Marc Bonnet

Huy Duong Bui

H D Bui

Regular BIE for three-dimensional cracks in elastodynamics

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

Introduction

The difficulty in solving 3D crack problem using B.I.E methods have been re cognized in the past [START_REF] Bui | An integral equation method for solving the problem of plane crac k of arbitrary shape[END_REF], [START_REF] Weaver | Three-dimensional crac k analysis[END_REF], [START_REF] Putot | Une nouvelle methode d'equations integrales pour certains pro hlcmes <le fissures planes[END_REF]. It might be explained by the conjunction of two serious mathematical difficulties : the singularities of the kernels which result in Cauchy principal value integrals, and the high variation of the crack displacement near the crack front.

To overcome these difficulties, R>lch et al. [START_REF] Polch | Buried crac k analysis with an advanced traction B.I.E algoritm[END_REF] proposed the improvement of the interpolation procedure, using for instance quadratic 8-nodes elements.

!lowever, considering mathematical formulations of B. I. E which use systemati cally non singular kernels and which we call the regularization technique appears to be a better approach. A review of methods of regularizations of integrals can be found in [S]. In recent works [START_REF] Polch | .E solutions for flat crac k s[END_REF], [START_REF] Rizzo | A boundary integral equation me thod for radiation and scattering of elastic waves in.three dimensions[END_REF], [START_REF] Bonnet | Methode des equations integrales regularisees en elastodyna mique[END_REF] accurate results have been obtained by the regularization technique which was combined with an advanced interpolation procedure.

The regularization method was applied to B.I.E for 3D plane cracks [START_REF] Bui | New trends in B.I.E. methods[END_REF], [START_REF] Polch | .E solutions for flat crac k s[END_REF].

In this paper, we reconsider the 3D curved crack problem which was formula ted in tenns• of sin& > Ular B. I. E by Levan and Royer [START_REF] Levan | Integral equations for three-dimensional problems[END_REF] in the static case, and by V. Sladek and J. Sladek [START_REF] Sladek | Dynamic stress intensity factors studied by boun dary integro-differential equation[END_REF] in the dynamic case. We shall present the regular integral equations, for smooth cracks as well as for kinked and crossed cracks. We will not discuss the nt.unerical aspects of the new equa tions, which are beyond the scope of the paper.

Notations

We consider the problem of a 3D curved crack embedded in an infinite elastic body. The crack surface S is assumed to be sectionally smooth and coincides with the lower face S of the crack, Fig. 1, whose unit normal is n(x). We i. e the crack displacement discontinuity, and f=n i (x). Therefore the deriva tives f . are meaningless, unless e. are the basis vectors of the plane of a ,] J crack. For curved cracks, we have to extend, in some way, the domain of definition of f, for example by introducing the function t (x)

t (x) = f(P(x)) x ED , P(x) ES ( 3
)
where D is :rneigbourhood of S, P(x) is the orthogonal projection onto S.

Clearly, the restriction of t to the surface S is equal to f. Moreover the normal derivative of t is equal to zero, i. e the vector grad t is tangent to S; therefore it may be used to define the tangential gradient grad s f of the function f(x). For the consistency of notations, we put grad s = grad -no/an.

f n grad t dS = J f (y)r (y) ds s " as Y ( 4
)
where r is the unit tangent vector to as. In the following, unless stated otherwise, we shall omit the symbol ( " ) keeping in mind when necessary the extension (3). The j-component of v Y f then reads

(V Y f). =e. v Y f(y) J JM pq v Y = n i._ -n i._ pq pay q qay p (5)
In some applications, f=<I>., the i-component of the crack discontinuity being 1 equal to zero along the crack border, the line integral in Eq. ( 4) vanishes identically. For the kinked crack , Eq. ( 4) applies to each domain s 1 or s 2 .

The line integral term also vanishes because the contributions to the line integrals, along the common boundary r of s 1 and s 2 , are of opposite sig n, due to the continuity of <I>. across r. For the crossed cracks, the line inte-

1
gral does not vanish, as ob served by Levan and Royer [START_REF] Levan | Integral equations for three-dimensional problems[END_REF] in their quasi statjc analysis of curved cracks .

Sin gu lar B. I. E for curved cracks

Our aim here is not to repeat in full details the derivation of the singular B.I.E, but rather to outline some key steps of the method of derivation used in [START_REF] Levan | Integral equations for three-dimensional problems[END_REF], [START_REF] Sladek | Dynamic stress intensity factors studied by boun dary integro-differential equation[END_REF], which we generalize for a complex geometry of cracks in elasto dynamics. Classically, the perturbation field u(x) caused by the crack which

The boundary condition on the crac k is t(x) = -F(ax,n(x))u inc (x). Fonnally, the traction vector t1(x) = -c1 k n (x) ( <1>. (y)n (y) -3 3 r."! k (x,y)dS p r p l s 1 s y r 1s ; y

(7)
is expressed hy a hypersingular integral, with a singular k ernel of order II x-yl l -3. To avoid a divergent integral, we consider as the first step, that x belongs to D, but not to S, and we replace n(x) by A(x). In the second step we integrate by parts, using Eq.( 4) and setting f=4i i r. � s; k " In the last step, we investigate the limit of the integral as x .... P(x). These steps lead to the Cauchy principal value integrals, which are indicated by the symbol ( * ) * = c1 k n (x) [ -J vY <1>.(y)r."! k (x,y) dS + p r p S rs 1 1s ; y

+ µ � J <I>. (y)n (y)E"!, k (x,y) dS ] , x € S S 1 r 1, y (8)
where µ �=pw 2 • Eq.( 8) have been derived by -µ � J <I>. (y)n (y)E".°'. k (x,y) dS

S 1 r 1, y * + yX <I>. (x) J r.� • k (x,y) dS ]+ line integral rs 1 S 1 s , y (10)
where the static k ernel � o is used for removing singularities. By virtue of the presence of the term �C.:�0, the first integrand is not singular, while the second one, if we assume a Holder condition on the tangential derivati ves Y� s � i (y), has a wea k integrable singularity. The third integral also con verges in the ordinary sense. The fourth integral over each subdomain S a is a Cauchy principal value integral only when x belongs to the same S • In a this case, using k nown values of the components of � o and ta k ing care of the singular nature of the integral when we perform the integration by parts, we arrive at an expression using regular integrals I. and J. k [8] the integrands, as well as the term r ar/an are of order 0(1/r). Therefore, any classical interpolation procedure, using for instance polar coordinates with the pole x, is suitable for the implementation of Eq.( 10) in a B.I.E code. Finally, it should be noticed that the line integral term of Eq.( 10) is regular for interior points x ES . With the normal vectors as indicated

1 ) 2 )

 12 shall use the same notation for the spatial position of a point and its vec fixed Cartesian basis vectors). The trac tion vector on S, associated with any continuously differentiable displace ment field u(x) in R3, is given by t(x)=F u(x), where F denotes the diffe rent i al operator F(ax, n(x)) Zµn. t grad + Xn div + µnACurl (In the above, (t) is the transposition of tensor, X and /1 are the Lame cons tants. The following notations are also used : L w = µA + (X + µ)grad div + pw 2 Id Elastic moduli tensor : Elastodynamic operator We shall consider the Stokes displacement tensor E w (x, y) which is the funda mental solution of the operator L( y) in which differentiations are made with respect to y. The case w=Q corresponds to Kelvin's tensor E 0 (x, y). We recall that E�• k (x, y) is the i-component of the displacement field at y, due to the harmoni� point force along e k -direction, acting on x. The tensor E w possesses the well-known symmetry properties E w (x, y)= t E w (x, y)=E w (y, x) and a weak singularity like llx-yll -l when X-+Y• A stronger singularity is found in the fundamental traction tensor T w (x, y, n(y)) = F(ay, n(y))E w (x, y), which behaves like the tensor T 0 as llx-yll -2 , or more precisely 11T w -T 0 1!=0(w2 ) as x-+y. Introducing the stress tensor 'f. w , one can write T w as follows (Tangential o p erators To reduce the singularities of kernels, we are led to perform an integration by parts of integrals over curved surfaces. In order to do so, several ma thematical identities using tangential operators are required. Consider first a field f(x) function of points x on S. The function f is not defined + outside S, as illustrated by the following examples f=<I>. (x)=u. (x )-u. (x ) ,

•h

  • • d inc ( ) E113 b d • t f th receives t e inc1 ent wave u x , x , , can e expresse in erms o e crack discontinuity as followsu k (x) = J <I>. (y) �. k (x,y,n(y)) dS

2 ,+

 2 J. Sl ade k and V. Slade k (11] for smooth surface. The above equations hold for the k in k ed crac k , as explained in the preceeding section. When there are more than two crac k branches s1, s s3 •• , a line integral term appears in the right hand side of the above equation and is given by (x € S , x � r) {line integral} = I !c l k n (x) r <I>. (y)r."! • k (x,y) T. (y)t. dS (9) a t. p r p la s 1 lS, J JrS y a Regular B.I.E for curved crac k s The technique employed in many wor k s is based on the singularity exclusion method. Applying this method to Eq.(8), we obtain t1(x) =-c1 k n (x)[J v Y <1>.(y)[r."! • k (x,y) -r.� s • k (x,y)] dS Y p J cvY <1>. (y) -r <1>. (x) ]r.� k (x,y) dS Y S rs 1 rs 1 1s ;

+

  y)..!.] dSr y J a s a ..!_ v. (y) dS r 1 y J is k (x) = Ii i k1 s (x) + Ii s k1 i (x) + + J {3r .r -2n.(y)n (y) } n k (y) _..!__ 2 � ar dS Y S • , 1 ,s 1 s r u n l y ; a + J {2n. (y)n (y) -r .r } n k (y)K(y) d � S 1 s , 1 ,s r a -J (o Y (n.n k ) + D�(n n k ) } dS Y + J fn. (y)n k (y)li (y) + n (y)nk(y)v. (y) -v k (y)r . r } ds y the above, differentiations of r =II x-yll are made with respect to y, K(y) stands for the sum of principal curvatures K=1 /R 1 + 1 /R 2 , and v = n A r • In Eq.(13) we put D�=e .• grad S . The integrals I. and J. k are regular because

a in Fig. 1 , 2 + 6

 126 there exists a relation between the crac k discontinuities at.r, (14) which constitutes, with Eq.(10), the regular B.I.E of 30 crac k problem. Fundamental t ensors a nd kernels i :?.k(x, y)=[(l+J3 2 )6.k+ (1-J3 2 )r .r k l /811µr kr .+li.kr .-li .. r k)+3(1-J3 2 )r .r .r k ]/411µr 2 k(x, y) = .--[ (1-213 )6 .. (G(k L r)) k + 6 . k (G(Lr)) . + . k (G(Lr)) . + -2 (G(Lr) -G( k L r)) . ' k ] k r)=exp(i k r) I r , k ={ k L 'k. f }, !<. i •= w p /µ, k L =w p I (>.. +2µ), J3 =µ/ (>.. +2µ).

  n

Fig. 1 .

