained analytically because the material coefficients are varying spatially through the overburden. It is only in the absence of gravity that such an analytical solution is derived. Sec ond, assessment is made of the influence on stability of the various parameters of the plasticity model adopted fo r the overburden. We have previously shown that the use of the flow theory version of Rudnicki and Rice 's [1975] model is appropriate to detect the onset of fo ld ing but not of localized faulting, even if a weak work hardening is adopted. The fact that these two modes of instability are not concomitant questions the rele vance of the flow theory version of Rudnicki and Rice's plasticity model for the problem treated. Indeed, it appears that a deformation theory of plasticity, which mimics the effects of pervasive small faults in accommo dating permanent bulk deformation, would be more ap propriate to trigger diffuse modes and localized faulting fo r realistic stress conditions [Triantafyllidis and Le roy, 1994] . This distinction between flow and deformation theory of plasticity deserves some clarification in the present context of a geological stability analysis at the tectonic scale and such a discussion is given herein and by [START_REF] Leroy | Onset of fo lding and fa ulting in density-stratified systems: an elastoplastic model[END_REF] .

The contents of this paper are as fo llows: In the sec ond section, the model problem and its stability varia tional formulation are summarized including a discus sion on the incremental response of the overburden ma terial. The solutions to the stability problem are also summarized, and the details of the numerical solution based on the finite element method and of the analyti cal solution obtained in the absence of gravity are post poned to the appendix. The third section is devoted to the application of these solutions to Rudnicki and Ri ce 's [1975] constitutive model. The range of validity of the asymptotic solution is established: its accuracy is excellent as long as the small parameter of the analysis, defined as the overburden wavenumber times the over burden thickness, is less than 0.4. That section also contains a comparison between predictions based on fl ow and deformation theories. The deformation the ory is found necessary to explain the initiation of lo calized faulting and fo lding under stress magnitudes of similar order. These solutions are then applied in the fourth section to a prototype, with a geometry iden tical to the one described above, proposed fo r a sec tion revealed by a seismic dip line through the Campos basin off the southeastern Brazilian coast [Cobbold and Szatmari, 1991]. An interpretation of the two stages of buckling in the absence of localized faulting, in the Albian and in the Te rtiary, is presented. It requires the application of the deformation theory of plasticity to explain fo lding fo r the stress conditions that must have prevailed in the field.

Model Problem

This section is devoted to the presentation of the model and to a summary of the hypotheses introduced during the derivation of the variational fo rmulation of the stability problem developed in paper 1. The so lutions to that stability problem, which are also dis cussed, provide the conditions for the onset of diffuse instabilities such as fo lding in compression and necking in extension (Figure 1 b).

Model Formulation

The stratified system considered is composed of three regions of infinite lateral extent referred to as the over burden, the substratum, and the rigid basement (Fig ure 1a). Thicknesses and densities of the overburden and substratum are denoted by Ha , Pa and Hb, Pb, respectively. In what follows, any material or geometric parameter or field variable related to one of the two layers has superscript or subscript a or b, indicating above and below, respectively. The coordinate system is such that the first and third axes are within the pla nar interface between substratum and overburden; the x2 coordinate axis is oriented vertically. The coordi nates e and (' introduced for convenience, are defined by normalizing x2 by the overburden and substratum thickness, respectively.

The in situ stress state in the stratified system is An dersonian , as indicated in Figure 1a. The sign con vention is that compressive stresses are negative. The stress component in directions 1 and 3 varies linearly with depth having a gradient k1 and k 3 times the litho static pressure, respectively. The stress at the interface between the two layers has magnitude u0 cos( t.p ) and uo sin( t.p ) along the 1 and 3 axis, respectively. A vari ation of t.p from zero to 1r /2 leads to a change from a state of uniaxial compression along the 1 axis to a sim ilar loading along the 3 axis. The stress state in the substratum results solely from the overburden weight and the hydrostatic pressure. Note that pore pressure effects are disregarded in this analysis. This hypothesis is common to paper 1 and is justified fo r systems fu lly drained on the geological timescale of interest.

Stability analyses require the knowledge of the ma terial's incremental response. Typically, the overbur den material results from the deposition of sediments which then sustain a mechanical and chemical com paction under varying tectonic fo rces, temperature, and fluid pressure. This history, often unknown, determines the incremental response measured in the laboratory fr om a core extracted from the field. To circumvent this complexity, it is proposed to determine the incre mental response at any point through the thickness of the overburden from the results of a single laboratory test fr om which a uniaxial stress-strain curve such as the one depicted in Figure 2 is obtained. As a conse quence, temperature effects on the material response, which could be accounted for in the constitutive model, are disregarded in this analysis. The curve in Figure 2 presents an equivalent stress measure as a fu nction of the accumulated equivalent strain; the exact definitions of these stress and strain measures will be provided later and are not important for the present discussion. It suf fices to know that at every point in the overburden the equivalent stress is uniquely determined fr om the stress parameterization discussed above. The curve in Fig ure 2 shows a linear relation if the equivalent stress is less than the yield stress Ty which plays a role similar to a cohesion. For stress larger than this critical value, Figure 2. The uniaxial stress-strain curve of the over burden material is linear for an equivalent stress less than the yield stress Ty which has for value either G x IQ-3 (solid curves) or 3G x IQ-3 (dashed curves). This relation becomes a power law with exponent m if the initial yield stress is exceeded. The two values fo r m of 1 and + oo corresponds to an elastic and an elastic, perfe ctly plastic response, respectively.

the relation is nonlinear due to a positive work harden ing but always one-to-one, such that a unique equivalent strain is associated to any value of the equivalent stress. With these hypotheses, and the fu rther assumption that the current stress state has been reached while keeping constant the directions and ratios of princi pal stresses (proportional loading) , the incremental re sponse of the material is then fully determined. It is repa resented by a fourth-order tensor L ijkl which relates the time derivative (denoted by an overdot) of the second Piola-Kirchhoff stress tensor IIji to the same derivative of the displacement gradient Ui,j . a IIji = L ijkl it1,k

(1)

The reader is referred to standard continuum literature fo r the definition of the Piola-Kirchhoff stress tensor and for its relation to the Cauchy stress tensor intro duced for the in situ stress parameterization [e .g., Oga de n, 1984]. Expressions for the moduli L ijkl are provided by Rudnicki and Rice [1975] for both flow and deformation theories. A derivation of these moduli plus a physical motivation fo r the use of deformation theo ries are given by Le roy [START_REF] Leroy | Onset of fo lding and fa ulting in density-stratified systems: an elastoplastic model[END_REF]. It should be stressed that the curve in Figure 2 is obtained fo r both the flow and deformation theory version of the same plasticity model. These two theories lead to the same moduli in the direction of proportional loading which is followed during the test conducted to obtain that curve.

Stability Analysis

The derivation of the variational fo rmulation of the stability problem presented in paper 1 is now summa rized to recapitulate the various hypotheses considered. The starting point of the derivation is the rate fo rm of the equilibrium equations in the overburden and sub stratum, the traction and velocity continuity equations at the interface of the overburden and substratum, the redistribution condition at the fr ee surface, and two types of boundary conditions at the basement. The two possibilities considered are, first, perfect bond and, sec ond, slip with no fr iction. The introduction of a simple erosion and deposition condition at the fr ee surface fo l lows the model of [START_REF] Biot | Theory of gravity instability with variable overburden and compaction[END_REF] which relates the normal component of the traction vector to the weight of the volume of material displaced

• a II2j = s P git20j2 at x2 = Ha , ( 2
)
in which Oji and g are the Kronecker delta (equal to one if i and j are identical and to zero otherwise) and the gravity acceleration. This relation indicates that a positive traction results from a top surface which is upheaved and thus should be eroded away, while a com pression is applied to the region which subsided due to the addition of new sediments of the same density as the overburden material. The scalars in (2) is set to one if redistribution is accounted for and leads to a stress-free boundary condition if a zero value is chosen.

