Ali Aghaei

Nicolas Bochud
email: nicolas.bochud@u-pec.fr

Giuseppe Rosi

Salah Naili

Wave propagation across a functionally graded interphase between soft and hard solids: Insight from a dynamic surface elasticity model

Keywords: Functionally graded interphase, Equivalent interface model, Enriched surface elasticity, Tendon-to-bone attachment, Elastic waves

Wave propagation across a functionally graded interphase between soft and hard solids: Insight from a dynamic surface elasticity model

Introduction

From a mechanical viewpoint, when materials having a dissimilar mechanical nature (e.g., soft and hard) are attached together, they typically display highly non-uniform deformations upon loading, eventually leading to stress concentration at their abrupt interface, which in turn increase the failure probability. An appealing solution to improve the integration between soft and hard materials consists in designing functionally graded interphases, which are typically conceived as multilayers whose composition, microstructure and material properties gradually vary in space, in order to reduce mechanical stresses [START_REF] Thomopoulos | The challenge of attaching dissimilar materials[END_REF]. Naturally present in the human body [START_REF] Sola | Functionally graded materials for orthopedic applications-an update on design and manufacturing[END_REF], graded materials represent a source of inspiration that offers technological solutions for general engineering purposes as well as for biomedical applications. This concerns for instance the skin, which is a complex multilayered system, where each layer has a specific and age-related biomechanical behavior [START_REF] Jor | Computational and experimental characterization of skin mechanics: identifying current challenges and future directions[END_REF], the cortex of long bones, which displays an increasing gradient of porosity from the periosteum to the endosteum that is, in turn, related to a gradual change in mechanical properties such as tensile strength and elasticity [START_REF] Bousson | Distribution of intracortical porosity in human midfemoral cortex by age and gender[END_REF], and the so-called entheses, which are specialized interfacial regions of the musculoskeletal system that allow joining connective tissues, such as tendon, ligament or cartilage, to bone [START_REF] Apostolakos | The enthesis: a review of the tendon-to-bone insertion[END_REF].

In particular, the tendon-to-bone interphase serves the challenging task of connecting two highly dissimilar tissues over a very small region, which is typically a few hundreds of micrometers wide (see Fig. 1a). This interphase has the remarkable ability to minimize stress concentration and related failure modes, with the possibility to withstand forces higher than the body weight for millions of loading cycles [START_REF] Mirzaali | Mechanics of bioinspired functionally graded soft-hard composites made by multi-material 3D printing[END_REF][START_REF] Tits | Joining soft tissues to bone: insights from modeling and simulations[END_REF]. These outstanding features, achieved by means of finely tuned gradients in structure, composition and biomechanical properties at different length scales (see Fig. 1b) [START_REF] Tits | Joining soft tissues to bone: insights from modeling and simulations[END_REF][START_REF] Rossetti | The microstructure and micromechanics of the tendon-bone insertion[END_REF], are currently playing a significant role in the design of bioinspired interphases [START_REF] Dunlop | Artful interfaces within biological materials[END_REF][START_REF] Seidi | Gradient biomaterials for soft-to-hard interface tissue engineering[END_REF][START_REF] Liu | Functional gradients and heterogeneities in biological materials: Design principles, functions, and bioinspired applications[END_REF]. To unlock those mechanisms, computational models were developed both to investigate fundamental anchoring strategies at the microstructural level [START_REF] Genin | Functional grading of mineral and collagen in the attachment of tendon to bone[END_REF][START_REF] Liu | Modelling the mechanics of partially mineralized collagen fibrils, fibres and tissue[END_REF][START_REF] Saadat | Effective elastic properties of a composite containing multiple types of anisotropic ellipsoidal inclusions, with application to the attachment of tendon to bone[END_REF][START_REF] Hu | Stochastic interdigitation as a toughening mechanism at the interface between tendon and bone[END_REF] or to address applied orthopedic strategies at the organ level [START_REF] Liu | Bi-material attachment through a compliant interfacial system at the tendon-to-bone insertion site[END_REF][START_REF] Quental | Full-thickness tears of the supraspinatus tendon: A three-dimensional finite element analysis[END_REF][START_REF] Mantovani | A 3D finite element model for geometrical and mechanical comparison of different supraspinatus repair techniques[END_REF]. Notwithstanding, from a modeling viewpoint, it is highly challenging to bridge the gap between these two levels [START_REF] Shafiei | The very hard and the very soft: Modeling bio-inspired scaled skins using the discrete element method[END_REF], and current models targeting reattachment procedures of the tendon-to-bone interphase should be enriched by including a more detailed description of the microstructure [START_REF] Avgoulas | Adhesive-based tendon-to-bone repair: failure modelling and materials selection[END_REF][START_REF] Kuznetsov | A structural-based computational model of tendon-bone insertion tissues[END_REF]. Indeed, to adequately capture the mechanical behavior of a graded interphase layer, the optimal choice would be to consider its exact geometry and varying mechanical properties across the two surrounding tissues [START_REF] Vayron | Finite element simulation of ultrasonic wave propagation in a dental implant for biomechanical stability assessment[END_REF]. However, this choice can be prohibitive when dealing with complex heterogeneous interphases. In particular, the finite size of such interphase, which is very small compared to that of the surrounding tissues, may cause computational burden when mesh refinements are required for convergence purposes. In contrast, a too basic model that would simplify the interphase to a large extent, or even ignore it, would fail in capturing its mechanics. To face these limitations, a possible solution consists in replacing the finite heterogeneous interphase (see Fig. 1c) by an equivalent interface with specific properties (see Fig. 1d) that retains the mechanical behavior of the original medium over a certain range of validity.

It is commonly accepted that such modeling approach can be satisfactorily addressed by enriching the equivalent model with additional fields [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF]. Initial attempts proposed to tackle this problem by replacing the interphase by an interface with null thickness and purely elastic properties. A general framework for such general elastic interface model was comprehensively described in [START_REF] Bövik | On the modelling of thin interface layers in elastic and acoustic scattering problems[END_REF][START_REF] Hashin | Thin interphase/imperfect interface in elasticity with application to coated fiber composites[END_REF], and was subsequently extended to account for the case where both the interphase and its surrounding media are anisotropic [START_REF] Benveniste | A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media[END_REF][START_REF] Gu | Interfacial discontinuity relations for coupled multifield phenomena and their application to the modeling of thin interphases as imperfect interfaces[END_REF] or the case of an elasto-plastic interphase [START_REF] Mishuris | Imperfect transmission conditions for a thin weakly compressible interface. 2d problems[END_REF][START_REF] Sonato | General transmission conditions for thin elasto-plastic pressure-dependent interphase between dissimilar materials[END_REF]. In the case of dynamic problems, however, and especially when dealing with wave propagation, a purely elastic surface model generally fails in accounting for the local interactions between the mechanical perturbation (e.g., a wave) and the interphase, even if its dimensions are much smaller than the involved mechanical characteristic length (e.g., the wavelength). Within this context, the inertial behavior of the interphase can have a considerable impact on its macroscopic dynamic response and should thus be included into the modeling strategy. This observation was also at the basis of the concept of a structural interface that possesses a finite thickness, which was introduced in [START_REF] Bigoni | Statics and dynamics of structural interfaces in elasticity[END_REF] and further developed in [START_REF] Brun | Dynamics of structural interfaces: filtering and focussing effects for elastic waves[END_REF], where the role of its inertial properties was highlighted.

In the case of resonant meta-interfaces, an alternative approach was to obtain effective jump conditions by applying a suitable homogenization process [START_REF] Touboul | Time-domain simulation of wave propagation across resonant meta-interfaces[END_REF][START_REF] Touboul | Effective resonant model and simulations in the time-domain of wave scattering from a periodic row of highly-contrasted inclusions[END_REF]. Moreover, some studies on elastic wave propagation showed that when the interphase is located between two surrounding media with microstructure, inertial properties also play a pivotal role in the modeling of the equivalent interface [START_REF] Dell'isola | Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua[END_REF][START_REF] Placidi | Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials[END_REF][START_REF] Scala | Effects of the microstructure and density profiles on wave propagation across an interface with material properties[END_REF].

Following these recent findings, and in line with a former study by our group [START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF], we introduce here an enriched equivalent interface model, whose properties are defined by means of surface kinetic and potential energy densities. Furthermore, to account for the nonlinear gradients in mechanical properties across the interphase found at the tendon-to-bone attachement [START_REF] Aghaei | Assessing the effective elastic properties of the tendonto-bone insertion: a multiscale modeling approach[END_REF], we hypothesize that the displacement field can be approximated using a piece-wise affine profile, whose characteristics depend upon an additional degree-of-freedom within the interface. The performance of our modeling approach is evaluated by calculating the frequency-dependent reflection coefficient of a plane pressure wave under normal inci-dence. First, based on energetic concepts, the identification of the specific surface properties of the equivalent interface model is achieved by comparison with the reference interphase model. Second, an optimization procedure is conducted to investigate the impact of the additional degree-of-freedom. Third, the performance of the optimal equivalent model is compared to different models available in the literature, which typically serve as a baseline in finite element (FE) simulations at the organ scale. Our numerical results show that this enriched model with specific interface conditions provides a very accurate approximation of the reference model over a broad frequency range, thus outperforming more simplistic models that fail in capturing the complex dynamics of the interphase. As a by-product, the link between the position of the additional degree-of-freedom and the microstructural features of the interphase (e.g., competing gradients in mineral content and collagen fibers organization) is also discussed, thus opening promising perspectives for characterizing the tendon-to-bone attachment status. Elastic waves indeed represent a relevant nondestructive means to probe the interphase quality, as they possess intrinsic sensitivity to the mechanical properties contributing to the tendon-to-bone attachment strength. Overall this modeling approach represents the first building block for developing more sophisticated models targeting reattachment procedures at the organ scale that incorporate a more detailed description of properties at lower length scales.

The paper is structured as follows: Sect. 2 introduces the theoretical fundamentals of our modeling approach. The numerical results are then presented in Sect. 3. Finally, the strengths and limitations of the proposed model are discussed in Sect. 4.

Theory

Based on the variational principles [START_REF] Dell'isola | On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface[END_REF][START_REF] Dell'isola | A phenomenological approach to phase transition in classical field theory[END_REF][START_REF] Dell'isola | Variational principles are a powerful tool also for formulating field theories[END_REF], this section first presents the governing equations and boundary conditions for the two models depicted in Fig. 1c-d: (1) A reference model, which consists in a finite thickness interphase with a gradient in mechanical properties, surrounded by two homogeneous media (see Subsect. 2.1), and (2) an equivalent model in which the interphase is replaced by specific interface conditions between the two homogeneous media (see Subsect. 2.2). This general framework is then reduced to the specific case of a pressure plane wave propagating under normal incidence (see Subsect. 2.3). Finally, the strategy to identify the coefficients of the equivalent interface model is described in Subsect. 2.4.

