B Faverjon

J-J Sinou

Identification of an open crack in a beam using an a posteriori error estimator of the frequency response functions with noisy measurements

This paper presents a robust damage assessment technique for the nondestructive detection and size estimation of open cracks in beams. The damage detection, based on the constitutive relation error updating method, is used for the identification of the crack's location and size in a simply-supported beam. The transverse open crack is modeled through the introduction of the flexibility due to the presence of the crack, i.e. by reducing the second moment of area of the element at the crack's location. This identification algorithm is illustrated through numerical examples involving different positions and sizes of a transverse open crack. We show that the detection of damage and the identification of the crack's size and position can be achieved with satisfactory precision, even if 20% noise has been added to the simulations and less than 5% of all displacements have been measured.

Introduction

The identification of the depth and position of cracks through nondestructive testing is important to ensure the integrity of structural components for a wide range of civil, mechanical and aeronautical engineering applications. Indeed, the importance of an early detection of cracks appears to be crucial for both safety and economic reasons and has led to the development of various methods. The most useful damage localization methods based on vibration measurements are probably those based on changes in natural frequencies and mode shapes. Reviews of research works dealing with the problem of crack detection based on changes in modal parameters can be found in [START_REF] Montalvao | A review of vibration-based structural health monitoring with special emphasis on composite materials[END_REF][START_REF] Doebling | Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review[END_REF][START_REF] Sohn | A Review of Structural Health Monitoring Literature : 1996-2001[END_REF][START_REF] Salawu | Detection of structural damage through changes in frequency: a review[END_REF]. Indeed, it has been established that the presence of a crack drastically affects the dynamic behavior of structures. Depending on the crack's size and location, the stiffness of the structure is reduced and, therefore, so are its natural frequencies compared to the original crack-free structure. This shift in natural frequencies has been commonly used to investigate the crack's location and size. For example, some research works [START_REF] Dong | Parameter identification of a rotor with an open crack[END_REF][START_REF] Nikolakopoulos | Crack identification in frame structures[END_REF][START_REF] Li | Detection of crack location and size in structures using wavelet finite element methods[END_REF] have been devoted to the identification of a crack's size and location through the determination of the intersection point of the superimposed contours corresponding to the measured eigenfrequency variations due to the crack. This damage identification technique is called the "frequency contour lines method". In order to avoid the problem of a non-unique damage location in the case of a structural symmetric beam, Swamidas et al. [START_REF] Swamidas | Identification of cracking in beam structures using Timoshenko and Euler formulations[END_REF] proposed to extend this crack identification method by adding an off-center mass to the simply-supported beam. Recently, Sinou [START_REF] Sinou | Numerical investigations of a robust identification of crack location and size in beams using only changes in ratio pulsations of the cracked beams[END_REF] developed an extension of the frequency contour lines method based on the changes of frequency ratios in the cracked beam, thus avoiding the need for accurate knowledge of the material properties and frequencies of the crack-free beam. Moreover, Owolabi et al. [START_REF] Owolabi | Crack detection in beams using changes in frequencies and amplitudes of frequency response functions[END_REF] proposed a crack identification technique based not only on the measured changes in the first three natural frequencies (i.e. the frequency contour lines method), but also in the corresponding amplitudes of the measured acceleration frequency response functions. Using the frequency contours and amplitude contours of the first three modes, they demonstrated that the crack's size and location can be determined uniquely. Recently, Dilena and Morassi [START_REF] Dilena | The use of antiresonances for crack detection in beams[END_REF][START_REF] Dilena | Damage detection in discrete vibrating systems[END_REF] proved that the measurement of an appropriate set of frequencies and antiresonance frequencies enables unique identification of damage. The theoretical results were confirmed by comparisons with numerical and experimental tests. The authors also suggested that the direction in which the nodal points move can point to the damage location [START_REF] Dilena | Identification of crack location in vibrating beams from changes in node positions[END_REF]. In these works, the identification techniques were based on a shift in natural frequencies. Even though experimental validations of the damage identification methods based on frequency variations mentioned above have led to some interesting results, all these methods require precise measurements of the natural frequencies of the cracked system and, therefore, their accuracy is affected by the presence of noise in these measurements. Indeed, incorrect measurements of natural frequencies, vibration modes and forced vibration amplitudes may result in erroneous identification of the crack's size and location. For example, Lele and Maiti [START_REF] Lele | Modeling of transverse vibration of short beams for crack detection and measurement of crack extension[END_REF] found errors in the crack's location and depth of about 10%. Cerri and Vestroni [START_REF] Cerri | Detection of damage in beams subjected to diffused cracking[END_REF] indicated that a small error in the data (a mean error of 0.5%) is usually amplified strongly and becomes more than 5% in identification of the damage parameters. Similarly, Sinou [START_REF] Sinou | Damage assessment based on the frequencies' ratio surfaces intersection method for the identification of the crack depth, location and orientation[END_REF] recently showed that identification of the crack's parameters for a simply-supported cracked beam using this type of damage identification technique can become very difficult if the noise level affecting the natural frequencies is greater than 4%. Consequently, to develop a robust damage assessment technique for the nondestructive detection and size estimation of open cracks in beams, we propose to use model updating methods that minimize the discrepancy between the test data and the model by modifying the numerical model. A state-of-the-art review can be found in [START_REF] Mottershead | Model updating in structural dynamics: a survey[END_REF]. On the one hand, "direct methods" [START_REF] Baruch | Optimal correction of mass and stiffness matrices using measured modes[END_REF][START_REF] Berman | Improvement of a large analytical model using test data[END_REF] apply corrections to the model's mass and stiffness matrices, but, unfortunately, do not take the physical meaning of these modifications into account. On the other hand, "indirect or parametric methods" update the model by changing its physical parameters. Three categories of cost functions can be considered: input residuals [START_REF] Berger | Updating methods for structural dynamics models : 9-20[END_REF][START_REF] Farhat | Updating finite element dynamics models using an element-by-element sensitivity methodology[END_REF], output residuals [START_REF] Piranda | Parametric correction of finite element modes by minimization of an output residual: improvement of the sensitivity method[END_REF][START_REF] Lammens | Application of a frf based model updating technique for the validation of a finite element model of components of the automotive industry[END_REF], and the residual called "Constitutive Relation Error" (CRE). In this study, the CRE is used and provides a measure of the quality of the updated model which is essential for model validation. This approach has been shown to be effective in structural dynamics with updated mass, stiffness and damping matrices obtained through eigenmodes or forced vibrations [START_REF] Ladevèze | Fe modeling and analysis: a localization method of stiffness errors and adjustments of fe models[END_REF][START_REF] Ladevèze | Updating of finite element models using vibrations tests[END_REF][START_REF] Ladevèze | Application of a posteriori error estimation for structural model updating[END_REF][START_REF] Chouaki | Updating structural dynamics models with emphasis on the damping properties[END_REF]. The method is based on the Drucker error and can also take into account nonlinearities due to material behavior or to contacts. Let us also mention two similar approaches used in the case of free vibration problems: the Minimum Dynamic Residual Expansion (MDRE) method [START_REF] Balmès | Review and evaluation of shape expansion methods[END_REF] and the Modeling Error in the Constitutive Equations (MECE) method [START_REF] Pascual | On the reliability of error localization indicators[END_REF][START_REF] Moine | Updating a non conservative finite element model: two modal methods[END_REF]. Finally, one should note that many other strategies for the resolution of crack identification problems have been proposed [START_REF] Bonnet | Inverse problems in elasticity[END_REF], such as the recent strategies based on the enclosure method or the virtual work principle, which enables the formulation of observation equations, with emphasis on the reciprocity gap concept. In this paper, we propose to investigate the effectiveness and robustness of the CRE method in the identification of cracks of various depths and locations based on noisy measurements. The paper is organized as follows: first, we give a brief presentation of the modeling of an open transverse crack and the associated model of a simply-supported cracked beam. Then, we introduce the damage identification technique based on the CRE updating method for the identification of the crack's location. Finally, we illustrate the effectiveness and robustness of the method through various numerical simulations corresponding to different physical configurations.