 1 Fig.1. Smooth crac k , k inked crac k and crossed crac k .


François Auslender

Michel Bornert

André Zaoui
email: zaoui@lms.polytechnique.fr

T Hoc

Renaud Masson

An "Affine" Micromechanical Approach for the Prediction of the Elastoplastic Behavior of Polycrystals at Finite Strain

Keywords: Polycrystals, elastoplasticity, finite strain, homogenization. IUTAM

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

Introduction

Until a very recent past, Hill's incremental method was unanimously consid ered the most appropriate micromechanical treatment to predict the elastoplastic behavior of polycrystals. Recently, Gilormini [START_REF] Gilormini | A critical evaluation of various nonlinear extensions of the self consistent model[END_REF] proved that this approach, ap plied to viscoplastic two-phase materials, can lead to an upper bound violation. This shows that the incremental treatment must be somewhat "softened" in order to yield more realistic predictions. This conclusion has motivated the concep tion of a new approach, the so-called "affine formulation", proposed by Masson et al. [START_REF] Masson | An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals[END_REF]. Initially defined, in the context of infinitesimal strain, for nonlinear elasticity or viscosity, it has been extended to history-dependent constitutive behavior such as rate-independent or rate-dependent elastoplasticity.

This paper aims at extending the affine elastoplastic approach to finite strain. In the first section, we briefly recall the small strain affine formulation for non linear elasticity and elastoplasticity. The second section is devoted to the finite strain extension: we first focus on the choice of the appropriate (lagrangian, eulerian ... ) description framework; we then address the question of the single crystal constitutive equations and the specific problems of the affine approach when finite deformations are considered and when the self-consistent scheme is adopted. Preliminary predictions of the overall response of polycrystals, com pared with those derived from Hill's approach, are reported in the last section.

Here, a polycrystal is considered as an aggregate of perfectly bound and ran domly distributed crystals which only differ by their lattice orientation: all the grains which have the same crystallographic orientation define a "phase" of this multiphase material. Plastic deformation is only due to glide on slip systems, the system a being defined by the unit normal to its slip planen a and its unit slip direction ma.

Affine formulation for small strains

NONLINEAR ELASTICITY OR VISCOPLASTICITY

In this section, the crystal constitutive behavior is derived from a convex stress energy function u( a) (a similar approach could be developed from a strain energy function). The strain response at any point x in the considered repre sentative volume element V is independent of the loading path:

OU e (x) = O
O" (x, a(x) ).

(1)

According to the affine approach and using the local tangent complianceM ( x) , the nonlinear constitutive equations ( 1) are linearized with respect to the current stress value at each point x . The linearization takes place once and for all without any incremental process:

s (x) = M (x) : a(x) + 17 ( x) , 8 2 u M (x) = o a 2 (a(x), x), 8 2 u 17(x) = o a 2 (a(x) , x) -M (x) : a(x).
(2)

These linearized equations may be considered as associated to a linear thermo elastic composite constituted of an infinite number of phases (the notationC : D means: Ci jk lDkz, where summation is done over repeated indices). To deal with such a composite, one has, as usual, to adopt the simplification of piecewise constant compliances and prestrains per phase. Accordingly, for each phase, we assign their value at some reference stress us to the tangent compliances and prestrains. In this paper, the reference stresses are the mean value of the stresses over the concerned phase. By this way we obtain a thermo-elastic linear composite with a finite number of phases. This composite represents the linear comparison medium (LCM) as developed by Ponte Casta"ieda [START_REF] Castaneda | The effective mechanical properties of nonlinear isotropic composites[END_REF]. The homogenization problem related to this composite can be solved by classical linear homogenization techniques for thermoelastic media, such as the self consistent scheme when one deals with polycrystals.

Various discussions and results (Masson et al. [START_REF] Masson | An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals[END_REF]) have proved the viscoplas tic affine formulation to offer an improved alternative to previous treatments such as the Hill-Hutchinson secant treatment (Hutchinson [5]) or the "tangent" model (Molinari et al. [START_REF] Molinari | A self-consistent approach of the large deformation polycrystal viscoplasticity[END_REF], Lebensohn and Torre [START_REF] Lebensohn | A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys[END_REF]). It differs from the second-order procedure (SOP) (Ponte Castaneda [START_REF] Castaneda | Exact second-order estimates for the effective mechanical properties of nonlinear composite materials[END_REF]) by making use of the direct stress-strain relations ( 1) instead of deriving the stresses from the overall potential: consequently, on the one hand, the SOP takes better into account the local intraphase heterogeneity and then gives better predictions for nonlinear elasticity or viscosity; on the other hand, the affine method, unlike the SOP, does not lead to any duality gap and can be extended to any constitutive behav ior which, such as rate-dependent or rate-independent elastoplasticity, does not derive from a single potential.

RATE-INDEPENDENT ELASTOPLASTICITY

In this section, rate-independent elastoplastic polycrystals are considered in the framework of small strains. The local strain rate is decomposed into its elastic and plastic parts:

e = ee + eP = K -1 : u + L .:y a Ra '
(3) a with K the elastic moduli and .:y a the non-negative plastic slip rate on system o: with the orientation tensor Ra = (na ® ma) 8 (A 8 denotes the symmetric part of the second-order tensor A). Associate plasticity theory is used with the Schmid law, with the critical resolved shear stressT� on a slip system o:. Such a system is a potentially active system (P.A.S.) if Ra : u = T�. It is active (A.S.) if, in addition, Ra : u = f�. We adopt the following hardening law: [START_REF] Hill | On constitutive macro-variables for heterogeneous solids at finite strain[END_REF] where the hardening matrixHa,a is assumed to be constant (we could also deal with a non constant matrix through an appropriate linearization procedure, but the associated technical developments are not essential here). Thus, the local elastoplastic constitutive equations read: iF= L:e, L = K : (I -I:: D°' f3 Ra ® R f3 : K) , a, (3 (5) whereD°' f3 = (Ha(3+Ra: K: R(3)-1 and the summation on(a, ,8)is limited to active systems.

The overall stress response at timet now depends on the whole previous loading path. Thus, a step by step procedure is needed (Masson et al. [START_REF] Masson | An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals[END_REF]). The steps can be defined as separating two consecutive "events": an event occurs at time tn when some new system becomes active or inactive in any phase of the polycrystal. The step by step procedure (which does not lead to an incremental approach!) can be described as follows: from the stress values obtained at the end of the former step, we determine, at the beginning of the current step (t = tn), which systems are potentially active. Among all these P.A.S., we select one branch, i.e., a given combination of A.S .. According to [START_REF] Hutchinson | Bounds and self-consistent estimates for creep of polycrystalline materials[END_REF], this choice helps to compute the tangent moduli in each phase, up to the next event. Meanwhile, according to (5), these moduli, say Ls, are constant. The local constitutive equations are then linearized in the usual affine manner: [START_REF] Iwakuma | Finite elastic-plastic deformation of polycrys talline metals[END_REF] where es and us are the strain and stress averages in phases obtained at the end of the previous step. From the known moduli and prestresses of the linear cm � parison composite, we derive t�e macroscopic elastoplastic tangent modulusL and the macroscopic pres tress (}, which characterize the macroscopic behavior during the current step. We can also calculate the concentration tensors and then the local variables during the current step and especially at the end of the step which is defined by the occurrence of the next nearest event. From these values, the consistency of the chosen branch can be checked, i.e., the selected A.S. must be still active and the initially non active P.A.S. still inactive at the end of the step. If this consistency is verified, we go on to the next step. If not, we go back to the beginning of the step and we select another branch until consistency conditions are satisfied. Resuming the same procedure at each new increment up tot provides the expected macroscopic response of the polycrystal at time t.

3.

The affine formulation at finite strain

CHOICE OF THE GENERAL DESCRIPTION FRAMEWORK

In view of applications to metal forming, which involve very large deformations, an extension of the affine formulation at finite strain is needed. We have first to specify the general description framework. For several reasons listed below, we have adopted a mixed description instead of the classical eulerian or lagrangian ones (Iwakuma and Nemat-Nasser [START_REF] Iwakuma | Finite elastic-plastic deformation of polycrys talline metals[END_REF], Lipinski et al. [START_REF] Lipinski | Elastoplasticite des metaux en grandes deformations[END_REF], Harren [START_REF] Harren | The finite deformation of rate-dependent polycrystals -I. A self consistent framework[END_REF]). This description makes special use of the deformation gradient tensor defined both on the initial and the current configurations. The small strains co i:i stitutive equation a = L : e of the single crystal now reads n = N : F where n is Hill's nominal stress tensor, F = 8 x/ 8X is the deformation gradient tensor and x and X are the position vectors on the current and the reference configuration respectively. The main advantage of this formulation relies on the fact that the macroscopic values of the nominal stress tensor and of the deformation gradient are the average over the initial configuration, which is known unambiguously, of their microscopic counterparts. Moreover, unlike for the Cauchy or the Kirchhoff stress tensors, this property also holds for their time derivatives (Hill [ 4]):

n = ( n )oo' F = ( F )oo' n = (n)oo' F = (F)oo. (7)
In the affine approach, we have rather to keep the reference configuration con stant all along the loading path because we do not know the current configuration at the beginning of a new step. The main implication of this choice concerns the implicit description of the microstructure evolution. Though the derivations are performed on a fixed geometry (e.g., the constraint tensor, considered in the following, is calculated on a fixed ellipsoid), the evolution of the phase dis tribution is taken into account through that of the local tangent moduli which depend on the per phase average of the total and the elastic deformation (see below).

SINGLE CRYSTAL ELASTOPLASTIC CONSTITUTIVE EQUATIONS

From the elastic-plastic decomposition F = Fe .FP, the plastic kinematic

Ya= F-P.(K: Aa + 2Ba).F eT , S e = F e-1 .F.n.F e -T .

Note that N in (8) exhibits diagonal symmetry (i.e., Ni jlk = Nk lji ).

The Schmid law reads at finite strain: [START_REF] Molinari | A self-consistent approach of the large deformation polycrystal viscoplasticity[END_REF] where r = det(F)u is the Kirchhoff stress. The resolved shear stress used in the Schmid law is the projection of the surface traction r. n � not on the initial (ma) but on the current slip direction m� . This choice is physically questionable and has significant consequences on the symmetry of the tangent modulus for the single crystal. We still use the hardening law ( 4) for the critical resolved shear stress. The slip rate)'°', which is a linear function of F, can be derived from the consistency condition and reintroduced into [START_REF] Lipinski | Elastoplasticite des metaux en grandes deformations[END_REF], so as to derive the expected constitutive equation for the single crystal:

n=N: F , N = N + L(Ha,6 + 2Ba: A ,a + A ,a : K: Aa)-1 Ya Q9 Y ,6. (11) a , ,6

Note that, even if Ha ,a is symmetric, the moduli N do not exhibit diagonal symmetry anymore. Since this fact can induce some difficulties for finite strain homogenization schemes which rely on the use of Green techniques, we decide arbitrarily to replace Ha ,a by Ha ,a such that [START_REF] Castaneda | The effective mechanical properties of nonlinear isotropic composites[END_REF] in order to save the diagonal symmetry of N.