All perturbations considered in this linear stability analysis are superposed to the fu lly three-dimensional stress state but satisfy the simplify ing conditions of plane strain. Any perturbed state fu lfills throughout the overburden the plastic flow conditions set by the in situ stress: if permanent deformation is necessary to accommodate the in situ stress prescribed at a given material point, then the perturbation can only induce further plastic straining. Possible elastic unloading in regions of the overburden upon perturbation are thus disregarded in this linear analysis. This choice is equiv alent, for the overburden, to the selection of a "com parison solid" usually employed for detecting the fi rst bifurcation in solids [Hi ll, 1958] and structures [Shan le y, 1947]. The validity of this hypothesis is verified from the results of a nonlinear stability analysis based on the finite element method [START_REF] Massin | On the stability of strain-rate dependent solids and structures[END_REF]Massi n et al. , 1996] .

The variational fo rmulation of the stability problem constitutes a generalized eigenvalue problem and takes the final form

1 a a 1 [( L1221 dU 1 a U ) 1 d6U 1 ( L1122 dU 2 0 wHa de + L1212 2 wHa df"-wHa de
The eigenvector of components U; together with the wavenumber w define the perturbation of interest. The eigenvalue of the problem is the stability exponent A which has, with the present notation, dimension of stress and whose real part is the rate of growth or de cay of the perturbation. If there is an eigenvector U; for which A has a positive real part, then the equilibrium is said to be unstable. Stability requires all admissible eigenvectors to have a A with negative real part. Neu tral stability is defined for a zero value of that quantity. The four functions /ij of the substratum's dimensionless thickness wHb introduced in (3 ) represent the influence of the viscous substratum assumed to be incompressible with an elastic moduli which can be disregarded compared to the in situ stress. These fu nctions depend on the type of boundary condition at the substratum basement interface. For a perfect bond, they are given by (5)

for the case of slip with no fr iction. Note fr om (3) that the solution to the stability variational formulation at neutral stability (A = 0) is independent of the func tion I ii (and thus of the depth of the substratum and the boundary conditions with the basement). Note also that for an infinitely deep substratum, these fu nctions are nothing but the Kronecker functions O;j fo und by [START_REF] Triantafyllidis | Stability of a frictional material layer resting on a viscous half-space[END_REF].

The predictions resulting fr om that variational for mulation are valid in the elliptic regime of the incre mental governing equations. The loss of ellipticity cor responds to the initiation of a discontinuity in the ve locity gradient which is interpreted as the onset of local ized faulting [R udnicki and Rice, 1975] . That condition is local in the sense that only the stress state and the incremental moduli are required for its application at a given point of the overburden. Equation (29) of Tri antafyllidis and [START_REF] Triantafyllidis | Stability of a frictional material layer resting on a viscous half-space[END_REF] is used for checking that condition in the following.

Solutions to the Stability Problem

See the appendix for the details of the derivations of two solutions of the stability problem presented in (3 ) to (5). The first is analytical and obtained in the absence of gravity effects. The second is numerical and based on the finite element method. Both contribute to the validation of the asymptotic solution which is now briefly reviewed to help us in analyzing the results presented in the next section.

To compensate for the absence of an analytical so lution to our stability problem, paper 1 proposed con struction of an asymptotic solution fo r a dimensionless number wHa small compared to one. The construction of the long-wavelength approximation starts from the observation that the dimensionless number !1pgfwG, interpreted as a function of A, admits a development of the type

!1 pg 2 3 0( 4) wG = 'Yo + f'Yl + f 'Y2 + f 'Ya + f , (6)
in which f , the small parameter, is the dimensionless number wHa. In (6), !1p and G denote the material a b density difference (P -P) and the modulus of elasticity in shear of the overburden material, respectively. The unknown scalars /a are fu nctions of A. The details of the derivation are found in the electronic supplement to [START_REF] Leroy | Stability of a frictional, cohesive layer on a substratum: Va riational fo rmulation , and asymptotic solution[END_REF]. Note that this develop ment is constructed for constant value of the ratio A/ f denoted by A1 in the fo llowing. Asymptotic develop ment for constant value of A, and not A1, is impossible due to the presence of singularities close to the neutral stability conditions (see paper 1 for further discussion). The expression proposed in (6) takes the more familiar fo rm A tl.pg

uo 3 1 G = wG -{ G - { 6{ 1-v ) ' (7)
in the restricted context of an elastic plate on an in finitely deep viscous foundation. The rate of growth of the instability is influenced by the density contrast, the tectonic stress, and the plate bending stiffness; these contributions being to the zero-, first-, and third-order in the small parameter f. The asymptotic develop ment in {6) extends this classical solution {7), discussed by [START_REF] Ramberg | Co mpression of float ing elastic and viscous plates affected by gravity, a basis fo r discussing crustal buckling[END_REF] and Sm oluch ovski [1909] for A equal to zero, to the case of an elasto plastic overburden sustaining a tectonic stress with an arbitrary linear distribution with depth and resting on a substratum of finite thickness.

Results

This section is in four subsections , with the first presenting the material and geometric parameters em ployed fo r all calculations. The second deals with the validation of the asymptotic solution for several sub stratum thicknesses and for the two types of boundary conditions at the basement . The third is devoted to a comparison between stability predictions based on flow and deformation theory versions of Rudnicki and Rice 's [1975] model. A short discussion on the relevance of the deformation theory of plasticity in this linear sta bility analysis is included. The fourth pertains to the sensitivity of the stability predictions obtained with the deformation theory to the material hardening law, the in situ stress gradient, and the redistribution condition.

Preliminaries

The in situ stress state in the overburden is param eterized by the stress u0 at the interface with the sub stratum, the orientation angle cp, and the two scalars k1 and k 3 which defi ne the gradients with depth of the two horizontal stresses (see Figure 1a). The angle cp as well as the stress gradient parameters k1 and k 3 are set to zero in the following, with the exception of the results in Figure 10 for which k 1 is set to 1 or 2.

The values chosen fo r all material parameters are sim ilar to those considered in paper 1 and are typical of sedimentary rocks. The modulus of elasticity of the overburden material is 10 1 0 Pa and its Poisson's ratio is 0.2. The initial yield stress T y is one thousandth of the elasticity modulus, unless otherwise stated. The yield criterion and plastic flow potential considered by Rud-nic ki and Rice [1975] depend on two constant scalars, the fr iction JL and the dilatancy coefficients. The dila tancy modeled here results fr om the opening of fracture during their sliding. Typically, during a triaxial test, the volume change is first negative and then positive prior to failure. A zero average value is thus chosen here in view of the simplicity of the micromechanism considered. Varying that dilatancy coefficient could be of interest for certain applications, such as in the next section fo r a sedimentary basin, only if a proper com paction law was proposed instead of the dilatancy in duced by shear considered here. The friction coefficient is set to 0.6 which is typical of the values discussed by [START_REF] Rudnicki | The inception of faulting in a rock mass with a weakened zone[END_REF].