2 The modeling of a simply-supported beam with a transverse crack

Equations of motion

In this paper, we consider a beam with a circular cross section, 1m in length and 0.1m in diameter, divided into 30 Timoshenko beam finite elements with four degrees-of-freedom per node (see Figure 1). The beam is assumed to be simply supported at each end. The equations of motion for the simply-supported cracked beam can be written as

[M] {X} + [C] {X} + [ K]{X} = {F d (t)} ( 1
)
where {X} is the vector of the nodal DOFs of the system, t the instant of time, [M] the mass matrix, [ K] the global stiffness matrix and {F d (t)} the external force vector, and where the dot represents the derivative with respect to time.

[C] denotes the proportional damping matrix, which can be expressed as

[C] = α[M] + β[K] (2)
where α and β are real constants.

[ K] contains the stiffness reduction [K crack ] at the crack's location and is given by

[ K] = [K] -[K crack ] (3)
where [K] is the stiffness matrix of the crack-free beam. The stiffness matrix [K crack ] of the cracked element will be given in the next section. Finally, let us define the external force vector as

{F d (t)} = {F }e iωt ( 4
)
where ω is the forcing frequency and {F } the amplitude vector.

The response vector can be written as {X(t)} = {X 0 }e iωt . Using Equation 1, the system governing the equation in the frequency domain is

-ω 2 [M] + iω[C] + [ K] {X 0 } = {F } (5)
We will perform the identification of the crack's size and location by considering the response simulated from Equation 5 to a given external force {F d (t)} and by using the CRE estimator described in the next section.

The cracked element

In order to represent the stiffness properties of the cracked cross section locally in an crack-free beam, the flexibility due to the presence of the transverse crack must be taken into account. For a comprehensive literature survey of various crack modeling techniques, see [START_REF] Dimarogonas | Vibration of cracked structures: a state of the art review[END_REF][START_REF] Wauer | Dynamics of cracked rotors: Literature survey[END_REF]. In this study, the stiffness properties of the cracked cross section in a beam are represented locally using Mayes and Davies' transverse crack model [START_REF] Davies | The vibrational behaviour of a multi-shaft, multi-bearing system in the presence of a propagating transverse crack[END_REF][START_REF] Mayes | Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor[END_REF]. This model incorporates the flexibility due to the presence of the transverse crack by reducing the element's second moment of area at the location of the crack by ∆I. As explained by Mayes and Davies [START_REF] Mayes | The vibrational behaviour of a rotating shaft system containing a transverse crack[END_REF], the change in the m th natural frequency of a system due to the presence of a transverse crack can be written as

∆ω 2 m = -g d 2 u m dx 2 x=sc ( 6
)
where u m is the m th mode shape of the beam and s c the location of the crack along the beam. g is a function of the geometries of the crack and of the beam. Expanding on their analysis and using dimensional analysis to describe the stress concentration factor at the crack's front, they obtained the following expression [START_REF] Davies | The vibrational behaviour of a multi-shaft, multi-bearing system in the presence of a propagating transverse crack[END_REF]:

∆ω 2 m = -4 EI 2 πR 3 1 -ν 2 F (µ) d 2 u m dx 2 x=sc ( 7
)
where I, R, E and ν denote respectively the second moment of area, the shaft's radius and the material's Young's modulus and Poisson's ratio. F (µ) is a nonlinear compliance function which can be obtained from a series of experiments with chordal cracks [START_REF] Davies | The vibrational behaviour of a multi-shaft, multi-bearing system in the presence of a propagating transverse crack[END_REF][START_REF] Mayes | Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor[END_REF]. µ denotes the nondimensional crack depth and is given by

µ = h R ( 8
)
where h is the depth of the crack in the shaft as illustrated in Figure 2.

Then, using the second derivatives of the deflection curves, the bending moments M 0 and M of the original and cracked systems are given respectively by

M 0 (s) = EI 0 (s) d 2 y dx 2 x=s ( 9
)
and

M (s) = M 0 (s) -∆M (s) = E (I 0 (s) -∆I (s)) d 2 y dx 2 x=s ( 10
)
where y and s denote the deflection of the beam and the axial position. I 0 is the second moment of area of the original system. In addition, assuming that the deflection curve is unchanged, the bending moments M 0 and M of the original and cracked systems are defined respectively by

M 0 (s) = ω 2 0 s 0 (s -z) m (z) y (z) dz (11)
and

∆M (s) = ∆ω 2 0 s 0 (s -z) m (z) y (z) dz (12)
where m (z) corresponds to the mass per unit length at location z along the beam. Using Equations 9-12 and Rayleigh's approach, Mayes and Davies [START_REF] Davies | The vibrational behaviour of a multi-shaft, multi-bearing system in the presence of a propagating transverse crack[END_REF] derived the relation:

∆ω 2 = E d 2 y dx 2 x=s ∆I 1 -∆I I 0 1 - ∆ω 2 ω 2 0 ( 13
)
By comparing Equations 7 and 13 and considering first-order changes in ∆ω 2 alone, Mayes and Davies obtained the following relation for a beam with a circular cross section:

∆I I 0 = R l 1 -ν 2 F (µ) 1 + R l 1 -ν 2 F (µ) (14)
where R and l are the shaft's radius and length respectively. Mayes and Davies [START_REF] Davies | The vibrational behaviour of a multi-shaft, multi-bearing system in the presence of a propagating transverse crack[END_REF][START_REF] Mayes | Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor[END_REF] proposed to obtain the evolution of F (µ) through a series of experiments with chordal cracks. This compliance function was later used by Sinou and Lees [START_REF] Sinou | Influence of cracks in rotating shafts[END_REF][START_REF] Sinou | A non-linear study of a cracked rotor[END_REF] to study the dynamic behavior of a rotating shaft with a crack, taking into account the opening and closing of the crack at different orientations of the shaft due to its weight. Then, using classical finite element formulation, the stiffness matrix [K crack ] at the crack's location modified by the transverse crack is:

[K crack ] = E l 3             12I X 0 0 6lI X -12I X 0 0 6lI X 12I Y -6lI Y 0 0 -12I Y -6lI Y 0 4l 2 I Y 0 0 6lI Y 2l 2 I Y 0 4l 2 I X -6lI X 0 0 2l 2 I X 12I X 0 0 -6lI X 12I Y 6lI Y 0 Sym. 4l 2 I Y 0 4l 2 I X             (15)
The moments of inertia I X and I Y with respect to the parallel centroidal axes are given by [START_REF] Sinou | Influence of cracks in rotating shafts[END_REF]

I X = R 4 4 (1 -µ) 1 -4µ + 2µ 2 γ + α 2 (16)
and For convenience, γ = 2µ -µ 2 . α is the crack's angle as shown in Figure 2 and is equal to α = 2cos -1 (1 -µ).

I Y = πR 4 4 + R 4 2 3 (1 -µ) γ 3 + 1 4 (1 -µ) 1 -4µ + 2µ 2 γ + sin -1 (γ) - 4 9 R 4 γ 6 (1 -µ) γ + α 2 -1 (17)

The constitutive relation error updating method

The Constitutive Relation Error (CRE) updating method, which uses an a posteriori estimator, has been widely and successfully used for validation problems in structural dynamics (see for instance [START_REF] Ladevèze | Fe modeling and analysis: a localization method of stiffness errors and adjustments of fe models[END_REF][START_REF] Ladevèze | Updating of finite element models using vibrations tests[END_REF][START_REF] Ladevèze | Application of a posteriori error estimation for structural model updating[END_REF][START_REF] Chouaki | Updating structural dynamics models with emphasis on the damping properties[END_REF]). It is presented in the next section in the general case. Its application to the simply-supported beam problem is described in Section 5.

The CRE estimator

We will only review the principles of the method. For further details, see [START_REF] Ladevèze | A modelling error estimator for dynamic structural model updating[END_REF].

Let Ω be the domain of the structure and ∂Ω its boundary (see Figure 3). The structure vibrates over a time interval [0, T ]. Displacements U d and forces F d are prescribed respectively at boundaries ∂Ω 1 and ∂Ω 2 , with ∂Ω 1 ∪ ∂Ω 2 = ∂Ω. Body forces f d are applied within Domain Ω.