Reference interphase model

Let us consider the Cartesian frame of reference with coordinates R(O; e 1 , e 2 , e 3 ), where O is the origin and (e 1 , e 2 , e 3 ) is an orthonormal basis for the space. The coordinates of a point M in R are specified by (x 1 , x 2 , x 3 ) and the time is denoted by t. As depicted in Fig. 2, the reference model consists of two homogeneous half-spaces

Ω -= {(x 1 , x 2 , x 3 ) | x 2 , x 3 ∈ R and x 1 < -h 2 } and Ω + = {(x 1 , x 2 , x 3 ) | x 2 , x 3 ∈ R and x 1 > h 2 }, separated by an heterogeneous interphase layer of thickness h, namely Ω I = {(x 1 , x 2 , x 3 ) | x 2 , x 3 ∈ R and - h 2 < x 1 < h 2 }
, where R is the set of real numbers. In what follows, the equations of motion are derived according to the least action principle. To this end, we introduce the action functional A ref of the considered system as in [START_REF] Placidi | Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials[END_REF],

A ref = ˆΩ×(0,t f ) T + + T -+ T I -U + -U --U I dA, (1)
where Ω = Ω ± ∪ Ω I spans the complete domain of interest, (0, t f ) is a time interval and dA represents the differential volume element. The kinetic and potential energy densities per unit volume, T and U , associated with each domain of interest are defined as,

T ± = 1 2 ρ ± u± . u± and U ± = 1 2 σ ± : ± , (2)
T I = 1 2 ρ I (x 1 ) uI . uI and U I = 1 2 σ I : I , (3)
where ρ ± are the mass densities of the domains Ω ± , and ρ I (x 1 ) is the mass density of the interphase Ω I , which varies along the x 1 -direction. The displacement vector associated with each domain is denoted by u, where the superimposed dot denotes the first derivative with respect to time. The stress and infinitesimal strain tensors associated with each domain are denoted by σ and , respectively, the latter being defined as = (1/2) ∇u + (∇u) T , whereby ∇ and the superscript T denote the gradient and transpose operators, respectively.

The operators dot (.) and colon (:) denote the scalar product of two vectors and the double contracted product of two tensors, respectively.

The least action principle implies that the variation of the action defined by Eq. ( 1)

should verify that δA ref = 0. This condition, along with the consideration of kinematic constraints on the boundaries, i.e., the continuity of displacements between the domains Ω ±

and Ω I , leads to the following boundary value problem for the reference model,

                   ∇.σ ± -ρ ± ü± = 0 ∀M ∈ Ω ± ∇.σ I -ρ I (x 1 )ü I = 0 ∀M ∈ Ω I σ -n -+ σ I n I-= 0 u --u I = 0 for x 1 = - h 2 σ I n I+ + σ + n + = 0 u I -u + = 0 for x 1 = h 2 , ( 4
)
where ∇. denote the divergence operator. The constitutive relation associated with each domain is defined in the frame of the linear elasticity, i.e., σ ± = C ± ± and σ I = C I (x 1 ) I , whereby C ± is the fourth-order elasticity tensor of the domains Ω ± , and C I (x 1 ) is the fourthorder elasticity tensor of the interphase Ω I , which varies with respect to the x 1 -direction.

Equivalent interface model

This section introduces the equivalent model based on surface elasticity, in which the finite thickness heterogeneous interphase is replaced by specific interface conditions. To this end, kinetic and potential energy densities of the interphase, T I and U I from Eq. ( 3), are now substituted by surface energy densities. The action functional A for the equivalent system can thus be defined in a similar manner as in Sect. 2.1,

A = ˆΩ×(0,t f ) T + + T --U + -U -dA + ˆ∂Ω×(0,t f ) T S -U S dS, ( 5
)
where ∂Ω is the mid-surface of the domain Ω I and dS represents the differential surface element, so that

T S = ˆxl -h 2 T I dx 1 + ˆh 2 x l T I dx 1 and U S = ˆxl -h 2 U I dx 1 + ˆh 2 x l U I dx 1 , (6)
where

-h 2 < x l < h 2 .
Owing to the continuity of displacements at the boundaries, we can state that u

I (-h 2 ) = u -(-h 2 ) and u I ( h 2 ) = u + ( h 2 )
, where, for the sake of conciseness, the dependence of the fields on the spatial coordinates x 2 and x 3 , as well as on time t, is dropped. We furthermore assume that the integrals in Eq. ( 6) are quadratic forms of the displacement and velocity fields evaluated on the x 2 x 3 -plane for x 1 equal to -h 2 , x l or h 2 . Altogether, these hypotheses allow deriving a general form for the surface energy densities as,

T S = 1 2 m + u+ . u+ + m l ul . ul + m -u-. u-+ 2γ 1 u-. ul + 2γ 2 ul . u+ , (7)
U S = 1 2 K 1 u l -u -. u l -u -+ 1 2 K 2 u + -u l . u + -u l , (8)
where u l = u I (x l ). Note that, by abuse of notation, we dropped the dependence on space for all displacements. These surface energy densities, which are concentrated in the mid-surface mass system depicted in Fig. 3. Within this frame, the second-order tensors m ± and m l represent masses concentrated at both sides of the interface and at position x l , respectively;

the second-order tensors K 1 and K 2 represent surface stiffnesses; and finally the secondorder tensors γ 1 and γ 2 account for the kinetic interactions between the displacement fields. It is worth pointing out that, in the most general case, all these constitutive tensors are function of space and time (or frequency in the harmonic regime). Nevertheless, for the model to be exploitable, i.e., to link these tensors to the physical characteristics of the interface rather than to its dynamic properties, a possibility is to turn them into constants.

Here the calculation of these tensors will be achieved by selecting specific profiles for the displacement and velocity fields across the interphase, u I (x 1 ) and uI (x 1 ), which fulfill the aforementioned constraints at the location x 1 given by the values -h 2 , x l or h 2 . The natural consequence of this choice is that all surface energy terms will be considered as approximations only.

Again, the variation of the action δA should verify that δA = 0, thus leading to the following boundary value problem,

           ∇.σ ± -ρ ± ü± = 0, ∀M ∈ Ω ± σ -n --K 1 (u l -u -) + m -ü-+ γ 1 ül = 0 σ + n + + K 2 (u + -u l ) + m + ü+ + γ 2 ül = 0 K 1 (u l -u -) -K 2 (u + -u l ) + m l ül + γ 1 ü-+ γ 2 ü+ = 0 (9)
where the first equation retains the same form than that of Eq. ( 4), whereas the other three now account for the specific interface conditions, in which, by abuse of notation, the stress tensors are now stated as σ -= σ Ih 2 and σ + = σ I h 2 . It should be noted that this boundary value problem could be easily implemented in a standard FE code, as it only involves a modification of the boundary conditions.

Pressure plane wave propagation under normal incidence

We now consider the problem of a plane elastic wave propagating under normal incidence across the considered functionally graded interphase. In this way, our modeling approach is reduced to an incident and a reflected longitudinal bulk wave in the domain Ω -and a transmitted longitudinal bulk wave in the domain Ω + . By assuming a general harmonic solution for the interphase Ω I , the general solution for a plane wave propagating along the

x 1 -direction reads as u -(x 1 , t) = A i exp (jk - p x 1 ) + A r exp (-jk - p x 1 ) exp (-jωt) u + (x 1 , t) = A t exp (j(k + p x 1 -ωt)) u I (x 1 , t) = g(x 1 )exp(-jωt) , ( 10
)
where ω represents the angular frequency and j is the unit imaginary number. The variable k ± p = ω/c ± p denotes the wave number, which depends upon the longitudinal bulk wave velocity c ± p in the domain Ω ± . The variables A i , A r and A t denote the amplitudes of the incident, reflected and transmitted plane waves, respectively. The function g(x 1 ) represents the unknown amplitude of the harmonic solution in the interphase, which accounts for the material heterogeneity along the x 1 -direction. Substituting the wave solutions from Eq.

(10) into the boundary value problem from Eq. ( 4), and working through these analytical equations to remove the variables A r and A t , yield the following boundary value problem for the function g(x 1 ) in strong form,

             C I 11 (x 1 )g ,1 (x 1 ) ,1 + ω 2 ρ I (x 1 )g(x 1 ) = 0 C I 11 - h 2 g ,1 - h 2 + jk - p C - 11 g - h 2 -2jk - p C - 11 A i exp -jk - p h 2 = 0 C I 11 h 2 g ,1 h 2 -jk + p C + 11 g h 2 = 0 , (11)
where the derivative with respect to x 1 is denoted by ( ,1 ) and C 11 is the stiffness coefficient along the x 1 -direction (in Voigt notation) associated with each domain. The first relation of Eq. ( 11) is an ordinary differential equation with respect to the coordinate x 1 , whereas the two last relations account for the boundary conditions at positions x 1 = -h 2 and x 1 = h 2 , respectively. Note that such boundary value problem in strong form must be solved numerically.

In the same vein, the boundary value problem associated with the equivalent interface model (recall Eq. ( 9)) can be simplified as follows in the case of a pressure plane wave under normal incidence,

           C ± 11 u ± ,11 -ρ ± ü± = 0, ∀M ∈ Ω ± C - 11 u - ,1 -K 1 (u l -u -) + m -ü-+ γ 1 ül = 0 -C + 11 u + ,1 + K 2 (u + -u l ) + m + ü+ + γ 2 ül = 0 K 1 (u l -u -) -K 2 (u + -u l ) + m l ül + γ 1 ü-+ γ 2 ü+ = 0 , (12)
where it should be noted that the second-order tensors of the surface energy terms are now all reduced to scalar coefficients. Furthermore, we hypothesize that the general solution for the displacement in the interface can be stated as

u l (t) = B exp(-jωt), ( 13
)
where B represents the unknown amplitude of the harmonic solution in the interface. Substituting Eq. ( 13), along with the wave solutions associated with the surrounding domains Ω ± from Eq. ( 10), into the boundary value problem from Eq. ( 12) leads to the following linear equation system

  -jk - p C - 11 + K 1 -ω 2 m - -K 1 -ω 2 γ 1 0 0 -K 2 -ω 2 γ 2 -jk + p C + 11 + K 2 -ω 2 m + -K 1 -ω 2 γ 1 K 1 + K 2 -ω 2 m l -K 2 -ω 2 γ 2     A r B A t   =   -jk - p C - 11 + K 1 -ω 2 m -A i 0 (K 1 + ω 2 γ 1 ) A i   , (14)
whose solution allows determining the unknown amplitudes, A r and A t , of the reflected and transmitted waveforms. Note that, in contrast to the boundary value problem associated with the reference interphase model (recall Eq. ( 11)), the resulting system for the equivalent interface can be solved analytically.

