A Guibert

O Chadebec

J.-L Coulomb

C Rannou

R P Nogueira

Ships hull corrosion diagnosis from close measurements of electric potential in the water

Keywords: Cathodic protection, boundary elements method, inverse problems, regularization techniques. 1

We present here an original application linking an electrochemical domain and the computational aspect of electromagnetic fields to make a corrosion diagnosis of a protected underwater steel structure. After a defined operating time, it is mandatory to check an underwater steel structure. This control may happen too early, if the protection has not been damaged; but it may happen too late, if some particularly bad environmental conditions have already damaged the structure. Moreover, current examination techniques require immobilizing the structure for a long time: they may be complicated to dry (pipe lines or tanks) or to place in dry docks for vessels. The efficiency of these techniques could be improved. The purpose of this paper is to replace this checking/examination by a series of close electrical measurements in the electrolyte which provides a corrosion diagnosis of the structure. This also allows computing the electric and magnetic fields caused by this phenomenon everywhere in the electrolyte. The new method introduced ensures a great time-saving but also an accuracy never reached before.

INTRODUCTION

WO MAIN electrochemical phenomenon grow up when a defect appears on the painting of an underwater steel structure. They are accelerated because of the seawater salinity which increases its conductivity. It first gives rise to an electrochemical corrosion: this is a reaction between the iron and the water leading to an iron oxidation and a creation of rust. But if the steel structure is also electrically linked to a more noble metal (like bronze, aluminum, etc…) in the electrolyte, a galvanic couple appears, making the steel become the anode of a reduction oxidation reaction. This will oxidize the iron until it is fully destroyed. This is called galvanic corrosion. In this configuration, the anode is the defect in steel, the cathode is the noble metal area and the electrolyte is the seawater. Currents spread in the seawater from the anode to the cathode and then pass along the steel structure.

To prevent these destructive reactions, two electrochemical techniques have been developed, called cathodic protection. They have to be coupled to a correct painting of the structure:

• Sacrificial Anode Cathodic Protection (SACP): the principle is to add a less noble metal than iron on the structure to protect (like zinc, magnesium, etc…) which will become the new anode of the reaction, steel becoming a cathode (which is not oxidized). The main drawback of this method is that the new anode has to be changed regularly.

• Impressed Current Cathodic Protection (ICCP): here a noble metal (often platinum) is placed on the structure, electrically linked to the steel, injecting current in the electrolyte. This will artificially decrease the electrochemical potential of the cathode. The more current is injected, the more the potential of the cathode is decreased, until it reaches its passivation domain where the iron is protected. These two techniques have in common the circulation of current in the electrolyte from the production of electrons due to the oxidation of the anode (in the case of the SACP only, electrons are directly provided by the anode with the ICCP) and their use by the reduction of the cathode. Like any conduction problem, this current circulation creates an electric field in the electrolyte. T Fig. 2. Cathodic protection current lines spreading in the seawater, under the ship. They go back to the hull through the defects and the noble areas.

It is important to see that with both cathodic protections, the current goes back to the structure through the damaged areas (cathodes). If it was possible to perform many electric potential (or current) measurements stuck to the structure, it would be easy to find the corroded areas: it would be the areas where currents go back to the hull or the ones with the less electric potentials. But in practice a user does not have the time to get so many measurements and cannot do them so close to the structure.

The purpose of this article is to show the possibility of such a diagnosis with a limited number of measurements, not so close to the structure. Some kind of studies have been made [START_REF] Cheng | An iterative BEM for the inverse problem of detecting corrosion in a pipe[END_REF] [START_REF] Adey | Predicting Corrosion Related Magnetics[END_REF] but in internal problems. The structure here can be a pipe line or any piece of painted steel fully drowned in the seawater. The conduction is then not bounded and the problem is called external.

First the forward modeling of this phenomenon will be presented: this means providing an electromagnetic prediction from the knowledge of some boundary conditions on the structure. Then an inverse method will be extrapolated to make a diagnosis. This means to find the boundary conditions of the structure (and so the corroded areas) from a series of electromagnetic measurements. Finally some experimental results will be presented to corroborate the method.

The example taken in the whole article will be a ship hull protected with an ICCP, but the extrapolation to get the case of SACP will be often reminded.

PREDICTION: THE FORWARD MODELING

As mentioned above, the considered case here will be a piece of painted steel equipped with defects and an ICCP, drowned in an infinite electrolyte; this means far from the sea floor and from the surface (also previously called external problem). It is also possible to consider the sea floor and surface (to model the immerged part of a floating vessel for example), then the problem becomes internal. This last case will be considered in the experimentation part.

The piece of steel shape studied is a ship mock-up with defects and ICCP protection on its lowest parts. Numerical simulations have shown that placing ICCP and defects there impose currents lines spreading under the steel. In this case, modeling the water surface is not useful: The infinite problem described can also be semi infinite. In this way, all the simulations made further in the article (in an infinite domain with the mock-up) are representative of the immerged part of a ship far away from the sea floor.

To examine such an external problem, it is necessary to think about the modeling method to use. First, the method has to easily take into account the infinite bound, case often faced. But as developing an inverse method is the main goal of the study, the method has to create a system linking directly the boundary conditions. The Boundary Elements Method (often called BEM) has both advantages and will be chosen for the study [START_REF] Brebbia | Boundary Elements Techniques, Theory and Applications in Engineering[END_REF].

2.1

Governing equations and BEM method The problem exposed above deals with some current injection in the resistive seawater which comes back to the hull through the defects. This is a conduction problem, whose main unknown is then the electrical potential ϕ in the seawater defined by the Faraday law:

( ) ϕ grad E - = (1)
E is the electric field defined by:

( ) ϕ σ σ grad E J . . - = = (2)
This also induces a very useful equation:

( ) n J n ∂ ∂ - = - = = = ϕ σ ϕ σ σ . . . . . . n grad n E n J (3)
The σ parameter is the seawater conductivity, related to to its salinity. Moreover, the seawater will be supposed homogenous and isotropic, with non additional charges so that the Laplace equation is verified:

0 = ∆ϕ (4)
Green's first identity gives:

( ) ( ) ( ) ( )dS n u v n v u dV u v v u S V ∫ ∫ ∂ ∂ - ∂ ∂ = ∆ - ∆ / / (5)
Replacing u by the electric potential φ and v by Green's function in 3D:

( )

PM r P M G PM / 1 / 1 , = = (6)
We obtain Green's 3 rd identity, linking the potential and its normal derivative on the whole surface:

( ) ( ) ( ) ( ) ( ) ( )dS n M P M G dS n P M G M P P h S S ∫ ∫ ∂ ∂ - ∂ ∂ = / . , / , . ) ( ). ( ϕ ϕ ϕ (7)
h coefficient is the solid angle the point P sees around. It is possible to extrapolate (4) for P on the boundaries of the problem (non singular ones i.e. P on a plane boundary) with the introduction of a potential for the infinity bound φ ∞ , (Fig. 3). Indeed, two choices remain for this infinity bound: to set the infinity bound potential or the current density. The choice made here is to set the current on the infinity bound to zero, letting a infinite potential φ ∞ exist [START_REF] Zamani | BEM simulation of cathodic protection systems employed in infinite electrolytes[END_REF]. This potential, taken as a reference potential will be then very useful during the resolution. The further potential results are then obtained regarding this reference. The Green 3rd identity on the boundaries of the problem becomes:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ∞ - ∂ ∂ - ∂ ∂ = - ∫ ∫ ϕ π ϕ ϕ ϕ π . 4 / . , / , . . 2 dS n M P M G dS n P M G M P S S (8)
The described problem can be illustrated by the following scheme: Equation ( 5) is applied on the whole surface of the problem. The values of the unknowns vary on the entire model and the geometries studied are not simple. That is why it is impossible to compute analytically the integral term in [START_REF] Dhatt | Méthode des éléments finis[END_REF] and the next step is to mesh the model.