The reference problem consists in finding

s = (M , t) = (U (M , t), σ(M , t), γ(M , t)) M ∈ Ω , t ∈ [0, T ] (18)
(respectively the displacement, the stress and the acceleration, M being the position vector) which verify a set of reliable equations and a set of less reliable equations. Here, since we are dealing with forced vibration problems, the equations will be expressed in the frequency domain. The reliable equations consist of the kinematic constraints and the equilibrium equations; the less reliable equations are the constitutive relations, which are written as

σ = K + iωC ǫ(U ) (19)
Ω F d ∂ 2 Ω U d ∂ 1 Ω f d
Γ = -ρω 2 U ( 20
)
where K is the Hooke's tensor (symmetric and positive definite), C the damping operator (symmetric, positive definite then ensuring the uniqueness of the solution and verifying Drucker's stability conditions, see [START_REF] Ladevèze | Mastering calculations in linear and nonlinear mechanics[END_REF]), ǫ the strain tensor and ρ the density. Drucker's stability conditions ensure the uniqueness of the solution and are verified by a large class of materials [START_REF] Ladevèze | A modelling error estimator for dynamic structural model updating[END_REF][START_REF] Ladevèze | Mastering calculations in linear and nonlinear mechanics[END_REF]. We seek a solution which is admissible (i.e. which verifies the equations considered to be reliable). In the context of model updating, there are additional data which come from measurements. For a structure solicited at one point and whose displacements are measured at different locations, these quantities are also subdivided into a reliable set and a less reliable set. The reliable set consists of the measured angular frequency ω and the positions and directions of the excitations and sensors; the measured amplitudes of the forces F d and displacements U d at the excitation and sensor points are considered to be less reliable. Finally, the problem to be solved consists in finding admissible fields s which minimize the modified CRE e 2 ω defined by

e 2 ω = ζ 2 ω D 2 ω + r 1 -r η 2 ω ( 21
)
with a term corresponding to a modeling error:

ζ 2 ω (U , V , W ) = Ω γ 2 tr[(K + T ω 2 C) (ǫ(V ) -ǫ(U ) ⋆ (ǫ(V ) -ǫ(U ))] + 1 -γ 2 ρω 2 (U -W ) ⋆ (U -W ) dΩ (22)
a term corresponding to a measurement error:

η 2 ω = ||U | ∂ 1 Ω -Ũ d || 2 || Ũ d || 2 + ||F | ∂ 2 Ω -F d || 2 || F d || 2 (23)
and displacements U , V , W which must verify

U c = U (24)
σ s = K + iωC ǫ(V ) (25)
Γ s = -ρω 2 W (26)
where the static quantities σ and Γ are denoted σ s and Γ s and the dynamic quantity U is denoted U c . The symbol ⋆ designates the complex conjugate of a quantity. These equations are written for a given frequency ω. r is a weighting coefficient, which we will take equal to 0.5 [START_REF] Deraemaeker | Model validation in the presence of uncertain experimental data[END_REF], representing the degree of trust one has in the experimental data. The denominator D 2 ω and the norms being used ensure that the two error terms have equivalent weights. The following expression can be used [START_REF] Deraemaeker | Model validation in the presence of uncertain experimental data[END_REF]:

D 2 ω = Ω γ 2 tr[(K + T ω 2 C) ǫ(U ) ⋆ ǫ(U )] + 1 -γ 2 ρω 2 U ⋆ U dΩ (27)
Finally, over the whole frequency range [ω min , ω max ], the modified error is calculated using a weighting factor z(ω) such that ωmax ω min z(ω)dω = 1 with z(ω) ≥ 0, e.g. z(ω) = 1/(ω max -ω min ). The error is given by

e 2 T = ζ 2 T + η 2 T ( 28
)
in which

ζ 2 T = ωmax ω min ζ 2 ω D 2 ω z(ω)dω (29)
η 2 T = ωmax ω min η 2 ω z(ω)dω (30)

Implementation of the method

The updating of the parameters is carried out by minimizing the modified CRE. Such an inverse problem with no additional manipulation is ill-posed. Our updating method can be viewed as a regularization process leading to a unique solution of the problem. The method is iterative, each iteration consisting of two steps. The first step consists in localizing the most erroneous regions, then the most erroneous structural parameters; the second step consists in correcting the parameters belonging to these regions. The updating process is stopped when the threshold error is reached (for further details, see [START_REF] Ladevèze | Application of a posteriori error estimation for structural model updating[END_REF][START_REF] Ladevèze | A modelling error estimator for dynamic structural model updating[END_REF][START_REF] Ladevèze | Mastering calculations in linear and nonlinear mechanics[END_REF]).

In practice, the solution of the problem described in Section 3.1 enables one to calculate ζ 2 T and e 2 T . The value of ζ 2 T yields the relative quality (in %) of the numerical model with respect to measurements over a frequency range, which is used to determine whether model updating is necessary. The first step of the model updating method consists in identifying which substructures have high model error values and can be achieved by calculating the model error of each substructure E and by choosing a level such as

ζ 2 ET ≥ δ max E∈E ζ 2 ET ( 31
)
where E is the set of all the substructures and δ is a chosen value. During the correction process, which is the second step of the model updating method, only the parameters from these substructures are updated. The final value of these parameters must correspond to a minimum of e 2

T . This problem, which is nonlinear with respect to these parameters, is solved using a BFGS minimization algorithm and the gradients of the parameters are calculated numerically. Besides, for each variation of the parameters, the mass, stiffness and damping matrices are reassembled. Once the correction has been made, the model error ζ 2 T is recalculated. If the new value is less than a given level, the updating process is terminated; otherwise, a new iteration consisting of a localization step and a correction step is performed.

Discretization of the problem

The discrete form of the modified error is written with respect to the vectors (denoted {U }, {V } and {W }) of the nodal values of the displacement fields U , V and W . In the case of a single excitation, the measured displacements are normalized by the amplitude of the force vector, so that only the amplitudes of the displacements appear in the expression of the error measure η 2 ω . Thus, Equation 23 depends on the displacements alone. Consequently, one has

e 2 ω ({U }, {V }, {W }) = γ 2 {U -V } T ⋆ ([K] + T ω 2 [C]){U -V } + 1 -γ 2 ω 2 {U -W } T ⋆ [M]{U -W } + r 1 -r {ΠU -U } T ⋆ [G]{ΠU -U } (32)
in which {U -V } and {U -W } denote {U } -{V } and {U } -{W } respectively. Moreover, Π is a projection operator which, when applied to a vector {Z}, yields the values of that vector at the sensors. The matrix [G] quantifies the error in the measurements. In our case, we used the following expression [START_REF] Deraemaeker | Reduced based for model updating in structural dynamics based on constitutive relation error[END_REF]:

[G] = γ 2 ([k] + T ω 2 [c]) + 1 -γ 2 ω 2 [m] (33)
where [m], [k] and [c] are respectively the reduced mass, stiffness and damping matrices of the system at the measurement points. In addition, the solution ({U }, {V }, {W }) must be admissible, i.e. it must verify

([K] + iω[C]){V } -ω 2 [M]{W } = {F } ( 34
)
{F } is the vector of the excitation forces. The minimization of the error e 2 ω under the admissibility constraints is obtained by introducing Lagrange multipliers, which leads to the resolution of a system of linear equations given by

[A]{Y } = {B} (35)
where [A], {Y } and {B} are written as

[A] =      γ 2 ([K] + T ω 2 [C]) 1 -γ 2 ω 2 [M] r 1 -r Π T [G]Π γ 2 ([K] + T ω 2 [C]) 1 -γ 2 ([K] -iω[C]) 0 -[K] -iω[C] ω 2 [M] [K] + iω[C] -ω 2 [M]      (36)
{Y } =   {U -V } {U -W } {U }   ( 37
) {B} =    r 1 -r Π T [G]{ U } 0 {F }    ( 38
)

Calculation of the crack's depth

The CRE updating method described in the previous section enables one to identify the crack's position within a particular element of the shaft. Then, the calculation of the crack's depth may be carried out using the following strategy. Considering Equations 15, 16 and 17, the stiffness matrix K crack modified by the transverse crack is expressed at the crack's location as a function of the nondimensional crack's depth µ = h R . Therefore, the identification of the crack's depth is achieved by minimizing the error e 2 µ in the crack's depth with respect to the nondimensional crack depth µ:

e 2 µ = 8 i=1 8 j=1 (K crack,ij -K ident,ij ) 2 (39)
where K crack,ij and K ident,ij denote the theoretical and identified flexibility coefficients respectively.

