Nathalie Limodin

Julien Réthoré

Jean-Yves Buffiere

Anthony Gravouil

François Hild

Stéphane Roux

Jean-Yves Buffière

Crack closure and stress intensity factor measurements in nodular graphite cast iron using three-dimensional correlation of laboratory X-ray microtomography images

Keywords: X-ray microtomography, 3D Digital Image Correlation, Stress Intensity Factors, Nodular graphite cast iron

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

Introduction

Fatigue crack closure was first described by Elber [START_REF] Elber | The significance of fatigue crack closure[END_REF] more than 30 years ago. Although it is a critical issue for life time predictions through Paris' laws (with the use of effective stress

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Linking powered by eXtyles intensity factor ranges, ΔK eff ), there is still no unified method of measurement that reaches general consensus. Crack closure occurs when the crack surfaces come into contact before the minimum load of the fatigue cycle is reached. The underlying mechanisms are diverse and may operate simultaneously; the reader is referred to ref. [START_REF] Suresh | Fatigue of materials[END_REF] for a more complete description. For example, a growing crack leaves behind a plastically deformed zone that is adjacent to the crack faces. This plastic wake induces early closure of the crack during unloading via a mechanism called Plasticity-Induced Crack Closure (PICC). Furthermore, asperities on the crack faces may also result in premature contact and induce Roughness-Induced Crack Closure (RICC). This mechanism becomes significant near the fatigue threshold when crack paths are highly crystallographic or under mode II or III loading that favours asperity contacts; in this case, closure concentrates in regions where the crack changed its direction of propagation [START_REF] Tanaka | Modeling of fatigue crack growth: mechanistic models[END_REF]. Therefore, ideally, to measure crack closure and identify the operative closure mechanism, one needs a way to measure precisely the crack mouth displacements or contact along the crack front knowing the topography of the crack.

Measurements at the surface of a sample during a loading/unloading cycle have been performed using for example scanning electron microscopy [START_REF] Zhang | A study of the crack wake closure/opening behaviour of short fatigue cracks and its influence on crack growth[END_REF]. Surface measurements allow one to visualize crack shapes and to analyze closure changes with the crack length [START_REF] Fleck | Crack closure -is it just a surface phenomenon?[END_REF].

However the relevance of this kind of measurement with respect to the bulk of the sample can be questioned. Bulk measurements of crack closure involve indirect observations from crack growth, e.g. with potential drop or acoustic emission, and with mechanical compliance curves [START_REF] Xu | Systematic assessment and validation of compliance-based crack closure measurements in fatigue[END_REF]. This last method is the most widely used. The loss of linearity of the displacement-load curve recorded during unloading of the specimen delimits the closure load. However, the result may depend on the positioning of the displacement gauge, namely, at the crack mouth, at the back face of the specimen, behind the crack tip within the specimen (push-rod gauge) or near the crack tip at the surface [START_REF] Xu | Systematic assessment and validation of compliance-based crack closure measurements in fatigue[END_REF]. With these techniques, no information on the relationship between crack topography and closure can be obtained as the detailed crack shape is not available.

Linking powered by eXtyles Three-dimensional visualisation of cracks in optically opaque materials such as metals is now possible using high resolution X-ray tomography [START_REF] Ludwig | Study of the interaction of a short fatigue crack with grain boundaries in a cast al alloy using x-ray microtomography[END_REF][START_REF] Mcdonald | X-ray tomographic imaging of Ti/SiC composites[END_REF][START_REF] Marrow | High resolution x-ray tomography of short fatigue crack nucleation in austempered ductile cast iron[END_REF][START_REF] Ferrié | 3d characterisation of the nucleation of a short fatigue crack at a pore in a cast al alloy using high resolution synchrotron microtomography[END_REF][START_REF] Buffière | Three-dimensional visualisation of fatigue cracks in metals using high resolution synchrotron x-ray micro-tomography[END_REF][START_REF] Verdu | Three-dimensional shape of the early stages of fatigue cracks nucleated in nodular cast iron[END_REF][START_REF] Toda | Direct measurement procedure for three-dimensional local crack driving force using synchrotron x-ray microtomography[END_REF]. So far, synchrotron X-rays have been used for this kind of studies as crack detection is enhanced by the high spatial coherence of X-ray beams available at modern (third generation) synchrotron facilities [START_REF] Cloetens | Observation of microstructure and damage in materials by phase sensitive radiography and tomography[END_REF].