The work hardening properties ofthe overburden ma terial are parameterized by a power law (Figure 2) with an exponent m. In Figure 2, the equivalent stress r + J.lP is the sum of the second invariant of the deviatoric part of the Kirchhoff stress T and of the fr iction coefficient times the first invariant of that same tensor p. It is the generalization of the equivalent stress introduced by [START_REF] Drucker | Soil mechanics and plastic analysis or limit design[END_REF] to large deformations. The equivalent strain is denoted by 'Y• Note that form equal to 1 and infinity, the material response is purely elastic and elastic, perfe ctly plastic, respectively. Selecting a representative value for m fr om laboratory test results is difficult. The desired data sets should be obtained prior to maximum load to make sure that strain local ization effects are not included [Le roy and Ortiz, 1989]. The selection of that exponent is also sensitive to the value taken for the first yield T y which is usually diffi cult to estimate. For these reasons , it is preferred here to chose an intermediate value for m of 5 for all calcu lations and to consider the influence of m by changing its value to 8 at a later stage. The latter val ue leads to large strain with little work hardening and is pro posed to model poorly consolidated sediments or dry sand under low confining pressure. Such materials are considered in the next section.

The overburden and substratum material densities are supposed to be 2.5 x 103 and 2.2 x 103kgfm3, re spectively. Those values are close to the ones considered by van Ke ke n et al. [1993]. Note that the substratum viscosity is not required explicitly here since equilibrium is assumed in the substratum: the viscosity scales with the stability exponent (see paper 1 for fu rther discus sion).

Before starting the discussion of our results, it is worthwhile to present the common fe atures of the calcu lations and of the figures to be presented. Every graph is spanned by the normalized stress u0 /r y and dimen sionless number wHa. Each curve relates the critical stress u0 /r y to the dimensionless overburden thickness or dimensionless perturbation wavenumber wHa neces sary for a perturbation to grow at a given fixed rate. The curves in all graphs are thus isocontours of the stability exponent. Note again that for the asymptotic solution, it is not the stability exponent A/G which is kept constant but the scalar At/G, which is defined by A = A1wHa and is referred to as the scaled stability ex-ponent. A zero exponent, A or At, corresponds to the condition of neutral stability which partitions every plot into an unstable and a stable region , marked by a plus and a minus, respectively. Note that the finite element solution, shown by solid curves in all graphs, approxi mates the exact solution with an accuracy which is of the order of the graph resolution. For that reason, it is sometimes referred to as the exact solution in what fo l lows. Calculations are always conducted for a constant value of one of the two dimensionless numbers tl.pg / Gw and tl.pgHafG. They differ by the characteristic length selected which is either the perturbation wavelength 21rjw or the substratum thickness Ha. In the case of a constant tl.pgjGw, the observer keeps the perturba tion wavelength constant and searches for the combi nations of overburden dimensionless thickness wHa and lateral stress which trigger instability at a given rate. In the case of constant tl.pgHa/G, the observer keeps unchanged the overburden thickness Ha and varies the dimensionless perturbation wavelength w H a. Results obtained by the two methods differ because of the pres ence of a stress gradient in the overburden which de pends on H a but not on w.

Validity of the Analytical and Asymptotic Solutions

The comparison of the analytical and asymptotic so lutions to the numerical results is conducted for the fl ow theory version of Rudnic ki and Rice's [1975] model. The case of an infi nitely deep substratum and a 27r km long perturbation wavelength is first considered with tl.pgfGw having the value 7.06 x 10-4. The isocontours of scaled stability exponent (At = Aft) are plotted in Figure 3a. It is seen that the asymptotic solution (dot ted curves) is indistinguishable from the exact solution (solid curves) even for values for wHa as large as 0.3. The range of validity of the asymptotic solution is sur prisingly large since wHa is the parameter which is as sumed to be small compared to one. The asymptotic solution thus captures perfectly the smallest magnitude of the critical stress u0 and the associated wavelength for neutral stability. These two quantities are the co ordinates of the critical point on the neutral stability curve (At = 0) where the tangent is vertical; it is the closest point to a vertical axis passing trough the origin of the u0 coordinate (not represented). The agreement between exact and analytical solutions (dashed curve) is satisfactory for large values of the dimensionless num ber wHa such as 0.8 but poor in the region of validity of the asymptotic solution. This is due to the improper account of the singular behavior for wHa close to zero of the analytical solution obtained in the absence of gravity. Note that the discontinuity in slope of the an alytical solution at neutral stability marks the limit of an elastic stress response.

\ A/G = 0 6 x w-2 \ ' \ ' \ s ' \ (-) \ t \ a \ b \ I \ I I ' I ' e I 3 x to-2 \ '
The parameterization adopted fo r the isocontours in Figure 3b is different: A is kept constant instead of A1. Interestingly enqugh, this difference improves the • validity of the analytical solution for small values of the dimensionless thickness w H a except close to neu tral stability. These results are well understood by con sidering the isocontours that can be drawn from equa tion (7) for an elastic beam on an infinitely deep sub stratum. For large values of wHa, the bending stiffness of the elastic beam dominates and the isocontours are quadratic in wHa and similar to the curves drawn for the exact elastoplastic solution in Figure 3a. For vanish ingly small values of wH a, the critical stress for the elas tic beam behaves like one over wHa with a coefficient of proportionality equal to (tl.pgf(wG)-A/G). Note that this coefficient is zero if gravity is disregarded and the conditions of neutral stability fulfilled. The same trend is observed in Figure 3b (dashed line, A = 0) for the elastoplastic overburden. Note also fr om the same equation ( 7) that this singularity does not exist in the absence of gravity if the scaled stability growth rate A1 is kept constant instead of A. This is exactly what hap pens for the dashed curves drawn for the elastoplastic overburden in the absence of gravity (Figure 3a).

The comparison between the three solutions of the stability problem is continued for the same 271' km long perturbation wavelength but now fo r a substratum of finite thickness, set to 1 and 5 km in Figures 4a and4b, respectively. A perfect bond with the basement is imposed. The results concerning neutral stability are identical to the ones obtained for an infinitely deep sub stratum in Figure 3a. This independence of the neutral stability conditions on the substratum thickness and the choice of boundary conditions at the basement can be seen fr om the structure of the generalized eigenvalue problem (A7 ) as already discussed. Comparing Fig ures 4a, 4b, and 3a, we confirm one of the conclusions of paper 1 on the influence of the substratum thickness Hb: it becomes important once wHb is less than 5. In deed, note that by decreasing the substratum thickness from 5 to 1 km, one reduces the instability exponent by a factor of 10. The confirmation of that earlier conclu sion is now possible because of the excellent agreement between the asymptotic and exact solutions, even fo r large values of wHa (up to 0.3 to 0.4).