Numerical studies

In this section, in order to verify the suitability of the proposed approach, we will present eight numerical example cases with different damage locations and crack sizes. The position and size of the crack for each case is defined in Table 1. These eight cases were chosen to demonstrate the robustness of the crack identification method with respect to the crack's depth and location. The crack sizes in the eight cases may be rearranged as follows:

• small crack depths: Cases 3, 6 and 7;

• intermediate crack depths: Cases 1 and 5;

• large crack depths: Cases 2, 4 and 8. It is well-known that an eigenfrequency is unaffected if the crack is located at a node of the associated mode (i.e. at the center of the beam for the second vertical and horizontal modes, and at one-third or two-thirds of the beam for the third vertical and horizontal modes). In order to illustrate the robustness of the identification method when the crack is located near a node, Case 1 concerns a crack situated at a node of the third vertical and horizontal modes (at one-third of the beam) and Case 2 concerns a crack situated at the node of the second vertical and horizontal modes (at the center of the beam). Cases 4, 5, 6 and 7 concern a crack located near a node of the fourth vertical and horizontal modes. Finally, for Case 3, the crack is located near one end of the beam. Moreover, since damage identification can be more or less difficult depending on the information available (e.g. the number, locations and directions of the sensors, or the presence of noise in the measurements), the robustness of the method was studied under all of these hypotheses. First of all, the effectiveness of the method was investigated for the eight cases by considering sensors all along the beam and in the two transverse directions, both without noise and with 5% noise. Then, damage identification was carried out for Cases 1 and 2 considering only five sensors along one direction and four noise levels (0%, 5%, 10% and 20%). The objective of all these numerical examples was to demonstrate that damage detection and the identification of the crack's size and position can be obtained with adequate precision even if the crack is small and located near a node. Moreover, it should be noted that the highest noise level (20%) was added to the simulations and that less than 5% of the displacements were measured. For each study, we first calculated the experimental data corresponding to the forced response of the cracked beam to a solicitation. The beam considered in this section was described in Section 2. The geometrical and physical parameters of the beam, the frequency range and the properties of the solicitation applied to the beam are summarized in Table 2. The discrepancy between the experiments and the data obtained from an initial model associated with a crackfree beam is high and can be observed by plotting the vertical and horizontal displacements of the crack-free and cracked beams for various crack depths and locations (see Figure 4). As explained by various researchers in the last few decades [START_REF] Sinou | Influence of cracks in rotating shafts[END_REF][START_REF] Liang | Detection of cracks in beam structures using measurements of natural frequencies[END_REF][START_REF] Adams | A vibration technique for non-destructively assessing the integrity of structures[END_REF][START_REF] Morassi | Crack-induced changes in eigenparameters of beam structures[END_REF][START_REF] Hearn | Modal analysis for damage detection in structures[END_REF], the presence of the crack reduces the stiffness of the structure and, therefore, the natural frequencies of the original crack-free beam. Basically, an increase in the nondimensional crack depth decreases the natural frequencies of the beam (see, for example, Cases 4 and 5 in Figure 4). The main changes in the natural frequencies occur in the vertical mode because of the beam's height and its influence on the crack, as indicated in Equation 16. The frequencies are unchanged when the crack is located at a node of the associated mode (for example, at the center of the beam for the second vertical and horizontal modes, see Case 2 in Figure 4), and the decrease in the natural frequencies is more pronounced when the crack is located at the antinodal points of the associated mode shape (for example, at the center of the beam for the first and third vertical and horizontal modes, and at one-third or two-thirds of the beam for the second vertical and horizontal modes; see the comparison between Case 2 and Cases 4-5 in Figure 4). Moreover, the presence of the crack induces changes not only in the natural frequencies, but also in the antiresonant frequencies [START_REF] Dilena | The use of antiresonances for crack detection in beams[END_REF]. Therefore, a change in natural frequencies and antiresonances is the common first step in the detection of a crack, as well as in the identification of both the crack's depth and its location.

Case

In each study, the model error was rather high. Then, the localization step was performed and all local errors were calculated. The most erroneous element of the beam, corresponding to the highest local error, was determined. The updating process was carried out by varying only the parameter associated with this element. In the end, the errors were significantly reduced. First, we considered Cases 1 to 8 with sensors located all along the beam and displacements measured in the two transverse directions. The measurements were assumed to be noise-free. Table 3 gives the modified and model errors for all these cases. Since the model error was rather high, we started with the localization step. Figure 5 shows the local errors along the length of the beam for the eight cases. One can see that since the most erroneous element corresponded to the location of the crack the localization step was successful in all cases. Following the updating process, the errors (given in Table 3) were relatively small. Then, the crack's size was obtained by minimizing the error in the crack's depth e 2 µ (defined in Equation 39) with respect to the nondimensional crack depth ratio µ. The assumed and estimated nondimensional crack depths match perfectly in the eight cases, as shown in Table 4. At this stage of the study, one may point out that the crack model considered in this paper is a simplified model (with a regular cross section and a transverse crack with its front axis in the X-direction) which is generally accepted and used for the validation of model updating methods. Had the crack's shape at the cross section been different or the cross section irregular, the identification of the crack's size would have been erroneous. However, the crack's location would have been correctly identified because the crack modifies only the stiffness properties of the cracked cross section.

Since experimental data are usually affected by measurement noise, we studied the influence of various noise levels on the method. The random noise was distributed uniformly in space and over the frequency range. Table 5 shows the results for the eight previous cases, this time with 5% random measurement noise. Again, for each case, the model error was rather large and yet the localization step was successful (see Figure 7). The errors following the updating process are given in Table 5. One can observe that the model error increases with the nondimensional crack depth ratio µ as well as with the level of measurement noise. The nondimensional crack depth estimated from the crack depth error function defined in Equation 39 correctly matched the assumed nondimensional crack depth in all eight cases as shown in Table 4. However, one can observe that noise in the measurements can make the identification of the size of small cracks more difficult (see Cases 6 and 7 in Table 4). Usually, measurements can be performed only at a few sensors, in one direction, and are unfortunately very noisy. In order to show the robustness of the method, we chose to calculate Cases 1 and 2 with only five sensors (located at Nodes 6,[START_REF] Dilena | The use of antiresonances for crack detection in beams[END_REF][START_REF] Sinou | Damage assessment based on the frequencies' ratio surfaces intersection method for the identification of the crack depth, location and orientation[END_REF]21 and 26), in one direction and with various levels of measurement noise. The noiseless case was also considered so results with and without noise could be easily compared. Table 6 shows the results obtained after one localization step followed by the updating process. Figures 8(a) and 9(a) illustrate the local errors in the 30 elements (normalized to the highest value). In all cases, the final identification of the crack's location appears to be in agreement with the assumed position of the crack. Then, the determination of the nondimensional crack depth was obtained by minimizing the crack depth error function. The results are given in Table 7. Figures 8(b) and 9(b) illustrate the evolutions of the error in the crack's depth. Even though the difference between the estimated and assumed nondimensional crack depths increases with the noise level, these results can be considered to be reasonably good. This proves that with 10% and 20% noise the crack's location obtained is perfect and the maximum errors in the estimation of the crack's size are about 7% and 13% respectively. Finally, the comparisons of the Frequency Response Functions (FRFs) in the vertical direction obtained using the estimated crack parameters (i.e. the crack's location and nondimensional depth after minimization, indicated with red dotted-dashed lines) with the measurements generated from the assumed theoretical crack locations and depths (black solid line) are shown in Figures 10 to 13 and 14 to 17 for Case 1 and Case 2 respectively. For each case, the four figures correspond to calculations with four random noise levels (0%, 5%, 10% and 20%) affecting the assumed FRFs. Perfect agreement between the estimated and assumed vertical FRFs is clearly observed in most of the frequency range. The crack-free vertical FRF (black dashed lines) is shown to indicate the initial configuration used for the CRE updating method. It appears clearly that the estimated vertical amplitudes are very close to their actual values, even though 20% noise was added to the numerical simulations and less than 5% of the displacements were measured. The damage assessment technique, based on the CRE updating method and a crack depth error function, was developed in order to identify the crack's location and its nondimensional size. The effectiveness of the proposed method is demonstrated through numerical simulations involving different crack locations and nondimensional crack depths. These examples show that the identification of the crack's parameters can be carried out with satisfactory precision even when the crack is located near a node of a vertical or horizontal mode or when the nondimensional crack depth is small. Moreover, these results demonstrate the robustness of the technique when 20% uniformly distributed noise is added to the simulations and less than 5% of the displacements are measured. The results obtained with this robust damage detection tool combining a model error estimator and model updating procedure (proposed by Ladeveze and co-workers [START_REF] Ladevèze | Fe modeling and analysis: a localization method of stiffness errors and adjustments of fe models[END_REF][START_REF] Ladevèze | Updating of finite element models using vibrations tests[END_REF][START_REF] Ladevèze | Application of a posteriori error estimation for structural model updating[END_REF][START_REF] Chouaki | Updating structural dynamics models with emphasis on the damping properties[END_REF]), and the use of antiresonant frequencies (proposed by Dilena and Morassi [START_REF] Dilena | The use of antiresonances for crack detection in beams[END_REF]) are encouraging. The procedure for determining the crack's location and depth appears to be both simple and general. Future works will focus on assessing the effectiveness and limitations of the method based on more complex illustrative examples (3D structures with beam networks).