However, the use of synchrotron radiation puts a strong limit on the number of experiments that can be carried out in situ.

3D tomography images can be used, in principle, to evaluate the 3D displacement field at the crack tip in a specimen under load. In practice, this can be achieved using two different methods, namely Particle Tracking (PT) and Digital Image Correlation (DIC). Particle Tracking consists in measuring the displacements of microstructural features or markers inside the material. These markers are first identified by their gravity centre. Then, they are individually tracked between 3D images of the same specimen under different loads in order to compute the global displacement field inside the specimen. The DIC method is well established for computing displacements and strains on the surface of samples or structures [START_REF] Sutton | Advances in two-dimensional and threedimensional computer vision[END_REF][START_REF] Roux | Stress intensity factor measurements from digital image correlation: postprocessing and integrated approaches[END_REF][START_REF] Mekky | The fracture toughness of Ni/Al 2 O 3 laminates by digital image correlation II: bridging-stresses and R-curve models[END_REF]. It consists in measuring displacement fields between image pairs of the same specimen at different loads. The displacement field is obtained, using the so-called brightness conservation, so that the image of the loaded sample is matched to the reference image when pixel locations are corrected for by the measured displacement field. Recent developments have made possible the extension of this Galerkin technique to 3D displacement measurements in the bulk of specimens using 3D tomography images [START_REF] Roux | Three-dimensional image correlation from x-ray computed tomography of solid foam[END_REF].

The uncertainty level of the latter is usually smaller than that of the former.

Both techniques have been applied to very different materials (Table 1). For example, the architecture in a trabecular bone [START_REF] Bay | Digital volume correlation: three-dimensional strain mapping using x-ray tomography[END_REF] or in an open-cell aluminium foam [START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF] has been proved a valid texture for 3D DIC while microporosity in an aluminium alloy has been successfully used for particle tracking [21; 22]. The resolution of both techniques depends directly on the presence of numerous and finely dispersed microstructural features inside the material and those features should not break nor deform heavily upon loading. If the material does not possess a natural texture that meets these requirements, artificial markers may be added. For example, tungsten particles were added to an aluminium matrix [23; 24] and microholes were laser-drilled in a dispersion-strengthened copper alloy [START_REF] Kobayashi | High-density three-dimensional mapping of internal strain by tracking microstructural features[END_REF] to allow for strain mapping in tensile testing for strain values ranging from 3 [23; 25] to 22% [START_REF] Kobayashi | High-density three-dimensional mapping of internal strain by tracking microstructural features[END_REF]. However, the influence of artificial markers on the matrix behaviour is difficult to assess. Increasing the number of artificial markers enhances the spatial resolution of strain mapping via tracking methods but it may also reinforce their potential perturbation effect. Whenever possible, the use of a material that possesses a natural texture suitable for DIC/PT is thus preferable. Measurement of crack closure has been carried out by manual tracking of microstructural features using high resolution synchrotron radiation (0.7 µm voxel size). It drew interesting results on the relationship between the crack shape and local values of the stress intensity factor [START_REF] Toda | A 3D measurement procedure for internal local crack driving forces via synchrotron x-ray microtomography[END_REF]. However, this technique has the disadvantage that crack tip displacements are not measured automatically. This paper is a first attempt to evaluate crack closure in a widely used structural material, i.e. a nodular graphite cast iron where the graphite nodules act as natural markers to determine 3D displacements and stress intensity factors by correlating images obtained with standard laboratory X-ray tomography equipment.