SPECIFIC FEATURES OF THE AFFINE APPROACH AT FINITE STRAIN

Linearization of the constitutive stress-strain relations. The tangent mod uli N are no more constant between two consecutive events because they depend on F(t , x) , Fe(t , x) , n(t, x). In view of an explicit prediction, we choose to linearize the constitutive equations at the beginning of the current step at time t = fn: n(t, x) = N(tn, x): F(t , x) + p(tn, x) , N ( tn, x) = N ( F ( tn, x), p e ( tn, x), n ( tn, x)) ,

p(tn,x) = n(tn,x) -N(tn,x): F(tn ,X) .

These are the constitutive equations of a continuously heterogeneous linear thermoelastic composite under finite strain.

Approximation of piecewise uniformity. Once again, the problem is simpli fied through the approximation of piecewise uniformity of the moduliN ( t n, x) and the prestresses p(tn, x ), which depend on x through F(tn , x ), Fe(tn, x) and n(tn,x): in phases, they are ascribed to take their value atF8(tn) = (F(tn, X ) )o s , F:(tn) = (Fe(tn, X ) )o s and ns(tn) = (n(tn, X ) )oJwith (A)o s the average of A in phases). This approximation results at stepn in the follow ing constitutive equations of the linear thermoelastic multiphase comparison material:

n(t, X ) = N 8(tn) : F(t , X ) + p(tn) , N8(tn) = N(Fs(tn), F � (tn), n 3 (tn)) , Ps(tn) = ns(tn) -Ns(tn) : Fs(tn).

(

) 14
Homogenization. At finite strain, no minimum potential energy principle is available. Ball [l] showed the existence of a solution to the problem of energy potential minimization for a polyconvex potential. However, uniqueness is not ensured. Thus, only variational principles (see Ogden [START_REF] Ogden | Non linear elastic defonnations[END_REF]) exist and estimates only can be derived. Here, we restrict ourselves to Hashin-Shtrikman type estimates, including the self-consistent scheme for specific application to polycrystals. They are based on the use of Green techniques which can be discussed in the context of finite strains through the 9-dimensional Eshelby inclusion problem. In terms of variables n and F, this problem is defined as follows:

n(X)i j,i = O VX,

n = N° : F + x 1 (X)n1, lim X -= F(X, t) = F, F _ ax(X,t) -ax ' (15)
where x 1 (X) is the characteristic function of the ellipsoidal inclusion! and n1 is constant. The solution for F(X) is known to be uniform in I (Iwakuma and Nemat-Nasser [START_REF] Iwakuma | Finite elastic-plastic deformation of polycrys talline metals[END_REF]) and given by FT(Xo) = P 0 : n1 + P VXo EI, Pgkz(Xo) = j uz j,ik(X -Xo) dVx = 1 r? jkl (X -Xo) dVx , (16) where u(X -X0) is the Green function for an infinite elastic medium with the moduli N° and r 0 is the associated strain-Green operator. Opposite to the case of small strains, the skew-symmetric parts ofuz j,ik with respect to (j, k) and (l, i) are taken into account in r 0 which does not exhibit anymore minor symmetries. As mentioned above, some difficulties, related to the use of Green techniques at finite strain, arise whenN ° does not exhibit anymore the diagonal symmetry. Although the uniqueness of the Green function is ensured, its existence is no more established, so that the derivation of P 0 becomes problematic. That is why such a situation has been excluded in (3.2). Therefore, the solution ( 16) of the inclusion problem may be used to establish estimates for elastic or thermoelastic linear composites in terms ofn and Fin a way strictly similar to the one used for small strains. For the elasto-plastic affine treatment, associated to the linear self-consistent scheme, the concentration equations at the current step n can be, according to (14 ), approximated by Fs(t) = As(tn): F + as(tn)

Vt E]tn, tn+1],

As(tn) = (1 + Psc: (N s(tn) -N sc))-1 , as(tn) = (Psc-1 + Ns(tn) -N sc)-1 : (jJ -Ps(tn))' Psc = f1 rsc(X -Xo) dVx , jJ = (p(tn) : A(tn)) ,

where r sc is attached to the infinite medium with moduli N sc .

(

) 17
Variables updating. Specializing equation ( 17) to the linear composite of the former step at time t = tn and proceeding similarly for the linear comparison composite of the current step at timet = tn+l lead to 6.. Fs = As(tn)6.. F+ (A8(tn)-A8(tn_i) )F(tn) +as(tn)-as(tn -1), (18) where 6.. Fs = F8(tn+i) -Fs(tn) and 6.. F = F(tn+1) -F(tn). The expres sion ( 18) of the deformation gradient increment clearly distinguishes the affine treatment from Hill's incremental one: according to Hill's method, this expres sion would reduce to 6.. F s = As ( tn) 6.. F. The macroscopic stress increment 6.. n and then the local stress increment6.. n8 can be derived from ( 18), ( 11) as a function of the macroscopic deformation. The lattice rotation is obtained as usual by subtracting the plastic deformation gradient increment from the total one.

"'

a.

e.

Preliminary results and conclusion

Both the affine and the incremental approaches have been applied to derive the tensile stress-strain response of an untextured elastoplastic face-centred cubic polycrystal. Hill's method has been implemented by using a true mixed formulation as explained in section (3.1). The elastic behavior is assumed to be isotropic (K = 44400 MPA andµ= 30750 MPA). The twelve octahedral slip systems are considered with the same critical resolved shear stressT; = 100 MPA Va. Self and latent hardening are taken into account (Ha(3 = (1 -q) Ma(3 + qh) with h = 200 and q = 0.3. An initial quasi-isotropic texture (with ten orientations in the standard triangle) is considered.

Figure 1 provides the stress-strain tensile curves for both these approaches, for small as well as for large strains. We can see that the affine approach still yields softer predictions than Hill's one at finite strain. Moreover, both finite strain formulations yield softer predictions than their small strain counterparts. This result stems from the softening effect of the lattice rotations which are taken into account at finite strain.

According to these preliminary results, the affine approach can be thought to be able to improve significantly on Hill's incremental treatment for both small and large deformations. Nevertheless, future work is necessary to validate the affine finite strain formulation and especially to extend it to more general anisotropic initial textures.

  relation F .(FP)-1 = "f a ma Q9 na and Hooke's law S = K: E , where Se is the second Piola-Kirchhoff stress tensor based on the configuration obtained after plastic deformation and Ee the elastic Green-Lagrange strain tensor, the following equation can be derived (see Harren [3]): pe K pP -l pe s: s: ijkl = ia jb abed kc ld + U ik Ujl ' Aa =(F e r .F e .ma Q9 na) 3 , Ba =(ma Q9 na.S e ) S ,

Figure 1 .

 1 Figure1. Tensile stress-strain curves of an untextured fee polycrystal according to the incre mental and affine models at small and finite strains.


M P Luong
email: luong@lms.polytechniqzre.fr

D Parganin

J Loizeau

INFRARE THERMOGRAPHY OF THERMOMECHANICAL COUPLINGS IN SOLIDS

The present paper aims to illustrate three advantages of infrared thermography as a nondestructive, real-time and non-contact technique to mechanically characterise solid materials. It permits observation of the macrostructural aspects of. thermoplasticity describing damage and failure processes in diverse and various engineering materials and their components subjected to monotonous, cyclic or vibratory loading. It usefully suggests the definition of a threshold of acceptable damage TAD for materials related to sport equipment such as leather shoe, leather-like composites or sail synthetics. Particularly in case of metallic products or automotive components subjected to fatigue loading, this newly proposed method could evaluate in a non-destructive manner the fatigue limit FL in a very short time compared to traditional fatigue testing techniques that are much more time-consuming and excessively expensive. In addition owing to the thermomechanical coupling, infrared thermography readily describes the damage location, the dissipative regime and the evolution of structural failure.

Introduction

The thermal effect due to thermomechanical coupling at the tip of a moving crack has been investigated [START_REF] Bui | Thennomechanical coupling in fracture mechanics, Thermomechanical Coupling in Solids[END_REF] within the framework of thermodynamics, taking into account stress and strain singularities. Internal energy dissipation was recognised by well-known scientists [1015- 16-25-26-28]. Carrying out experiments on the cyclic twisting of cylindrical bars, Dillon [START_REF] Dillon | Coupled thermoplasticity[END_REF] identified the work done to the system by plastic deformation as the major contribution to the heat effect, and proposed an internal dissipation rate D. Well-developed empirical theories of plastic deformation in metal allowed engineers successfully to predict the behaviour of a variety of structures and machine elements loaded beyond the elastic limit for purposes of design.

This paper emphasises the application of infrared thermography to detect the macrostructural aspects of thermoplasticity describing the occurrence of damage in engineering materials under monotonous loading and metallic products subject to fatigue testing. 1

Heat production mechanism

Damage theories rely on assumed discontinuous phenomena at the microscopic scale [START_REF] Bui | Damage theories for brittle and ductile materials, Fracture of Non-Metallic Materials[END_REF]. Kachanov [START_REF] Kachanov | Time of rupture process under creep conditions[END_REF] introduced a continuous variable D, related to the scalar density of defects. Taking into account the most fundamental aspects of inelastic deformation and neglecting the details at the microstructural level, Mroz [START_REF] Mroz | Phenomenological constitutive models for metals, Modelling small deformations of polycrystals[END_REF] suggested phenomenological constitutive models for engineering applications. On the basis of micro-and macro-scale relationships, Dang-van [START_REF] Dang-Van | Sur la resistance à la fatigue des métaux[END_REF] proposed a multiaxial fatigue criterion with a realistic physical interpretation of fatigue phenomena. During a polycyclic fatigue test, the stress at the macroscopic scale remains elastic. However, at the microscopic scale, the metal is neither isotropic nor homogeneous; it is constituted of randomly oriented crystals. This induces local fluctuations of the microscopic stress and defines the macroscopic stress. Thus the local microscopic stress can locally exceed the yield strength in certain unfavourably oriented grains, whereas the macroscopic stress remains elastic. If the cyclic plastic response of the grain is not an elastic shakedown, then some microcracks will be nucleated. They could coalesce to form a crack of detectable size. Since fatigue is generally caused by cyclic plastic strain, the plastic strain energy plays a very significant role in the damage process [9-12-22-23]. The significance of such an energy approach is in its ability to unify microscopic and macroscopic testing data. Therefore, the idea of relating fatigue to the intrinsic dissipation detected by infrared thermography seems to be highly relevant. Metal fatigue of aircraft components was detected very early in the history of powered flight [START_REF] Mann | Aircraft fatigue with particular emphasis on Australian operations and research[END_REF], and it was also of major importance in determining the durability of ground vehicles. The failure mode was identified early by railway engineers. Even very low failure rates can produce an appreciable number of failures [START_REF] Mclester | Railway component fatigue testing, Full-scale Fatigue Testing of Components and Structures[END_REF]. Fatigue phenomena, leading for example to the failure of diesel engines [START_REF] Habib | Failure analysis of a broken diesel engine[END_REF], were poorly understood until recently because they occur unexpectedly at relatively low stress levels. Some progress has been made recently in design methods [START_REF] Ballard | High cycle fatigue and a finite element analysis[END_REF].