Identification of the equivalent model coefficients

This section presents the adopted strategy for identifying the coefficients of the surface kinetic and internal energy densities, i.e., m -, m l , m + , γ 1 , γ 2 , K 1 , and K 2 . Towards this goal, we hypothesize that the displacement field is directed towards the surface normal, i.e., u I = u I (x 1 ) e 1 , and that it can be approximated using a piece-wise affine profile with respect to the coordinate x 1 (see Fig. 4).

In such a case, the kinetic and potential energy densities across the interphase (recall Eq. ( 3)) can be written as

T I = 1 2 ˆh 2 -h 2 ρ I (x 1 ) v I (x 1 ) 2 dx 1 and U I = 1 2 ˆh 2 -h 2 C I 11 (x 1 ) u I ,1 (x 1 ) 2 dx 1 , (15)
where in which u -, u l and u + are the values of the displacement field u I (x 1 ) at the location x 1

u I (x 1 ) =          2 (u l -u -) h + 2x l x 1 + u l h + 2u -x l h + 2x l for -h 2 < x 1 < x l 2 (u + -u l ) h -2x l x 1 - 2u + x l -u l h h -2x l for x l < x 1 < h 2 , (16)
given by the values -h 2 , x l or h 2 . Note that, since we assumed a general harmonic solution for the displacement field in the interphase, the velocity field v I (x 1 ) will have the same piece-wise form, modulo the constant complex factor jω, than that of Eq. ( 16), with v -, v l and v + being the values of the velocity field at the location x 1 given by the values -h 2 , x l or h 2 . According to the pressure plane wave assumption, the kinetic and potential energies per unit surface, T S and U S , can now be written as

T S = 1 2 [m + (v + ) 2 + m l (v l ) 2 + m -(v -) 2 + 2γ 1 v -v l + 2γ 2 v l v + ] , U S = 1 2 K 1 (u l -u -) 2 + 1 2 K 2 (u + -u l ) 2 , (17)
where the necessary conditions for the kinetic energy density, T S , to be a positive definite quadratic form are that m ± > 0, γ 2 2 < 1 2 m + m l , and γ 2 1 < 1 2 m -m l . By inserting the displacement and velocity fields, u I (x 1 ) and v I (x 1 ), into Eq. ( 15), the outcome can then be compared to Eq. ( 17) by making use of the relations defined in Eq. ( 6), thus allowing for the identification of the coefficients belonging to the surface kinetic and potential energy densities T S and U S as

m -= 4 ˆxl -h 2 ρ I (x 1 ) x 1 -x l h + 2x l 2 dx 1 m + = 4 ˆh 2 x l ρ I (x 1 ) x 1 -x l h -2x l 2 dx 1 m l = ˆxl -h 2 ρ I (x 1 ) h + 2x 1 h + 2x l 2 dx 1 + ˆh 2 x l ρ I (x 1 ) h -2x 1 h -2x l 2 dx 1 γ 1 = 2 ˆxl -h 2 ρ I (x 1 ) (h + 2x 1 )(-x 1 + x l ) (h + 2x l ) 2 dx 1 γ 2 = 2 ˆh 2 x l ρ I (x 1 ) (h -2x 1 )(x 1 -x l ) (h -2x l ) 2 dx 1 K 1 = 1 x 2 l ˆxl -h 2 C I 11 (x 1 )dx 1 K 2 = 1 (h -x l ) 2 ˆh 2 x l C I 11 (x 1 )dx 1 . (18)
In our earlier study [START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF], the equivalent interface model was associated with an affine displacement field (dashed blue line in Fig. 4). To serve as a comparison with Eqs. ( 17)-( 18), the related expressions for the kinetic and potential energy densities per unit surface, along with the identified coefficients, are briefly recalled in Appendix A.

Numerical results

The performance of the proposed enriched model is assessed by evaluating the frequencydependent reflection coefficient in a reflection/transmission problem between two homogeneous solid half-spaces, which are separated by a finite thickness and heterogeneous interphase. First, we identify the specific surface properties of the equivalent interface by comparison with the reference interphase. Second, we perform an optimization procedure to investigate the impact of the position x l of the additional degree-of-freedom on the modeling performance. The optimal equivalent model is subsequently compared to different baselines.

Towards these goals, we consider here an interphase whose stiffness and density vary along the x 1 -direction. This configuration is inspired by our recent multiscale modeling results [START_REF] Aghaei | Assessing the effective elastic properties of the tendonto-bone insertion: a multiscale modeling approach[END_REF], which showed that the tendon-to-bone interphase can be seen as a continuous functionally graded material. The selected model parameters for the two surroundings tissues, i.e., tendon Ω -and bone Ω + , together with the finite thickness h of the interphase, are summarized in Tab. 1.

The gradients in mechanical properties across the interphase Ω I , which are used as further input in the reference model, are depicted in Fig. 5. The effective stiffness variation at the mesoscale is the result of competing gradients in mineral content and collagen fibers organization at lower length scales. Indeed, it has been shown that the linear increase in mineral content caused a stiffening of the interphase that became significant beyond a certain percolation threshold (approximately located between 0 -50 µm in Fig. 5a), whereas the decreasing collagen fiber organization across the interphase led to a reduced tissue stiffness along the main fibers direction [START_REF] Liu | Modelling the mechanics of partially mineralized collagen fibrils, fibres and tissue[END_REF][START_REF] Aghaei | Assessing the effective elastic properties of the tendonto-bone insertion: a multiscale modeling approach[END_REF]. In contrast, the effective mass density evolves linearly between the two surrounding tissues (see Fig. 5b).

Evaluation of the equivalent interface model

To validate the proposed equivalent interface model, we first evaluate the reflection coefficient R = |A r /A i | 2 as a function of frequency, in which the amplitude A i of the incident plane wave was set to unity. For a better interpretation, the reflection coefficient R was calculated as well as a function of the ratio between the average wavelength λ and the thickness h of the interphase. The average wavelength λ was defined as

λ = cI p f , with cI p = CI 11 ρI , (19)
where cI p , CI 11 and ρI are the average phase velocity, the average stiffness and the average mass density, respectively.

The reference model from Eq. ( 11) is solved using a standard FE code such as Comsol Multiphysics R , from which the reflection coefficient, R ref can be derived as

R ref = A r A i 2 = g - h 2 -A i exp -jk - p h 2
A i exp jk - p h 2 2 . ( 20
)
In contrast, the reflection coefficient associated with the equivalent interface model can be calculated analytically using Eq. ( 14). Note, however, that this enriched modeling approach now depends upon the position of the additional degree-of-freedom x l , whose optimal location is a priori not known. To serve as an example, the coefficients of the kinetic and potential energy densities were first calculated for two arbitrarily selected values of x l using Eq. ( 18). The numerical results are depicted in Fig. 6, in which the proposed model (dashed red line) with different values of x l is compared to the reference interphase model (continuous black line), to a homogeneous interphase Ω I with averaged mechanical properties (dashed gray line), and to an abrupt transition (continuous gray line) corresponding to the case where the surroundings tissues Ω ± are placed directly in contact. As can be observed in the enlargement of Fig. 6, for a low frequency regime below 0.3 MHz (i.e., high ratio λ/h), the equivalent interface model satisfactorily captures the dynamics of the reference model, independently of the position of the additional degree-of-freedom x l . At an intermediate frequency regime around 0.3-2.5 MHz, the choice of x l has a significant impact on the ability of our model to match the reference one, and thus account for the gradient in mechanical properties, which in turn reflect microstructural features at lower length scales. As expected, when the ratio of the average wavelength to the interphase thickness decreases towards values below 3, the deviation from the reference model increases, and our approximated model fails in capturing the complex dynamics of the interphase, which is associated with stronger wave-interphase interactions. Nonetheless, it is worth mentioning that, overall, the deviation of this model from the reference ones is much lower than that obtained using a simple abrupt transition or a homogeneous interphase with averaged mechanical properties, which represent the commonly used baselines in FE simulations at the organ scale [START_REF] Wakabayashi | Mechanical environment of the supraspinatus tendon: a two-dimensional finite element model analysis[END_REF][START_REF] Debski | Stress and strain in the anterior band of the inferior glenohumeral ligament during a simulated clinical examination[END_REF][START_REF] Pena | A three-dimensional finite element analysis of the combined behavior of ligaments and menisci in the healthy human knee joint[END_REF].

Optimization of the equivalent interface model

Second, an optimization procedure was conducted to further investigate the impact of the additional degree-of-freedom position x l on the performance of the proposed model. To this end, the relative cumulative error on the reflection coefficient (denoted by err) between the one of the reference model (denoted by R ref ) and the one of the equivalent interface model (denoted by R) was calculated by sweeping the frequency and the position of x l across the interphase as

err(x l , f ) = 1 N max Nmax n=1 R ref (f n ) -R(x l , f n ) R ref (f n ) 2 , (21)
where N max is an integer value corresponding to the maximal frequency position. Figure 7a represents the dynamics of the error as a function of the position of x l and the frequency f . As can be observed, at a low frequency regime below 0.3 MHz, the equivalent interface model 21)) between the one of the reference model and the one of the equivalent interface model as a function of the frequency and the position of the additional degree-of-freedom x l across the interphase. The dashed red line represents the optimal x l -path as a function of the frequency, whereas the black dots depict the local minima x

(1) l and x

(2)

l . The region around the second local minima is displayed in the insert. (b) Relative cumulative error along the optimal x l -path as a function of the frequency (dashed red line). The error between the reference model and the model presented in [START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF] is shown for comparison (dashed blue line). The gray line represents the frequency threshold (0.3 MHz) below which the error is independent of the position x l .

provides an accurate approximation of the reference model independently of the position of the additional degree-of-freedom x l . Towards higher frequencies, the performance of the equivalent model slightly decreases, but there is a specific path along x l (displayed as a dashed red line) for which the error remains reasonably low. Interestingly, there are two local minima located at around 1 and 2.6 MHz along this path (black dots), for which the optimal values of xl are equal to 32 and 1.2 µm, respectively (i.e., thicknesses that fall within the range corresponding to the percolation threshold in Fig. 5a). A more advanced insight is provided in Fig. 7b, which depicts the relative cumulative error along the optimal x l -path as a function of the frequency, thus clearly highlighting the presence of two local minima, i.e., x (1) l and x

(2) l . It is worth mentioning that the error associated with x

(2) l at 2.6 MHz is of the same order than that obtained using the model from [START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF] at 0.3 MHz (low frequency regime represented as a continuous gray line). Therefore, fine-tuning the position of the additional degree-of-freedom x l in our modeling approach allows enlarging nearly 8 times the validity range with respect to that model.