Material and experimental methods

Material

The material of this study is a ferritic cast iron whose composition is 3.4 wt% C, 2.6 wt% Si, 0.05 wt% Mg, 0.19 wt% Mn, 0.005 wt% S, and 0.01 wt% P. After casting and heat treatment (ferritization at 880 °C followed by air cooling) the microstructure obtained is a ferritic matrix (98 wt% ferrite and 2 wt% perlite; average grain size 50 µm) with a 14 % volume fraction of graphite nodules. Carbon (nodules) and iron (matrix) have atomic numbers that are different enough to give a strong X-ray attenuation contrast. Therefore, the spherical nodules, which Linking powered by eXtyles are homogeneously distributed inside the matrix, are easily imaged by tomography and they can be used as natural markers for image correlation in the 3D reconstructed images. The average nodule diameter of 45 µm is large enough to allow for a relatively low resolution voxel size to be used for tomography. This also enables one to observe millimetric samples, the only limitation for the sample size being the overall attenuation of the material that has to allow for, at least, a 10 % transmission of the incoming X-ray beam. The fatigue crack interactions with the microstructure gives a roughness of the crack path of the order of the grain / nodule size. The material Young's modulus, yield stress and Poisson's ratio are equal to 175 GPa, 315 MPa, and 0.27, respectively. More details on the microstructure and mechanical properties of the material can be found in ref. [START_REF] Adrien | Optimisation des cycles thermiques appliqués aux fontes g.s. ferritiques vis à vis des propriétés de fatigue[END_REF].

Fatigue pre-cracking

A notched fatigue specimen with cross-section 6 × 4 mm (see Figure 1a) was pre-cracked at room temperature using a maximum load of 3,217 N (stress ratio: 0.1, frequency 10 Hz). The specimen faces were mirror polished prior to the pre-cracking experiment and an optical travelling microscope was used to detect crack initiation and monitor crack propagation in situ. Fatigue testing was periodically interrupted and the specimen held at mean load to ease crack length measurement. When a crack longer than 350 µm was observed, a loadshedding technique was applied to maintain a maximum stress intensity factor, K max , less than 12.8 MPa√m in order to make sure that a small-scale yielding condition prevailed at the crack tip (estimated plastic zone size in plane strain: 170 µm). Fatigue cycling was stopped as soon as a 1.5 mm long crack was observed. For the tomography experiment, a smaller specimen with a square cross-section of 1.6 × 1.55 mm 2 (Figure 1b) was cut by electro discharge machining in the larger sample (Figure 1a) so that it contains only a part of the total fatigue crack tip with a crack size greater than the grain / nodule size. One face of the small specimen used for the present study contains the crack tip that was at the surface of the larger sample while the other face contains the crack tip that was at the centre of the larger sample.

X-Ray Microtomography

The X-ray microtomography experiments were performed using a computed tomography system V-Tomex (Phoenix X-ray) fitted with a Nanofocus tube whose acceleration voltage can be adjusted from 10 to 160 kV. In the present work, a 95 kV acceleration voltage was selected to ensure a 10 % transmission of the X-ray beam through the 1.6 × 1.55 mm 2 crosssection of the sample. The specimen was placed on a rotating stage in the tomography chamber between the X-ray source and an amorphous Si diode array detector of dimensions 1920 × 1536 voxels. A set of 900 radiographs (scan) were taken while the sample was rotating over 360° along its vertical axis. With an acquisition time per image of 500 ms, one scan lasted about 45 min. A miniature loading device 1 was used to load the specimen (see Figure 1c). The distance between the sample and the tube could thus be reduced to 15.9 mm resulting in an image voxel size of 3.5 µm (divergent beam). The specimen was loaded from 13 N up to 217 N in six incremental steps. A scan was started as soon as stability of the load was reached, i.e. at 13, 44, 82, 130, 185 and 217 N. Reconstruction of the tomographic data was performed with a filtered back-projection algorithm using datos|xacquisition software developed by Phoenix. It provides a 3D image with a 16 bit greyscale colormap that is proportional to the X-ray attenuation coefficient. Definition of a Region Of Interest (ROI) that is focused on the crack reduced the image size to 500 × 700 × 480 voxels; a 2D section of this image is shown in Figure 1b. For the sake of clarity, all the images are presented using the same coordinate system represented in Figure 1 with loading axis along x, crack propagation in the y direction and sample thickness along z.