The influence of the substratum can also be judged fr om Figure 5 drawn for a constant dimensionless num ber tl.pgHa/G of 7.06x 10-4 obtained for an overburden thickness of 1 km. The substratum thickness has the same value in Figure 5a and is set to 5 km in Figure 5b, with the same perfect bond condition at the basement as in Figure 4. A comparison between the various iso contours in Figures 5a and5b reveals that the stability exponent is 10 times larger for the 5 km thick than for the 1 km thick substratum. The dominant wavelength at a given compressive stress, defined by the value of wHa having the largest rate of growth, is varying, ap proximately, fr om 0.45 to 0.95 in Figure 5a dominant wavelength is thus 14 km long and requires a minimum stress of 20ry. If the tectonic stress has a magnitude 3 times larger than this minimum, then the dominant wavelength becomes either 8.4 km or 6.6 km long for thick or shallow substratum, respectively. The substratum thickness thus influences the dominant per turbation wavelength by as much as 30% and its initial rate of growth by a factor 10. Note from Figure 5b that the analytical solution (dashed curves) is precise for a dimensionless rate of growth larger than or equal to 2 x 10-2 • The comparison between analytical and numerical so lutions is finalized with some comments on Figure 6, which illustrates the influence on stability of the con dition of slip with no friction at the basement contact for a substratum thickness of 1 km. In Figure 6a, it is the perturbation wavelength which is kept to a constant length of 271' km, while the overburden has a constant thickness of 1 km in Figure 6b. These results should be compared with those presented in Figures 4a and5a, respectively. That comparison reveals the stabiliz ing influence of the perfe ct bond at the basement . For example, the minimum stress to initiate an instability at a rate A!/G of 10-2 is decreased approximately by a factor 2 fr om the perfect bond case of Figure 4a to the slip with no fr iction of Figure 6a. However, the dom inant wavelength for a given stress u0 does not seem to be affected by the type of boundary condition (Fig ures 5a and6b). The accuracy of the asymptotic so lution is confirmed again in Figure 6a for the condi tion of slip with no fr iction. The analytical solution (dashed curves) provides the general trend of every con tour, with the exception of the singularity for vanishing wHa for constant At.

It should be noted in concluding this subsection that all stability calculations reported so far fall within the elliptic regime of the incremental governing equations. Thus the onset of localized faulting was not detected even at compressive stresses 3 to 5 times larger (in magnitude) than the minimum one required to initi ate folding. The next subsection questions this absence of localized faulting from our predictions and proposes to remedy this situation with the use of a deformation theory of plasticity. mation theory in promoting diffuse modes of stability is now discussed. The stability predictions for the flow and deformation theories are presented fo r a constant value for .6.pg / Gw of 1.41 x IQ-4 obtained for a perturbation wavelength of 40011" m. Results for an infinitely deep and a 200 m thick substratum are presented in Figures 7a and7b, respec tively. Perfect bond at the basement is assumed. Three important results emerge fr om that comparison. The first is due to the presence of diamonds at the termi nation of the curves drawn fo r the deformation theory. They marked the onset of localized faulting which is now predicted fo r a stress which is only twice as large in magnitude as the minimum one required for fo lding (of the order of 80 MPa) . The second main result con cerns the position of the isocontours in the (uo-wHa) plane. The minimum stress to trigger an instability at a given rate of growth is decreased by 25 to 40% as one changes the plasticity theory fr om flow to defor mation. The critical dimensionless thickness wHa asso ciated with this minimum is multiplied by a factor of up to 2 during the same operation. The third result in Figure 7 is the surprisingly good predictions of the asymptotic solution (dotted curves) fo r values of wHa of the order of 0.6.

The deformation theory version appears to provide more realistic results than those obtained with the flow theory version of the same plasticity model since fo ld ing and localized faulting are now predicted for similar stress magnitudes. For that reason , it is the deforma tion theory which is adopted in the rest of this section to explore the sensitivity of the predictions to the work hardening, to the in situ stress distribution, and to the redistribution condition at the top surface.

Influence of Work Hardening, in Situ Stress Gradient, and Redistribution

One of the tenets of our plasticity model is the recog nition of a work hardening phase after first yield and prior to the approach of a maximum load at which fail ure would be observed in an experiment . Wo rk harden ing is parameterized by a power law shown in Figure 2 fo r three values of the exponent m. The value of 1 and of infinity corresponds to the elastic case and to the perfe ctly plastic case, respectively. A value of 5 was chosen for all previous calculations. That value is now changed to 8, tending more toward the perfe ct plasticity case, which is likely to be realistic fo r most sedimentary rocks under low confining pressure. Results, presented in Figure 8, compare with those on deformation the ory in Figure 7 if a general shift of the isocontours to the right is applied. The reduction in material stiffness resulting from an increase in m leads to a reduction in compressive stress required to trigger the same in stability (same rate of growth and wavelength). That reduction in stress magnitude is of the order of 30% but the dominant wavelength fo r a given rate of growth does not seem to be affected. Note again the agreement between asymptotic (dotted) and exact solution which is, however, not as excellent as in Figure 7.

The influence of the in situ stress gradient with depth is now discussed from the results presented in Figure 9. The in situ stress gradient can be close to the litho static gradient ( k1 = 1) in sedimentary basins [ Breckels , 1978]. The results are presented fo r !:l.pgHa/G set to the constant value of 1.41 x 10-3 , obtained fo r an overburden thickness of 2 km. The cohesion is assumed to be twice as large as be fo re fo r that thick layer . The substratum thickness is 1 km with a condition of slip with no fr iction on the base ment. To understand the results presented in Figure 9 fo r the three values of k1 of 0, 1, and 2, one should recall the in situ stress parameterization depicted in Figure 1.

The compressive stress at the interface overburden to substratum is uo in direction 1 since cp is set to zero. The stress magnitude in the same direction is reduced a by the amount k1 P gHa at the fr ee surface due to the linear gradient with depth. The value taken by that last expression must be of the order of the compres sive stress u0 fo r the in situ stress gradient to have an influence on the stability predictions. It is for that rea son that no influence was fo und fo r shallow overburden (thickness of order 100 m ) and why a thick layer of 2 km was considered here. It is also for that reason that no influence of the stress gradient was detected in pa per 1 since instability predictions, based on flow theory, required a large compressive stress (20 times Ty ).

This order of magnitude calculation justifies the ge ometry and parameter chosen but does not explain the trend observed in Figure 9. There, it seems that the larger the in situ stress gradient the smaller is the mag nitude of the compressive stress uo fo r instability. In-deed, all curves presented in Figure 9 for k1 = 1 and 2 are shifted to the right fr om the curves obtained for k1 = 0 by an amount of, approximately, 30 and 60 MPa, respectively. That trend is counterintuitive if one esti mates the average stress magnitude across the overbur den depth from our parameterization. Increasing k1 reduces that average and should result in an increase and not a decrease in u0 if the average compressive stress magnitude was the determinant parameter. It appears that the in situ stress gradient results in an overall bending moment which promotes instability and is more important than the reduction in compressive force. Grounds fo r that physical argument are fo und in the asymptotic solution obtained previously for an infinitely deep substratum [Triantafyllidis and Leroy, 1994, equation (79)]. There it is seen that the in situ stress gradient acts on the bending stiffness which en ters the term of third order in the asymptotic develop ment. That action can only be effective if the stress magnitude is comparable to the tangent moduli. Note that this sensitivity cannot be inferred from the clas sical beam solution which disregards stress quantities compared to the elastic stiffness.

The last topic of discussion in this section is the role of the redistribution condition at the top surface. That condition, first proposed by [START_REF] Biot | Theory of gravity instability with variable overburden and compaction[END_REF], re places the action of erosion and deposition by assuming that any section of the top surface that is upheaved sustains a positive traction which is equivalent to the mass displaced and which is supposedly removed by ero sion. On any area that subsides, the traction is com pressive to replace the weight of new sediments, of same material density as the original overburden material, which should be deposited. The influence of that re distribution condition was discussed by [START_REF] Triantafyllidis | Stability of a frictional material layer resting on a viscous half-space[END_REF] fo r the case of an infinitely deep sub stratum. It was found that shallow overburdens were destabilized even fo r tectonic stress in the elastic range of deformation. These shallow overburdens, under a zero stress u0 , must have a dimensionless thickness w H a less than [6(1-v)ApgfGw)pl3, fr om the results fo r an elastic beam with redistribution at its top [Lero y and Triantafyllidis, 1996] . That number is 0.088 for our data set.