Despite this wealth of methods developed by engineers over the years, the principal question that remains unanswered is this: what is the physical phenomenon called fatigue damage? The ASTM STP 811 meeting in 1982 on fatigue mechanisms (advances in quantitative measurement of physical damage) concluded that fatigue damage can be defined as either: i) a chemical-physical process whereby irreversible degradation of a specific property results from the application of cyclic stress and strain, or ii) a physical separation of the material (cracks, cavitation, etc.). Understanding fatigue damage requires making a clear distinction between the physical damage, the process of damage, and the manifestation of damage. Various heat production mechanisms have been proposed by various authors and are now discussed. According to Moore and Kommers [START_REF] Moore | Fatigue of metals under repeated stress[END_REF], the temperature test was suggested and to some extent used by C.E. Stromeyer of Manchester in England as early as 1913. However, their search for some short-time test to predict fatigue resistance, based on the use of thermocouples, was not conclusive since they could not identify a critical temperature with the endurance limit of steel specimens subjected to cyclic loading. This paper proposes using differential infrared thermography to quantitatively evaluate the evolution of temperature generated by the specimen under reversed stresses applied for a few minutes or less. Infrared thermography has been successfully used as an experimental method to detect the plastic deformation of a steel plate under monotonic loading [START_REF] Bui | Etude experimentale de la dissipation dans la propagation de la fissure par thermographie infrarouge[END_REF], or as a laboratory technique for investigating damage, fatigue, and failure mechanisms occurring in engineering materials [START_REF] Luong | Detection par thermographie infrarouge du seuil caractéristique d'un sable cisaillé en vibrations[END_REF][START_REF] Luong | Infrared thermovision of damage processes in concrete and rock[END_REF]. Plastic deformation is not homogeneous and the stress acting on a plastic inhomogeneity embedded in an elastic surrounding is a function of its plastic strain, diminishing with increasing strain. Due to the thermomechanical coupling, the generated plastic dissipation is readily detected by infrared thermography.

Thermomechanical coupling in solids

The development of the thermo-elastic-plasticity equations [l-7-10-14] leads to the following coupled thermomechanical equation:

where β denotes the coefficient of the thermal expansion matrix, r e 0 the heat supply, E the elastic strain tensor, E I the inelastic strain tensor, C v the specific heat at constant deformation, the fourth-order elasticity tensor, e the specific internal energy, K the thermal conductivity, and finally T the absolute temperature.

This equation shows the potential applications of the infrared scanning technique in diverse engineering domains: detection of fluid leakage, non-destructive testing using thermal conduction phenomena, elastic stress measurements, and localisation of dissipative phenomena. Thus the detected temperature change, resulting from four quite different phenomena, must be correctly discriminated by particular test conditions and/or specific data reduction. This is the main difficulty when interpreting the thermal images obtained from experiments under the usual conditions.

The last term on the right-hand side of the thermomechanical equation defines the energy dissipation due to plasticity and/or viscosity. The work done by plastic deformation per unit volume can be evaluated by integrating the material stress-strain curve. Infrared thermography readily detects the internal dissipation term, which constitutes a significant part of the non-linear coupled thermomechanical analysis. The proposed thermographic technique is mainly concerned with the differences in temperature (or thermal gradients) that exist in the material rather than with the absolute values of temperature. It conveniently detects the dissipation evolution of the material under loading. The reported work considers intrinsic dissipation as a highly accurate indicator of damage manifestation. It highlights the advantages of the infrared thermographic technique, used for the detection and the discrimination of physical phenomena involved in this non-linear coupled thermomechanical effect.

Infrared thermographic technique

Infrared thermography is a convenient technique for producing heat pictures from the invisible radiant energy emitted from stationary or moving objects at any distance and without surface contact or any perturbation of the actual surface temperature of the objects viewed. A scanning camera, analogous to a television camera, uses an infrared detector in a sophisticated electronics system. It detects radiated energy and converts it into a detailed real-time thermal picture in a video system either in colour or monochromatically. Response times are shorter than one microsecond. Temperature differences in the heat patterns are discernible instantly and are represented by several hues. The quantity of energy W emitted as infrared radiation, is a function of the temperature and emissivity of the specimen. The higher the temperature, the greater the emitted energy. Differences of radiated energy correspond to differences of temperature. The AGA 782 SW infrared scanner unit in use comprises the following: (a) a set of infrared lenses which focuses the electromagnetic energy radiating from the object being scanned, into the vertical prism; (b) an electro-optical mechanism which discriminates the field of view into 10 4 pixels by means of two rotating vertical (180 rpm) and horizontal (18,000 rpm) prisms with a scanning rate of 25 fields per second; (c) a set of relay optics containing a selectable aperture unit and a filter cassette unit which focuses the output from the horizontal prism onto a single-element point detector located in the wall of a Dewar chamber; (d) a photovoltaic SW short-wave infrared detector composed of Indium Antimonide InSb which produces an electronic signal output varying in proportion to the radiation from the object within the spectral response 3.5 µm to 5.6 µm; (e) a liquid nitrogen Dewar that maintains the InSb detector at a very low temperature of -196 °C, allowing a very short response time of about one microsecond; and (f) an electronic control with preamplifier that produces a video signal on the display screen.

The received radiation has a non-linear relation with the object temperature, can be affected by atmosphere damping, and includes reflected radiation from the object's surrounding. In consequence, calibration and correction procedures have to be applied. Knowing the temperature of the reference, we can calculate the view-field temperature with a sensitivity of 0.1 °C at 20 °C. This infrared device is used to scan specimens in the following two test programs.

Threshold of acceptable damage

On August 22, 1851, the yacht America raced a fleet of English cutters and schooners around the Isle of Wight. According to the legend, America easily prevailed. America's sails were unusual [START_REF] Doyle | Strong fabrics for fast sails[END_REF]. Through innovative technology, engineers have continually found ways to make stronger, lighter sails. The resistance of the fabric to stretching is especially crucial, as is its breaking strength. Also, yield strength, or the load beyond which the material is permanently elongated, is relevant, because even if they do not break, sails can be ruined by deformation. There is thus a need to detect damage using a very sensitive and non-invasive technique [START_REF] Parganin | Infrared scanning of damage in sail material[END_REF].

Before 1924 the whole sport footwear (upper and sole) was made in leather, thanks to its ability to absorb the sweating that is 6 to 8 times higher during a sport activity than as usual. Today leather is still well present in the sport footwears. Annually France produces 320 000 pairs for tennis, basket, training and footing, 200 000 pairs with staples and toes, 930 000 pairs for touring, 24 000 pairs for riding, and 180 000 pairs for dancing. Leather is a product made by stabilising through tanning the proteins of animal skins that are by-products of the meat industry. While about five-sixths of all the leather produced is used by the shoe industry, special uses of leather fill many real industrial needs. Through its many uses of boots, shoes, clothing, straps, harness or handbags, wallets, billfolds, leather novelties, etc., leather has become a most essential commodity of man. Softness, damage and strength behaviour is an important consideration in connection with designs or regulations. An other concern is to produce leather of uniform good quality, for example, which is acceptable to the shoe manufacturer and the wearer of the shoes. Balls for football games were made in leather. Since World Cup 1986 in Mexico, synthetic composites substituted leather. However they are mechanically considered as leather-like materials. A specimen of soccer ball envelope was tested under uniaxial tension test. Figure 1 shows the load-displacement curve of this leather-like specimen. The Agema SW-782 infrared scanner, incorporating a hightemperature filter and equipped with a real time system Discon (Digital Infrared System for Coloration) converting thermal images into a ten-colour isotherm display, was used to record the bulk of the heat emission data. That device displays a ten-colour calibrated surface-temperature picture of the specimen. Infrared thermography readily detects intrinsic dissipation announcing the occurrence of damage, preceding the failure of leather-like specimen. Figure 2 presents the evolution of intrinsic dissipation when loading increases up to failure and thus defines a threshold of acceptable dissipation or damage (TAD), separating low and high regimes of dissipation or damage occurrence.

The process of tailoring leather-like products involves the transformation of a twodimensional structure (planar sheet fabric) into a three-dimensional form. It necessitates the formation of extreme curvatures by sewing or fusing and pressing. The joining of different pieces of leather components is achieved by sewing with sewing threads. There are many problems associated with the performance of sewing threads during the sewing process that can reach very high speeds. During the stitching process, thread is subjected to complex kinematic and dynamic conditions. Tensile and bending stresses, that take place at very high speeds, localised heating that may cause reduction of strength of thermoplastic sewing threads by as much as 60 %. In addition the mechanics of the thread structure and their properties in the seams after sewing include seam strength. Stretch and pucker are also extremely important. Current design procedures for leatherlike seams are based upon simplified assumptions of connection behaviour. The increased use of diverse types of seams by sewing and/or glueing suggests that a more thorough understanding of the behaviour of seams would be beneficial for upgrading design procedures. As infrared thermographic scanning offers new information on leather-like connection behaviour, refinement of design procedures can be made as needed and new, more effective, and specialised seam configurations can be contemplated. Figure 3 presents a tension test on a soccer envelope seam. A threshold of acceptable dissipation or damage TAD for the seam appears on Figure 4. This experimental result shows that the process of leather-like seam could be optimised in order to meet a higher value for its TAD limit. This means that the envelope seam fails if this load is exceeded.

Evaluation of fatigue strength

The material for the rotating bending tests was XC55 steel, quite extensively used in automobile construction. To minimise scatter, the 6.74 mm diameter specimens have been prepared from a single melt. The number of load cycles was 10 7 , corresponding to automobile fatigue damage. Cyclic fatigue tests were performed on a 4-point loading rotating bending Schenck machine, running at approximately 100 Hz. A total of 18 specimens was step-tested according to the Staircase or Up-and-Down method. This procedure means that each specimen was run for 10 7 cycles at 370 MPa for example, whereupon the stress was raised in 10 MPa steps in the absence of failure, and lowered otherwise. Using standard methods of statistical analysis, we estimated the fatigue limit to be 399 MPa with a standard deviation of 41 MPa.