In a first approximation an order 0 approach of the unknowns is made: all quantities are set constant on each element. The unknowns are then located at each elements barycenter: this is also called a point matching method. The error induced by this approximation is insignificant in our case, so more complicated modeling will not be exposed. Moreover, the measurements made hereinafter are not very close to the model: this explains why the approximation provides quite good results. Finally, this approximation avoids singularities on the further computations. After meshing, the system obtained is:

[ ] 0 / . 4 =           ∂ ∂ - ∞ ϕ ϕ ϕ π n T H (9)
The corresponding H and T equal to:

( ) ∫ ∫ ∫ Ω Ω Ω = - =         ∂ ∂ - = j j j j M ij PM ij j M ij PM ij PM j M j M ij PM j M ij dS r T dS r r n dS r n H ) ( 3 ) ( ) ( ) ( ) ( ) ( 1 . 1 r r (10)
Indeed the unknowns can be taken out of the integral term as they are considered constant on the integration surface (order 0). The first step of the system creation is to build the left matrix, containing all surface integrations computed at each barycenter. This part relatively complicated: analytically integrate would spend too much time so the solution chosen is to proceed to a numerical integration. The method used is the Gauss integration process. The integration term T becomes, for a sufficient number m of Gauss points:

q m q q ij PM j M ij PM ij W r dS r T j . 1 1 1 ) ( ) ( ∑ ∫ = Ω ≈ = (11) r (ij)
PMq is the distance between the P point (where the integration is computed) and each Gauss point, Wq is the weight given to the q-Gauss point. The only variable to study is the number of Gauss points to place on the surface to get integrations close to the analytical ones. One difficulty comes from the choice of the number m of Gauss points: for an element P, the influence (i.e. the integration) of a far element is not preponderant and it needs only a few Gauss points but it is not the case of a neighbor element which needs many ones. A fixed number is not well optimized, but with simple geometries meshed uniformly, this will work quite well.

In our studies where geometries do not exceed one or two meters, and where meshing elements can reach 10 square centimeters, around 25 to 64 Gauss points are set. [START_REF] Brebbia | Boundary Elements Techniques, Theory and Applications in Engineering[END_REF] An important error occurs when computing numerically the influence of an element on itself. This error is erased by an analytic correction: the integral term in H ii and T ii are computed analytically.

As the integral computation is done, the separation of variable can be done. The boundary conditions available for an active protection (PCCI) are the active anode (A), the polarizable cathode (C) and the isolated part (I). The systems can be finally written:

[ ] 0 . 0 0 0 0 4 2 4 2 4 2
=                             ∂ ∂ ∂ ∂ ∂ ∂             - + - + - + ∞ ϕ ϕ ϕ ϕ ϕ ϕ ϕ π π π π π π n n n A A A T T T H H H T T T H H H T T T H H H I C A I C A (12)
The A i is the area of the element i. div(J) = 0 is included in the Laplace equation (starting point of the system) but the introduction of φ ∞ does not ensure this equality anymore. The last line is then added to ensure this condition. It is interesting to remark in (9) that, if the problem is meshed in N elements, the system has N+1 equations for 2N+1 unknowns.

2.2

Application for a PCCI case Since the problem studied here deals with PCCI, the boundary conditions physically mean:

• Anode regions (non-polarizable electrode), have their current density J A known. ∂φ A /∂n is linked to J A by multiplying it by σ, the conductivity of the electrolyte.

• Cathode regions (polarizable electrode), have their polarization law ∂φ C /∂n=f(φ C ) known and are most of the time non linear.

• On insulator regions the current density is null:

∂φ I /∂n=0.
The previous system (6) can then be written, by separating known and unknown variables: (13) The only information not used is the polarization law. Its non-linearity makes the resolution problematic: Fig. 4. Example of a non linear polarization law of steel. The steel is the reference in our problem so the abscissa scale is not a potential gap but only the electrolyte potential close to steel.

                    ∂ ∂ - ∂ ∂ - ∂ ∂ - ∂ ∂ - =                   ∂ ∂             - + - + - + ∞ n A n T n T n T n A T H H H T H H H

T H H H

As said in the legend, the steel is the reference. The curve gives then the relation f between the current density received by the metal and it electric potential:

( ) C PL C f n ϕ ϕ =       ∂ ∂ (14)
This knowledge simplifies the (13) system which can then be solve by an iterative solver; the one chosen is a Newton Raphson [START_REF] Dhatt | Méthode des éléments finis[END_REF] algorithm. The varying parameter is φ C , which is noted φ C NR below. The previous system becomes: Considering the size of the left matrix, it is interesting to observe that if the problem has N meshing elements; this matrix has N+1 equations (lines) and N+1 unknowns (rows). The new system is square and can be easily solved.

                      ∂ ∂ - ∂ ∂ - - ∂ ∂ - + - ∂ ∂ - - =                         ∂ ∂             - + - - + ∞ n A n T H n T H n T H n A T H H T H H T H H A A NR C A NR C A NR C BEM C I A ϕ ϕ ϕ ϕ ϕ π ϕ ϕ ϕ ϕ ϕ ϕ π π π π π . . .
In the Newton Raphson algorithm, the residual norm to minimize is the difference between the ∂φ C /∂n obtained with the system (9) and the one got with the polarization law:

( ) ( ) C BEM C PL C BEM C C f n n n R ϕ ϕ ϕ ϕ ϕ -       ∂ ∂ =       ∂ ∂ -       ∂ ∂ = (16)
The two quantities (∂φ C /∂n) BEM and f(φ C ) can be computed from the knowledge of φ C . The Newton Raphson algorithm make the φ C parameter vary to minimize the residual. When the algorithm has correctly converged, the two (∂φ C /∂n) are close to equality and the solution obtained has a physical behavior.

With a good starting point for the algorithm (often situated at the middle of the polarization law), the system converge in less than a dozen iterations. It is possible to insert the infinite potential in the varying parameters, adding a new line in the residual norm system (div(J)=0 again), which makes the algorithm numerically converge faster: in 3 to 6 iterations. The accuracy of algorithm's convergence does not have to be very low, as the quantities ∂φ C /∂n have values close to the unit. For example, in the further cases, a precision of 10 -4 is in most of times sufficient.

One example of resolution is shown below with a ship hull model under cathodic protection (ICCP) meshed in 1052 elements: The algorithm presented above is used to compute the boundary conditions on the whole surface. The results are presented in the following Fig. 5: The scale of the figure is in volts and the values are centered on the infinite potential. The white areas correspond to the active anodes and, as said in the introduction, they have the higher potential. The paint defects are colored in dark and as expected they have the lowest potential. The simulation also gives current densities, represented as arrows coming out of the hull through anodes and get back through cathodes. Using equation (12) for the residual norm, results are obtained with a maximum error of 10 -4 A/m² between injected and received currents.

2.3

Extrapolating the electromagnetic field The previous step gives all the boundary conditions on the model, that is to say each ∂φ/∂n and φ at the elements barycenter. Physically, equivalence is possible between those results and virtual electrostatic sources in conduction problems. To deduct the electrical potential somewhere in the electrolyte, it is possible to use Green 3 rd identity. The solid angle a Q point in the electrolyte see is -4π (the whole problem) so the equation becomes:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ∞ - ∂ ∂ - ∂ ∂ = - ∫ ∫ ϕ π ϕ ϕ ϕ π . 4 / . , / , . . 4 S S dS n M Q M G dS n Q M G M Q (17)
In this expression, the potential computed at the Q point will have the same offset than Fig. 5 due to the presence of the ϕ ∞ term. It is possible to remove this term to get the electric potential centered in zero. The boundary conditions are set with a point matching approximation, the values can be extracted from the integral terms and the meshing gives:

( ) ( ) ( ) ( ) ∑ ∫ ∑ ∫ = Ω = Ω -       - + ∂ ∂ - = N i M QM QM M i N i M QM i i i j i i i j i i dS r M dS r n M Q 1 3 1 . . 4 1 1 . 4 1 r n ϕ π ϕ π ϕ (18)
Like the previous resolution, the first step will be to evaluate the integral terms and then use the boundary conditions to get the potential at point Q. Moreover the Maxwell Faraday gives the well known equation [START_REF] Cheng | An iterative BEM for the inverse problem of detecting corrosion in a pipe[END_REF].