Digital image correlation

A C8-DIC multiscale algorithm with eight node cubic element was used to compute the displacement fields in the loaded specimen. Details about the method can be found in ref. [START_REF] Roux | Three-dimensional image correlation from x-ray computed tomography of solid foam[END_REF]. The image acquired at minimum load, i.e. 13 N, is used as the reference image of the computation (Figure 2). A ROI of 592 × 416 × 400 voxels was centred on the original Linking powered by eXtyles image of 700 × 500 × 480 voxels. 8-node cubic elements were used. A resolution analysis was performed to choose the relevant size of the element for correlation. Different displacement values in the [0,1]-voxel range are numerically prescribed to the reference picture. Then, the correlation algorithm is applied to this pair of images, thus allowing for an evaluation of the average error and uncertainty, which is equal to the standard deviation of the displacement about its mean value, for different sizes of elements (Figure 3). An increase in the element size results in a reduction of the mean error and standard uncertainty.

However, even though the displacement uncertainty is the lowest for large elements, the heterogeneity of the displacement prompts one to use small sizes. Therefore an element size of 16 voxels was chosen as good compromise resulting in a mean displacement error of 3.3 × 10 -3 voxel and a standard uncertainty of 8.7 × 10 -2 voxel.

The image of the loaded specimen corrected by the displacements measured with DIC gives the "corrected image" shown in Figure 2. The presence of a discontinuity, i.e. the crack itself, impedes perfect matching of the image of the deformed sample to the reference image when the deformed image is corrected by the estimated displacement field. The difference between the reference and the deformed-and-corrected images is the residual error map that provides valuable information about the crack geometry.

Stress Intensity Factors extraction

From the measured displacement field, it is proposed to extract SIFs. First, least squares techniques are implemented by using 2D solutions [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: application in cyclic fatigue[END_REF][START_REF] Yoneyama | Evaluating mixed-mode stress intensity factors from full-field displacement fields obtained by optical methods[END_REF][START_REF] Abanto-Bueno | Investigation of crack growth in functionally graded materials using digital image correlation[END_REF][START_REF] Mcneill | Estimation of stress intensity factor by digital image correlation[END_REF]. An alternative technique to determine SIFs is to evaluate directly the J-integral along a given contour [START_REF] Huntley | Measurement of crack tip displacement field using laser speckle photography[END_REF]. However, only few measurement data are used and strains and stresses are to be computed. To avoid this drawback, interaction integrals [START_REF] Parks | A stiffness derivative finite element technique for determination of crack tip stress intensity factors[END_REF] are used for extracting SIFs from a discrete displacement field [33; 34]. This last technique can be extended to the analysis of 3D cracks [START_REF] Gosz | Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks[END_REF]. It will not be used in the following. Instead, the technique proposed by Hamam et al. [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: application in cyclic fatigue[END_REF] will be followed.

It is assumed that each of the node layers orthogonal to the mean direction of the crack front ((x,y) plane in Figure 2) can be treated separately. For each of these planes (nearly orthogonal to the crack front), the 3D displacement field is projected in a least squares sense onto a basis that includes 2D solutions for a cracked solid (mode I and mode II) as well as mode III asymptotic solutions. Following Williams [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF], a closed-form solution for the in-plane displacements v is derived from Kolossov-Muskhelishvili's potentials [START_REF] Muskhelishvili | Some basic problems of the mathematical theory of elasticity[END_REF] 

      n n n n n r r r v ) , ( ) , ( ) , (      (1)
where ω n and υ n are real numbers, n the number of terms of Williams' series. The reference field for mode I is

                                                  2 exp 2 ) 1 ( 2 2 exp 2 2 exp 2 2 2 / n i n n i n n i r n n n (2)
and for mode II

                                                   2 exp 2 ) 1 ( 2 2 exp 2 2 exp 2 2 1 2 / n i n n i n n i ir n n n (3)
where r and θ denote the in-plane coordinates associated to the usual crack tip frame, μ Lamé's shear modulus, and κ a dimensionless elastic (or Kolossov) coefficient related to Poisson's ratio ν according to (3 -ν) / (1 + ν) for plane stress conditions or 3 -4ν for plane strain conditions (used in the sequel).