To further assess the performance of our enriched model, Fig. 8 now depicts the obtained frequency-dependent reflection coefficient R for the optimal position x 4) the model (dashed blue line) associated with a displacement field with an affine profile (Ref. [START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF]); and ( 5) the equivalent interface model (dashed red lines) for an optimal value of x l equal to 1.2 µm.

optimization, our equivalent model with specific interface conditions provides a very accurate approximation of the reference model over a broad frequency range up to around 2.6 MHz.

It thus outperforms the results obtained with the equivalent model from [START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF], which still remains valid but for a much narrower frequency regime. In contrast, an equivalent model that only accounts for elastic interface conditions totally fails in capturing the complex dynamics of the interphase.

Discussion

Computational modeling of functionally graded intephases between soft and hard materials has provided important insights on underlying anchoring mechanisms at the microscale and on surgical repair strategies at the macroscale. To unravel additional mechanical characteristics related to the functioning of such complex interphases, as well as to enhance the characterization of bioinspired attachment procedures or tissue engineered implants, current models should include a precise description of the graded mechanical behavior of the interphase across several length scales. However, such modeling strategy raises several challenges from a computational viewpoint, especially in the case of dynamic loading.

In this study, we proposed an equivalent model based on surface elasticity, in which the finite thickness, continuous functionally graded interphase between the tendon and bone can be substituted by a zero-thickness interface. This was achieved by replacing the kinetic and potential energies of the interphase by specific surface energy density terms, whose coefficients could be identified in a straightforward manner using known profiles for the effective stiffness and density across the interphase. With respect to previously reported models, this novel formulation introduces an additional degree-of-freedom, which allows for a more refined description of the mechanical fields (e.g., displacements and velocities) across the interphase. The performance of the model was evaluated in the context of wave propagation.

The main findings from this numerical study were as follows: (1) as the performance of enriched models generally depends upon a proper selection of the kinematics across the interphase, our numerical results showed that the displacement field could be adequately modeled as a two-piece profile associated with an additional degree-of-freedom; (2) the assessment of the frequency-dependent reflection coefficient showed that our optimized enriched model can provide a very accurate approximation of the reference model over a broad frequency range up to around 2.6 MHz, thus outperforming the results obtained with more simplistic models; and (3) this model has a much lower computational cost than the reference one, as the interphase must not be incorporated (i.e., meshed) explicitly. Overall, our numerical results

showed that the proposed approach is well-suited for the solution of a complete elastodynamics problem in the frequency-domain, as it captures the complex mechanical behavior of the tendon-to-bone interphase over a broad frequency range. Interestingly, all these significant outcomes are reached based on the optimal position of the additional degree-of-freedom, whose choice may depend on several parameters, among which the underlying behavior of the gradient in mechanical properties across the interphase is thought to be a determinant factor.

To test this hypothesis, we also investigated the impact of microstructural damage at the tendon-bone interphase on the optimal position of the additional degree-of-freedom (see Fig. 9). Clinical burden related to physical overloading, disuse or systemic diseases may manifest as a delayed onset of mineralization and an increasing angular deviation from the main collagen fibers direction across the tendon-to-bone insertion [START_REF] Thomopoulos | Tendon to bone healing: differences in biomechanical, structural, and compositional properties due to a range of activity levels[END_REF][START_REF] Benjamin | Where tendons and ligaments meet bone: attachment sites ('entheses') in relation to exercise and/or mechanical load[END_REF][START_REF] Deymier | The multiscale structural and mechanical effects of mouse supraspinatus muscle unloading on the mature enthesis[END_REF], which in turn lead to modified graded mechanical properties at the macroscale [START_REF] Liu | Modelling the mechanics of partially mineralized collagen fibrils, fibres and tissue[END_REF][START_REF] Aghaei | Assessing the effective elastic properties of the tendonto-bone insertion: a multiscale modeling approach[END_REF]. A can be observed, for such a damaged configuration the gradient in stiffness displays a slightly shifted percolation threshold with respect to the nominal case (see Fig. 9a), whereas the gradient in mass density now has a piece-wise profile (see Fig. 9b).

Using these modified graded profiles as input of our enriched model and applying the optimization procedure described in Subsect. 3.2 yield different values for the optimal position xl and for the identified coefficients associated with the surface energy density terms (see Tab. 2). Interestingly, the optimal position xl has now raised to around 9 µm with respect to its nominal value, and this shift is of the same order than that related to the percolation threshold observed in the stiffness profile (see Fig. 9a). Beside the shift of xl , the identified coefficients are also impacted, but their interpretation is not straightforward and we postpone this analysis to future works. Nevertheless, our model proves to be sensitive to slight changes of the graded properties across the interphase (see Fig. 10) and could potentially be used for characterization purposes. Although the proposed enriched framework was established to account for a general 3-D formulation, the investigated configuration was subsequently reduced to the rather simple case of a plane elastic wave propagating under normal incidence across a functionally graded interphase, whose mechanical properties vary with respect to a single direction only. In a such case, the reflection/transmission problem is limited to the analysis of longitudinal bulk waves. Therefore, to further tackle the complexity of the tendon-to-bone attachment, whose geometry, for instance at the Achilles tendon, is characterized by a shallow attachment angle and an optimized shape of the outward splay, as well as by interdigitation mechanisms (recall Fig. 1a-b) [START_REF] Liu | Mechanisms of bimaterial attachment at the interface of tendon to bone[END_REF][START_REF] Abraham | From meniscus to bone: a quantitative evaluation of structure and function of the human meniscal attachments[END_REF], future modeling efforts should consider 2-D geometries using plane wave propagating under oblique incidence (i.e., both longitudinal and shear waves) or surface waves [START_REF] Eremeyev | Transverse surface waves on a cylindrical surface with coating[END_REF]. As a further drawback, the performance of the enriched model was here only assessed based on the analysis of the frequency-dependent reflection coefficient, thus only displaying information on the amplitude spectrum of the propagating waves. However, the phase of the waves plays a crucial role as well, especially for the analysis of time-domain signals traveling across thin interphases. Further studies are warranted to precisely address the impact of the equivalent modeling approach on the phase spectrum. Overall this modeling approach represents the first critical step towards the development of more sophisticated models targeting mechanical strategies for improving diagnostic and reattachment procedures, which will not only be valuable for evaluating the tendon-to-bone insertion, but also other multiscale and graded biological attachments such as the bone-implant interphase [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF][START_REF] Le Cann | Spatio-temporal evolution of hydroxyapatite crystal thickness at the bone-implant interface[END_REF] and new generations of engineered composite materials [START_REF] Xu | A review on functionally graded structures and materials for energy absorption[END_REF].

Conclusion

The original equivalent interface model presented in this study aimed at mimicking the elastodynamic behavior of a finite heterogeneous interphase, while reducing the overall computational burden related to the consideration of its complete geometry. It has been shown that, in the case of a dynamic experiment, i.e., plane pressure waves under normal incidence, the proposed enriched model remains valid over a broad frequency range and outperforms more simplistic models that typically serve as a baseline in FE simulations at the organ scale. The key feature of the model was the introduction of an additional degree of freedom in the interface, which allowed for a finer description of the kinematic fields within the interphase that accounted for both its inertial behavior and the kinetic interactions with the surrounding media. Furthermore, it has been shown that the model was sensitive to slight variations of the graded mechanical properties across the interphase, thus opening potential applications in the field of nondestructive testing and characterization. Finally, it should be noted that such modeling approach is straightforwardly extendable to account for more complex interphase geometries.

Figure 1 :

 1 Figure 1: (a) Organ level illustration of the tendon-to-bone attachment at the insertion site corresponding to the Achilles tendon (image adapted from Ref. [12]); (b) Schematic of the attachment at the microstructural level, highlighting the gradients in composition and biomechanical properties across the tendon-to-bone interphase (image adapted from Ref. [8]); (c) Finite thickness interphase model with varying mechanical properties across the interphase; and (d) Equivalent interface model with specific properties.

Figure 2 :

 2 Figure 2: Modeling configuration of the reference model: A finite thickness interphase Ω I with varying mechanical properties along the x 1 -direction is surrounded by two homogeneous half-spaces Ω ± . Superscripts +, -and I denote the variables associated with the domains Ω + , Ω -, and Ω I , respectively.

Figure 3 :

 3 Figure 3: Modeling configuration of the equivalent model with specific interface conditions. The generalized spring-mass system illustrates the role played by the additional degree-of-freedom l to account for the dynamic interactions across the interphase.

Figure 4 :

 4 Figure 4: Illustration of the displacement field across the interphase for an arbitrarily selected frequency: Comparison between the exact profile (continuous black line) used in the reference model and the piece-wise affine profile (dashed red line) used in the equivalent interface model. The affine profile (dashed blue line) proposed in [38] is shown for comparison.

Figure 5 :

 5 Figure 5: Graded mechanical properties across the tendon-to-bone interphase: (a) effective stiffness coefficient C 11 and (b) effective mass density ρ. These trends are based upon the multiscale model reported in [39].

Figure 6 :

 6 Figure 6: Frequency-dependent reflection coefficient R obtained using: (1) the reference model (continuous black line); (2) a homogeneous interphase (dashed gray line); (3) an abrupt transition (continuous gray line); and (4) the equivalent interface model (dashed red lines) for two arbitrarily selected values of x l equal to -50 and 50 µm. The black arrow indicates the resulting trend with respect to increasing values of x l .

Figure 7 :

 7 Figure 7: (a) Relative cumulative error on the reflection coefficient (Eq. (21)) between the one of the reference model and the one of the equivalent interface model as a function of the frequency and the position of the additional degree-of-freedom x l across the interphase. The dashed red line represents the optimal x l -path as a function of the frequency, whereas the black dots depict the local minima x

  line), which is compared to the reference model (continuous black line), an abrupt transition (continuous gray line), the equivalent model from[START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF] (dashed blue line) and an equivalent model possessing only elastic properties (dotted red line), that is to say non-inertial interface conditions (i.e., m + = m -= m l = 0 kg.m -2 and γ 1 = γ 2 = 0 kg.m -2 ). As expected from the

Figure 8 :

 8 Figure 8: Frequency-dependent reflection coefficient R obtained using: (1) the reference model (continuous black line); (2) an abrupt transition (continuous gray line); (3) a non-inertial equivalent interface model (dotted red line); (4) the model (dashed blue line) associated with a displacement field with an affine profile (Ref.[START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF]); and (5) the equivalent interface model (dashed red lines) for an optimal value of x l equal to 1.2 µm.