Results similar to those discussed above are presented in Figure 10 fo r the case of a 40071" m perturbation wave length and for an infinitely thick (Figure lOa) or fo r a 200 m thick (Figure lOb) substratum. Slip with no fric tion is assumed at the basement. The main fe ature of Figure lOa is that the isocontours do not have the sin gular behavior fo r wHa close to zero that was observed previously. As wHa decreases, the critical stress on a given isocontour does not have a minimum in magnitude but continues to decrease and enters the region of elastic response, marked by an abrupt change in slope of the isocontour. In Figure lOb, fo r the 200 m thick substra tum, this description has to be amended to recognize that for a large enough rate of growth the shape of the isocontours is the one discussed so far for all previous figures. That change in behavior occurs fo r a dimension less rate of growth between 4 and 5 x 10-5 (Figure 10c) . For the simplified elastic beam solution with redistribu tion, this transition is an isocontour intersecting the ori gin of the coordinate system for a rate of growth A/G equal to ApgfGw. That number is 1.41 x 10-4 and is thus 3 times larger then the one estimated by the exact solution. Continuing our analysis of the redis tribution condition influence, we note from Figure lOb that a.ny overburden thinner than 400 m is unstable if the tectonic stress distribution generates only elastic deformation. Furthermore, any dimensionless thickness wHa below 0.087 is unstable fo r zero compressive stress. That critical value is very close to the 0.088 fo und with the plate analysis and provides a non trivial check to our finite element analysis.

Coming back to Figure lOa for a fi nal comment , we note that the asymptotic solution is also used in the ex tension regime to reveal the initiation of a necking-type instability for a fo rce very close to the one required for the initiation of localized faulting. This remark illus trates the potential usefulness of the asymptotic and of the exact solution in the extension regime which is not explored fu rther in this paper.

Application

The stability predictions are now applied to a proto type fo r a folded sequence in a sedimentary basin. The prototype has the geometry shown in Figure la. The objective of this section is to explain the two stages of fo lding which have been reported in the field. In the process, it will be shown that a deformation theory of plasticity is required to explain the first fo lding event in the Albian.

Field Case Description and Motivation

The field structure of interest was revealed by a seis mic dip line and is situated 200 km offshore southeast from Cap Sao To me , southeastern Brazilian coast [Cob bold and Szatmari, 1991, Figure 3, line E]. That 45 km long line, oriented radially in the Campos salient and presented by Demercian et a/., [1993, Figure 5] is sketched in our Figure 11. It has been suggested that gravitational gliding of sediments in the uppermost part of the passive margin results in a downslope con traction which is acting continuously in time. Folding occurred in two stages . The lowermost layers of Cre taceous, 250 m thick, buckled first during the Albian period with a wavelength of 2.5 km. From the end of Cretaceous and through the Te rtiary, an anticline de veloped as the result of buckling on a wavelength of 30 km of the entire sequence which was then 2.5 km thick. The fo ld development was possible because of a passive inflow of salt. Despite the difficulty of seismic inter pretation below the thick salt layer, the base of the salt layer can be assumed to be flat [D em ercian et a/. , 1993] .

Two reasons motivate our selection of this field case. First, it is believed that the geometry of the field struc ture prior to buckling is rather similar to the one adopted fo r our stability analysis. The prototype proposed for the field case is thus composed of layers of infinite lat eral extent with an overburden thickness of 250 m in the Albian or of 2500 m at the end of Cretaceous and during the Tertiary. The use of this prototype for the Campos salient is justified as fo llows. It is assumed that the instability occurred within the plane of observation, corresponding to the plane strain mode of deformation considered herein. This hypothesis is justified by the ra dial orientation of the seismic line and the small length of the section compared to the radius of the salient. The thickness of the salt is thought to be of the order of 600 m if the salt inflow in the present dome has been possible by in-plane salt movement only. The salt to basement interface is flat and the absence of deforma tion in the basement seems to justify its description as a rigid medium. The second reason for our interest in this section of the Campos salient is that folding occurred without any localized faulting. This absence, which we shall try to explain, renders the use of our linear sta bility analysis valid since the governing equations must be in the elliptic regime. The elastoplasticity model for the overburden material is thus necessary, first to assess the possibility fo r, or to rule out, localized faulting and second to quantify the importance of the two driving fo rces for folding, density contrast, and tectonic fo rces. Note that the lateral contraction is modeled by a lateral compressive stress uo which is an outcome of our sta bility analysis and can be compared with an estimate o(_the fo rces acting in the field that will be provided. [1993]. The vertical dimension is in seconds (two-way travel time). The solid lines correspond to the sea floor (SF) and the boundaries between the overburden (0), the substratum (S), and the basement (B). The dashed line shows the limit between the Te rtiary and the Cretaceous stratigraphic units.

The in situ stress parameterization presented in Fig ure 1a and adopted fo r our prototype is assumed to have prevailed in the field. We first concentrate on the substratum and try to justify this hypothesis. The salt movement during the contraction in the Campos salient is assumed to have been slow enough to produce a hy drostatic stress of state in the salt layer. The validity of that hypothesis is checked by comparing the hydro static pressure, which is at least of the order of 5 MPa in the Albian, to the equivalent shear stress, which we now estimate for a Newtonian model. Consider a salt viscosity of 1017 Pa/s [Carter and Hans en, 1983] and a strain rate of order 10-1 3 due to the simple shear of the 600 m thick salt layer by a displacement of its top of 1 km/Myr. The shear stress is then found to be of the order of a few percent of the pressure and can be disregarded in the analysis. We now consider the over burden and try to motivate the adoption of our stress parameterization. Plane strain deformation is assumed along the seismic line described. If the elastic deforma tion is disregarded, compared to the permanent strain, and the assumption of zero dilatancy is recalled, then we can conclude that the out of plane stress must be equal to half the vertical stress plus the lateral radial stress. This result is approximated by assigning to the stress angle cp the value 1r /8. The two stress gradient parameters k1 and k3 are given the value of 1, typical of sedimentary basins. An order of magnitude for the acting compressive stress u0 is nc:>w proposed. This in fo rmation is not an input for our predictions but will be used to justify our findings. Assume that the sedi ments in the radial direction of the salient were gliding toward the structure of interest as a rigid block 100 km long without resistance from the substratum. The slope fo llowed by that block is inferred fr om Figure 3 of Cob bold and Szatmari [1993] to be on average inclined by 1.5°. The fr action of the block weight that generates the compressive stress is thus of the order of 60 MPa.

As a final step in this preliminary part, it is fo und nec essary to provide an explanation for the absence fr om our discussion of any timescale related to the develop ment of a fold. Such information could permit us to fu rther constrain the conditions that prevailed during fo lding in the Campos salient. However, they are not available from our analysis for two reasons. First, nei ther the rate of loading, that governs the contraction, nor the non-Newtonian properties of the salt are ac counted fo r. These two factors are important to quan tify the fold development [START_REF] Sherwin | Wavelengths of single layer folds: A comparison between theory and observa tion[END_REF]Fletch er, 1974] but not fo r our fold initiation predic tions. Second, the linear stability analysis is based on perturbation of infinitesimal size. The observable evo lution of the instability, and in particular the fo ld finite amplitude, can only be estimated fr om the results of nonlinear analysis. This research direction is under in vestigation [START_REF] Massin | On the stability of strain-rate dependent solids and structures[END_REF][START_REF] Massin | Stability of density-stratified two-layer system[END_REF] and the set of complete results will be reported later.