By using the opposite gradient of the potential calculation in (14), a new formula provides a simple electric field calculation: The scale is still in volts, without caring about the infinite potential (φ ∞ = 1.3762 V). That is why the values are centered on zero. The field arrows are also represented but not well highlighted.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑ ∫ ∑ ∫ = Ω = Ω                     - - + ∂ ∂ = N i M QM M QM M QM QM i N i M QM QM i i i j i i i i i i j i i i dS r r M dS r n M Q
To get such results, a process can be imagined, like:

Fig. 8. Electric potential measurements principle: a ship passing over a series of 20 electric sensors at a determined depth, making 70 set of measurements. The final measurements number is then 1400.

DIAGNOSTIC: INVERSE METHOD

The purpose of this part is quite simple: the previous study provides a method which, from a set of boundary conditions of a defined geometry, predicts the electromagnetic field in the electrolyte. The inverse method consists in a boundary conditions reconstruction from a set of measurements. This method is often represented by the following simple equation:

B X A = . (23)
B is a measurements vector, which can be electric or magnetic here. X is the unknown vector and A the interaction matrix between each measurement point and each unknown.

The main problem that appears in such methods is the condition number matrix A, directly linked to the difficulty to inverse it. Indeed matrix A does not often have the same number of rows and lines or contains values that are close to the computer zero (10 -16 ). These reasons give a bad condition number and, with a noisy B vector, lead to important difficulties to inverse the system (23). [START_REF] Hadamard | Lectures on Cauchy's Problem in Linear Partial Differential Equations[END_REF]

Naive inversion

Although matrix A has a bad condition number, it is possible to solve the normal equation to get a solution, which implies to find a minimum:

2 . min B X A = (20)
This means to solve:

( ) B A X A A t t . . . = (21)
This is possible by using a LU solver but some other exists. An example is developed hereinafter with the ship hull used before (N=1052 elements). Vector B is entirely described so the position of each barycenter of the grid is known. The A mes matrix can be then easily built with ( 9) and (14). The unknown vector X will contain the potential and its normal derivative at the barycenter of each hull element (except for the anode (A in the further equations) whose current density is known). This method only gives a mathematical convenient solution but often not physical. To focus these difficulties, an example Indeed vector B will first be simulated with the direct method. The 1400 electric potentials computed on a grid below the hull are taken adding 10% of the maximum potential random noise. This will be vector B. This induces a number of 2098 unknowns bringing the system under determined. In this case the system to inverse is: x 10 14 Fig. 9. Boundary conditions (electric potential and current densities) obtained on the hull with a "naive" inversion. The scale is in volts.

We obtain non physic boundary conditions with maximum potential values reaching 10 14 (the target values are shown in Fig. 5). The first clue for the inversion problem is the bad condition number which is here equal to 4.1279 10 16 . It is interesting to notice that the (23) equality is verified: the inversion is mathematically correct. The problem here is very ill-posed. A work on this system is necessary before inversing it to get a better solution.

One possible method would consist in 3 steps: first try to orthogonalyze the system. Then an injection of additional information can make the system have a better condition number. Finally a regularization technique allows imposing a physical behavior to the solution [START_REF] Chadebec | Recent improvements for solving inverse magnetostatic problem applied to thin hulls[END_REF]. The first step purpose, not described in this paper, is to change the basis description. The unknowns introduced (potential ϕ and its normal derivative) are representative of the physical phenomenon. There is no need to change them, as the other steps exposed before give good results.

Injection of information

The first step of pre conditioning is to add some information to the system. A smart way to do it is to use the straightforward matrix defined in [START_REF] Tikhonov | Solution of Ill-Posed Problem[END_REF] which links all unknowns of the inverse system. With the new meshing (no separation of cathodes and isolated parts), it becomes:

[ ] 0 . ' 0 0 0 ' 4 2 ' ' 4 ' 2
=                     ∂ ∂ ∂ ∂           - + - + ∞ n n A A T T H H T T H H hull A hull A ϕ ϕ ϕ ϕ ϕ π π π π (23)
H 11 , H 21 , T 11 and T 21 are the same as in [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion[END_REF], because the anodes are in the same location. On the other hand the location of the cathodes is the main unknown. Under the notation "hull" we gather the former cathodes and isolated parts. This explains the new terms appearing: H' 12 , H' 22 , T' 12 and T' 22 .

Remembering the system (25) does not take into account the infinite potential, this unknown and the equation associated can be deleted. In our case the infinite potential is supposed known as the forward modeling gave it. This potential could be evaluated in practice by measuring the electric potential far away from the structure toward the steel potential (set to reference here). The anode current is known, that is why the previous system can be written: (24)

      ∂ ∂ - ∂ ∂ - =           ∂ ∂       + + n T n T n T H H T H H A Green A Green
This new system will be then summarized:

[ ] [ ] n T n A A Green hull hull A Green ∂ ∂ - =           ∂ ∂ / . / . ϕ ϕ ϕ ϕ (25)
The reader should see the link with equation ( 25) which has the same unknowns. This allows building a new enhanced system with the same number of unknowns and more equations: This system can be simplified, as the number of equations brought by Green system is equal to the number of potential unknowns' ϕA and ϕhull. The potentials can be eliminated from the system and the final system to inverse can be written:

      ∂ ∂ - ∂ ∂ - =           ∂ ∂       n T n T B n A A A Green
[ ][ ] [ ] ' / . ' B n A hull = ∂ ∂ϕ (27)
The only unknowns are the current densities on the hull, which will allow computing the potentials by forward modeling. The new number of unknowns is then 1046 for 1400 equations from measurements: it is an over determined case.

This injection of information has also brought a physical dimension to the problem because it avoids solution diverging. But moreover, the number of unknowns has been really decreased, permitting us to work with higher mesh density.

In the case of a SACP system, the only known is the anode potential and not the anode current density. The only difference with the previous procedure is that Green equations eliminate the ∂φ/∂n and the system to inverse becomes:

[ ][ ] [ ] ' ' . ' ' B A hull = ϕ (28)

Regularization

The new system to inverse has been previously presented, simpler than it could first appear. But its direct inversion does not give good results, because of the noisy behavior it has.

Remembering that the case has become over determined, the next step will be to choose the best solution close to the analytical one (injective). It is interesting to use a regularization technique here.

The one studied here is Tikhonov regularization, whose principle is to minimize a new quantity [START_REF] Tikhonov | Solution of Ill-Posed Problem[END_REF]:

||A'.X -B'||+λ.||L.X || (29)
This means here:

||A'. ∂φ hull /∂n -B'||+λ.||L. ∂φ hull /∂n || (30)

The main goal is to bring again some physical behavior to the solution and the impact the user wants to give. Indeed, L is the regularization matrix which impose the behavior. The elementary one is the identity matrix (zero order): minimizing ||L.X|| amounts to minimize ||X|| and gives a minimum norm solution. It gives a physical solution but keep the noisy behavior. In the example current densities are found on every element, even if it is an isolated one. Consequently the corroded areas do not appear clearly and higher order regularization is needed. A first order matrix focuses on a continuous solution; the second order minimizes the Laplace operator of the solution: Fig. 10. Representation of a triangle mesh: the black element has 14 neighbors which needs to be taken into account to minimize the Laplace operator.