Figure 9 :

 9 Figure 9: Graded mechanical properties across the tendon-to-bone interphase representative of a damaged configuration (dashed lines): (a) effective stiffness coefficient C 11 and (b) effective mass density ρ. The nominal case from Fig. 5 is shown for comparison (continuous lines).

Figure 10 :

 10 Figure 10: Comparison between the reference model (continuous black lines) and the optimal equivalent interface model (dashed red lines) for two different gradients in mechanical properties across the tendon-tobone interphase (recall Fig.9).

Table 1 :

 1 Model parameters for the two surroundings tissues Ω ± and the interphase Ω I .

	Mass density (g.cm -3 ) Stiffness coefficient (GPa) Thickness (µm)
	ρ -	ρ +	C -11	C + 11	h
	1.22	1.88	4.37	27.48	300

Table 2 :

 2 Comparison between the optimal positions of the additional degree-of-freedom and the coefficients of the enriched model for the healthy and damaged cases.

		Position Surface mass densities Kinetic interactions Surface stiffnesses
	Case	xl	m -	m +	m l	γ 1	γ 2	K 1	K 2
		µm		kg.m -2		kg.m -2	kg.(s.m) -2 ×10 13
	Healthy	1.2	0.0653 0.0881 0.1525 0.0346	0.0418	3.08	8.05
	Damaged	10.3	0.0658 0.0818 0.1427 0.0339	0.0384	2.72	7.57

Acknowledgements

This work was partially funded by the BEST-AMUS project (IIN program, CNRS-INSIS), the "Support for research for newly appointed Associate Professors" and the "Bonus Qualité Recherche" (Faculté des Sciences et Technologie, Université Paris-Est Créteil).

Appendix A. Equivalent interface model used for comparison

Here we briefly recall the underlying equations of the equivalent interface model introduced in [START_REF] Rosi | Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties[END_REF]. In that study, the displacement field was associated with an affine profile. As such, the general forms for the kinetic and potential energies per unit surface were stated as

where the coefficients to be identified are

in which M α represents the α-th order moment of the density profile of the interphase M α = ˆh 0 x α 1 ρ I (x 1 )dx 1 , for α = 0, 1, 2. (A.4)


Glenn Roe

A SHEEP IN WOLFF'S CLOTHING: ÉMILIE DU CHÂTELET AND THE ENCYCLOPÉDIE

Émilie Du

in which Du Châtelet's work is used by the encyclopédistes, who, as we shall see, both amplified and muted her authorial voice over the pages of their philosophical dictionary.

Conceived initially as an introduction to Isaac Newton's physics for her young son, Du Châtelet's Institutions de physique became a defining text for the intellectual history of the 1740s. Like Voltaire before her, Du Châtelet's attempt at explaining Newtonian physics came at a time when, in France at least, most members of the Académie des sciences had come to accept much of Newton's description of the universe, from his mathematics and optics to his more controversial notion of universal gravitation. [START_REF] See | The Newton Wars and the Beginning of the French Enlightenment[END_REF] There was, however, still much debate concerning Newton's mathematical formulas and how, exactly, gravity worked among the planets. Writers who concerned themselves with the topic thus had to, at some point, make a choice as to how fully they would subscribe to the Newtonian model. Most followed Voltaire's lead in his Eléments de la philosophie de Newton (1738)-written while living with Du Châtelet at her Château in Cirey from 1734 to 1749-and were content to sound sufficiently Newtonian while simply describing the mechanics of the universe. D'Alembert's Traité de dynamique (1743) and Alexis Clairaut's Théorie de la lune (1752) are obvious examples of the sort, wherein larger questions of causation, or, what would later come to be labeled derisively as "metaphysical" questions, are left largely untreated. Voltaire shared this aversion to the metaphysical and indeed derided Du Châtelet's efforts in the Institutions de physique to go beyond pure description and to look for causes-i.e., to explain what, exactly, made gravity work on such a large scale and how such a system came into being. The great innovation of Du Châtelet's Institutions, then, is to be found in her bold willingness to leverage Gottfried Wilhelm Leibniz's metaphysical system in order to better explain Newtonian mechanics. [START_REF] Shank | [END_REF] A contemporary of Newton, Leibniz filtered into France with the publication of the Leibniz-Clarke correspondence in 1720. [START_REF]Recueil de pièces diverses sur la philosophie la religion et l'histoire par Leibniz, Clarke et Newton[END_REF] For Du Châtelet, Leibniz, along with his disciple Christian Wolff, offered answers to the big metaphysical questions she wished to explore in the Institutions, leading her to do with Leibniz and Wolff what others had previously done with Descartes and Newton: i.e., to create her own philosophical synthesis based on separate parts of their ideas. [START_REF]On the intellectual importance of the Institutions, see the "Editor's Introduction[END_REF] The aim of Du Châtelet's work was thus to move beyond the black-and-white, either/ or manner in which Newton and Descartes were discussed in France, and to serve as a necessary corrective to the general rejection of Leibniz by the French scientific establishment of the period. In her desire to find some truth among all these dissenting voices, it was important for Du Châtelet to understand how individuals can be certain about what it is they know. She thus adapted Leibniz and Wolff's ideas in order to establish the "certainty" of the Newtonian model of the universe over the pages of the Institutions. These broad metaphysical questions touched on a variety of different categories of thought and modes of knowing, from metaphysics, mathematics, and mechanics, to geometry and physics, which were all, for Du Châtelet and her contemporaries, part and parcel of the "natural philosophy." [START_REF] Shank | [END_REF] It is surprising then, given the importance of Du Châtelet's magnum opus and the big questions it sought to answer, that it is does not figure more prominently in the pages of the Encyclopédie. In his article NEWTONIANISME, ou Philosophie Newtonienne, d'Alembert briefly praises Du Châtelet for her significant contribu-tions as a commentator and translator of Newton's ideas, though he stops short of suggesting any original contribution on her part. And yet, as Judith P. Zinsser has recently claimed, "others who wrote for them [Diderot and d'Alembert] simply took whole passages from the Institutions for subjects as general as 'Time' and 'Hypothesis' and as specific as the descriptions of the metaphysical concepts of Leibniz and Wolff. Few identified the original author, and so Du Châtelet acquired the title of so many other women before and after her: 'anonymous.'" [START_REF] Zinsser | Émilie Du Châtelet: Daring Genius of the Enlightenment[END_REF] In fact, there are only two articles we know of in the Encyclopédie authored by a woman, both on the subject of fashion-FALBALA and FONTANGE. In each case attribution is given only anonymously by Diderot in the "Avertissement" to volume six. [START_REF]Une Femme que nous n'avons pas l'honneur de connoître, nous a envoyé les articles FALBALA, FONTANGE, & autres[END_REF] The accepted wisdom over the past 250 years, then, was that of Du Châtelet as just another missing, if not muted, female voice in the pages of the Encyclopédie. This over-simplified view has recently been brought into question thanks to renewed interest in Du Châtelet not merely as a translator, commentator, or companion of great men, but equally as a significant intellectual force in her own right. [START_REF]Émilie Du Châtelet: Between Leibniz and Newton[END_REF] More specifically, recent articles by Koffi Maglo and Khan Dao Duc have succeeded in challenging what had for centuries been assumed as Du Châtelet's decidedly minor role in the encyclopedic enterprise. [START_REF] Koffi Maglo | Madame Du Châtelet, l'Encyclopédie et la philosophie des sciences[END_REF] The present study thus aims to build productively off of this growing base of Du Châtelet scholarship through the use of new digital humanities tools and resources. By uncovering traces-some hidden and others in plain sight-of Du Châtelet's influence and intellectual voice in the Encyclopédie, my goal is to resituate and re-evaluate the status of Émilie Du Châtelet both as an encyclopédiste in her own right and as a key authorial contributor to the philosophic conversation that the Encyclopédie enacts.

AUTHORSHIP AND AUTHORITY IN THE ENCYCLOP ÉDIE

Authorship is a thorny issue when it comes to the Encyclopédie. With over 130 contributors and a far from linear publication history, the exact dimensions of the Encyclopédie's authorial space has been heavily debated by scholars over the past half century, if not longer. [START_REF] Schwab | Inventory of Diderot's Encyclopédie and Kafker, The Encyclopédistes as Individuals. On the Encyclopédie's publication history[END_REF] While this article is concerned with Émilie Du Châtelet's presence and participation in the Encyclopédie, it is first necessary to think a bit more broadly about the very category of authorship itself in the Encyclopédie and how it, in many cases, becomes a function of the multifaceted system of references (both acknowledged and unacknowledged) that are woven together by its various contributors. The interplay of these intertextual references makes the Encyclopédie a much more dialogical space than one normally assumes, a site of conversation and controversy in which source material and editorial commentary are given equal, though by no means equitable, weighting.

From a practical standpoint, this underlying multivocality means that there are often at least three separate authorial, or rather dialogical, functions at play in the Encyclopédie for any given article. The first of these functions concerns the article as such, which includes the contribution of one, or sometimes several, writers on a particular subject. Second, there are the various outside sources with which these contributors marshal evidence for and against the topic of their article. As Dan Edelstein, Robert Morrissey, and I have recently shown, there is a complex system of citation and non-citation at play in the Encyclopédie: references fall anywhere on the spectrum from full chapter-and-verse citations; more "worldly" indications that attest to a common shared culture ("comme dit un beau génie de ce siècle," for example); and finally, a fully-formed politics of non-citation for dangerous or subversive works. [START_REF] Dan Edelstein | To Quote or Not to Quote: Citation strategies in the 'Encyclopédie[END_REF] Alongside these layers of authorship and authority, there is often an additional editorial layer at work in many articles, wherein Diderot and d'Alembert at times intervene, offering correctives or commentary, and engage in a sort of conversation. This form of dialogism is often aimed not at the contributors themselves, but rather at their various sources, or indeed with the general ideas these sources are seen to embody.