Figure 1 :

 1 Figure 1: Finite-element model of the beam with the cracked cross section

Figure 2 :

 2 Figure 2: Cross section of the cracked beam

Figure 3 :

 3 Figure 3: The domain being studied and the applied loads

Figure 4 :β 1 . 2 × 10 - 6

 412106 Figure 4: Frequency response functions of the beam for various crack depths and locations at Element 5 of the beam (-crack-free, --Case 2, • • • Case 4,-.-Case 5)

Figure 5 :

 5 Figure 5: Local errors in the 30 elements (normalized to the highest value) for the eight cases with no measurement noise

Figure 7 :

 7 Figure 7: Local errors in the 30 elements (normalized to the largest value) for the eight cases with 5% random measurement noise

Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 15 :Figure 16 :Figure 17 :

 8910111213151617 Figure 8: Local errors in the 30 elements (normalized to the highest value) and evolution of the error in the crack's depth for Case 1 with noisy measurements and only with five sensors in one direction

Table 1 :

 1 Specific data for Cases 1 to 8

		Crack's location (element)	Nondimensional crack depth µ
	1	10	0.5
	2	15	0.7
	3	28	0.3
	4	7	1.0
	5	7	0.5
	6	7	0.2
	7	7	0.1
	8	5	0.8

Table 2 :

 2 Geometrical and physical parameters of the beam, frequency range of the study and nature of the excitation

	Case % noise	ζ 2 T	i	e 2 T	i	ζ 2 T	f	T e 2	f
	1	0	0.92	1.40	0.108	0.109
	2	0	1.53	2.24	0.109	0.111
	3	0	0.68	0.92	0.108	0.110
	4	0	3.66	4.98	0.110	0.112
	5	0	0.94	1.40	0.109	0.110
	6	0	0.28	0.40	0.108	0.110
	7	0	0.14	0.18	0.108	0.110
	8	0	2.42	3.41	0.107	0.108

Table 3 :

 3

Errors (in %) before (subscript i) and after (subscript f ) the updating process for Cases 1 to 8 with no measurement noise

Table 4 :

 4 Comparison of assumed vs. identified nondimensional crack depth µ for Cases 1 to 8

	2	x 10 12
		1 and 5
	1.8	2
		3
	1.6	4
		6
		7
	Crack depth error	8
		Nondimensional crack depth µ
	Figure 6: Error in the crack's depth for the eight cases with no measurement noise
		14

Table 5 :

 5 Errors (in %) before (subscript i) and after (subscript f ) the updating process for Cases 1 to 8 with measurement noise

Table 6 :

 6 Errors (in %) before (subscript i) and after (subscript f ) the updating process for cases 1 and 2 with sensors in 5 displacements and with various levels of noise

	6 Conclusion

Table 7 :

 7 Comparison of the assumed and identified non-dimensional crack depth µ for cases 1 and 2 with sensors in 5 displacements and with various levels of noise

										1		8	x 10 11
										0.9		7
										0.8		6
	Local errors	30 0 0.5 1	25	20				10%	20%	0.3 0.4 0.5 0.6 0.7	Crack depth error	2 3 4 5
				15	10			5%		0.1 0.2		1
		Beam element	5	0	0 %	Level of noise			0 0	0.1	0.2	0.3 Nondimensional crack depth µ 0.4 0.5 0.6 0.7	0.8	0.9	1
					(a) Local errors					(b) Crack depth error with noise
													(• • • 0%, --5%, -.-10%, -20%)

This research deals with the nondestructive detection of cracks in a simply-supported beam. The cracked beam is modeled by taking into account the flexibility due to the presence of the open transverse crack and by reducing the second moment of area of the element at the crack's location.


domain: sdv sde

text:
Pierre-Olivier Cheptou
email: pierre-olivier.cheptou@cefe.cnrs.fr

Pollination strategies in the face of pollinator decline

Keywords: Pollinator decline, Plant pollination strategies, Adaptation, Evolutionary rescue, Plant-pollinator network

The recent pollinator decline in developed countries has been documented in many parts of the world and its anthropogenic causes are now identified. Because pollinators contribute to the sexual reproduction of many plant species, pollinator decline constitutes a potential threat on insect-pollinated plant populations. Moreover, it is susceptible to impede pollination services to crops in agriculture. In this paper, I review the empirical evidence that pollinator decline translates (or not) into pollen limitation for plants and alters plant-pollinator networks.

I also analyse the possibility of short-term evolution of plant pollination strategies. In particular, I discuss how changes in pollinator fauna constitutes new selection pressures for plants and how plant reproductive traits are able to respond rapidly to such new selection regime. Mating system evolutionary theory thus predicts that short-term evolution may allow plant populations to adapt to pollinator decline, potentially rescuing populations, as predicted by the evolutionary rescue theory. Nevertheless, mating system theory and empirical data support the idea that evolutionary rescue through mating system adaptation is not always possible. Finally, I discuss how plant evolution may disturb plant-pollinator interactions potentially breaking major trophic links in ecosystems, which could in turn reinforce pollinator's extinction in the future.

Introduction

A recent study reported that more than 75% of insects declined during the last thirty years [START_REF] Hallmann | More than 75 percent decline over 27 years in total flying insect biomass in protected areas[END_REF]). Among them, pollinator decline has become a feature of particular interest because of their well-identified role in ecosystems: they allow sexual reproduction of many plant species. Moreover, by pollinating crops, pollinators have been identified as a major ecosystemic service in agriculture. About 85% of crops rely at various degrees on insect pollination, which concerns about 40% of crop production.

Pollinator decline is sometimes assimilated to bee decline called the "Colony collapse disorder" [START_REF] Goulson | Bee declines driven by combined stress from parasites, pesticides, and lack of flowers[END_REF]. This is however not the unique group of pollinator declining.

Figure 1 illustrates the contraction of Bombus distiguendus in the United Kingdom. According to the National Biodiversity Network data (UK), the species was recorded in the whole United Kingdom and Ireland in the past. Since 2000, the species is restricted to the northern parts of Scotland. Changes in land use (e.g. loss of flowering grasslands, intensive agriculture,…), parasites and pesticides have been identified as important causes of decline in developed countries [START_REF] Goulson | Bee declines driven by combined stress from parasites, pesticides, and lack of flowers[END_REF]. Because plant depends on pollinators to produce their seeds and thus to maintain a sustainable demography, pollinator decline has the potential to disturb plant reproduction and to induce plant population extinction, with cascading effects on other species linked by trophic relationships. Furthermore, experiments have shown that it may alter the pollination of crops and reduce crop yield (Perrot al. 2018).