Material Properties

This subsection is devoted to the selection of mate rial parameters. A word of caution is necessary since obtaining precise values fo r the material properties of the sedimentary rocks in the Campos salient is beyond the scope of this paper. The estimates provided are in tended to illustrate the potential of our method in ex plaining the folding events and the absence of localized faulting. With these estimates of the material proper ties, it is already possible to show how the magnitudes of the stress necessary for instability depend on the model chosen (flow or deformation theory, redistribution or not). These magnitudes are sufficiently different for our conclusions to be insensitive to the exact values of the material parameters.

A short description of the sedimentary record in the Campos basin is provided by [START_REF] Demercian | Style and pattern of salt diapirs due to thin-skinned gravitational gliding, Campos and Santos basins, offshore Brazil[END_REF]. The sedimentary rocks properties adopted are those found in the previous section with two exceptions. First, the two material densities are now set to 2100 and 2200 kg/m3. Second, the elastic properties of the sedimen tary rock during the Albian are an elastic modulus of 5 GPa whereas a value 10 seems to be more appropriate for the whole sequence during the second stage of the folding.

The Two Fo lding Events

The stability analysis to be presented is partitioned in two parts. First, it is desirable to understand the initiation of the instability during the Albian. For that purpose, our predictions are proposed for a constant value of tl.pgHa/G set to 1.2 x 10-4 since the overbur den is known to be 250 m thick. The stress ITo necessary for the dominant wavelength to be 2.5 km is then cal culated. Second, why the buckling development ceased is explained. For that purpose, the observer keeps the wavelength selected previously constant and varies the overburden thickness as it occurred during sedimenta tion (.D.. pgfwG is constant) . With a similar analysis, it is shown that the second phase of the buckling pro cess could not be initiated without some redistribution at the top surface. All results to be reported in this section are based on the finite element method.

The analysis starts with the prototype for the field structure and the 250 m thick overburden which initi ated fo lding on a wavelength of 2.5 km in the Albian. The stability predictions are presented in Figure 12 for the deformation theory version of Rudnicki and Rice 's [1975) model and accounting fo r (dotted curves) or dis regarding (solid curves) redistribution. The action of a redistribution superposed to the continuous sedimen tation is possible from the observation in Figure 5a of [START_REF] Demercian | Style and pattern of salt diapirs due to thin-skinned gravitational gliding, Campos and Santos basins, offshore Brazil[END_REF]: the layering on the crest of the small folds appears to be thinner than at the trough. If the dominant perturbation had a wavelength of 2.5 km (w Ha = 0 .63) , then the tectonic stress ITo must have been close to -4.3r11, according to the results in Fig ure 12. Indeed, it is seen that the isocontour fo r A/G equal to 1.9 x 10-3 has an infinite slope at the point of coordinate (-4.3,0.63) corresponding to the dominant wavelength. The redistribution condition could affect the rate at which the perturbation is initially growing (5 x 10-3, dotted curve) but seems to be of less im portance for the critical stress. That conclusion would be completely different if the perturbation wavelength was twice as large since then erosion would change the stability verdict. Note that the critical stress proposed is not sufficient to initiate localized faulting which is marked by diamonds in Figure 12. To summarize the first part of the analysis, the elastoplasticity model pre dicts that the instability was initiated for a compressive stress of 43 MPa while localized faulting required as much as 50 MPa of compressive stress. The estimated value for ITo in the field of 60 MPa is close to these two values. Note that these results were obtained with the deformation theory. Indeed, predictions with the flow theory indicate that a compressive stress of 140 MPa is necessary for the dimensionless wavenumber 0.63 to be dominant.

The question now debated is the reasons fo r the ar rest of the folding initiated during the Albian and for the generation during the late Cretaceous of the fold with a long wavelength of the order of 30 km. A first explanation for the arrest of the fold development can be fo und in the nonlinear stability analysis of M assin et al. [1996), who showed that an initially unstable equilibrium state will evolve toward a new equilibrium state in the form of a fo ld of finite amplitude. The am plitude is a fu nction of the compressive stress ITo . We use another approach here and assume that the dom inant wavelength selected was the only possible mode to develop and that the compressive action of the glid ing sediments was unaltered with time. Furthermore, we assume that the linear stability argument is still ac ceptable despite the finite amplitude of the fo ld . Fig ure 13 plots the conditions for neutral stability for a wavelength of 2.5 km in the absence (solid line) and in presence of redistribution (dotted curve). If the tec tonic compressive stress is kept to the suggested value, then the system is unstable as long as the dimensionless thickness wHa is less than 0.73 or 0.80 depending on the action of redistribution. These results suggests that the system became stable once the thickness reached the critical values of 290 or 320 m, the latter value account ing for redistribution.

The onset of the fold with a 30 km long wavelength is now discussed. Results are presented in Figure 13 by a dashed curve obtained with the redistribution condi tion. The equivalent curve in the absence of erosion

1.0 0.8 � 0.6 � 0.4 0.2 �\ 2 rrlro = 2.5 km • .. •-•-•-•-•-• • ,, -. "\ \ ... • \. • . .. . . \\ ( -) ... .. 2rrlro = 30 km , .. ( -) ' ••.l... --- • •• ••• • "" --••::. •• • • • � . .. .. 0 ( -) Redistri bu tioru -5.0 -4.0 -3.0 -2.0 Figure 13.
For a constant perturbation wavelength of 2.5 km and 30 km, the range of dimensionless over burden height in the Campos salient for instability is presented as a fu nction of the compressive stress. The curves correspond to neutral stability in the absence (solid ) and the presence of redistribution (dotted and dashed ) at the top surface. In the absence of redistri bution and for the long wavelength perturbation, neu tral stability requires localized faulting (not presented here ). Redistribution was thus necessary to trigger such a long wavelength mode of instability.

is not presented in Figure 13 since the critical stress required for neutral stability is 40 times larger than Ty . That finding suggests that it was not possible to initiate folding on such a long wavelength without the action of redistribution. The dashed curve of Figure 13, obtained in the presence of redistribution, reveals that any overburden with a thickness less than 3 km is destabilized for a compressive stress which is amended from the previously suggested value to account fo r the new lithostatic pressure (u0 = -80 MPa). The action of redistribution during the long-wavelength fo ld develop ment is confi rmed by the observation made by Cobbold and Szatmari [1991] that the syncline had been filled by sediments during the Tertiary, as it is illustrated in Figure 11.

Conclusion

Two solutions for the stability of the stratified system composed of an elastoplastic overburden, a viscous sub stratum, and a rigid basement are proposed. The fi rst is analytical and obtained in the absence of gravity effects, and the second, based on the finite element method, ap proximates with the desired accuracy the exact solution which cannot be obtained analytically. The stratified system studied could be seen as a prototype for study ing the initiation of modes of instability such as fo lding and localized faulting in the context of salt tectonics or at the lithospheric scale. Note that such stability anal ysis could also be applied to analogue models as it is done by Lero y and [START_REF] Leroy | Onset of fo lding and fa ulting in density-stratified systems: an elastoplastic model[END_REF], who discussed the difficulty in reproducing the onset of both localized faulting and of fo lding with granular materials.

Results are presented in the form of stability expo nent isocontours A in a plane spanned by the com pressive stress u0 and the dimensionless wavenumber or overburden thickness wHa . The general structure of the solution is reminiscent of the classical solution for an elastic plate on an infinitely deep substratum [Sm oluchowski, 1909] with the difference that the in situ stress gradient, the nonlinear overburden stiffness, and a possible redistribution condition at the top surface are also accounted for. The asymptotic solution, derived in paper 1, is fo und to be accurate in all cases consid ered for values of the small parameter wHa as large as 0.4. The analytical solution, obtained disregarding gravity, is also accurate for large value of the dimen sionless wavenumber wHa . This accuracy is improved for large values of the dimensionless stability exponent A/G compared to the dimensionless number !J.. pgfGw , in which !J.. p is the density contrast and G the modulus of elasticity in shear.