A series of five rotating bending tests has been scanned by the infrared system at different stress levels that were chosen in accordance with the step-by-step procedures for using statistical methods in usual fatigue testing. All specimens were coated with matte spray paint to maximise their emissivity. The paint coating raises the surface emissivity to a uniformly high level. In addition, it allows examination of the surface which is undertaken at oblique angles, without degradation of sensitivity. The recommended paint coating also reduces surface reflections and ensures uniformity of response across the specimen. With a loading frequency of 100 Hz and after stabilisation of the temperature rise, the load duration chosen was 30 seconds and 60 seconds corresponding to 3,000 load cycles and 6,000 load cycles respectively.

Computer-aided thermography software TIC 8000 allowed data reduction of thermal images that shows heat generation after 6,000 load cycles at 390 MPa (Figure 5). These thermal images provided quantitative values of the intrinsic dissipation of a steel specimen in rotating bending caused by 3,000 and 6,000 load cycles. The same procedure has been applied for each load step. The manifestation of the fatigue damage mechanism is revealed by a break in the intrinsic dissipation regime of the loaded specimen. Experimental results are summarised in Figure 6 which shows how fatigue limit is determined using a graphical procedure. The threshold of critical thermal dissipation is roughly the same for the 3,000 and 6,000 cycle curves. It corresponds to the value deduced from standard procedure. The star point at co-ordinates (600 MPa, 180 °C) represents the last load step at which failure occurred.

These experiments showed that infrared thermography can provide the fatigue limit of XC55 steel within a few hours instead of the several months required when using the standard staircase method.

Concluding remarks

This work has demonstrated that material dissipativity is a highly sensitive and accurate manifestation of damage, owing to the thermomechanical coupling. Infrared thermography provides a non-destructive, real-time and no contact technique to observe the physical processes of degradation and to detect the occurrence of intrinsic dissipation. It readily evaluates the threshold of acceptable damage under monotonous loading and the fatigue limit in cases of fatigue testing.

Figure 1 .

 1 Figure 1. Tension-deformation of a specimen of football envelope.

Figure 2 .

 2 Figure 2. Corresponding tension-dissipation curve for the same specimen. The two straight lines define a threshold of acceptable damage (TAD). This means that the material fails if this load is exceeded.

Figure 3 .

 3 Figure 3. Tension testing on a soccer envelope seam.

Figure 4 .

 4 Figure 4. Tension-dissipation curve for a football envelope seam The two straight lines define a threshold of acceptable damage (TAD).This means that the envelope seam fails if this load is exceeded.

Figure 5 .

 5 Figure 5. intrinsic dissipation of the specimen under 390 MPa after 6,000 load cycles (Temperature scale is given in °C).

Figure 6 .

 6 Figure 6. Graphical determination of fatigue limit (FL) of XC55 steel specimen.


Emmanuelle Olivier

G Simmel

Olivier

Individual career paths and multiple connections. Portraits of three arrangers and sound engineers in Abidjan (Ivory Coast)

Keywords: Situated multiculturality, Cosmopolitanism, Ethnonationalism, Popular Music, Recording Studios, Ivory Coast

This paper examines the carrier paths and the connections of three arrangers and sound engineers of popular music in Abidjan (Ivory Coast). Through their family history, the multiple skills they have acquired in the music and sound professions, and the professional opportunities they have been able to seize, we will see how their artistic identity is constructed at the crossroads of a territorial attachment and an experience of the world. The paper starts with a portrait of these three 'makers of Ivorian music', before examining their multiculturality through the prism of an Ivorian history pulled in the opposite directions of cosmopolitanism and ethno-nationalism. Finally, we will show how their individual trajectories and creative resources embody a series of geographical and artistic shifts, that are characteristic of a situated multiculturality.

Introduction

Abidjan, 7 March 2020: I am on my way to the Abidjan Market for Performing Arts (MASA) located in the Palais de la Culture in the working class district of Treichville, on the banks of the lagoon. Created in 1990 in Liège (Belgium) during the second Conference of Ministers of Culture and Francophonie, MASA aims at "strengthening the capacities of African professionals in the performing arts (music, theatre, dance) and allow access to the international market for African productions and their artists". 1 It also facilitates "the circulation of artists and their production in Africa and in the world", and contributes to "the professional training of artists and operators of the production chain of shows". On its website, MASA prides itself of having become "a showcase for contemporary creation in The audience accesses new songs through audio recordings and videoclips posted on social networks, streaming platforms, radio networks or television channels. And live performances are meant to play the songs as they were recorded in the studio, which is what the audience primarily expects. Thus, there emerges a new dialectic between recorded music and stage.

I am particularly interested in the figures of the arranger and the sound engineer, who have largely remained outside the scope of African popular music studies. At the same time, this allows me to challenge the notion of the almighty composer/creator, which is far too culturally situated (Olivier 2014 7 ). These arrangers, whom the Ivorian press readily describes as "star makers", 8 or even "geniuses in the shadows", 9 occupy a central place in the music making, to the point of often being credited as "composers" on albums they work on. In fact, far from being confined to the role of technician, they compose the instrumental parts of the songs, often contribute to the lyrics and vocals, and form a particular style to the song. Some of them also do the mixing, thus combining the roles of arranger and sound engineer. But many recording studios prefer to use the services of specific sound engineers whose mixing and possibly mastering work are expected to provide the music with a specific 'shade', as well as a sound balance that conforms to international standards, thus ensuring the largest possible distribution of the music.

When I reached MASA, Eliezer had almost completed setting up the sound system for the Street Art stage. He was talking to Adolphe Yacé, a musician, arranger, sound engineer and teacher at the Institut National Supérieur des Arts et de l'Action Culturelle (INSAAC) in Abidjan. I have come along Fatoma Diarra, an Ivorian keyboard player whose family is originally from Burkina Faso, and whom I met in Bamako, where he used to accompany Islamic pop music singers during the annual Mawlid festival. 10 Thanks to Eliezer, I could talk to Adolphe Yacé, who turned out to be an eminent figure in the Ivorian music scene. Fatoma Diarra introduced me to arrangers and sound engineers who are part of his professional network and circle of friends, including Doyoh Théodore Cyriaque Djé, a.k.a. Patché, and Gabriel Adjé.

From this visit to MASA in Abidjan and the encounters I made, I will in this paper draw the portraits of Adolphe Yacé, Patché and Gabriel Adjé, who made a particular impression on me, by examining their trajectory and their "connections" (Amselle 2001 11 ) as arrangers and sound engineers of Ivorian popular music. 12 Through their family history, the multiple skills they have acquired in the music and sound professions, and the professional opportunities they have been able to seize, I will show how their artistic identity is constructed at the crossroads of a territorial attachment and an experience of the world, involving context and projection (Agier 2013 13 ).

I will pay particular attention to the musical and sound resources that Adolphe Yacé, Patché and Gabriel Adjé mobilize and connect, evidence of a multiculturality that I describe as banal, emerging from below, so much so that it permeates their daily life and their professional practice. This banal multiculturality is the hallmark of a "musical omnivorousness" (Peterson 1992 14 ) that is specific to these great arrangers and sound engineers. Beyond that, it reflects the cosmopolitanism of Abidjan and its music industry, characterized by populations mix and multiple influences, be they African, French or North American. This multiculturality is also situated, referring to imaginaries of tradition and modernity produced in the Ivory Coast for an essentially African audience. The vicinity of Ivorian music with other African and Afro-American music is the strongest expression of this situated multiculturality, which evokes an implicit 'Afro' identity. This multiculturality at work in the Abidjan recording studios ultimately produces a sense of national belonging, which makes sense in a global context. The aim of most arrangers and sound engineers is to produce a music that is both anchored in an Ivorian reality and exportable within a globalized industry, i.e. music defining itself resolutely Ivorian towards the outside world.

In this text, I will first draw up a portrait of Adolphe Yacé, Patché and Gabriel Adjé, before examining their multiculturality through the prism of an Ivorian history pulled in the opposite directions of cosmopolitanism and ethno-nationalism. Finally, we will see how the individual trajectories and creative resources of these three arrangers and sound engineers embody a series of geographical and artistic shifts, that are characteristic of this situated multiculturality 15 .

11 Jean-Loup Amselle, Branchements. Anthropologie de l'universalité des cultures (Paris: Flammarion, 2001). 12 By popular music, I mean all musical genres that enter a media regime and mass consumption. 13 Michel Agier, La condition cosmopolite : l'anthropologie à l'épreuve du piège identitaire (Paris: La Découverte, 2013). 14 Richard A. Peterson, "Understanding audience segmentation: From elite and mass to omnivore and univore," Poetics: Journal of Empirical Research on Literature 21(1992): 243-258. 15 I would like to extend my warm thanks to Fatoma Diarra for accompanying my fieldwork in Abidjan and to all the arrangers, sound engineers and musicians who participated in this ethnography for their welcoming me, giving me their time, and for letting me attend their recording sessions. I would like also to thank Sébastien Le Pipec for his English language review and translation. This research is funded by a 2019-2024 partnership grant

Individual career paths

Adolphe Yacé: the figurehead of musical cosmopolitanism

A few days after the MASA event, I conducted a first interview with Adolphe Yacé, who received me at his home, in a villa in the wealthy district of Deux Plateaux. I would then meet him several times during my two field trips to Abidjan, in March 2020 and July 2021, notably at his recording studio in the Cité des Arts or at the B&B where I was staying in Cocody.

Adolphe Yacé spoke bluntly: "We are musicians from father to son, so from a young age we live in a musical world." 16 His father was a gendarme, but a saxophonist gendarme who conducted a band and then taught at the National Institute of Arts. 17 From an early age, in the 1970s, Adolphe and his two brothers, Marcellin and Évariste, listened to and played Afro-Cuban music, French Variety and Congolese Rumba, which were all trendy in Ivory Coast at the time. Among the Cubans, Adolphe recalled the Matamoros trio 18 and the Los Compadres ensemble. 19 He also mentioned the Dominican musician Johnny Pacheco, 20 Joseph Kabasele alias Grand Kallé 21 from Congo (DRC), the music played by his father's band, with its Paso Doble, Waltzes and Marches, the traditional Ivorian repertoire and the Evangelical songs imported from Ghana and Nigeria. Adolphe explained that his family hailed from a coastal village near San Pedro, in the west of Ivory Coast, which was very early on in contact with the Portuguese, English and French European trading posts: "That's how they [the people of the coast] inherited this culture [of the brass band or Fanfare], a bit like the American Blacks in the sugar cane fields who watched the masters playing Waltzes, Ball music in the evenings." It was precisely this Fanfare music that his elder brother Marcellin would promote and modernize in the early 1980s with his Woya 22 ensemble, which soon became famous in Ivory Coast. from the National Research Agency of France (ANR) as part of AFRINUM "Digital Cultures in West Africa: Music, Youth and Mediations" programme. 16 The quotes from Adolphe Yacé are from three interviews conducted on March 10, 2020, July 6 and 13, 2021. 17 The National Institute of Arts became the National Higher Institute of Arts and Cultural Action (INSAAC) in 1991. 18 See: https://www.montunocubano.com/Tumbao/biogroupes/matamoros,trio.htm. 19 See: https://www.montunocubano.com/Tumbao/biogroupes/compadres,%20los.htm. 20 See his biography on the Universal website: https://www.universalmusic.fr/artistes/10000011864. 21 Grand Kallé is an emblematic musician of the Democratic Republic of Congo (ex Zaïre), composer and singer of Indépendance Cha Cha (1960) which will be played by most new independent African countries. 22 See: https://rti.info/culture/7590.