In the 1980's, evolutionary ecologists and population geneticists have developed an important theoretical framework analyzing plant mating system evolution. Coupled with a large corpus of empirical studies, the major selective factors have been identified and in particular, the role of pollination environment. While mating system evolution models have been mainly used as a way to interpret patterns in natural populations, the pollinator's crisis has stimulated studies analyzing microevolution on mating systems. To date, empirical studies are scarce but several experimental studies have shown convincingly that floral traits and mating system traits can evolve rapidly [START_REF] Roels | Rapid evolution caused by pollinator loss in Mimulus guttatus[END_REF][START_REF] Gervasi | Real-time divergent evolution in plants driven by pollinators[END_REF].

This paper aims at exploring the impact of pollinator decline on entomophilous plants. More specifically, I discuss (1) how pollinator decline is susceptible of modifying plant communities (2) how pollinator decline translates into pollen limitation for plant populations and (3) the potential for plants to adapt to pollinator decline and its consequences on population demography and trophic relationships in ecosystems.

1-Pollinator decline and plant communities

Plant pollinator interactions are in most cases mutualistic, the plants provide food for the pollinators (nectar and pollen), while the pollinators allow sexual reproduction through pollen transport. According to the Intergovernmental Science Policy Platform on Biodiversity and Ecosystem Services (IPBES), more than 300 000 plant species benefit from insects for their reproduction i.e. 87.5% of the total flora [START_REF] Rhodes | Pollinator decline -an ecological calamity in the making?[END_REF]. We can thus make the prediction that changes in pollinator composition should affect plant community. Curiously, such interactions have traditionally been little studied in community ecology (E-Vojtkó et al. 2020).

Community ecology has rather focused on functional traits (morphology, physiology, phenology) with little interest on reproductive traits and pollination. Recent empirical studies have however demonstrated that reproductive traits are linked to community structure. Using a large dataset of more than 600 000 trees and historical information documenting recently colonized areas, Rejou-Mechain and Cheptou (2015) showed that sexual system (dioecy) was the best predictor of young successional communities in Central Africa. Importantly, functional traits classically associated with young successional areas (e.g. N-fixing) were not found to be associated with dioecy. This thus suggests that dioecy does not influence plant community composition through trait association (at least those measured) but that dioecy, on its own, advantageous in young successional stages.

In 2006, [START_REF] Biesmeijer | Parallel declines in pollinators and insect-pollinated plants in Britain and the Netherlands[END_REF] published an important paper about the mutual influence of pollinators and plants in the context of pollinator decline. In their study, the authors compared plant composition and bee and hoverfly assemblages in Britain and the Netherlands several decades apart (pre-versus post-1980). The authors showed a parallel decline of insect pollinated plants and pollinators, thus demonstrating that plant and pollinators composition are interdependent in ecosystems. While it is tempting to conclude on the causal role of pollinator decline (caused by external factors) on plant species composition, the authors rigorously conclude that their data do not allow to conclude whether the decline of plants precede the loss of associated pollinators or the reverse. Using historical surveys in Britain, [START_REF] Baude | Historical nectar assessment reveals the fall and rise of floral resources in Britain[END_REF] have shown that nectar resources have greatly reduced between the 1930s and 1970s. These trends are consistent with pollinator declines but, again, does not allow to determine the causal connection of the parallel decline. The difficulty of attributing a causal connection in such a parallel decline relies on the structure of plant pollinator networks and more specifically whether pollinators or plants are limiting in ecosystems. Since 2000's, mutualistic networks have been subject to intensive interests in theoretical community ecology and synthetic metrics have been defined to characterize networks [START_REF] Bascompte | The nested assembly of plantanimal mutualistic networks[END_REF][START_REF] Vazquez | Uniting pattern and process in plant-animal mutualistic networks: a review[END_REF]. While network structures can vary according to ecosystemic specificities, the general feature is the imbalanced number of pollinator species and plant species: there are approximately four times more pollinator species than plant species. The interactions are asymmetric, which means that specialist plant species tend to interact with generalist pollinator species. [START_REF] Astegiano | The Robustness of Plant-Pollinator Assemblages: Linking Plant Interaction Patterns and Sensitivity to Pollinator Loss[END_REF] studied the robustness of networks to pollinator loss. Studying 13 plant-pollinator networks from various parts of the world, they showed that networks are globally robust to moderate pollinator loss. Such community ecology studies highlight the fact that plant populations may be resilient to pollinator decline and that it will not necessarily generate pollen limitation for plants.

2-Pollinator decline and pollen limitation in natural populations

At the population level, pollination ecology has for a long time revealed that most of plant populations are pollen-limited for seed production in natural conditions. In a survey of more than 250 plant species, Burd (1994) showed that more than 150 were significantly pollenlimited. Leaving ovules unfertilized may appear at first sight as non-adaptive since producing more seeds, by self-fertilizing the remaining ovules would a priori increase individual fitness.

Pollen limitation is classically estimated by experiments comparing seed (or fruit) production under natural pollination and when plants are pollen supplemented. More refined techniques have been developed to estimate reproductive assurance by manipulating the possibility of selfing through emasculation [START_REF] Eckert | Plant mating systems in a changing world[END_REF]. The question here is to asses if pollinator decline translates into higher pollen limitation for plants. The few studies that have explored the effect of pollinator decline on pollen limitation have revealed mixed conclusions. [START_REF] Pauw | Reconstruction of historical pollination rates reveals linked declines of pollinators and plants[END_REF] have produced an interesting study using herbarium specimens in the oilsecreting orchid species Pterygodium catholicum that depends on the oil-collecting bee Rediviva peringueyi (Melittidae) for its pollination. They used the pollinarium removal rates to estimate pollination activity from herbarium specimen. They established a pollination time series and showed that pollinarium removal declined in time. Though the lack of pollinarium removal does not equate to pollen limitation, this study suggests a temporal decrease in pollinator activity. Another herbarium-based study analyzed a century of pollination success through seed production in outcrossing Fabaceae in China [START_REF] Duan | A century of pollination success revealed by herbarium specimens of seed pods[END_REF]. Overall, the data does not reveal a clear decline in seed production. Over the 109 species analyzed (4637 specimens), four out of 95 species (with at least ten samples) exhibited a significant decline. Surprisingly, nine species exhibited an increase in seed production over time. The data were however highly variable and noisy, which limits the ability to detect a clear trend. In Erythronium grandiflorum, [START_REF] Thomson | Flowering phenology, fruiting success and progressive deterioration of pollination in an early-flowering geophyte[END_REF] reported results from a longitudinal survey using natural and supplementary pollination. In the 2010's paper, [START_REF] Thomson | Flowering phenology, fruiting success and progressive deterioration of pollination in an early-flowering geophyte[END_REF] concluded to a pollination deterioration from 1993 to 2009 over 17 years. However, after completing the time series, [START_REF] Thomson | Progressive deterioration of pollination service detected in a 17-year study vanishes in a 26-year study[END_REF] concluded that this trend was not significant anymore over 26 years (from 1993 to 2018). The results are illustrated in Figure 2. This study underlines the difficulty to capture long term trends, given inherent year-to-year variation of pollination, and the need to gather data on the long run.

A recent study by [START_REF] Martin | New indices for rapid assessment of pollination services based on crop yield data: France as a case study[END_REF] analyzed pollen limitation in France using crop yield in agrosystems. The authors classified crops according to their dependence to pollinators. It is expected that highly dependent crops will have reduced yield in poor pollination environment while such reduction is not expected in non-dependent crops. Using a 10-years dataset, the authors were able to estimate a relative pollination index at the level of French administrative department. They found that the South of France has a high values index while North of France (specifically departments around Paris) had the lowest pollination services. While the study does not directly address pollination of wild plants, it provides for the first time a pollination map at a large scale that may impact wild plants.