The validation of the asymptotic solution permits us to confi rm some of the results presented in paper 1. The substratum thickness influences by as much as a factor of two the critical stress for instability and by up to 30% the wavelength of the dominant mode. If the substratum thickness is less than 4 times the perturba tion wavelength (wHb � 4), then the type of boundary condition at the basement infl uences the stability pre dictions. The dominant wavelength is not affected, but the magnitude of in situ stress necessary to trigger an instability at a positive rate can be increased by up to a factor of 2 if the condition of slip with no fr iction is replaced by a perfect bond. Another conclusion of that comparison is that fo lding modes of instability are always found fo r the smallest magnitude of the com pressive stress for all practical cases considered herein. However, surface modes of instability [ Triantafyllis and [START_REF] Triantafyllidis | Interfacial instability of density-stratified two-layer systems under initial stress[END_REF] ] cannot be ruled out fo r very thick over burden.

The comparison between predictions for fl ow and de formation theories shows the sensitivity of the stabil ity conditions to the details of the constitutive model adopted for the overburden. The presence of small, per vasive faults in the overburden which accommodate part of the deformation by slip, modeled here by the defor mation theory, leads to a reduction in the magnitude of the compressive stress required for fo lding instabil ity. The same trend is observed for the initiation of localized faulting, as already known since the work of Rudnicki and Rice [1975 ]. With that deformation the-ory, it is found that the in situ stress gradient has a destabilizing effect: increasing the stress gradient leads to a reduction of the average lateral compressive stress magnitude necessary fo r instability. This influence is ex plained from the asymptotic solution which reveals that the stabilizing role of the bending stiffness is reduced by the presence of a stress gradient. This result could not be inferred from previously published asymptotic so lutions which systematically disregard stress compared to tangent moduli. It is also shown with that deforma tion theory of plasticity that the redistribution condi tion at the top surface drastically changes the structure of the stability conditions. In that instance, it is pos sible to destabilize shallow overburden even with small compressive stress.

The stability predictions are subsequently applied to a folded structure revealed by a seismic survey in the Campos salient, 200 km southest fr om the Cap Sa.O Tome on the Brazilian coast [Cobbold and Szatmari, 1991]. That structure is selected because our model could be considered as a prototype to study the onset of folding, which occurred in the absence of localized faulting and is due to the compressive lateral stress and to the density contrast. It is shown that the compres sive stress predicted to initiate buckling in the Albian is close to the one estimated fr om the field description, if a deformation theory of plasticity is adopted. Fur thermore, as sedimentation continues, the stiffness of the overburden increases and the instability is shown to cease if one accepts our stability predictions which disregard the presence of finite amplitude perturbation. The long-wavelength fo ld triggered during the late Cre taceous is fo und to result fr om the combined action of the tectonic compressive force, the density contrast, and an uneven sedimentation which is modeled as a redis tribution condition at the top surface [START_REF] Biot | Theory of gravity instability with variable overburden and compaction[END_REF] . That uneven sedimentation is confirmed by the observation that the syncline was filled by Te rtiary sed iments [D emercian et al. , 1993].

The sensitivity of the stability predictions to the de tails of the elastoplastic model shows the need to de velop, in the future, plasticity models which are ap propriate for studies at the tectonic scale. These mod els should be constructed fr om a mechanical descrip tion of the dominant microscale deformation mechanism that leads to bulk deformation. For example, the mi cromechanism could be the opening or the sliding along a population of diffuse fractures, initially randomly dis tributed as discussed by Leh ner and Kachanov [1995] . Such constitutive relations will improve our modeling capabilities, compared to classical phenomenological laws inherited from the geotechnical literature, and will also provide the means to relate the anisotropic development of pervasive fr actures to bulk deformation. The impor tance of that link fo r predicting fr actured reservoir per meability is discussed by [START_REF] Leroy | A plasticity model for discon tinua[END_REF].

Figure 1 .

 1 Figure 1. Geometry of the problem, in situ stress distribution, and illustration of some of the instability modes studied. (a) The overburden, composed of a fr ictional, cohesive material, lies on a layer of viscous fluid. The two strata are above a rigid basement. The overburden sustains its own weight and a tectonic stress distribution characterized by the stress parameter u0, the gradient coefficients k1 and k 3 , and the orientation angle <p. The state of stress in the substratum is hydrostatic. (b) Two structural modes of instability are illustrated together with the mode of localized faulting (LF).

Figure 3 .

 3 Figure 3. Comparison between stability predictions in a space spanned by the dimensionless thickness wHa and normalized stress u0 /u y . Predictions obtained with the analytical solution in the absence of gravity (dashed curves), the fi nite element solution (solid curves), and the asymptotic solution for small values of w H a (dotted curves), the latter two accounting fo r gravity effects. The iocontours are drawn (a) for a constant value of the scaled stability exponent At and (b) fo r a constant value of the stability exponent A. The two exponents are related by A= AtwHa. Results are obtained for the flow theory version of Rudnicki and Rice's [1975] model for an infi nitely deep substratum and fo r tl.pgfGw 7.06 X 10-4•

Figure 4 .

 4 Figure 4. Influence of the substratum thickness on the stability of the stratified system according to the fi nite element (solid curves), asymptotic (dotted curves), and analytical (dashed curves) solutions for tl.pgfGw = 7.06 x 10-4• The substratum thickness is set (a) to 1 km and (b) to 5 km. A thicker substratum implies a fas ter rate of growth of the instability fo r the same in situ stress. Results are obtained for flow theory and a perfect bond between substratum and basement.

Figure 5 .

 5 Figure 5. Influence of the substratum thickness for tl.pgHa/G = 7.06 x 10-4 obtained for an overburden thickness of 1 km. The isocontours of constant stability exponent A/G are drawn for a ratio of substratum to overburden thicknesses of (a) 1 km and (b) 5 km. Fi nite element, analytical, and asymptotic results are the solid, dashed, and dotted curves, respectively. Results are obtained for flow theory and a perfect bond between substratum and basement.

FlowFigure 6 .

 6 Figure 6. Influence of the boundary condition between substratum and basement. (a) The substratum thick ness is 1 km and (b) the overburden thickness takes that same value. These results are obtained for a condition of fr ictionless slip and should be compared with those obtained for a perfe ct bond presented in Figures 4a and 5a. Finite element, analytical, and asymptotic results are the solid, dashed, and dotted curves, respectively. Results are obtained for flow theory.

Figure 7 .Figure 8 .

 78 Figure 7. Comparison between the flow and defor mation theory versions of Rudnickz and Rice's [1975] model for !:::. . pgfGw = 1.4 1 x 10-4. (a) Results obtained with the finite element method (solid curves) and the asymptotic analysis (dotted curves) fo r the same three values of the scaled stability exponent A!/G and for an infinitely deep substratum. (b) The stability exponent AJG is kept constant and the substratum is 200 m thick. Diamonds indicate the onset of localized faulting.

Figure 9 .