The factor  2 is introduced to match the usual definitions of K I and K II SIFs [START_REF] Irwin | Analysis of the stresses and strains near the end of a crack traversing a plate[END_REF], which are thus equal respectively to the amplitudes ω 1 and υ 1 associated with the fields Ω 1 and Υ 1 .

Let us note first that the status of the fields Ω n and Υ n is very different for n less than or greater than 1. For n > 1, the fields are called subsingular. They have no impact on the crack tip. Conversely, the attached stress fields increase with the distance to the crack tip. Such functions are thus useful to match the vicinity of the crack tip with the remote geometry, or Linking powered by eXtyles boundary conditions. However, since their impact on the crack tip process zone is negligible, one does not insist here on the structure of these fields. For n < 0, the fields are referred to as supersingular. Traditionally, the supersingular fields are ignored because their asymptotic behaviour near the crack tip is non-physical (i.e. diverging energy density). However, for the present purpose, since the crack tip process zone is cut out from the domain of analysis, one does not have to reject these solutions [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: application in cyclic fatigue[END_REF]. However, it makes no sense to extrapolate them inside the process zone. They rather characterize the (vanishing) influence of non-linearities present in the process-zone in the outer elastic domain.

Out of plane motions w induced by mode III displacements whose canonical field Ξ reads

        2 sin 2 2 2 / 1    r ( 4
)
allow one to determine K III .

Using least squares techniques, the measured displacement in each plane orthogonal to the crack front is projected onto the considered basis function. In practice, in-plane solutions are considered for n ranging between -3 and 5. As mentioned above, negative n values correspond to supersingular solutions whose coefficients are related to a shift of the crack tip position in the plane with respect to the tip of the equivalent "elastic crack" for n = -1. By cancelling out its contribution, the equivalent crack tip position is estimated [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: application in cyclic fatigue[END_REF]. A length scale proportional to the process zone size [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: application in cyclic fatigue[END_REF] is obtained when ω -3 is related to ω 1 = K I . As all odd n < 0 functions lead to singular displacements at the crack tip, the displacement data at nodes whose distance to the crack tip (or respectively to the crack mouth) is less than 20 (resp. 16) voxels are not considered in the analysis.

A key feature of this study is to use a laboratory tomograph. Consequently, special care is to be paid to the post-processing of the displacement fields. For DIC applications, it is essential that the two images to be registered are directly comparable. Due to thermal expansion of the x-ray tube, the distance between the beam source and the specimen may vary by tiny Linking powered by eXtyles amounts during the experiment. The divergent beam geometry used for laboratory tomographs is such that this small distance change may result in significant (and spurious) image dilation. To decrease the voxel size and give a good crack description, the smallest possible sample / tube distance is used and even a modest thermal expansion in the tube results in an "apparent strain" of the order of 10 -3 , i.e. comparable to elastic strain levels in most metals. This apparent strain should not be considered as a real one. Even though consequences of this artefact are limited in some cases (e.g. when large strain levels are observed [START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF]), it is not true in the present analysis. The strategy is to account for dilation by introducing the corresponding displacement field in the basis on which the DIC displacement is projected. The additional field varies linearly with the distance to the centre of the specimen and thus it is similar to a T-stress field. For conditioning purposes, the T-stress field is replaced by the dilation field and the least-squares analysis is performed with this modified basis. In other words, it is assumed that the actual T-stress is negligible as compared to the artifactual dilation to be extracted. The geometry of the mode I test used herein is consistent with this assumption.