The Yacé family lived in the Cité des Arts, in the wealthy district of Cocody, where the Abidjan bourgeoisie at the time of Houphouët-Boigny 23 resided. There lived Georges Diby, one of Abidjan's town planners and a friend of Adolphe's father's, who was also a musician.

In 1978 he enabled Les Mewlessel ensemble (which included Marcellin) to record the album "La marche du Bélier" in New York. There was also François Konian, the founder and owner of Radio Jam, one of the most popular commercial radio stations in Ivory Coast. Upon his return from the USA where he spent some time, he brought back audio equipment with which he set up a vinyl record manufacturing company, the Société Ivoirienne de l'Industrie du Son. 24 Adophe Yacé explained: "We were in contact with friends who were often travelling: ministers' sons, ambassadors' sons who were our neighbours. [...] We even had friends who were flight attendants. Kanté, I remember him, he was a flight attendant. He used to bring us the latest news. When he travelled, he flew to New York, he flew to London, he came back with the latest reggae, the latest folk. We had all the new stuff, even before it landed here. So we were ahead of the game! We had all the hits! Kanté, every 2-3 days, he would travel around the world and bring us back all the new stuff. So, there were all the sounds that we heard and that we could Houphouët-Boigny (1905-1993) was the first President of the Republic of Ivory Coast from 1960 to 1993. 24 About François Konian, Adophe Yacé added: "Konian François was always on the look out for the new the most efficient, the most awsome thing. "What do they broadcast in LA? What does this radio station in New York broadcast? What does this engineer in London use?" These were his references. He said to himself: "If I want to run a radio station, it must be up to international standards." that he regularly performed in the United States from 1986 to 1991. When he was in Detroit, he seized the opportunity to visit the famous Motown studios, which produced the greatest African-American Soul, Funk, RnB and Pop Music artists, and where he came across sound engineering that did not yet exist in Ivory Coast.

In Abidjan, Adophe Yacé recalled the "swinging" 1980s and 1990s, when the major recording studios (JBZ, Nefertiti, Séquence) operated and welcomed musicians from all over the region: "The Malian producers would bring in six artists at a time. They used to book JBZ for two months." For Adophe, the 1980s also means the arrival of the French band Kassav, which chose Abidjan for its first concert in 1985. Thanks to Jacob Desvarieux, the band's lead singer and a friend of François Konian's, Kassav offered its audio equipment to the Woya band conducted by Marcellin Yacé. Kassav thus introduced the Yacé brothers to the world of digital music which, in the mid-1980s, aroused amazement and wonder in Abidjan.

In the 1990s, Adolphe Yacé set up the Yacé's Brothers studio, while at the same time being a professor and then the director of INSAAC. His studio was one of the first in Abidjan to switch from analog to hybrid production (analog and digital), using an Atari computer as a midi sequencer, plugged to a mixer and a multitrack tape recorder. From 1994 to 2000,

Adolphe also worked as an audio production manager at Population Services International

Gabriel Adjé, Universal's sound sculptor

I first met Gabriel Adjé, a.k.a. Gab, precisely at Patché's studio. When I arrived there, Gab was about to mix the track "Tudi" by Griot Guéré, an Ivorian singer based in France, whose latest album Patché arranged and recorded. 33 Born at the end of the 1980s to a father who is "half Nigerian and half Ghanaian" and a mother from Benin, Gabriel grew up in Lagos (Nigeria) and then in Accra (Ghana). Unlike Adophe Yacé and Patché, there were no musicians in his family. But his father, who made his career in the military, practiced DJing as a hobby and was fond of the Afrobeat music of Fela Kuti. At the age of 17, Gabriel was in Accra and began to visit recording studios. Then, he wanted to become an "artist", i.e. a professional singer, but he was already interested in the work of sound engineers. He attended recording sessions, while teaching himself how to use mixing software on his computer with the help of online tutorials. 31 The "backeur" is the singer who does the secondary voice. 32 Patché has notably worked with Satchela Evrard Djédjé alias Vegedream, a famous French rapper of Ivorian origin produced by Universal. 33 The interviews with Gabriel Adjé took place on 14 March 2020, 6 and 10 July 2021.

Gabriel's father left the army in 2009 and set up a transport company based in Abidjan.

Gabriel moved with him before joining his mother in Lomé, as Ivory Coast was undergoing political crisis anew. Between 2010 and 2012, he travelled between Lomé, Accra and Lagos to train as a sound engineer in several recording studios, notably those of Nollywood. Back in Abidjan in 2012, once the political crisis was over, Gabriel studied for a degree in computer networking/programming in an engineering school. At the same time, he learnt to play the piano at the Evangelical church in his district and soon started to play professionally at the many religious services during the week. In addition to earning a living, religious music allowed him to become acquainted with the harmony of Western music. He also began to buy studio equipment, which he ordered in Togo 34 and brought to Abidjan via his father's transport company.

After completing his degree, Gabriel was employed as a sound engineer in two important audiovisual production companies in Abidjan, where he recorded and mixed TV shows and series, as well as advertising clips. At the same time, he collaborated with prominent Ivorian arrangers such as Patché or Shado Chris, whose tracks he mixed. In May 2021, Gabriel was hired by Universal as the resident sound engineer of their Abidjan recording studio. He has since then recorded many Ivorian, Malian and Congolese artists who were produced by Universal.

Gabriel listens to many genres of music: Nigerian and Ghanaian Afrobeat, but also RnB, Soul, especially Motown artists, and American Rap. Alongside these artists, he follows the work of Nigerian, Togolese and Ivorian sound engineers. He mobilizes all these resources in his work as a sound engineer. Gabriel needs to immerse himself in different styles of music in order to be able to mix them. "You don't mix Afrobeat like Trap," he explained. He considers that mixing is a "point of view" on the track; it depends on the sound engineer's own creativity and sensitivity, which stems from both his or her own musical tastes and his or her intimate knowledge of the music.

Ivory Coast between foreign cultural influences, cosmopolitanism and ethnonationalist temptations

What appears clearly from these three portraits of arrangers and sound engineers is that their multiple connections stem from their family history and their individual carrier paths, and 34 In Togo, audio equipment is cleared through customs, which significantly lowers its cost. from the professional opportunities they have seized. What is also striking is their ability to move from one music to another, also when it is new, and their knowledge of the codes for accessing the international music market. But these connections are anchored in an Ivorian context, which we need to return to in order to understand the "space of experience" (Kosseleck 1990 35 ) of Adolphe Yacé, Patché and Gabriel Adjé.

Whether in the political or economic, artistic or media world, everyone agrees on the cosmopolitan character of the Ivorian capital. By virtue of its geographical position, a harbour and on maritime routes, Abidjan seems destined to be a cosmopolitan city (Boutin and N'Guessan 2016: 174 36 ). But it is above all the international integration of Ivory Coast by means of colonization, the continuation of this process after independence and its amplification through the regime's "liberal migration policy" (Bredeloup 2003: 25 37 ), which was to foster the formidable cosmopolitan character of Abidjan. In fact, the project promoted by President Houphouët-Boigny advocated the opening of Ivory Coast's borders to capital and people, and the maintenance of close links with the former colonial power. Houphouët-Boigny thus imagined a kind of "pan-Africanism à l'ivoirienne" (Dozon 2000: 60 38 ), according to which the Ivorian territory was to be "the hub for neighbouring African populations, and [...] its population it were to reinvent themselves in view of this necessary cosmopolitanism." (ibid.). 39 Abidjan thus imposed itself as the top metropolis in French-speaking West Africa, the one that "sets the tone [...], that broadcasts information and fashions." (Konaté 2005: 321 40 42 The vinyl records were distributed by the major international labels of the time: Columbia, HMV, La Voix de son maître, Odéon and Pathé Marconi. American music, Jazz, Ghanaian Highlife, Nigerian Juju and Congolese Rumba (Collins 2012 43 ).

In the wake of independence, Ivorian singers such as Amédée Pierre, Ernesto Djédjé and François Lougah emerged on the national scene. However, the 'liberal' Ivory Coast (Losh 2000: 14 44 ) did not develop a cultural policy 45 , unlike its 'socialist' neighbours Guinea and Mali, which made music one of the strongest components of their national identity [START_REF] Counsel | Cultural Policy and Music in Mali[END_REF]Djebbari 2013 46 ). Abidjan's cultural autonomy was thus based on materials from the local Akan 47 culture, French colonization mixed with other popular music, whether African, Latin American or Western ones. The Ivorian government considered music as a commercial product and an instrument for promoting its development policy as any other good on the market: what mattered was attracting foreign investors to a country in development, where it was pleasant to live (Bahi 2021: 81 48 ). Moreover, when in the early 1970s the Ivorian state set up the Radio-Télévision Ivoirienne (RTI) orchestra, which can be considered the first national orchestra, musicians from abroad were appointed to conduct it: Nigerian Lionel Fibbs, followed by Malian Boncana Maïga and Cameroonian Manu Dibango (Ketchiemen 2020 49 ).