Nowhere, perhaps, is this tripartite dialogical system more prominent than in the multiple contributions of Johann Heinrich Samuel Formey (1711-1797) to the Encyclopédie, contributions that for the most part predated the editorial engagement of Diderot and d'Alembert and thus represented inherited material not wholly in phase with their philosophical ambitions. [START_REF] Georges | Samuel Formey et son projet d''Encyclopédie réduite[END_REF] In order to unravel Émilie Du Châtelet's role as an encyclopédiste, and in particular the extent to which her Institutions de physique was incorporated into the text of the Encyclopédie, one must first turn to Formey's articles on metaphysics. Indeed, Formey, according to Zinsser, "is presumed to have written many of the articles in the Encyclopédie that used the Institutions," 16 an assessment that is in harmony with Koffi Maglo's more recent assertion that if one follows "les traces de Formey et vous serez en compagnie de Mme Du Châtelet" ["the traces of Formey […] you will find yourself in the company of Mme Du Châtelet."] 17 The possible connection between Formey and Du Châtelet was first explored by Sonia Carboncini through the bias of Christian Wolff's presence in several anonymous Encyclopédie articles, though here the treatment of Du Châtelet is somewhat secondary. [START_REF] Carboncini | L'Encyclopédie et Christian Wolff. A propos de quelques articles anonymes[END_REF] This article is thus an attempt to verify and expand upon these statements, evaluating to what extent Formey used Du Châtelet as a source-at times acknowledged, and at others not-and then examining how Du Châtelet's textual presence, whether mediated through Formey or others, can be considered an important, albeit largely overlooked, contribution to the philosophical debates enacted in the Encyclopédie.

SAMUEL FORMEY, ENCYCLOP ÉDISTE ANTI-PHILOSOPHE

Formey was a Huguenot pastor of French extraction who spent his entire life in Berlin, most notably as the historiographer and then perpetual secretary of the Academy of Berlin after its reorganization by Frederick the Great in the 1740s. [START_REF] Kafker | Un Encyclopédiste oublié: Formey[END_REF] Appointed to the Royal Society in 1750, from 1741 to 1753 he gained some renown through the extended publication of his pedagogical work La Belle Wolfienne, which was meant to promote the philosophy of Christian Wolff to young women. While far from a success either commercially or philosophically, Formey's work was nonetheless important for diffusing-along with Du Châtelet's Institutions de physique published in 1740-the ideas of both Wolff and Leibniz to the general French reading public. [START_REF] Johann | La Belle Wolfienne[END_REF] Formey's desire to write about Wolff for French readers in the style of letters to a young woman was almost certainly inspired by the earlier successes of Fontenelle's Entretiens sur la pluralité des mondes (1686), which explained the Cartesian universe, and Algarotti's Neutonianismo per le dame (1737), that set out to explain Newton's, both of which equally serve as precursors to Du Châtelet's intended pedagogical goals for her Institutions de physique. As we shall see, Formey relies heavily on Du Châtelet's work for his contributions to the Encyclopédie, a fact that suggests a more direct relationship of influence between the Institutions de physique and La Belle Wolfienne, though this is a connection that lies beyond the confines of this study. The scope and scale of Formey's direct involvement in the Encyclopédie project will be touched upon later, but first it is important to sketch briefly his role in the various other philosophical conversations-enacted in this case on the larger scale of the transnational Republic of Letters-to which he contributed in the second half of the eighteenth century.

A prodigious letter writer-his correspondence supposedly rivalled that of Leibniz [START_REF] Krauss | La Correspondance de Formey[END_REF] -Formey was in many ways representative of the moderate or mainstream strain of the Enlightenment, opposed to the more "radical" currents that would shape philosophical thought in the years following the Encyclopédie's publication, according to Jonathan Israel. [START_REF] See | Comment écrit-on l'histoire intellectuelle des Lumières? Spinozisme, radicalisme et philosophie[END_REF] Formey's idea of philosophy was thus, like Émilie Du Châtelet's, more expansive than the narrow definition we tend to give it today. In his particular case, Formey sought to enlighten his readers with a deft admixture of Wolffian metaphysics, philosophical Protestantism, and Cartesian rationalism, positions that would all, to some degree, come under attack by the encyclopédistes. [START_REF] Smith | In defence of an eighteenth-century academician, philosopher and journalist: Jean-Henri-Samuel Formey[END_REF] As such, and despite his involvement in the encyclopedic enterprise, however tepid his engagement may have been, from the mid-eighteenth century onwards he was increasingly active in decrying what he saw as the more radical and morally dangerous positions of the French philosophes. [START_REF] See | Journalismus und Aufklärung: Jean Henri Samuel Formey und die Entwicklung der Zeitschrift zum Medium der Kritik[END_REF] This specific subset of philosophical writers-those driving the Radical Enlightenment forward, according to Israel-are designated here by the French term in order to distinguish them from the broader field of natural and experimental philosophy to which Formey, Du Châtelet, and countless others belonged. [START_REF]On the persistence and impact of natural philosophy on the Enlightenment[END_REF] Formey's anti-philosophe campaign began as early as 1749 with the publication of his Pensées raisonnables, a response to, and refutation of, Diderot's Pensées philosophiques published in 1746. This first pamphlet was followed closely by his Lettre de M. Gervaise Holmes contre l'auteur de la Lettre sur les aveugles in 1750, another riposte aimed at the young editor of the Encyclopédie. [START_REF] Formey | Pensées raisonnables opposées aux "Pensées philosophiques[END_REF] A few years later, Jean-Jacques Rousseau, still firmly in the philosophe camp at this point, comes under fire for his Discours sur les sciences et les arts, which Formey counters with an Examen philosophique de la liaison réelle qu'il y a entre les sciences et les moeurs, published first as part of the Mémoires de l'Académie de Berlin in 1753, and then in book form in 1755. [START_REF] Formey | Examen philosophique de la liaison réelle qu'il y a entre les sciences & les moeurs…[END_REF] Rousseau next publicly attacks Formey in 1759 for what he perceived as an unauthorized printing in Berlin of his 1756 letter to Voltaire "Sur la providence"his famous response to Voltaire's "poème sur le désastre de Lisbonne." [START_REF] Havens | Voltaire, Rousseau, and the "Lettre sur la Providence[END_REF] Formey responds in kind with another unauthorised collection of excerpts taken from Rousseau's Nouvelle Héloïse (all the while voicing significant moral reservations about the work in his "averstissement") in his Esprit de Julie of 1762. [START_REF] Formey | ou extrait de "la Nouvelle Héloïse[END_REF] Formey moves beyond unauthorized reprinting to an outright attack on Rousseau, and in particular his 1762 publication Émile, which Formey quickly counters with the publication of his Anti-Émile (1763), a stern rebuke not only of Rousseau's pedagogical theories, but also what Formey considered his many religious and political aberrations. [START_REF] Formey | Anti-Émile[END_REF] A year later, Formey will move his critique of Émile forward, going so far as to "remake" Rousseau's work in his own image, publishing an Émile chrétien in 1764, a text that notably replaces the "Confession de foi du vicaire Savoyard" with a more doctrinally appropriate section. [START_REF] Formey | Le Bon Sens face à l'utopie: Formey juge de Jean-Jacques, ou l'Emile revisité[END_REF] Given the above quarrels with Diderot and Rousseau, it is perhaps surprising that Voltaire-who in the 1760s and 1770s was by far the most publically active and universally targeted of the philosophes-is spared, seemingly, the brunt of much of Formey's public critique. This will change, however, with the publication of his memoires-Souvenirs d'un citoyen-on the eve of the Revolution. Here Formey revisits the 1750s and makes it a point to offer a rather unflattering and highly critical description of Voltaire's visit to Potsdam, which allows him to furthermore condemn most of the other leading encyclopédistes, who like Voltaire, promoted unorthodox and theologically dangerous views in the pages of the great mid-century dictionary. [START_REF] Formey | Souvenirs d'un citoyen[END_REF] This retrospective return to the early days of the Encyclopédie thus brings us back to the topic of Formey's uneven involvement with the enterprise. Indeed, the use (he would say "abuse") of Formey's original source material at the hands of Diderot, and also notably that of d'Alembert, would remain a point of contention for the last forty years of his life. [START_REF] François | L'Encyclopédie d'après les correspondants de Formey[END_REF] In order to rectify these perceived slights to his character, Formey went so far as to envision a corrective "encyclopédie réduite" that was planned but never published, although much of his post-1750 encyclopedic work would find its way into several of the Encyclopédie's successors, including the Protestant Encyclopédie d'Yverdon. [START_REF] Roth | Samuel Formey et son projet d''Encyclopédie réduite[END_REF] One may well ask then, how, exactly, the anti-philosophe Formey come to be a contributor at all to Diderot and d'Alembert's philosophic machine de guerre? As it so happens, in 1742 he had begun working on his own philosophical dictionary, one that would purportedly do for France what Ephraïm Chambers' Cyclopaedia had accomplished in England. [START_REF]See Correspondance passive de Formey: lettres adressées à Jean-Henri-Samuel Formey[END_REF] By 1747, however, Formey had heard rumour of a French encyclopaedia project, one that took, coincidentally enough, as its starting point a translation of Chambers into French, and decided to approach its editor, then the Abbé Gua de Malves, offering his completed articles to the new enterprise. [START_REF] Kafker | Gua de Malves and the 'Encyclopédie[END_REF] In early 1749, the deal-executed by the "libraires associés" who controlled the project-was finalized, and Formey sent the editors (by then Diderot and d'Alembert) some 1800 manuscript pages (petit in folio) in exchange for 300 livres, with the added stipulation that the manuscript be returned to the author and that his contribution be acknowledged in the work's preface. Thus, Formey is dutifully mentioned in both Diderot's 1750 "Prospectus" as well as d'Alembert's 1751 "Discours préliminaire": Voilà les richesses sur lesquelles nous pouvions compter ; mais il nous en est survenu d'autres que notre entreprise doit, pour ainsi-dire, à sa bonne fortune. Ce sont des Manuscrits qui nous ont été communiqués par des Amateurs, ou fournis par des Sçavans entre lesquels nous nommerons ici M. FORMEY, Secrétaire perpétuel de l'Académie Royale des Sciences & des Belles-Lettres de Prusse. Cet habile Académicien avoit médité un Dictionnaire, tel à peu près que le nôtre, & il nous a généreusement sacrifié la partie considérable qu'il en avoit exécuté, & dont nous ne manquerons pas de lui faire honneur. [START_REF] Le Rond D'alembert | Discours préliminaire[END_REF] [Such are the resources upon which we could count; but thanks to the good fortune of our enterprise, so to speak, other resources have fallen to us. They include some manuscripts sent to us by amateurs or submitted by scholars, among whom we will name here M. Formey, perpetual secretary of the Royal Academy of Sciences and Belles-Lettres of Prussia. That illustrious academician had planned a dictionary almost like our own, and he has generously turned over to us the considerable part of it which he had completed, for which we will not fail to pay him due honor.] It stands to reason, then, that the second condition of his bequest, that the manuscript be returned, was also carried out. This is in many ways a great shame for the historiography of the Encyclopédie and its publication history, as it seems most of Formey's manuscripts from this period have been lost. As such, we have no real way of comparing the original articles to their edited versions in the final text of the Encyclopédie.