In the previous scheme, the k element has 14 neighbors. To minimize the Laplace operator, the values of the k current density must not too much differ from the i current densities. To minimize this difference, the line corresponding to this k element in the L matrix applied to X is then: The X vector presented above contains current densities, but they are simply linked to the ∂φ/∂n by [START_REF] Adey | Predicting Corrosion Related Magnetics[END_REF]. The presented L matrix brings more homogeneity to the values obtained and provides a noise smoothing without removing gaps of values. Moreover, the Green equations avoids the results from diverging, which is main goal of an order zero regularization. An order 2 regularization with Green equations cumulates the two benefits.

                                      - - - - ... ..
The coefficient λ introduced is the regularization parameter, representative of the weight given to the regularization term ||L.X|| in Tikhonov regularization. In the less noisy cases this term can have a very little value. On one hand, if a very high λ is taken, the solution is only regularized. On the other hand, if λ is very low, nothing changes from the previous solution [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion[END_REF].

Empirically, for a random noise reaching 10% of the maximal potential computed, λ is often chosen between 10 -2 and 10 -3 . If the λ term chosen has a value more than 5.10 -1 , the solution obtained is often too smooth, erasing the discontinuity of the boundary conditions, which is mainly interesting here. Indeed the goal is to find the lowest potential areas, so that smoothing potential results makes the diagnosis inefficient. The regularization has then a too important influence. As a small conclusion, this parameter choice is the most important part of the inversion, because it is the only choice to do.

A good tool to choose a correct λ is the L-curve [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion[END_REF] which shows on a graph the solution semi norm ||L.X || toward the residual norm ||A'.X -B'|| for different λ values. In the ship case, for a zero order regularization matrix, the L-curve obtained is shown below: This curve has often a corner which is the best area to choose the parameter. In this area, the solution semi norm and the residual norm have both values close to their minimum ones. It was the goal exposed before when introducing the problem in (31).

3.4

Final result and comparison with the target When chosen, the inversion is possible, and the ∂φ hull /∂n vector is obtained. As said above, the potential ϕA and ϕhull are then computed by forward modeling. The boundary conditions obtained on the hull are shown below: These results are very important because they give two boundary conditions which have to fit. Indeed corroded areas are situated where currents flow back into the hull, results got from the minimization of (32). As said above, potential on the hull are then obtained by forward modeling. The lowest ones are significant of a corroded area. The two previous figures show that the clues for corrosion are situated at the expected locations compared to the target on Fig. 4 with the 1400 measurements. The diagnosis has well succeeded.

A last remark can be made about the measurements locations. It is obvious that measuring far away from reactions does not provide good results. But how guess the best locations? In the case presented here, the reactions are expected on the lowest parts of the hull; this is why the grid presented is placed below the hull. A better solution is to make measurements like a semi cylinder shape, to get more precisions on the sides of the hull. The other question is the maximum depth the measurements can reach to stay relevant. This needs a study on the seawater conductivity. The more conductive the electrolyte is, the closer the measurements have to be. Like any conduction problem, current lines spread more in a more resistive domain. A smart study is necessary before making any measurements.

The three steps (take well situated measurements, inject information and regularization) allow making a well corrosion diagnosis with noised measurements. For a clear visibility, the approach does not take the infinite potential into account, but it just can be added in (25). The next part deals with experimentations to check the results got with the presented algorithm.

EXPERIMENTATIONS

The previous parts have focused on an infinite problem, to fit with marine applications for example. Making directly a scale 1:1 experimentation would firstly face to memory consuming problems so that the first trial will be held in laboratory. The main idea is to reproduce a real case but at different scale, so that a one square meter PMMA bowls has been built with a 25cm depth. To fit the scale, the only thing to L-Curve corner, best λ area do is to control the electrolyte salinity linked to the conductivity (in S.m). PMMA bowl for first experimentations. The appropriate measurements system is also on the picture with a piece of protected steel in the middle.

A system of electrical potential measurements is held on the bowl: the principle is to measure electrochemical potentials in the electrolyte through a reference electrode (Calomel electrode) toward the iron potential. This can easily be with a multimeter whose positive side is connected to the reference electrode and the negative one directly to the iron in the electrolyte.

The model drown in the bowl should be representative of the immerged part of a hull; it will be a piece of painted iron with paint defects (lack of paint) on it. To make this structure protected from corrosion, an ICCP is set on it with a platinum mesh (2x2 cm) set on its middle. The iron and platinum are electrically linked to a computer with adapted electrochemical equipment which can set and measure the current flowing into the electrolyte due to the platinum anode. The model meshed edge is shown below, with dimension 40x40x2 cm: The anode in the middle is represented by the current densities it delivers with arrows. The two grey areas are the defects with 8x4 cm dimension. The white areas are the well isolated part.

As the model geometry is well known, it can be drown in the electrolyte. Seawater has conductivity close to 5.6 S.m, here it will be chosen equal to 0.1 S.m which is 56 times less. The current injected by the anode is 160 A/m² which makes 64 mA injected in the electrolyte. The reaction begins with a release of hydrogen on the iron and platinum. It needs to work on for a couple of hours depending on natural oxidation state of the iron. Then an electric potential measurement is made on a grid 2 cm above the model as shown: The interaction matrix between measurement locations and the model is built, the Green equations are added and Tikhonov regularization is applied. The corresponding L-Curve is set below: The corner is not as simple to see as in numerical cases. Indeed the problem comes from the salinity: to get a low conductivity (0.1 S.m), the salinity have to be very low. In the abacus, such salinities are subject to huge errors on the corresponding conductivity because of two main factors: the non homogenous temperature in the electrolyte and the salt purity (NaCl quantity in the salt) [START_REF] Keller | Electrical methods in geophysical prospecting[END_REF]. The information injected with the known current density J A through ∂φ A /∂n depends on the conductivity, so it induces errors on the inversion. The more noisy information the system has, the less significant the L-curve is. But the vertical dotted line gives us a clue of the inflection point. The λ parameter is taken equal to 0.0024 and the inversion gives the following electric potential results: To get visible results, only lowest potentials are represented. It is an arbitrary criterion: as the corroded areas have the lowest potential ones, it is decided that here only the 10% lowest ones are kept. The target is also calculated with the same criterion by forward modeling: The lowest potential areas are the same. The diagnosis is successful: the corroded areas have been explicitly recovered by the algorithm. The current density results give the same diagnosis and have not been exposed.

CONCLUSION

As a conclusion, the method explained in this article can be used in several domains such as ships hulls, oil platforms, underwater pipe lines, offshore wind energy platforms, etc… It can be also very useful to monitor water tanks, sewers pipes, silos, etc… Moreover other scientific topics such as thermal studies use the same equations and could take advantages of this method.

It provides a global tool to get clues about the state of an underwater iron structure with an appropriate cathodic protection. It must be said that this method works with internal or external problems with any cathodic protection (impressed current or sacrificial). An industrial application has been developed to examine the state of a hull at each passage in a harbor.

The next step of the approach is to constraint the inversion of the system by evaluation of the solution we want to find (current densities with order 2 regularization) or the permissible error due to measurement error prediction.

Other measurements have been made with scale 1:1 seawater (conductivity close to 5.6 S.m) in the bowl which works, but the current injection has obviously to be increased. Those results will be presented in further papers.

Fig. 1 .

 1 Fig. 1. Principle of impressed current cathodic protection applied on the hull of a ship. The current is provided by a DC power source in the vessel.

Fig. 3 .

 3 Fig. 3. Schematic representation of the boundary problem, a closed structure filled with air is drowned in an infinite domain. An infinite bound is represented to make the problem external.

Fig. 5 .