To date, data on the impact of pollinator decline on pollination limitation are scarce but we see from the few examples that the conclusions are not clear-cut. Several lines of interpretations can be proposed. First, it is possible that, because plant-pollinators interactions are in general imbalanced, a pollinator lost can be replaced by another one. In the peculiar case of the oil-collecting bee Rediviva peringueyi (Melittidae) and the guild of oil-secreting orchids [START_REF] Pauw | Reconstruction of historical pollination rates reveals linked declines of pollinators and plants[END_REF] a single pollinator species interacts with the plant species and the loss of pollinator translates into a decline in pollinarium removal over time. In accordance with this interpretation, Jacquemin (2020) found that specialist plant species have declined more than generalist plant species during the 20th century in Belgium. Though is difficult to predict the future effects of pollinator decline, we may fear that the ongoing decline may produce sudden effects on plant populations (tipping point) when pollinator redundancy will become critical. Second, there may be a potential experimental problem when pollen limitation is inferred from seed production directly. If plants are plastic in their ability to self-fertilize ovules, a reduction in outcrossed pollen will not necessarily translate into a total seed production decrease because outcrossed seeds may be replaced by autonomous selfed seeds [START_REF] Eckert | Plant mating systems in a changing world[END_REF]. Ideally, self-incompatible plant would be good model to test the impact of pollinator decline on pollen limitation because they prevent the possibility of outcrossing seeds replacement by selfed seeds. We may also expect that the effect of pollinator decline on pollen limitation will be more pronounced in "simplified ecosystems" where plant-pollinator networks are simplified and the redundancy is lower. This is typically what we expect in agrosystems where pollinators are threatened by human activities (low plant diversity, pesticides, habitat fragmentation,…), in line with the large scale study of [START_REF] Martin | New indices for rapid assessment of pollination services based on crop yield data: France as a case study[END_REF]. In accordance with this interpretation, Perrot et al. (2018) found that sunflower yield is positively correlated to the total bee abundance and bee genera diversity. Considering crops as phytometers in agrosystems, this indicates that weed pollination has likely suffered from pollinator decline.

3-Can populations respond to changes in pollinator environment?

Elements of theory. Pollination ecology has developed metrics to estimate pollen limitation in natural populations. [START_REF] Eckert | Plant mating systems in a changing world[END_REF] have summarized the pollination parameters and pollination manipulation to achieve them. The proportional decrease in seed production from emasculated flowers (relative to non-manipulated flowers) allow to evaluate how autonomous selfing contribute to seed production (reproductive assurance). Such techniques have been intensively used in pollination experiments as reproductive assurance is one of the key parameters in the evolution of self-fertilisation (see [START_REF] Lloyd | Some reproductive factors affecting the selection of self-fertilization in plants[END_REF]. Indeed, the evolution of selffertilization is basically governed by three factors. Pollen limitation (reproductive assurance) and the Fisherian cost of outcrossing select for higher selfing, while inbreeding depression selects for higher outcrossing. It is tempting to predict that pollen limitation will necessarily increase selfing but rigorous analysis of the model shows that is not necessarily the case.

Evolution can drive pollination strategy towards outcrossing, even in the presence of pollen limitation. This means that evolving a fraction of unfertilized ovules can be evolutionarily stable, which provides a rationale for natural plant populations to be pollen limited (Burd 1994). However, if pollinator decline translates into higher pollen limitation, we can make the general prediction that the strength of selection towards selfing should increase.

Microevolution of reproductive systems. In plants, many traits are involved in mating system strategy. For instance, anther-stigma distance has been shown to be tightly linked to outcrossing rate. Floral scents and flower colours are involved in pollinator attraction defining the guild of pollinator as well the intensity of pollinator visits [START_REF] Ashman | Genetic constraints on floral evolution: a review and evaluation of patterns[END_REF].

Because floral traits are under strong selection, they are a priori not expected to exhibit large genetic variance. However, compiling quantitative genetics data on floral traits in a set of 41 hermaphroditic species, [START_REF] Ashman | Genetic constraints on floral evolution: a review and evaluation of patterns[END_REF] found substantial heritability for floral traits associated with mating systems. For instance, they found h² = 0.40 for anther-stigma separation and positioning, h² = 0.45 for corolla size, and average h² = 0.20 for nectar production. As a consequence, the conditions for mating system traits to evolve after a change in selection regime are fulfilled, in particular in the context of pollinator decline.

Experimental evolution has indeed revealed that plant mating system traits can shift in a few generations. Applying various intensities of pollination, [START_REF] Roels | Rapid evolution caused by pollinator loss in Mimulus guttatus[END_REF] showed in Mimulus guttatus that anther-stigma distance has been reduced in the low pollination treatment after only five generations, resulting in a higher propensity to produce seeds by automous selfing. Thus, this experiment clearly shows that self-fertilisation has the potential to evolve in the context of pollinator decline. In another experimental evolution study, [START_REF] Gervasi | Real-time divergent evolution in plants driven by pollinators[END_REF] investigated the evolution of floral traits in Brassica rapa under three different pollination treatments (hand-pollination, hoverfly and bumblebees) over nine generations. Plants pollinated by bumblebees evolved (among other traits) higher plant sizes, a more pronounced floral display and higher floral scent, while plants pollinated by hoverflies have evolved higher selfing (Figure 3). This study demonstrates that plants can rapidly adapt to a change in pollinator fauna. Interestingly the traits under selection are not necessarily floral traits in the manner of plant size. This illustrates that pollination strategy encompasses a variety of co-adapted traits that must be considered altogether.

Rapid evolution of reproductive systems in natural populations has been less studied and only a few examples have been reported. In Centaurium erythraea, [START_REF] Brys | Effects of human-mediated pollinator impoverishment on floral traits and mating patterns in a short-lived herb: an experimental approach[END_REF] compared fragmented urban populations (where pollination activity is low) to non-fragmented rural populations (where pollinator activity is high). Plants from fragmented environments exhibited less herkogamy and a higher capacity for autonomous selfing than plants from nonfragmented populations (Figure 4). Assuming that urbanity is a proxy for pollinator decline, this study provides a convincing demonstration that mating systems can evolve under pollinator decline in natural populations. In agrosystems, [START_REF] Thomann | Contemporary evolution of plant reproductive strategies under global change is revealed by stored seeds[END_REF] studied adaptation to pollinator decline and climate warming in the self-incompatible species Centaurea cyanus (Asteraceae) using resurrection ecology methodology. Using seeds sampled 18 years apart in North of France, they showed an increase in capitula size and in floral display i.e. no evidence of evolution towards selfing but rather the evolution of traits reinforcing the plant-pollinator interactions. Because the species has not evolved selfcompatibility, the authors interpreted the evolution towards higher attractiveness as a strategy to improve their fitness in an impoverished pollination environment.

Altogether, these empirical results show that rapid change in floral traits is susceptible to evolve in the face of pollinator decline. While experimental evolution has rather confirmed expectations, the direction of selection in natural populations is less clear, which suggests that the selection caused by pollinator decline is probably variable and needs to be properly characterized to make predictions about floral traits evolution.