X-FEM computation

An elastic eXtended-Finite Element Method using ELFE3-D software was performed to validate the results obtained with the SIF extractor described above. Details about X-FEM applications to tomographic images can be found in refs. [40; 41]. The computation was carried out using the true crack geometry described by level sets computed from the correlation residuals. The material model is elastic and defined by Young's modulus and Poisson's ratio reported in Section 2.1. From the previous analysis, an experimental displacement field is considered with rigid body and dilation corrections. The three components of this corrected displacement field are then used as a remote displacement on the top and the bottom face of the extended finite element model. From the resulting numerical displacement field, the same procedure as in Section 2.5 (except that the dilation correction is not searched for) is applied to extract SIFs.

Linking powered by eXtyles 3 Results

Identification of 3D crack shape

The grey level tomographic image of the cracked area can be thresholded in order to obtain a 3D rendering of the crack. However, for the cast iron used herein, similarity of grey values between the crack and the nodules makes it impossible to visualize the crack alone using this method (see Figure 4a). As reported in ref.

[42], the growing crack is seeking out the nodules, which explains that the crack path deviates from a flat surface and that many nodules are embedded between the crack faces as shown in Figure 4a. To overcome this difficulty, the previously described residual error map was digitized on an 8 bit colormap and was thresholded using a 3D region growing algorithm from Amira software [43] to separate the voxels that belong to the crack from those that can be ascribed to noise such as computation error or image artefacts. Figure 4b shows the crack geometry as observed from the thresholded residual field under maximum load, i.e. at 217 N. The crack plane and tip position can be matched with those obtained from thresholding the tomographic image (Figure 4a) but the nodules are now erased so that the 3D topography of the crack is easier to observe. It is flat at a macroscopic scale but tortuous at the nodules scale. As this small sample was machined from one half of the larger pre-cracked specimen, the right-hand side of Figure 4 corresponds to the centre of the larger specimen, which underwent conditions close to plane strain during fatigue pre-cracking, while the other side corresponds to the free surface, i.e. closer to plane stress conditions. It is noticeable in Figure 4 that, due to tunnelling effect, the crack is longer in the former plane strain zone. For the sake of simplicity, left and right-hand sides of the specimen in Figure 4 will be referred to as short crack side and long crack side hereafter.

Crack tip position

The 3D rendering of the crack surface from the residual error map largely depends on the threshold value that is used for the region growing algorithm. A threshold value that is too small might result in noise embedded in the thresholded crack while a value that is too large might underestimate the crack front location especially at the crack tip where the thickness of the 3D object to threshold is the smallest. It is to be stressed that the residual field is markedly different along the crack path for a crack opening of order of half a voxel. Hence, the estimate of the crack front position is slightly underestimated as compared to its actual position where the crack opening displacement vanishes, and becomes undetectable by this procedure. For small crack opening (below 217 N), the crack tip visibility is better in the tomographic image than in the residual error probably because of some residual crack opening. Consequently, the crack tip was measured directly via visual inspection of the tomographic image in grey levels to check the position that can be deduced from the residual error field at maximum load (217 N) and validate the thresholding method. The methodology uses Image J software [44]. At a given load, the crack is viewed along the z direction and artificially coloured to enhance contrast between the crack and the cast iron matrix. A plug-in allows one to retrieve the (x,y) coordinates of the voxel that is manually selected at the crack tip. Thus, the crack tip position is determined in the thickness of the specimen and the resulting points are reported in Figure 5a. A rather large scatter of data is obtained at small loads when small crack opening results in limited crack tip visibility. This scatter is reduced at maximum load. Experimental points were fitted to analyse the variation of the apparent crack tip position, as observed via image analysis, with loading (Figure 5b). There is no visible evolution of the crack front between 13 and 44 N. Only at 82 N does the crack seem to go deeper inside the specimen. This level probably corresponds to a full opening of the crack front. The crack front positions measured at 82 N and above on both sides of the sample are consistent with the crack lengths of 636 and 1084 µm measured with an optical microscope.