We do, however, have a fairly good notion of which articles draw upon Formey's material, although there is still some disagreement as to the exact number of articles one can count as "belonging" to Formey from an authorial standpoint. The editors of the digital edition of the Encyclopédie at the University of Chicago's ARTFL Project have marked some 111 articles as being authored by Formey, for instance, though this differs somewhat from both Richard Schwab's Inventory (114 articles) and Koffi Maglo's count of around 120 ("cent vingt articles environ qu'il a signés"). [START_REF] Schwab | Inventory, 100-101 and Maglo[END_REF] There are multiple reasons for this discrepancy. First and foremost, it is not exactly correct to say that Formey "signed" any of the articles that have been attributed to him, but rather that his material was used by the editors in a variety of different manners. This hearkens back to the three layers of dialogism at play in the Encyclopédie alluded to earlier, all of which are exhibited to some degree when looking at Formey's articles. He is, at times, simply the author of an article: this is the case with "DÉFINITION" in Tome 4, for example, as the final reference "article de M. Formey" attests. This is not to say, however, that even these seemingly unambiguous attributions are simple; quite the contrary, in this case almost the entire article is drawn from Condillac's Essai sur l'origine des connaissances humaines (1746), a source used without any citation whatsoever. [START_REF]My thanks to Alexander Bocast for alerting me to Condillac's presence in this particular article[END_REF] In a second instance, Formey's papers are used by other authors in the Encyclopédie more or less as source material, which leads to a host of seemingly multi-authored articles, such as Mallet's ABRAXAS, which includes the note "Cet article est en grande partie tiré des Mémoires de M. Formey, Historiographe de l'Académie royale de Prusse" directly before Mallet's signature (G). [START_REF] Mallet | ABRAXAS[END_REF] In this case, the article should logically be attributed solely to Mallet, rather than being identified as co-authored with Formey, though Richard Schwab tends to assign at least partial authorship to Formey for every article in which he is mentioned. 41 Another example of this genre of authorship is found in the Abbé Yvon's article ATTRIBUT, which includes the more concise note "Cet article est tiré de M. Formey" preceding Yvon's authorial sign (X).

At times the use of Formey is more blatantly dialogical, in that he is used strategically to move a specific argument forward or otherwise introduce new concepts, inclusions that are often marked simply with a "M. Formey" at the end of a paragraph. This is precisely the case with d'Alembert's article AREOMETRE, in which he engages directly with Formey as a source, or rather as a co-signatory of the article. After a short section that is presumably drawn from Formey's papers (though we have no structural indication of what is Formey and what is d'Alembert other than the brief reference "M. Formey"), d'Alembert launches into the main crux of the article that acts as both commentary and critique of the previous section. This is by and large the same short-hand citation scheme we find peppered across the Encyclopédie with regards to Ephraim Chambers, wherein paragraphs, or even whole articles, are terminated with a simple "Chambers"-though again, with no indication of what, exactly, or even approximately, comes from the English Cyclopaedia.

In a more playful way, not surprisingly, Diderot is even more pronounced in this sort of call-and-response editorial engagement, often voicing his opinion not only with the article at hand, but also on the whole class of knowledge from which it is drawn. This is the case for Formey's metaphysics article CONSERVA-TION, which, after an authorial reference to Formey (Article de M. Formey), ends with the unsigned remark: "C'est ainsi que dans les questions métaphysiques fort élevées, on se retrouve après bien des détours au même point d'où l'on étoit parti, & où on auroit dû rester" [And thus it is with these sorts of high metaphysical questions, one finds oneself after many detours at the same point from which one departed, and where one should have remained]-an intervention that almost certainly originated from Diderot's acerbic pen. A similar situation can be found in the article ETERNITÉ, a text whose first half is signed by the Chevalier de Jaucourt, and whose second ends with the same attribution as CONSERVATION: "Article de M. Formey." Diderot here again intervenes after Formey's contribution, asserting that he includes the above description of the Thomist and Scotist positions on eternity: "1° parce qu'elles appartiennent à l'histoire de la Philosophie, qui est l'objet de notre ouvrage: 2° parce qu'elles servent à montrer dans quel labyrinthe on se jette, quand on veut raisonner sur ce qu'on ne conçoit pas" [firstly because they belong to the history of Philosophy, which is the object of our work; secondly because they serve to demonstrate the labyrinth into which one throws oneself when trying to reason with the inconceivable]. [START_REF] Jaucourt | ETERNITÉ[END_REF] Formey thus functions in the Encyclopédie as: a) an author; b) a source for other authors; c) a dialogical partner; and then, finally, d) a source of other sources, by which I mean that the editors (and here d'Alembert is especially active) sometimes indicate the sources that Formey is drawing upon in his articles. At the end of the article AREOMETRE, for instance, d'Alembert indicates that "nous devons ces remarques à M. Formey, qui les a tirées de M. l'abbé Nollet Lect. Phys." [we owe these remarks to M. Formey, who drew them from M. l'abbé Nollet's Lect. Phys.]. [START_REF] D'alembert | AREOMETRE[END_REF] Here again, as Formey's original articles have been lost, it is difficult to say exactly how Formey uses his various sources, nor the means by which the encyclopédistes were able to identify them (were they cited in the original? deduced from reading?, etc.). Much like Ephraïm Chamber's Cyclopaedia, which served as both an inspiration and originary source for the Encyclopédie, Formey's contributions are woven throughout this massive compilation in a variety of different manners, and for seemingly different means. [START_REF]On Chambers' use as an intermediary source for the Encyclopédie, see Alain Cernuschi[END_REF]

ÉMILIE DU CHÂTELET, ENCYCLOP ÉDISTE?

This brings us then to the case of Madame Du Châtelet, who, as Maglo and Zinsser argue, in many ways enters the Encyclopédie through Formey's mediation. This is certainly the case with the seven articles that explicitly mention Du Châtelet's Institutions de physique. Though, while Maglo asserts (or rather overstates to some degree) that "practiquement dans tous les cas, il est precisé quels passages des Institutions sont copiés ou servent d'insipration" ["in virtually all cases it is indicated which passages of the Institutions are copied or serve as inspiration"], [START_REF] Maglo | Mme Du Châtelet[END_REF] until now it has remained unclear to what extent Du Châtelet's text was used in these articles, or how exactly she participates in the larger dialogical space of the Encyclopédie outlined above. To redress this gap in our knowledge, we need an indication of where the Institutions de physique is used not only when it is mentioned, but also when it is not.

In order, then, to find references to Du Châtelet's work, both acknowledged and unacknowledged, I deployed a new computational method developed in the field of Digital Humanities that compares the entire text of the Encyclopédie with a version of the Institutions de physique taken from the BnF's Gallica digital library. [START_REF] Le Tonnelier De Breteuil | Institutions de physique[END_REF] And, while the text in this case is far from perfect-it includes, for instance, many errors generated by the automatic optical character recognition (OCR) process used to digitize the book-I was nonetheless able to identify a further six articles that all borrow extensively from Du Châtelet's work, but in this case with no citation or authorial mention (see Table 1). One notices immediately that all of these newly identified articles, save SUFFISANTE RAISON, are signed by Formey or Formey and d'Alembert. The above identifications are made using a technique known as "sequence alignment," a computational approach borrowed from the field of bio-informatics. [START_REF]Automatic detection of reuses and citations in literary texts[END_REF] For this project I used the PhiloLine sequence alignment package developed by the ARTFL Project at the University of Chicago, an algorithm that works in tandem with its full-text search and retrieval software PhiloLogic. [START_REF] Russell Horton | Something Borrowed: Sequence Alignment and the Identification of Similar Passages in Large Text Collections[END_REF] Basically, the system treats two texts as long potential sequences, and then looks for common subsequences with a high degree of flexibility between the two (think of missing genetic information that must be skipped over in the sequencing of the human genome, for example). This technique allows us not only to identify borrowed passages that are uncited, but also to see exactly how these passages are deployed in the Encyclopédie even when cited. The article "Continuité, (loi de)" serves as a prime example of how these new approaches can help us unravel the often-complicated interplay of authorial gestures at work in the Encyclopédie. The article is signed by d'Alembert, who then claims "Nous devons cet article à M. Formey" [we owe this article to M. Formey], and which furthermore includes the reference "Lisez le chap. j. des instit. de Physiq. de Mad. Duchatelet, depuis le § 13 jusqu'à la fin" [Read chapter I of Mme Duchatelet's Institutions de physique, from paragraph 13 until the end], an imperative command that seems, in form, much more like a renvois, or cross-reference, than a citation. [START_REF] Formey | Continuité, (loi de)[END_REF] Using the sequence alignment comparison technique, we can now, for the first time perhaps, see exactly to what extent Du Châtelet is used in this article as well as which passages are shared between the two texts. Remarkably, almost the entire article is in fact drawn from the Institutions de physique, though certainly not in the word-for-word or copy-paste manner one might assume. Instead, the article incorporates parts of paragraph §13, to which the reference alludes, modifies the text, draws on a sentence from paragraph §14, before moving to paragraph §17, and then backwards to paragraph §15 to conclude. 50 Here again, this manner of manipulating source material is much more like a conversation than a rote citation, wherein Mme Du Châtelet (or is it Formey?) is used as a proxy or conceptual standin for both "les Leibnitiens" and "les partisans de ce principe [de continuité]." [START_REF]See D'Alembert and Formey[END_REF] The granular level of this sort of text comparison allows us to think more broadly about the proportionality of Du Châtelet's text against the rest of the article in which it occurs. In the figure below, the total word count of the articles in blue is compared to Du Châtelet's contribution in red (see Figure 1). Obviously, longer articles such as MOVEMENT and PENDULE will yield a much lower proportion of included text, and indeed in these cases Du Châtelet can be considered one among many different sources used to construct the longer articles (though in the case of PENDULE there is no reference to her work).