 5 Fig. 5. Geometry meshed of an existing mock-up with: 3 uniform active anodes under the hull (6 elements in black), 4 different cathodes (31 elements in grey), and the hull completely isolated (1015 elements in white)

Fig. 6 .

 6 Fig. 6. Boundary conditions (electric potential and current densities) obtained on the hull with φ∞ = 1.3762 V. The scale is in volts

Fig. 7 .

 7 Fig. 7. Result of electric potential measurements made on the grid (1400 points) below the hull

  the same form as (23). The results of a direct inversion are shown below:

  Fig. 11. L-Curve obtained for the considered problem. The highlighted corner points out the λ parameters giving the best compromise for the regularization.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Current densities obtained on the hull after inversion (scale in A/m²) -0.02 0 0.02 0.04 0.06 0.08

Fig

  Fig. 14.PMMA bowl for first experimentations. The appropriate measurements system is also on the picture with a piece of protected steel in the middle.

Fig. 15 .

 15 Fig. 15. Meshed model drown in the bowl with 1 anode and two defects.

Fig. 16 .

 16 Fig. 16. Measurements locations set on a square grid above the piece of steel Measurements are made at each barycenter of the grid; the bowl meshed representation has been set off for clarity.The interaction matrix between measurement locations and the model is built, the Green equations are added and Tikhonov regularization is applied. The corresponding L-Curve is set below:

  A x -b || 2 solution semi-norm || L x || 2 L-curve, Tikh. corner at 0.00084156

Fig. 17 .

 17 Fig. 17. Corresponding L-Curve of the system to inverse.

Fig. 18 .

 18 Fig. 18. Electric potential boundary conditions for a λ parameter equal to 0.0024 (scale in Volts)

Fig. 19 .

 19 Fig. 19. Target of boundary conditions (electric potential) in the same conditions (scale in Volts)

  .

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Interfacial adhesion assessment in flax/epoxy and in flax/vinylester composites by single yarn fragmentation test: correlation with micro-CT analysis

Keywords: A. Natural fibres, B. Interface/Interphase, B. Fragmentation, D. CT analysis

Introduction

It is very well-known that the global mechanical properties of composite materials are strongly dependent on the adhesion quality of fibre/matrix interface and this is even more stringent for natural fibre composites due to the inherent hydrophilic nature of plant fibres. As a result, the good specific mechanical properties of the fibres are far from being efficiently exploited, thus leading to an unsatisfactory mechanical performance of the resulting composites that cannot be reliably used in semi-structural applications. Therefore, it is not surprising that a great deal of research efforts has been directed towards the improvement of natural fibre/polymer matrix interface compatibility [START_REF] Kabir | Chemical treatments on plant-based natural fibre reinforced polymer composites: An overview[END_REF] [START_REF] Li | Chemical treatments of natural fiber for use in natural fiber-reinforced composites: A review[END_REF]. However, these efforts need to be supplemented with a reliable and quantitative assessment of interfacial adhesion in order to optimize the surface treatments and broaden the applications of natural fibre composites. For the analysis of traditional composites based on synthetic fibres such as glass or carbon, several tests have been developed, including the pull-out test, the microbond test, the push-in test and the single fibre fragmentation test (SFFT) [3][4]. The applicability of these methods has been the subject of extensive debate in literature and some of them have been also

extended to the characterisation of the fibre/matrix interactions of lignocellulosic fibre-reinforced polymers [START_REF] Torres | Study of the interfacial properties of natural fibre reinforced polyethylene[END_REF]- [START_REF] Zafeiropoulos | On the use of single fibre composites testing to characterise the interface in natural fibre composites[END_REF]. The results clearly show that when dealing with natural fibre composites, which are highly heterogeneous in nature, additional care and caution should be paid in assessing the interfacial adhesion through single fibre model composites. Single fibre fragmentation test is one of the most widely used tests mainly because of practical advantages related to the simplicity of sample preparation and the possibility of producing many fragments in a single specimen and then, through statistical analysis, a valid distribution can be obtained from a limited number of test samples. Nonetheless, this test method applied to natural fibre composites suffers from several limitations including the need to estimate accurately the fibre strength at the critical fibre fragment length and the significant variation of the fibre diameter not only among fibres but also within the same fibre. It is worth noting that an adequate test configuration should involve a stress distribution similar to that in a real composite and in natural fibre composites the potential of plant fibres can be only explored by using yarns, where short fibres are twisted with an angle to the yarn axis in order to provide axial strength to the yarn [START_REF] Madsen | Hemp yarn reinforced composites -I. Yarn characteristics[END_REF]. Therefore, the interfacial adhesion should be assessed at the yarn scale and not only at the technical fibre scale. In this regard, Spārniņš et al. [11][12] proposed a method for IFSS estimation using the pulled-out fibre length distribution of oriented flax fibre reinforced composite incorporating a modified Weibull strength distribution. The authors obtained values of IFSS lower than those determined by fragmentation tests, which were ascribed to a defective impregnation of flax yarns and low mutual adhesion of elementary flax fibres. To partially address some of the mentioned issues, it could be possible to extend the SFFT at the yarn scale, thus embedding one single yarn and eliminating problems related to imperfect impregnation of plant yarns. This approach has been investigated in previous works [13][14], where detailed experimental, numerical and full-field strain measurements by digital image correlation helped in getting insight into the strong heterogeneity of strain fields that develop with the applied stress and in providing a first step towards the assessment of local behaviour. The complexity of such approach stems also from the cross-sectional area of the yarn that typically includes voids between the fibres which are then replaced by matrix during the impregnation of the single yarn. In this framework, the purpose of this work is to investigate further the possibility to use fragmentation test to assess the adhesion of flax yarns to different thermoset resins. In particular, epoxy and vinylester matrices have been selected. A specific metallic mold has been designed to manufacture monofilament composite samples, where the filament is represented by a single flax yarn embedded in the polymer resin. The adhesion quality between flax yarn and both epoxy and vinylester matrices has been assessed in terms of critical fragment length, debonding length and interfacial shear strength. This experimental characterization has been supplemented for the first time, to the best of authors' knowledge, with microcomputed tomography. High-resolution microtomography (1.5 µm) allowed the measurement of the debonding length between flax yarn and polymer matrices. Using the AVIZO 9.0 software, it was also possible to perform a volumetric reconstruction of the breaking area of the flax yarn.

1 -Materials and methods A specific metallic mould has been designed to manufacture single yarn composite specimens with the yarn aligned in the loading direction. Some little slots were designed at each end of the negative samples to facilitate the flax yarn placement in the middle of the specimen. The whole surface of the mould was prepared by applying a release agent. Before casting, flax yarns were conditioned at 45 °C for 24 h for moisture elimination. Each flax yarn was positioned in the mould and held in place by adhesive tape. In order to avoid the leakage of the resin from the mould and simultaneously ensure the right quantity of resin to reduce as much as possible the formation of a meniscus due to the shrinkage during the crosslinking process of the resin, the mould has been filled up with resin using a syringe. The metallic mould used for the sample manufacture and the specimen geometry can be seen in figure 1.

Test set-up

The fragmentation testing was performed with an Instron E1000 ElectroPuls test machine with a load cell of 2 kN and using a crosshead speed of 0.005 mm/min. The test was carried out at a rather low loading rate and the loading phase was stopped if the specimen failed, or when the fragmentation saturation level was achieve which, in the present case, was defined as occurring when no new yarn breaks appeared during a subsequent strain increase by 0.5%, as frequently reported in literature (R. Joffe, J. Andersons, and L. Wallström, "Interfacial shear strength of flax fiber/thermoset polymers estimated by fiber fragmentation tests," J. Mater. Sci., vol. 40, no. 9-10, pp. 2721-2722, 2005). Some epoxy-and vinylester-based specimens were loaded up to failure at strains higher that 9 and 11%, respectively, in order to be sure that the saturation had occurred. The average flax yarn diameter was evaluated by optical microscopy before each test. For each specimen, the yarn diameter has been measured in at least ten points along the gauge length (15 mm).