In the 1960s and 1970s, Abidjan's musical life concentrated in the Treichville district, where local musicians performed, along with Congolese orchestras which were very popular in Abidjan. Musical recordings were made at the RTI studio, but also in Nigeria, at the EMI studios in Lagos, in Togo at the OTODI 50 studios in Lomé, or in Benin at the SATEL 51 studio in Cotonou. As for distribution, it was operated by French firms (Philips, Fiesta, Africavox, Vogue and Le Chant du Monde). But from the 1970s onwards, a proper music industry was established locally, with recording studios, producers, record labels, distributors and vinyl pressing companies. According to Flemming Harrev, "by 1983 producers and labels had grown to around 75" (2015 [START_REF] Harrev | Afrodisc. African discographies[END_REF] ). This music industry was highly cosmopolitan, with Nigerian 53 , Senegalese 54 , Malian 55 , French 56 , North American 57 and Lebanese 58 professionals. Some Ivorian labels implemented an international strategy. This was the case of Disco Stock which expanded its catalogue by establishing licenses with Congolese, Ghanaian, Malian and Senegalese producers. Its albums were then sold in France by Sonodisc and in the USA, thanks to a partnership with the renowned Makossa label (Harrev 2015). This music industry, unique in French-speaking Africa at the time, attracted numerous musicians from the region who came to Abidjan for concerts and to record in the studios. Some even settled down in Abidjan, such as Salif Keita (Mali), Mory Kanté (Guinea), Laba Sosseh (Senegal), Sam Mangwana (Congo) and Sammy Lartey (Ghana). Thanks to a relatively stable political situation and a flourishing liberal economy, Abidjan became "the hub of African variety music" (Konaté 2005: 325), where many artists launched their careers internationally in the 1980s, while some proceeded to recording in France and in the United States. As for the music recorded in the major studios in Abidjan, it was often mixed and mastered in France, England or the United States, [START_REF]Travelling from Ivory Coast to North countries did not require a visa, which made it easier to travel around[END_REF] which can be interpreted as a sign of fruitful collaboration between Ivorian and Western studios, or conversely as "industrial patronage" (De Beukelaer and Eisenberg 2020 [START_REF] De Beukelaer | Mobilising African music: how mobile telecommunications and technology firms are transforming African music sectors[END_REF] ). RTI broadcast music from all continents and all influences, more particularly Afro-American music. Indeed, the role of television-mediated music was strong in the Americanization of young Ivorians (Akindes 2002 61 ). While being receptive to African musical genres, but also to Reggae and Zouk for their implicit 'Afro' identity, young Ivorians favoured Funk. They positioned themselves on a transatlantic axis, mediated however by Paris where Funk music was very popular in clubs. Some people viewed this inclination for African American music to the detriment of local music as "cultural imperialism" (Land 1992 62 ). They blamed Ivorian youth for being "entirely under the yoke of American fashion in terms of music, dance, and sometimes even language, in a word, the American way of life" (Ouattara 1985: 84 63 ).

Conversely, the Ivorian authorities considered this opening up to foreign, and more particularly American, music and television programmes as powerful means for modernizing the country and opening it up to the world.

The end of the 1980s was marked by the weakening of the socialist regime in the USSR that was rocked by a deep crisis, and consequently the loss of relevance of developmentalist ideologies. In addition, President Houphouët-Boigny's regime, known as 'The Old Man', who had been in place for decades, was too starting to lose its lustre. The issue of 'Ivorianness' crept into the public debate, and culminated in a major political crisis at the end of the 1990s (Dozon 2000 64 ).

In the music world, this led to artists and sound engineers leaving the country to France or the United States. One example is Georges Kouakou, Alpha Blondy's arranger, who left for the USA where he worked with the Wailers and other Jamaican artists. A famous Ivorian producer, David Monsoh, moved to France where he worked with Congolese music stars Koffi Olomidé and Fally Ipupa. It was also in his Paris studio that the Coupé-décalé movement emerged in the early 2000s. In Ivory Coast, the crisis did not imply reviving 'traditions', but resulted in music that could both unite Ivorians and impose itself on a West African market were foreign music, particularly Congolese music, had so far prevailed (Akindes 2002;Kohlahgen 2005 65 ). As World Music emerged and Rap arrived in Ivory Coast, Zouglou music played this role, followed a few years later by Coupé-décalé. These two types of music have very different projects: Zouglou is clearly a protest music (Konaté 2002 66 ), while Coupé-décalé is an ambient, depoliticized music (Kolé 2021 67 ). However, they both quickly managed to unite the Ivorian youths. The national identity of these two musics is constantly highlighted, but their creative resources and their production methods rely on a greater complexity of belonging, not to mention their inclusion in the Ivorian music industry, which remains highly international.

There is a tension between an "ethno-nationalist temptation" (Dozon 2000 68 ) of Ivorian society and a musical world that is not limited to national borders. This is particularly true of Coupé-décalé music, which draws its inspiration on both Ivorian Zouglou The international success of Zouglou and Coupé-décalé, with the starization of Magic System and DJ Arafat, had an impact on the entire Ivorian music industry, which gathered renewed momentum after the crisis. Many musicians, arrangers and sound engineers who had left the country returned to Abidjan. The major music companies were back in Abidjan too, which was attracting musicians from across French-speaking West Africa and beyond. This is evidenced by the numerous concerts of African and international stars 72 regularly held in Abidjan, as well as the festivals and performing arts markets organized there. But the context has changed since the 1980s.

In the wake of the 'digital turn,' 73 a new musical ecosystem is taking shape. Most of the analogue recording studios have disappeared, while Digital Audio Workstation (DAW) studios are sprouting and are now at the heart of this new music ecosystem. The reproduction and distribution companies of vinyl records, analogue cassettes and CDs have been replaced by local and international digital aggregators and distributors. 74 New actors and prescribers of culture are appearing: traditional media (private radio stations, cable television channels, 75 68 Dozon op.cit. 69 Zouglou itself derives from the Alloukou rhythms of the Bété of Ivory Coast, but also from Congolese Zaïko and West Indian Zouk (Akindès 2002: 99). 70 Ndombolo is a music and dance that originates from Congolese Rumba and Soukous. 71 Kohlahgen op.cit. 72 Since 2019, Damso, Fally Ipupa, Koffi Olomide, Maître Gim's, Sidiki Diabaté, Singuila, Wizkid, etc. have performed in Abidjan. 73 The Internet has been available in Abidjan since 1996, but the digital turn really started with the arrival of 3G technology and the development of mobile telephony in 2012. 74 Universal and Believe are the most important digital music distributors in Ivory Coast. 75 In particular, Canal + Africa and Trace Africa broadcast numerous African and international music video clips.

DTT), but above all digital media (social media on the Internet) and international phone operators (Orange, MTN, Moov-Etisalat) are carving an increasingly spacious niche for themselves in the music industry by developing their own music content services or by allowing local and international streaming platforms to deliver content via their mobile payment devices. 76 The digital music production, distribution and broadcasting chain has been set up without severing its links with the analogue era, allowing the Ivorian music industry to keep its influence over Francophone Africa.

Geographical and artistic shifts

The individual and professional career paths of these 'makers of Ivorian music', Adolphe Yacé, Patché and Gabriel Adjé, show both an "original subjectivity" (Heinich and Schaeffer 2004: 75 77 ) and a shared history, characterized by populations mixing and multiple influences.

But what emerges more particularly from these three portraits are shifts. First of all, there are geographical shifts: that of Gabriel Adjé, whose family originates from Nigeria, Ghana and Benin, and who is a product of economic immigration. His choice to live in Abidjan bears witness to the multiple waves of immigration that have marked the history of Ivory Coast since the beginning of the twentieth century and to the importance of immigrants in the Ivorian economy and culture, despite the identity crisis that the country experienced at the turn of the twenty-first century. The artistic milieu cannot be compared to that of the cocoa or coffee economy, but let us note that many musicians from West and Central Africa have come to Abidjan since the 1970s, for political or economic reasons. Thanks to a well-structured music industry, Ivory Coast has a strong appeal and clout over a large part of Africa.

Conversely, let us observe Adolphe Yacé's shift, who performs as a musician in the United States and leaves Ivory Coast to graduate in musicology at Boston University. While these transatlantic circulations are in fact marginal (Ndiaye and Robin 2010 78 ), they do reveal real connections between Ivory Coast and the United States in musical circles, which have resulted in numerous artistic vicinities. Moreover, the impact of this type of shift was all the more important for the Ivorian music industry as it concerned the makers of music, the music technicians. From the USA, Adolphe Yacé brought not only high-tech equipment that was not 76 Several local and international streaming platforms are available through Orange phone operator: Deezer, Waw Musik, Digster and Zikplay. 77 available in Ivory Coast, but also digital audio knowledge and an understanding of American show business that he adapted to the Ivorian context once he had returned to Abidjan.

Finally, Patché too embodies geographical shifts, but some of them are part of networks of cooperation and cultural development between France and West Africa, while others are part of a globalized economy of music performances. Throughout this career path, Patché incorporated music, sounds, and techniques destined to constitute reservoirs of knowledge then combined, merged, and metamorphosed to give rise to original compositions and arrangements. The acquisition of this plural, and sometimes even heterogeneous, knowledge allowed him to access freedom, in the sense of a form of musical subjectivity. For Patché, these shifts constitute so many spaces for capturing the knowledge of other artists, whether they are close to or more distant from him, leading to the acquisition of a unique "know-howto-create" (Andrieu 2012: 39 79 ).

In addition to these geographical shifts, there are also artistic shifts. As these three portraits show, Adolphe Yacé, Patché and Gabriel Adjé have an extensive and plural musical culture. This testifies to the cosmopolitanism of the Ivorian musical milieu, to the international networks for the distribution and dissemination of music in Ivory Coast, and also to their musical tastes, which stem from their family heritage and their own career paths. They are eager to discover new music, which is very similar to "omnivorousness" (Peterson 1992 80 ).

The listening practices of Adolphe Yacé, Patché and Gabriel Adjé are constantly evolving to nourish and renew their professional practice, but also to compete with that of other musicians, arrangers or sound engineers. This is a highly challenging music environment, where the various actors are in competition with one another. These listening practices combined with their own practices as musicians provide a wide space of experiences that allows them to work with artists from highly diverse backgrounds, both geographically and artistically. This is clear in Gabriel Adjé's case who collaborates at Universal with musicians from West and Central Africa, in musical styles ranging from Rap to RnB, Afrobeat, Reggae, Zouglou, Coupé-décalé and Manding. But this is also true of Adophe Yacé and Patché, whose clientele is just as eclectic. Moving from one musical style to another, cross-breeding them, even creating new styles, and moving from one cultural universe to another are part of their daily routine. In fact, we are dealing here with individuals who skillfully, even strategically, handle several artistic and cultural registers at the same time, depending on the situation. These practices reveal composite musical identities that allow them to adapt to all kinds of artistic individualities, but also to respond to commercial demand, which is part of a highly competitive market based on constant renewals. In such a context, the pitfall may well be "standard creativity" (Nowak 2017 81 ) adaptable to all kinds of musical styles and cultures, with which many musical productions in Ivory Coast is fraught. But it is also true that the composite character of these musical identities is an asset that allows some arrangers and sound engineers such as Adolphe Yacé, Patché and Gabriel Adjé to be singled out for their 'sound signature'. In return, this increases their attractiveness, their artistic capital within the Ivorian music market.

Conclusion: The sound of Ivorian music in a globalized market

If Adolphe Yacé, Patché and Gabriel Adjé embody "figures of success" (Banégas and Warnier 2001 82 ) in the milieu of Ivorian arrangers and sound engineers, it is eventually accounted for their mastery of the digital audio knowledge of music production and their experience of local and international sound standards. They are thus able to produce a sound that will find its place in a very localized market or a much wider one. Indeed, it is through the arrangement and adjustment of sound effects that aesthetic distinctions appear in the digital age and that the insertion of a music in the global market is negotiated.

Most of the music produced in Abidjan, whether recorded in an independent DAW studio or in the studio of a major label, will in reality only reach out to a local or regional audience.