We can also think about this proportionality in terms of a "score" or percentage of any given article that belongs to Du Châtelet-those of us who use plagiarism software like Turnitin to mark essays will be all too familiar with these sorts of metrics. Below we have in ascending order all of the articles identified as containing at least some of the Institutions de physique, along with their corresponding "Du Châtelet scores" (See Table 2). To return to an example that we have already seen, d'Alembert and Formey's article "Continuité, (loi de)" scores as approximately eighty-one percent Du Châtelet. This fact necessarily raises the following question: to what extent is Formey (or even d'Alembert for that matter) really the "author" of this article. Table 2. Articles that contain at least some text of the Institutions de physique and their percentages in ascending order. Some of these proportionality scores are variable (indicated by the *), and this is again due to the multi-vocal nature of many of the articles Formey's papers were used to construct. About forty-five percent of d'Alembert's CONTINU article, for instance, comes from the Institutions de physique (without citation). However, if we look closely, we see that CONTINU is in fact two articles, wherein the first paragraph, d'Alembert tells us, can be considered by itself an "Article de M. Formey." [START_REF] D'alembert | CONTINU[END_REF] Thus, if we score only Formey's part, the Du Châtelet percentage almost doubles, and we are again left with the obvious question as to the validity of attribution concerning Formey's inclusion as an "author" of this part of the article. A similar pattern of re-use is found in the longer articles such as PESANTEUR, which rates at an impressive eighty-two percent Du Châtelet, albeit in this case there is at least a reference to Formey's use of Du [START_REF] Formey | PESANTEUR[END_REF] We are thus left to wonder if this information was included in Formey's original article, or whether it was added by the editors post hoc, and therefore to what extent this article should be considered an article by Madame Du Châtelet with a brief introduction by Samuel Formey, rather than its current attribution to Formey alone.

Finally, to conclude, I would like to consider the two articles with the highest Du Châtelet scores: SUFFISANTE RAISON (eighty-seven percent score) and CONTRADICTION (ninety-five percent score); the former unsigned in the Encyclopédie, and the latter attributed again to Formey. SUFFISANTE RAISON is interesting in that it quite clearly participates in the dialogical system of the Encyclopédie I lay out above, wherein the only part of the article not taken from Du Châtelet's text is in effect a counter-argument to the principle of sufficient reason as elaborated by Leibniz and Wolff and filtered through Madame Du Châtelet, although the article includes no mention of her or the Institutions de physique. Clearly, we know that one of Du Châtelet's aims in writing the Institutions was to present Leibniz's (and to a lesser degree Wolff's) ideas to the French-speaking world, and this is perhaps the case of her being too successful in her execution, as her text becomes (in this particular case at least) the unacknowledged point of departure from which the Encyclopédie can criticise Leibniz's and Wolff's metaphysics. [START_REF] Leibniz | Du Châtelet in the Encyclopédie, see Khanh Dao Duc[END_REF] It is highly probable that the penultimate paragraph of SUFFISANTE RAISON is another instance of editorial intervention, most likely from Diderot, who ceased using his authorial mark (*) in the final suppressed volumes of the Encyclopédie published en masse in 1765. Responding to the text of the Institutions-quite literally in this case, as the main body of the article includes Madame Du Châtelet's first person pronoun 'Je' in its exposition-the interlocutor suggests that, if taken to its logical extreme, the doctrine of sufficient reason would lead to "spinosism." A loaded term, for sure, the evocation of Spinoza here as a counterweight to Leibniz and Wolff moves the reader subtly (and perhaps ironically) from the metaphysical to the materialistic: Au reste, on peut faire une espece d'argument ad hominem contre le principe de la raison suffisante, en demandant à Messieurs Leibnits & Wolf comment ils peuvent l'accorder avec la contingence de l'univers.

La contingence en effet suppose une différence d'équilibre. Or, quoi de plus opposé à cette indifférence que le principe de la raison suffisante? Il faut donc dire que le monde existe, non contingemment, mais en vertu d'une raison suffisante, & cet aveu pourroit mener jusqu'aux bords du spinosisme. 55 [Furthermore, one can make a kind of ad hominem argument against the principle of sufficient reason by asking Messieurs Leibniz and Wolff how they can harmonize it with the contingency of the universe. A contingency implying, in effect, a difference in equilibrium. But, what is more opposed to this indifference than the principal of sufficient reason? One must then say that the world exists, not contingently, but by virtue of a sufficient reason, and this admission could very well lead to the edges of Spinozism.]

The article CONTRADICTION is less conversational in tone, primarily because there are only eighteen words that are not drawn from Du Châtelet, comprising all or most of paragraphs §4 and §7 from chapter one of the Institutions de physique. There are, however, two interesting cross-references included-and we know that Diderot held the system of renvois to be the most important (and indeed most dialogical) of the Encyclopédie's organisational schemes. [START_REF] Blanchard | Le système de renvois dans L'Encyclopédie: une cartographie de la structure des connaissances au XVIIIème siècle[END_REF] There is first an indirect renvois (Voy. son Article) that refers the reader to the article SUF-FISANTE RAISON (which we have just seen belongs largely to Mme Du Châtelet) along with a direct renvois to the abbé Yvon's article AXIOME, directly following the attribution of the article to Formey: "Cet article est de M. Formey, sur quoi voyez l'article AXIOME" ["This article is from M. Formey, whereupon see the article AXIOME"]. [START_REF] Formey | CONTRADICTION[END_REF] Following this renvois thus opens the reader up to further links through the metaphysical articles CONNAISSANCE, attributed by Schwab to Formey, with citations to John Locke, Claude Buffier, and Chambers; ABSTRAC-TION by César Chesneau Dumarsais; and finally ANALYSE by d'Alembert, which is decidedly less Leibnizian/Wolffian in scope and therefore acts as a critical counterbalance to Formey's (and Du Châtelet's) articles. This brings us back, finally, to the notion of Formey as a dialogical actor, or pivot, in the Encyclopédie-an author that in many cases is used as a pretext for an argument a contrario with Leibniz and Wolff and their respective metaphysics. This is clearly Diderot's stance in CONSERVATION, as we have seen earlier, as well as the article ETERNITÉ, in which the editor effectively undermines Formey's entire contribution, noting that the article's propositions "servent à montrer dans quel labyrinthe (a favourite Diderotien metaphor) on se jette, quand on veut raisonner ce qu'on ne conçoit pas" [serve to demonstrate the labyrinth into which one throws oneself when trying to reason with the inconceivable]. [START_REF] Jaucourt | ETERNITÉ[END_REF] Though more circumspect than Diderot, d'Alembert too is not immune from this sort of editorial banter, as evidenced by his presumed intervention at the end of the Formey article ESPACE, which, as we have seen above, is in reality fiftyone percent Du Châtelet. In this case, the shared nature of the article's authorship is at least suggested at by its unusually long and precise statement of attribution: "Cet article est tiré des papiers de M. Formey, qui l'a composé en partie sur le recueil des Lettres de Clarke, Leibnitz, Newton, Amsterd. 1740, & sur les inst. de Physique de madame Du Châtelet" [This article is drawn from the papers of M. Formey, who composed it in part using the collected letters of Clarke, Leibniz and Newton (Amsterdam, 1740), and with the Institutions de physique of Madame Du Châtelet]. [START_REF] Formey | ESPACE[END_REF] Directly following this intervention, which logically can be seen as an editorial attempt at clarifying some of Formey's uncited sources, the article closes with a unusual flourish: "Nous ne prendrons point de parti sur la question de l'espace ; on peut voir, par tout ce qui a été dit au mot Élémens des Sciences, combien cette question obscure est inutile à la Géométrie & à la Physique" [We take no position on the question of space; one can see, by all that has ben said in the article Élémens des Sciences, how much this obscure question is useless to Geometry and Physics]. [START_REF] Formey | ESPACE[END_REF] The implied superiority of geometry and physics above Formey's (i.e., Leibniz's, Wolff's, and Du Châtelet's) metaphysics is here quite clear, and wholly in line with d'Alembert's philosophical outlook. Furthermore, the implicit renvoi to the article ELEMENS DES SCIENCES-one of d'Alembert's most important contributions-and the explicit cross-references to several more of his articles in the domains of mechanics and physics leave little doubt as to the authorship of this particular comment, and demonstrates forcefully the various layers of authorial interaction at play in the Encyclopédie.

CONCLUSION

Lost, perhaps, in the midst of these conversations for and against the metaphysical systems of Leibniz and Wolff is the active role that Madame Du Châtelet plays in their elaboration over the entirety of the Encyclopédie. The Institutions de physique is in this light much more than a mere source that Formey draws upon among others. Indeed, it is safe to say that in the majority of the cases we have outlined above, the ideas with which Diderot and d'Alembert are conversing, the clear and concise crystallizations of Wolff and Leibniz they wish to combat, come directly from the Institutions de physique and only secondarily belong to Formey, if at all. Thus, if we think again about the particular dialogical space of the Encyclopédie I have outlined above, and the philosophical conversations that are enacted therein, the traditional (and unsurprisingly male-dominated) image of Diderot and d'Alembert debating the ideas of Leibniz and Wolff through the mediation of Formey is now in need of serious reconsideration. This article is intended as a first step in this process of recalibration, and as a testament to the growing recognition of Madame Du Châtelet's importance to both the encyclopedic enterprise and the intellectual history of the eighteenth century.

What is perhaps most significant about the findings presented here, beyond Du Châtelet's role as an unacknowledged source for the Encyclopédie, is how, in a matter of just ten years, Du Châtelet had become the authority on Leibniz and Wolff in French, first for Formey and later for both Diderot and d'Alembert. Unattributed authors were used for a variety of reasons in the Encyclopédie, as we have previously demonstrated, [START_REF] Edelstein | To Quote or Not to Quote[END_REF] so uncovering previously unseen passages of Du Châtelet's Institutions, while important, is by no means surprising. Nor is the fact that Diderot and d'Alembert, and behind them most other French (and English) philosophers, rejected the metaphysics of Leibniz and Wolff as a means for acquiring "certainty" about the physical universe. What is surprising, however, is the important and hitherto unacknowledged role that Du Châtelet played in framing those big questions concerning the nature of knowledge, the limits of human intellect, and the workings of the universe that the Encyclopédie itself tried to answer.

Figure 1 .

 1 Figure 1. Total number of words compared to number of words drawn from the Institutions de physique.

Table 1 .

 1 Encyclopédie articles that draw from Émilie du Châtelet's Institutions de physique with and without citation.

  Châtelet as a source text at the end of the article (though with no indication of what part of Du Châtelet's text is used and where): "Cet article est de M. Formey, qui l'a tiré en partie des Inst. de Phys. de Mad. Du Châtelet" ["This article is from M. Formey, who extracted it in part from the Mad. Du Châtelet's Inst. de Phys."].