Optical and FE-SEM observations

The flax yarn diameters and all the fragment length measurements were carried out by using a ZEISS Axio Imager optical microscope. The morphology of the single yarn composite fracture surfaces was investigated by a field-emission gun scanning electron microscope (JEOL JSM-7000F). All specimens were sputter coated with gold prior to FE-SEM observations.

Micro-CT apparatus

Image acquisition has been performed using an UltraTom CT scanner manufactured by RX Solutions (France). The system consists in a Hamamatsu open type microfocus X-ray tube operating at 20-100 kV / 0-200 µA, within a maximum power of 20 W (Configuration with a Lanthanum hexaboride filament). A precision object manipulator with two translations and one rotation facilitates rotating the sample for acquisition of tomographic data, and displacement along the optical axis to adjust the magnification. X-rays generated by the source diverge at an angle providing a cone-beam, thus samples can be imaged at various geometric magnifications by moving the sample close to the source to provide high resolution mode or close to the detector to provide low resolution measurements. The generator and the detector are also mobile to cover a large range of magnification. A 1.5 μm resolution has been used in this work, with an accelerating voltage of 50 kV and a beam current of 157 µA. This X-ray detector consists in an X-ray CsI scintillator screen which is settled on an amorphous silicon layer. For 3D reconstruction, X-ray images were acquired from 1120 rotation views over 360° (approximately 0.32° rotation step). The reconstruction was performed using an algorithm based on the filtered back-projection procedure for Feldkamp cone beam geometry. The analysis of the micro-CT pictures has been performed by using the Avizo 9.0 software.

2-Results and discussion

Characterization of composite single constituents

A mechanical characterization for both types of polymer resin has been done by performing tensile tests. Table 1 summarizes the mechanical results. High failure strain values were found for both epoxy and vinylester resins. This behaviour may be explained by the low crosshead speed used during tensile tests. In particular, vinylester resin displayed a markedly ductile behaviour. Unlike synthetic fibres, one of the main characteristics of the plant ones is their extreme heterogeneity.

One of the parameters of greatest variability of this type of reinforcement is surely the diameter. In this study, a statistical analysis based on measurements at a hundred points was performed to define the diameter value of flax yarn. The Gaussian probability distribution of flax yarn diameter is defined by the equation (1) [START_REF] Guillebaud-Bonnafous | Experimental and numerical investigation of the interface between epoxy matrix and hemp yarn[END_REF]:

𝑃(𝑑) = √ • 𝑒 ( ) • (1)
where d is the diameter value of flax yarn, μ is the mean of the distribution, and σ 2 is the variance of the distribution. In figure 2 is reported the Laplace-Gauss distribution results for the yarn diameter. A very large diameter distribution was found for this kind of flax, with an average value of 327 m and a standard deviation of 96m. These results confirm the significant scattering in the measured diameter values and were used for evaluating the yarn cross-sectional area assuming a circular cross-section. Table 2 presents the tensile strength of flax yarns evaluated at three different gauge lengths, namely 20 mm, 30 mm and 40 mm. As reported in table 2, the strain to failure value for all the yarns was between 3.4 % and 3.8 %. It can be seen that with longer gauge length there is a small decrease in the yarn strength, from 272 to 236 MPa. The scatter of the tensile strength of flax yarns was statistically analysed using a two-parameter Weibull distribution, according to the equation (2) [START_REF] Sarasini | Influence of thermal conditioning on tensile behaviour of single basalt fibres[END_REF]:

𝑃𝑟(𝜎) = 1 -exp - (2)
Where Pr(σ) is the cumulative probability of failure, m is the Weibull modulus (related to the dispersion of the data) and σ0 is a scale parameter (characteristic strength). The probability of failure was estimated by the estimator F reported in equation ( 3):

𝐹 = . ( 3
)
where N is the number of filaments tested and i is the rank of data point for each filament. The graphs in figure 3 represent the results of the two-parameter Weibull distribution applied to all the gauge lengths. The Weibull modulus and the characteristic life values found for the flax yarns are reported in table 2. A comparison between the stress-strain curves of flax yarn and polymer resins may be observed in figure 4. It can be seen that flax yarn is characterized by a clearly higher tensile strength in comparison to both epoxy and vinylester resins, and a far smaller strain at failure. These differences in mechanical behaviour between the flax yarn and the polymer matrices are essential for carrying out the fragmentation tests. As reported by Zafeiropoulos [START_REF] Zafeiropoulos | On the use of single fibre composites testing to characterise the interface in natural fibre composites[END_REF], only in monofilament composites characterized by a very ductile matrices and a brittle filament, it is possible to reach a significant strain higher than the strain to failure of the filaments that ensures a full debonding/yielding of the filament fragments. This will provide a constant shear at the filament/matrix interface and so a full satisfaction of the Kelly-Tyson requirements.

-Determination of critical fragment length

According to the Kelly and Tyson model [START_REF] Kelly | Tensile properties of fibre-reinforced metals: Copper/Tungsten and Copper/Molybdenum[END_REF], if a long filament is embedded in a resin matrix and the resultant system is elongated, the stress transferable to the filament at a distance x from its end may be evaluated by the equation ( 4):

𝜎 = • • 𝑥 ( 4
)
where d is the diameter of the filament and τ the shear strength at the filament/matrix interface.

This equation comes from a simple force balance at filament/matrix interface and it is valid only under the condition of assuming a constant shear stress τ at the interface. When the monofilament composite is elongated further, the tensile stress transferable to the filament increases until it reaches the ultimate tensile strength of the filament, equation ( 5):

𝜎 = • • 𝑥 ( 5
)
where x0 is the value of the distance x from the end of the filament at which the tensile stress reaches the ultimate tensile strength of the filament. If a broken piece of filament exceeds 2x0 in length, the breakage process repeats itself. On the contrary, if the broken piece is characterized by a length shorter than 2x0, the tensile stress transferable to the filament is not able to reach the ultimate tensile strength of the filament and hence no further fragmentation occurs. As reported by Ohsawa et al. [START_REF] Ohsawa | Temperature dependence of critical fiber length for glass fiber-reinforced thermosetting resins[END_REF], the length of filament fragments (l) follows the uniform distribution characterized by a range x0 ≤ l ≤ 2x0, and the average value of the fragment length is given by the equation ( 6):

l = (𝑥 + 2𝑥 ) = 𝑥 (6)
Since x0 is the length needed for the stress on filament to reach the ultimate tensile strength, the critical filament length lc is equal to 2x0. By introducing this relationship into eq. ( 6), it is possible to evaluate the critical filament length from equation ( 7):