However, the aim of arrangers and sound engineers such as Adolphe Yacé, Patché and Gabriel Adjé is to produce music that is exportable, i.e. that primarily meets international sound standards. What is obvious in their work is their rejection of an inward-looking attitude, as much as that of an identity assignment. In this sense, the music produced today in Abidjan is both a "music from here and elsewhere" (Kohlahgen 2005: 103 83 ), a music with which young Ivorians identify and which at the same time participates in a global sound system.

Bibliographical references

81 Florence Nowak, "Co-créer dans le marché de la fusion : suivi croisé de deux chanteuses," in S. Andrieu and E. Olivier (eds.), Création artistique et imaginaires de la globalisation (Paris : Hermann, 2017), pp. 151-176. 82 Richard Banégas and Jean-Pierre Warnier, "Nouvelles figures de la réussite et du pouvoir," Politique africaine 82(2001): 5-23. 83 Kohlahgen op.cit.

  reproduce. There was equipment that he could carry too, without customs problems or anything. He came up: "Hey you, I brought you such a thing! You play the guitar, I'll bring you something next time. He'd bring us Jimmy Hendricks records, he's always bring us something. The whole foyer of the Cité des Arts was a tremendous cultural melting pot! So you had access to a lot of things, without necessarily buying them. And without travelling, because everyone who'd travel brought us what they could carry." Adolphe Yacé attended the National Institute of Arts, then the École Normale Supérieure and ended up at University to become a music professor, while leading several University orchestras and a Reggae band. Finally, he engaged in what is known in Ivory Coast as "Research Music", which draws on traditional repertoires to modernize them, which entailed 23 Félix

(

  PSI-Ecodev), a major American non-profit social marketing organization fighting AIDS and advocating family planning in West Africa. Although this position may appear remote from the Ivorian music scene, it nevertheless offered him the opportunity to build skills to operate computers and to become familiar with the emerging Internet in Ivory Coast: new skills that he very swiftly harnessed in his studio work. He thus purchased high-tech equipment and acquired a solid knowledge of new audio production software (Soundforge, Vegas, Protools, etc.). In 2002, at the time of the Ivorian crisis, Adolphe moved to the United States where he stayed 16 years, first as a Master's and PhD student, then as a lecturer at the College of Fine Arts at Boston University. Back in Abidjan in 2018, after the conflict ended, he was approached by Universal to set up its recording studio. And renowned Malian Toumani Diabaté, whom Eliezer worked with in March 2020, was the very first musician to record in this studio. Then, Adolphe Yacé resumed his teaching at INSAAC and at the Houphouët-Boigny University, 25 while setting up recording studio in the Cité des Arts where he grew up. Impressed by the artists he met in the USA, for they can carry out several professional activities at the same 25 Teaching bass guitar, computer-assisted music (CAM) and musical English at INSAAC; teaching musical acoustics, CAM and collective creation at the University. time, he too sets up recording studios and concurrently manages the Pentatonic Sounds & System company, which produces commercials, institutional films, film and show soundtracks. Patché, the musical omnivore Like Adophe Yacé, Patché comes from the San Pedro coastal area. He too was born into a family of musicians in the mid-1970s. Actually, his father modernized a dance, the Bolo (from the English word 'ballroom'), based on a repertoire imported from Ghana at the beginning of the 20th century by the San Pedro sailors (Guiro 1975 26 ). Patché himself is a percussionist and a singer. At his recording studio located in the Marcory Résidentiel district where I met him several times, 27 he explained that he grew up in Treichville. This is the district of Abidjan where, in the 1970s and 1980s, most of the musical life concentrated, in its clubs, concert halls and recording studios. His childhood was filled with music: that his father listened to, especially French Variety -he quoted Charles Aznavour, Joe Dassin, Claude François... -and the music he liked: Reggae, RnB, New Jack and Gospel, all of which were to become the resources for his work as an arranger. In 1997 he formed the RnB group Nexane with his younger brother, and the following year he took part in the 'Sono de Villes' 28 project initiated by Grenoble-based NGO 'Culture et développement', which supported cultural industries in the South. In this context, he participated in workshops and exchanged what he calls "sketches" with Burkinabè and French musicians, with the aim of producing 'fusion music'. This collaborative experience with musicians (and an NGO) from very different backgrounds sharpened his sense of dialogue, which allowed him to adapt to all kinds of music and to be multi-skilled. Sono de Villes led him to perform in Ouagadougou, but also in France and Germany where he discovered musical genres unknown in Ivory Coast, such as Punk and Metal music. In the early 2000s, Patché engaged in training as an arranger and sound engineer in one of Abidjan's major recording studios, Studio Project, run by Gustave Oulahi a.k.a. Tonton Momo. Patché worked there with the famous Ivorian Rap band R.A.S. 29 At the same time, he joined Gbonhi Yo-Yo-Yo 30 a band created by the Ivorian singer Nash in 2005, of which he became one of the main composers and arrangers, as well as the "backeur." 31 Along with Nash, he engaged in Rap Nouchi (Abidjan slang) with multiple influences: from Ivorian Zouglou to Zouk, via RnB. Together, they performed several times at the Eurockéennes festival in Belfort, but also in West Africa and in Europe in many Hip Hop festivals. When he was in France, Patché worked with artists from the Ivorian diaspora, whose tracks he recorded in studios in Montreuil. After RnB and Rap, Patché joined the Coupé-décalé scene, a new Ivorian music that was already very popular among the youth in Abidjan. He composed and arranged tracks for several popular singers, such as DJ Alan or Prince Bardy DJ, and thus established his reputation as an arranger in Ivory Coast, until he was nominated in 2018 for the International Prize for Urban Music and Coupé Décalé (PRIMUD Awards). In 2012, he ended up setting up his own recording studio, Matrix, where he recorded West African artists who have very different musical styles, including Ivorian Zouglou, Manding and Rap groups, but also Congolese artists and French rappers. 32 Since 2019, Patché has been collaborating with Gabriel Adjé for the mixing of the tracks he arranges and records. He used to do the mixing himself, but he recognizes Gabriel's expertise and added value in terms of the sound quality of the recordings.

  69 and Congolese Ndombolo,70 while incorporating a number of globalized references, mainly from American Rap and DJ culture. Shaped by young Ivorians who emigrated to Paris and chose to return to Abidjan, Coupé-décalé praises "a new cosmopolitanism, a resolutely African influence onto the outside world" (Kohlahgen 2005: 105 71 ). The Coupé-décalé singers rose to become emblematic figures of Ivorian youth, whose success and clout testified to Africa's projection into Europe and North America, "where economic and social mobility is possible and where subjects succeed in asserting themselves as individuals" (ibid.).

79

  Sarah Andrieu, "Capture de flux esthétiques et imagination d'une danse intime : itinéraire d'un jeune chorégraphe burkinabè," in E. Olivier (ed.), Musiques au monde. La tradition au prisme de la création (Sampzon: Delatour, 2012), pp. 29-46. 80 Richard A. Peterson op.cit.

  ). Musical life testifies to the cosmopolitanism of the Ivorian capital, starting in the 1950s with the popularization of radio 41 and vinyl records 42 that broadcast French Variety as well as Latin 35 Reinhardt Kosseleck, Le Futur passé. Contribution à la sémantique des temps historiques (Paris:Éditions de l'EHESS, 1990). 36 Akissi Béatrice Boutin and Jérémie Kouadio N'Guessan, "Abidjan, une métropole de plus en plus francophone ?," Le français en Afrique. Revue du Réseau des Observatoires du Français Contemporain en Radio Abidjan was created in 1951 by two technicians from Radio Dakar. It covered the whole country from 1954. In 1961, Radio Abidjan became Radio Côte d'Ivoire, the national radio. See: http://raymondadrienne.blogspot.com/2013/09/lepopee-de-radio-abidjan-2.html

	Afrique Noire 30(2016): 173-186.
	37 Sylvie Bredeloup, "Abidjan sous haute tension," Les Annales de la recherche urbaine 94(2003): 24-32.
	38 Jean-Pierre Dozon, "La Côte d'Ivoire entre démocratie, nationalisme et ethnonationalisme," Politique
	africaine 78(2000): 45-62.
	39 This cosmopolitanism has, however, provoked a feeling of loss of identity among certain Ivorian social
	groups, leading to a policy of ivoirisation of jobs from 1975 onwards (Dozon ibid.: 60).

40

Yacouba Konaté, "Abidjan: malentendu, poésies et lieux propres, " Outre-terre 11(2005): 319-328.

41

  the number of local record 43 John Collins, "Contemporary Ghanaian Popular Music," in E. Charry (ed.), Hip Hop Africa: New African Music in a Globalizing World (Bloomington: Indiana University Press, 2012), pp. 211-231. 44 Losh op.cit. 45 For a history of the Ministry of Culture in Ivory Coast, see: http://culture.gouv.ci/historique/. 46 Graeme Counsel, "Cultural Policy and Music in Mali," Africa Quarterly 43/3(2003): 36-51; Élina Djebbari, Le Ballet National du Mali : créer un patrimoine, construire une nation. Enjeux politiques, sociologiques et esthétiques d'un genre musico-chorégraphique, de l'indépendance du pays à aujourd'hui (Paris, École des Hautes Études en Sciences Sociales, 2013).47 The Akan are a group of populations spread between Ivory Coast and Ghana, who speak the same language with many dialectal variants. 48 Aghi Bahi, 'Il n'y a pas de crocodile à Cocody'. Anthropologie de la communication musicale en Côte d'Ivoire (Mankon: Bamenda, 2021).49 Arol Ketchiemen, Manu Dibango. La plus grande icône de la musique camerounaise (Douala: Éditions du Muntu, 2020).50 Office Togolais du Disque.51 Société Africaine des Techniques Électroniques, which was also a record pressing company and later an analogue cassette reproduction company.

  Nathalie Heinich and Jean-Marie Schaeffer, Art, création, fiction. Entre sociologie et philosophie (Nîmes : Éditions Chambon, 2004).78 Mandiogou Ndiaye and Nelly Robin, "Les migrations internationales en Afrique de l'ouest. Une dynamique de régionalisation renouvelée," Hommes & Migrations 1286-1287(2010): 48-61.

The history of MASA is recounted on its website at: http://www.fr.masa.ci/information-generale/

Clément Guiro, La fanfare krouman (Abidjan: ORSTOM, 1975).

Several interviews were conducted with Patché on March 11 and 14, 2020, and on July 1, 7 and 12, 2021.

https://www.culture-developpement.asso.fr/a-propos-de-c-et-d/nos-actions/ingenierie-culturelle/sono-devilles/

 29 For a presentation of R.A.S., see: https://www.afrisson.com/r-a-s-

903/.30 On Gbonhi Yo-Yo-Yo, see http://forleaders.afrikblog.com/archives/2007/02/09/3752927.html