 9 Figure 9. Influence of the stress gradient parameter k1. Curves of similar type (solid, dotted, or dashed ) correspond to the same stability exponent. The finite element results are obtained fo r deformation theory and fo r !:l.pgHa/G = 1.41 x 10-3 corrresponding to an overburden thickness of 2 km. The substratum thickness is 1 km, and the boundary condition at the basement is one of slip with no friction. The diamonds, which indicate loss of ellipticity, are not shown for k1 = 2.

Figure 10 .Figure 11 .

 1011 Figure 10. Influence of the redistribution condition at the top surface fo r a value of Cl.pg fGw =

Figure 12 .

 12 Figure 12. Instability predictions for a prototype of the Campos salient in the Albian with an overburden thickness of 250 m. Results obtained accounting (dot ted curves) and disregarding (solid curves) redistribu tion at the top surface and with the deformation theory version of Rudnicki and Rice 's [1975) model. For the wavelength of 2.5 km to be dominant, the compression stress ITo had to be close to -45 MPa. That value is not sufficient to trigger localized faulting identified by diamonds.

Acknowledgments. This work was initiated while one of us (Y.M.L.) was working for Shell Research KSEPL (now RT S), Rijswijk, Holland. Helpful discussions with J.L. Urai (University of Aachen, Germany) and D. Nieuwland (RTS) are gratefully acknowledged.

The field case con sidered was kindly suggested by P. Cobbold (University of Rennes, France). The continuous encouragements and the contribution of F.K. Lehner (University of Bonn, Germany) was essential to the realization of this work. This work was completed during the visit in 1996 of Y.M.L. at the Uni versity of Michigan. The hospitality of the Department of Aerospace Engineering is gratefull y acknowledged. Permis sion to publish this paper was granted by Shell lnterna tionale Re search Maatschappij BV.

Appendix: Two Solutions to the Stability Problem

This appendix contains the derivation of two solu tions to the stability problem: the first is analytical, obtained in the absence of gravity effects and the sec ond is numerical, based on the finite element method.

Analytical Solution Disregarding Gravity

The motivation to disregard gravity comes fr om the observation that the zero-order term on the right-hand side of the plate solution (7), .tl.pgfwG, which has typ ical values of order IQ-4 to IQ-5 , becomes negligible compared to sufficiently large rate of growth of the per turbation A/G. It is then the tectonic stress and not the density contrast which controls the initiation of the instability. The absence of gravity renders the stress distribution presented in Figure 1 The four scalars .Ba introduced in (A 8) take the values -Cu (Za)/C12(Za)•

The consecutive resolution of (A 7) ( det[ M tJ a( A)] = 0) and (A 1) permits to construct the fo ur eigenvectors and to determine the stability exponent A, which are at the basis of the general solution to the stability problem in the absence of gravity.

The Finite Element Solution

The finite element solution provides the means to as sess the accuracy of the asymptotic solution, to de termine the range of validity of the analytical solu tion in absence of gravity, and to check if modes of instability other than bending modes could be relevant. Such modes include interfacial instabilities [Than tafyl lidis an d [START_REF] Triantafyllidis | Interfacial instability of density-stratified two-layer systems under initial stress[END_REF] which require a fine spatial res olution close to the interface with the substratum.

The overburden thickness is partitioned in N fi nite elements, with the last node being at the interface with the substratum to simplify the presentation of the ar gument that fo llows. These elements are chosen to be three noded with the same interpolation fo r the eigen vector U and the variations 8U The problem presented in (A 10) is a generalized al gebraic eigenvalue analysis for a non-symmetric system. Its solution computation is facilitated by expressing the (N -1) first unknown eigenvectors Um in terms of the last ones positioned on the interface, iJ N. This opera tion is done with the use of the first equation in (A 10) and the simplified eigenvector problem then reads fr om which the critical A is fo und as the eigenvalue with the largest real part of the 2 x 2 system.


Title : Interactions in Collective Fish Swimming

Abstract : Fish school is the classical example of self-emergent system of collective behavior in animal locomotion, which involves complex interactions between individuals and has drawn fascination of numerous multidisciplinary analytical, numerical and experimental researches.

Various reasons have been cited for the formation of fish school such as evading predators, enhancing foraging success or advantage in terms of energy consumption. However, most of the works primary focus on what fish should do in a school instead of what fish really do, leading to discrepancies between the works of theoretical physicists and engineers and what is observed in the nature by biologists. This thesis is an attempt to bridge this gap.

In this work, we investigate the swimming dynamics of the red nose tetra fish Hemigrammus bleheri in a controlled experiment. The tetra fish are observed to swim using a burst-and-coast strategy, which could be for the purpose of active flow sensing.We also address the case of red nose tetra fish Hemigrammus bleheri swimming in groups in a uniform flow, giving special attention to the basic interactions and cooperative swimming of a single pair of fish. We first bring evidence of synchronization of the two fish, where the swimming modes are dominated by "out of phase" and "in phase" configurations. At last, we challenge the

  Université Sorbonne Paris Cité Préparée à l'Université Paris Diderot Ecole doctorale [n • 564 : Physique en Île-de-France] Laboratoire de physique et mécanique des milieux hétérogènes (PMMH) UMR 7636 Interactions in Collective Fish Swimming Auteur : ASHRAF Intesaaf Spécialité doctorale "Mécanique des Fluides" Directeurs de thèse : Ramiro Godoy-Diana & Benjamin Thiria Présentée et soutenue publiquement par ASHRAF Intesaaf le 4 Avril 2018, PMMH Jury : Président du jury: José Halloy (Professseur, Université Paris Diderot) Rapporteur: Charlotte K. Hemelrijk (Professseur, University of Groningen) Rapporteur: Christian Jost (Maitre de Conférences -Université Toulouse III) Examinateur: Francisco Huera-Huarte (Professseur, Universitat Politécnica de Catalunya) Directeur de thèse: Ramiro Godoy-Diana (Chargé de recherches, CNRS, ESPCI Paris) Directeur de thèse (invité): Benjamin Thiria (Maitre de Conférences, Université Paris Diderot, ESPCI Paris) Except where otherwise noted, this work is licensed under http://creativecommmons.org/licenses/by-nc-nd/3.0/ i Titre : Interactions des poissons en nage collective. Résumé : Le banc de poisson est un exemple typique d'auto organisation de groupe. Ce système implique des interactions complexes d'individus dans un milieu fluide et fait l'objets de nombreuses recherches multidisciplinaire théoriques, numériques ou expérimentales. Plusieurs hypothèses ont déjà été formulées pour expliquer la formation de ces bancs: la fuite face à des prédateurs, l'optimisation dans la recherche de nutriments ou encore l'économie d'énergie. Dans la plupart des recherches, les travaux se concentrent sur ce que les poissons devraient faire pour optimiser une tâche plutôt que sur des observations directes. Cela conduit à des différences dans les conclusions issus du travail des physiciens théoriciens ou ingénieurs avec ce qui est observé dans la nature par les biologistes. Dans ce travail, nous apportons, basées sur des observations directes tetra Hemigrammus bleheri, de nouveaux éléments sur les interactions des poissons lors de la nage collective comme le phénomène de burst-and-coast ou la synchronisation des cinématiques entre plus proches voisins. Enfin, nous démontrons que les bancs de Hemigrammus bleheri privilégient la configuration en ligne plutôt que celle en diamant souvent mise en avant dans la littérature. Mots clefs : Economie d'énergie, Nage collective, Poisson.

question of energy benefit by discussing the channelling effect versus the vortex interaction hypothesis (Weihs, Nature 241:290-291, 1973) about diamond shape. We provide the experimental observation that fish prefer inline formation or phalanx formation over the diamond shape while swimming in a school.

Keywords :

Collective fish, Fish, energy saving.