𝑙 = • l (7)
In figure 5 is reported an example of a typical optical micrograph obtained for the VF_3 fragmented flax yarn. The yarn diameter and fragment lengths were measured in the gauge length region of the specimen. The fragment lengths have been determined as the average values between the internal and the central points of the breaking zones (reported in dotted red lines). Moreover, around each yarn break, the length of the black zone (debonding length) observed on micrographs (figure 5) has also been measured (reported in blue line). As it will be demonstrated by the micro-CT analysis in section 2.5, this black zone corresponds to the debonding length. Therefore, the black zone length values are noted as ldebonding in figure 5 and table 3. Regarding the fragmentation tests, the two thermoset systems exhibited different results. All the critical length measurements are reported in table 3. A larger number of fragments was found for flax yarn embedded in epoxy matrix than in the case of vinylester matrix. Flax yarn has suffered a greater number of cracks with critical fragment length value lower than that found in the case of vinylester resin. The critical filament length plays an essential role in determining the adhesion quality in a composite. In fact, the measurement of the critical filament length is directly correlated to the shear strength value transferred at filament/matrix interface. As reported by Zafeiropoulos [START_REF] Zafeiropoulos | On the use of single fibre composites testing to characterise the interface in natural fibre composites[END_REF][18], a very interesting approach to assess the interface properties in a composite material is the model proposed by Fraser et al. [START_REF] Fraser | Evaluation of surface treatments for fibers in composite materials[END_REF]. In this approach, the adhesion quality is not estimated from the measurement of the interfacial shear stress but rather by the fragment length values. The principle of the approach introduced by Fraser and co-authors is that, at saturation level, the critical filament fragment length reflects the stress transfer efficiency at the interface and thus it offers a qualitative value of the interfacial bond. However, it must be emphasized that, with this method, it is possible to compare the adhesion properties of different filament/matrix systems through the comparison of the filament fragment length values only in the case that the filament strength values are the same for all the systems. In this study, both epoxy and vinylester composites have been realized using the same type of flax yarn. This allows us to use the approach proposed by Fraser and co-authors to determine the differences in interfacial bond in terms of critical yarn fragment length. From the results reported in table 3, it is possible to see that the flax/epoxy system is characterized by smaller values of critical length. This indicates a better adhesion quality between flax yarn and epoxy matrix contrary to what has been observed for the vinylester matrix. As previously reported, flax yarn is a highly heterogeneous material, especially for its diameters values (figure 2). For this reason, the lc/d ratio has been calculated in order to compare normalized values (see in Table 3).

Also in this case the flax/epoxy system displays the best results in terms of interfacial adhesion.

-Calculation of interfacial shear strength

The filament/matrix interface adhesion quality is usually described by determining the interfacial shear strength value (IFSS). According to Kelly and Tyson [START_REF] Kelly | Tensile properties of fibre-reinforced metals: Copper/Tungsten and Copper/Molybdenum[END_REF], the IFSS may be estimated using equation ( 8):

𝐼𝐹𝑆𝑆 = ( )• • ( 8
)
where d is the filament diameter, lc is the critical fragment length and σr(lc) is the filament strength at a length equal to the critical filament length. Because of the small lc values, it is impossible to measure experimentally the tensile strength of a single filament at a length equal to the critical one.

To overcome this problem, an extrapolation of strength at critical length lc was realized. In this study, two different ways of extrapolation have been taken into consideration. The first method has been proposed by Asloun et al. [START_REF] Asloun | On the estimation of the tensile strength of carbon fibres at short lengths[END_REF]. According to this study, it is possible to extrapolate the maximum stress of flax yarn at small gauge lengths from tensile tests performed at several large gauge lengths. It is possible to do this extrapolation by taking the logarithm of stress in equation ( 9):

𝑙𝑛 𝜎 = -𝑙𝑛(𝐿 ) + 𝑙𝑛[𝜎 𝛤 1 + ] ( 9
)
where mi and σ i 0 are the shape and the scale parameters of the equivalent Weibull distribution respectlively, L0 the gauge length, σf the tensile strength and Γ the Gamma function. Thus, the tensile strength of a flax yarn at a given gauge length can be easily estimated by linear extrapolation using equation [START_REF] Zafeiropoulos | On the use of single fibre composites testing to characterise the interface in natural fibre composites[END_REF]. As it is possible to see in figure 6, the graph plotting ln(σf) against ln(L0) is linear with a slope of ̵ = ̵ 0.2. The second extrapolation method that has been taken into account in this study is based on the Weibull cumulative distribution function determined for only one given filament gauge length L0. This method is widely used in literature [START_REF] Zafeiropoulos | On the use of single fibre composites testing to characterise the interface in natural fibre composites[END_REF], [START_REF] Guillebaud-Bonnafous | Experimental and numerical investigation of the interface between epoxy matrix and hemp yarn[END_REF], [START_REF] Ramirez | Evaluation of water degradation of vinylester and epoxy matrix composites by single fiber and composite tests[END_REF]- [START_REF] Joffe | Strength and adhesion characteristics of elementary flax fibres with different surface treatments[END_REF]. Using the obtained two-parameter Weibull distribution, the average filament strength may be evaluated by equation [START_REF] Madsen | Hemp yarn reinforced composites -I. Yarn characteristics[END_REF]:

< 𝜎 > = 𝜎 𝐿 𝛤 1 + (10)
where Γ is the Gamma function and σ0 and m are the characteristic life and the shape parameter of Weibull distribution for the tested gauge length L0, respectively. In this study, the chosen value for L0 is 40 mm. Therefore, the used Weibull parameters for this second method are those obtained with tensile tests on flax yarns of 40 mm length (Table 2). By applying equation [START_REF] Madsen | Hemp yarn reinforced composites -I. Yarn characteristics[END_REF] to L0 and to the critical length lc, it is possible to obtain the following equation [START_REF] Spārniņš | Interfacial shear strength of flax fibers in thermoset resins evaluated via tensile tests of UD composites[END_REF]:

𝜎 (𝑙 ) = 𝜎 (𝐿 ) (11)
For comparison purposes, in this study both the linear extrapolation method with equation ( 9) and the Weibull weakest link theory with equation ( 11) have been used to determine the yarn strength at a length equal to the critical yarn length, σf (lc), and so the IFSS values. In table 4 are reported the results obtained with both methods. Comparing the results of the two methods, a negligible difference has been found for the flax yarn σf (lc) values. These results prove the validity of both calculation methods. Concerning the analysis of the yarn/matrix adhesion quality, from table 4 it is possible to see that the IFSS values reflect the results already found by comparing the critical fragment lengths. In fact, from the IFSS values, it is possible again to conclude that the flax yarn/epoxy system is characterized by a better interface quality than the flax/vinylester one.

-Fractographic analysis

After fragmentation tests, a morphological investigation of the fracture surface has been performed using the FE-SEM. The fracture surface of both single flax yarn samples with epoxy and vinylester resins was investigated. From the direct analysis of the fracture surfaces (figure 7), it can be concluded that the yarn failure is concentrated mainly in the peripheral zone of the yarn. This phenomenon is particularly evident for the flax/epoxy samples, figure 7-A. Concerning the vinylester resin, a partial tearing of the yarn, occurring at the moment of failure was found for most of the samples. This different behavior can be considered as a further confirmation of the lower adhesion quality between the flax yarn and the vinylester matrix. At a smaller scale, in figure 8, it is possible to see that there are differences between the two systems at the fibre/matrix interface.

Indeed, for the flax/epoxy sample the presence of little resin ligaments, albeit minimal, was found between the matrix and the flax fibres. It has not been observed for the flax/vinylester composite.

These little connections may explain the better adhesion that the flax yarn displayed in presence of the epoxy resin. The 3D micro-CT images also allow the measurement of the debonding length between the flax yarn and the polymer matrix. Indeed, the debonding zone is detectable by the black voxels (corresponding to voids) between the periphery of the yarn and the matrix. This measurement was possible through the high resolution micro-CT observations (voxel size of 1.5 µm). In figure 10 are reported two examples of x-slices obtained for both flax/epoxy and flax/vinylester single yarn samples. Debonding length values of 530µm ± 70µm and 624µm ± 76µm were found for the flax/epoxy (EF_1) and the flax/vinylester (VF_3) monofilament composites, respectively. These values of the debonding lengths are consistent with the black zone lengths measured by optical microscopy on the same samples: 504µm ±102µm for EF_1 and 599µm ±181µm for VF_3 (Table 3). This demonstrates that the observed black zone on micrographs coincides with the debonding zone between the flax yarn and the polymer matrix. According to Kim et al. [START_REF] Kim | Observations of Fiber Fracture and Interfacial Debonding Phenomena Using the Fragmentation Test in Single Fiber Composites[END_REF] and Ramirez et al.