4-A little prospective for the evolution of plant communities

Can we predict the evolution of floral traits in the face of pollinator decline? Though it would be tempting to assimilate pollinator decline to pollen limitation for plants, we see from the few empirical studies that the results are contrasted (see section 2). This question is however central and worth to be studied in the near future. In particular, we need to clarify whether pollinator community composition has changed or/and whether pollination intensity has changed. We may expect that evolutionary consequences will not be the same (see [START_REF] Roels | Rapid evolution caused by pollinator loss in Mimulus guttatus[END_REF][START_REF] Roels | Rapid evolution caused by pollinator loss in Mimulus guttatus[END_REF][START_REF] Gervasi | Real-time divergent evolution in plants driven by pollinators[END_REF]Schiestl 2017 for comparison). In this respect, predicting evolutionary response to pollinator decline is not an easy task. Assuming that pollinator decline translates into pollen limitation, evolutionary models would predict that plants should evolve towards higher selfing but the few empirical results have revealed that evolutionary trajectories may differ. The evolution of larger capitula and larger floral display in the selfincompatible cornflower ( Cyanus segetum; Thomann et al. 2015) appears contradictory with other patterns of evolution such as the reduced anther stigma distance in urban populations of Centaurium erythrea resulting in higher selfing propensity demonstrated by [START_REF] Brys | Effects of human-mediated pollinator impoverishment on floral traits and mating patterns in a short-lived herb: an experimental approach[END_REF]. In a review, [START_REF] Thomann | Flowering plants under global pollinator decline[END_REF] analysed the literature of adaptation under low pollination and proposed that both evolution towards increased selfing or evolution toward reinforced interactions can actually occur. The lack of standing variation for some traits (e.g. no self-compatible variant in self-incompatible species) will prevent evolution, while in contrast, available standing variation will allow a rapid evolutionary trajectory. A recent study on Impatiens capensis illustrates that the two roads described by [START_REF] Thomann | Flowering plants under global pollinator decline[END_REF] can evolve within population. [START_REF] Panique | Simulated pollinator declines intensify selection on floral traits that facilitate selfing and outcrossing in Impatiens capensis[END_REF] have shown that reduced pollination selected both for increased outcrossing traits in chasmogamous flowers (larger flowers) and for increased selfing through higher rate of cleistogamy. Does mating system adaptation face to pollinator decline rescue populations? If change in pollination regimes is likely to produced evolutionary changes, whether these changes can allow plant populations to escape extinction -i.e. will provide evolutionary rescue -is worth to be studied. The evolutionary rescue is a conceptual model proposed by [START_REF] Gomulkiewicz | When does the evolution by natural selection prevent extinction[END_REF]. It posits that, after an environmental change, evolution may help to recover a viable population that without evolution would have been doomed to extinction. This model has been tested in laboratory in microorganisms and the results show that organisms can increase their population vital rates by adapting to new conditions (temperature, pH,…). One can wonder if adaptation of reproductive traits in plants in the face of pollinator decline can provide such a rescue. Though it would be tempting to assert that adaption (for instance via increased selfing) would rescue populations, [START_REF] Cheptou | Does the evolution of self-fertilization rescue populations or increase the risk of extinction?[END_REF] showed that this is not necessarily the case in the evolution of self-fertilisation. This is due to the inherent frequencydependence selection acting on selfing traits. Analogously to the classical Hawk-dove game [START_REF] Smith | Evolution and the theory of games[END_REF] where the hawk strategy inevitably evolves but is deleterious once established, self-fertilization evolves through the higher replication of selfing genes (see [START_REF] Fisher | Average excess and average effect of a gene substitution[END_REF]) but this does not equate to higher per-capita growth rate. In other words, the population vital rates are not maximised at the evolutionary stable selfing rate. It is thus possible, at least theoretically, that pollen limitation favours selfing causing a lower per capita growth rate i.e. allows deterministic suicide (see Figure 5). Then, if a few studies have shown that selfing can increase seed set and per capita growth rate under certain pollination circumstances [START_REF] Lennartsson | Extinction thresholds and disrupted plant-pollinator interactions in fragmented plant populations[END_REF], models for the evolution of self-fertilisation warns on the fact that conditions does not always lead to rescue. Besides, on the long run, self-fertilization is often associated to high extinction rates as it has been demonstrated in Solanaceae [START_REF] Goldberg | Species selection maintains self-incompatibility[END_REF].

Ecosystems structure after pollinator decline. The pollination biology and plant mating system is deeply rooted in the population biology paradigm centred on the plant species, the crisis of pollinators worldwide has however recalled that the realised mating system is in most cases the results of a pollination strategy in a given pollination environment. Moreover, if pollinator decline is expected to feedback on plants (the purpose of this article), plant adaptation will feedback on pollinator through changes in pollen and nectar production. It is now admitted that pollinator decline has been caused by many factors, among others, pesticides and pollution i.e. factors that are external to the plant-pollinator interactions. The possibility for plants to adapt rapidly to impoverished pollination environment (as shown by experimental evolution) may be considered as a way to tone down the deleterious effects of anthropic activities. Several lines of arguments suggest that evolution may, in certain circumstances, not alleviate the effects of anthropic activities. First, the possibility of evolutionary suicide by selfing suggests that some plant populations could go extinct in communities. Second, we may naively expect that plant and pollinator populations will recover if sanitary conditions improve in the future (e.g. though the interdiction of the use of pesticides). However, if plant strategies have evolved reduced pollination interactions (e.g. selfing) because of the current pollinator crisis, we may fear that pollinators will not find enough food (pollen nectar,…) to maintain viable populations and that plant evolution will not be reversible as selfing is classically considered as a dead-end evolution [START_REF] Takebayshi | Is self-fertilization an evolutionary dead end? Revisiting an old hypothesis with genetic theories and macroevolutionary approach[END_REF]. More than pollination strategies, the consequences of pollinator decline could be the loss of major trophic relationships in ecosystems.

5-Conclusion

The aim of this paper was to analyse if pollinator decline translates into pollen limitation for plants and eventually, how plants can adapt to such environmental change. Literature survey indicates that empirical studies are scarce and, to date, we do not have a clear picture of the effect of pollinator decline on plants. It is likely that such effects vary from ecosystems to ecosystems and that human-altered ecosystems are more affected than more natural ones.

Nevertheless, we crucially lack of long-term data. On the other hand, the capacity of plants to evolve in the face of reduced pollination is now clearly established in experimental evolution.

Given the current threat on pollinators, we can then conjecture that plant evolution will occur and that plant-pollinator interactions will be durably affected. [START_REF] Lloyd | Self and cross fertilization in plants. 2-The selection of self fertilization[END_REF] where evolution is driven by the cost of outcrossing, inbreeding depression (δ) and pollination rate (e). Evolution of selfing: Lloyd's model has established that higher self-fertilization (respectively higher outcrossing) evolves if (respectively if ). In ). In the figure, the threshold is indicated by the bold line ( ) above which the per capita in higher and below which, the per capita is lower.

Conclusion:

Because the evolution of selfing increases or decreases the per capita growth rate, the evolution of selfing can either provide evolutionary rescue (below the thin line) or evolutionary suicide (above the thin line) (from Cheptou, 2019).

Figure 1 :Figure 2 :

 12 Figure 1: An example of pollinator range contraction in the last decades. The range limits of the species Bombus distinguendus have drastically reduced during the last fifty years (data from the National Biodiversity Network, illustrations courtesy of Dave Goulson).

Figure 3 :

 3 Figure 3: Experimental evolution by manipulating pollination environment with fast cycling Brassica rapa plants. Plants were submitted to (1) hand pollination, (2) hoverfly pollination, and (3) bumblebee pollination for 11 generations. The graph below reports trait shifts in the two dimensions of the linear discriminant function analysis (bumblebee: blue circles, hoverfly: green squares, control hand pollination: black triangles; filled symbols are group centroids of replicates). The results show that pollination treatments have led to significant evolutionary changes. Though not directly visible on this graph, plants having evolved under bumblebee pollination were taller, with increased ultraviolet reflection of flowers. Plants having evolved under hoverfly pollination had reduced emission of some floral volatiles, but a higher self-pollination ability (after Gervasi and Schielst 2017, illustrations courtesy of Florian Schielst).

Figure 4 :

 4 Figure 4: Reduction of anther-stigma distance in Centaurium erythreae in urban populations. On the left (a), in rural populations, the shape of the flower shows a spatial separation of the male (stamens) and female (pistil) organs favoring outcrossing, while in urban populations (on the right, b) the proximity of male and female organs favors self-fertilization (modified from Brys and Jacquemyn 2012).

Figure 5 :

 5 Figure 5: Evolution of self-fertilization and its consequences on demography. The evolution of self-fertilisation is based on the seminal model developed by David Lloyd[START_REF] Lloyd | Self and cross fertilization in plants. 2-The selection of self fertilization[END_REF]

  the figure, the threshold for selfing/outcrossing evolution is indicated by the bold line (). Demography of selfing populations:[START_REF] Cheptou | Does the evolution of self-fertilization rescue populations or increase the risk of extinction?[END_REF] established that the per capita growth rate increases (respectively decreases) with selfing if (respectively if