[21], debonding represents a very important phenomenon in studying the interface quality of polymer composites. These authors have performed fragmentation tests on single carbon and glass fibre composites. They measured the debonding length around each fragment by using photoelastic birefringence patterns. As explained by Kim and Nairn [START_REF] Kim | Observations of Fiber Fracture and Interfacial Debonding Phenomena Using the Fragmentation Test in Single Fiber Composites[END_REF] ,the interfacial zone between the filament and the polymer matrix may be considered as a set of bonding lines. When the monofilament composite is loaded, the molecular bonded lines become tense until filament fractures and some lines near the filament break are disconnected. They defined the debonding zone as the region containing both the filament breaking gap and the breaking zone of the molecular bonding lines. They showed that a large value of the debonding zone is typical for weak interfacial bonds. Thus, the higher debonding length value found for the system constituted by flax yarn and vinylester matrix displays, once again, the lower adhesion quality of flax yarn with this type of polymer matrix. This result reflects perfectly the calculated IFSS values. A comparison between the debonding length measurements, performed by optical microscopy and micro-CT, and the inverse of the IFSS values obtained for both EF_1 and VF_3 samples has been carried out and reported in figure 11. As it is possible to see in table 3 and in table 4, the IFSS values are directly linked to the debonding length between the flax yarn and the polymer matrix, and thus to the adhesion quality at the yarn/matrix interface. The greater the adhesion interfacial properties and the IFSS values, the lower the debonding lengths. It is also worth noting that micro-CT is able to provide a more accurate measurement of the debonding length with a reduced dispersion of data.

-Conclusions

This study has been focused on the characterization of the adhesion properties between flax yarns and polymer (epoxy and vinylester) matrices. Experimental investigation was performed on single flax yarn composite specimens. Fragmentation tests allowed the determination of the interfacial shear strength at yarn/matrix interface. The adhesion quality between flax yarn and both epoxy and vinylester matrices has been assessed in terms of critical fragment length, interfacial shear strength The red dotted lines show the debonding length measuraments obtained by micro computed tomography.

Tables

  Interfacial adhesion assessment in flax/epoxy and in flax/vinylester composites by single yarn fragmentation test: Correlation with micro-CT analysis M C Seghini, F Touchard, F Sarasini, L Chocinski-Arnault, D Mellier, J Tirillò To cite this version: M C Seghini, F Touchard, F Sarasini, L Chocinski-Arnault, D Mellier, et al.. Interfacial adhesion assessment in flax/epoxy and in flax/vinylester composites by single yarn fragmentation test: Correlation with micro-CT analysis. Composites Part A: Applied Science and Manufacturing, 2018, 113, pp.66 -75. 10.1016/j.compositesa.2018.07.015. hal-03026038

2. 5 -

 5 Fragment debonding analysisAn in-depth analysis of the fracture zone distribution along the flax yarn was also carried out. A post-mortem examination of the monofilament composites has been performed through the use of micro computed tomography. It is important to point out the great novelty of the use of this type of analysis for the assessment of the interfacial properties in monofilament composites. For both EF_1 (flax/epoxy) and VF_3 (flax/vinylester) specimens, microtomographic image acquisitions have been made with a resolution of 1.5 µm. Using the AVIZO 9.0 software, it was possible to perform a 3-D reconstruction of the breaking area of the flax yarn (figure9). Raw images were pre-treated using smoothing filters and an image segmentation was carried out. Black voids (breaking zone) were identified from the background (the rest of flax yarn) on the basis of the different grey threshold values. The analysis carried out on both types of samples has shown that the breakage of flax is mainly concentrated in the peripheral zone of the yarn. This result is clearly visible in the 9-E and 9-F images that show the 3-D reconstruction of the breaking zone, and it is totally in accordance with what was observed by FE-SEM analysis on fracture surfaces, figures 7-A and 7-B.

Figure 1 (

 1 Figure 1 (A) Metallic mould used for sample manufacture: the resin casting stage using a syringe; (B) Specimen geometry for the single yarn fragmentation tests.

Figure 2

 2 Figure 2 Gaussian distribution of measured diameters of flax yarn.

Figure 3

 3 Figure 3 Cumulative distributions of the failure probability values measured for three different gauge lengths (20, 30 and 40 mm) for flax yarn.

Figure 4

 4 Figure 4 Stress-strain curves for flax yarn (gauge length of 40 mm), epoxy and vinylester resins obtained at 0.005mm/min.

Figure 5

 5 Figure 5 Typical optical micrograph of a fragmented flax yarn. In particular this image shows the yarn diameter d (green colour), the fragment length l (red colour) and the debonding length ldebonding (blue colour) measurements for the VF_3 sample.

Figure 6

 6 Figure 6 Evolution of tensile strength of flax yarn versus gauge length.

Figure 7

 7 Figure 7 FE-SEM-micrographs showing the fracture surface for flax/epoxy (A) and flax/vinylester (B) single yarn composites.

Figure 8

 8 Figure 8 FE-SEM-micrographs showing the different interfacial morphology of flax/epoxy (A) and flax/vinylester (B) single yarn composites after testing.

Figure 9

 9 Figure 9 Volumetric reconstruction of the flax yarn and the fracture zone for both flax/epoxy (A-C-E) and flax/vinylester (B-D-F) systems. The flax yarn is represented in yellow, the fracture zone in blue and red respectively for flax/epoxy and flax/vinylester samples.

Figure 10

 10 Figure 10 High resolution micro-CT images for EF_1 flax/epoxy (A) and VF_3 flax/vinylester (B) single yarn samples.

Table 1

 1 Tensile properties of epoxy and vinylester resins obtained at 0.005mm/min

		σf [MPa]	εf [%]
	Epoxy Resin	58.19 ± 0.75	8.29 ± 0.15
	Vinylester Resin	34.99 ± 0.47	21.21 ± 2.89

Table 2

 2 Tensile properties of flax yarns as a function of gauge length and the corresponding Weibull parameters. Tests have been carried out at 2 mm/min.

		Gauge length	σf [MPa] εf [%]	Characteristic	Weibull Modulus
		L0 [mm]			Strength σ0 [MPa]	m
	Flax yarn	20	272 ± 46	3.78 ± 0.57	292	6.83
		30	248 ± 46	3.62 ± 0.62	265	6.36
		40	236 ± 57	3.40 ± 0.42	257	5.02

Table 3

 3 Fragment length values for flax/epoxy and flax/vinylester single yarn composites.

	Flax/Epoxy Resin	Yarn diameter d	Number of	Average l [µm]	lc/d	ldebonding [µm]	ldebonding/d
		[µm]	fragments
	EF_1	210 ± 35	8	1924 ± 639	12.18	504 ± 102	2.39
	EF_2	222 ± 32	8	1473 ± 532	8.83	356 ± 155	1.61
	EF_3	253 ± 43	7	2345 ± 372	12.35	460 ± 96	1.82
	EF_4	247 ± 21	8	1847 ± 486	9.95	465 ± 9	1.88
	EF_5	308 ± 25	5	2786 ± 508	12.04	446 ± 128	1.45

(IFSS) and debonding length. A link between these parameters was demonstrated. The best adhesion properties have been obtained for the flax/epoxy system. In fact, this composite is characterized by the lowest values of critical and debonding lengths and the highest values of IFSS.

For the first time, micro computed tomography has been used as analysis technique in the study of the interfacial properties after performing fragmentation tests. High-resolution microtomography allowed to measure the debonding lengths and to correlate it with the black zone lengths observed by optical microscopy. Using the AVIZO 9.0 software, it was possible to perform a 3-D reconstruction of the breaking area of the flax yarn. For both epoxy and vinylester samples, it was found that the breakage of flax has been mainly concentrated in the peripheral zone of the yarn, which confirms that fragmentation tests on single yarn composites allow the analysis of the interface adhesion quality between yarn and matrix in view of the optimization of woven composite materials.