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It is also noticeable that the apparent crack "advance" is substantially higher on the short crack side than on the long crack side.

Displacements fields in the bulk of the specimen

Only a limited amount of information is made available by such "simple" image analysis while image correlation provides the full displacement fields in the bulk of the specimen. The discontinuity induced by the crack is clearly visible in the U x displacement field (Figure 6) corresponding to mode I offering the possibility to study the crack opening change with load at a local scale. A slight discontinuity is also detectable in the U y and U z displacement fields, i.e. in Mode II and III. Mode II displacement or in-plane shear is localized on the right-hand side of Figure 4 where the crack deviates from the overall mean crack plane; the difference in U y displacements between upper and lower crack faces remains less than 0.8 voxel (i.e. less than 3 µm) even under maximum load. Mode III displacement on both sides of the crack plane is distributed through the specimen thickness along the crack mouth with localization towards the short crack side at maximum load. U z displacement is less than 1.4 voxel (i.e. 5 µm).

Crack opening displacement

The displacement field in mode I, U x , is smooth and continuous away from the crack as shown in Figure 6. Thus, the displacement jump (ΔU x ) on the crack surface is estimated from the computed difference between the U x values in two slices taken parallel to the average crack plane on both sides of the crack (Figure 6). The resulting COD map in the specimen cross-section is plotted for increasing load levels in Figure 7 with a colormap that is proportional to the value of crack opening. Values of ΔU x greater than 3 times the standard uncertainty, i.e. 0.27 voxel or 1 µm, are assumed to correspond to a "real" crack opening so that black areas in Figure 7 (depicted by a white line in Figure 7). Under maximum load, i.e. 217 N, the opening at the crack mouth amounts to 19 µm, i.e. more than five voxels, and it decreases towards the crack tip making the tip positions more difficult to assess. It is visible that at 130 and 217 N, the displacement jump goes farther than the crack tip position and that the opening values behind the crack tip are well above the opening level so that they could not be attributed to measurement noise only.

Profiles of ΔU x from crack mouth to crack tip were extracted for each loading step on both sides of the specimen, i.e. along the dotted lines drawn in Figure 7, to obtain a more quantitative information. The resulting points are plotted in Figure 8 for the short and long crack side. Crack lengths reported in Figure 8 (indicated with arrows) were measured on the thresholded residual error map in Figure 4 so that they also correspond to the crack front position reported in Figure 7. They are shorter than the displacement jump extent by about 200 and 75 µm, respectively for the short and long crack side. At the smallest load (44 N), the crack is not completely open and ΔU x shows variations from maximum values that are below the voxel size (3.5 µm) to zero. The values of COD measured should be a square root function of the distance to the crack tip if a linear elastic behaviour prevail at the vicinity of the crack tip [START_REF] Anderson | Fracture mechanics: fundamentals and applications[END_REF]. Thus, the solid line curves in Figure 8 correspond to the best fit curves that could be obtained for each profile. Above 130 N, these fitted curves fail to represent experimental data close to the crack tip, probably because of the large plastic zone that develops at the crack tip at these load levels.

Estimation of the Stress Intensity Factors

The K I SIF could be computed directly from the curves shown in Figure 8. However, this computation method does not give a full description of K I values along the crack front and it does not exploit the full 3D displacement field information. Last, the dilation artefacts have to be incorporated. Therefore, the above mentioned extraction method (cf. Section 2.5) was used to measure SIF values along the crack front for the different loading steps. The results obtained for K I , K II and K III are reported in Figure 9. The short and long crack sides correspond to the left and right hand sides of Figure 9a to c respectively. In Figure 9a, the measured values of K I on both sides of the sample emphasize the importance of the 3D effect. On the side where the crack is shorter, K I is enhanced while it is reduced on the side where the crack is in advance due to the constraint exerted by the surrounding uncracked ligament. A 2D analysis would be unable to represent such an effect. The 3D X-FEM analysis predicts the same trend as the experimental values. Up to a 130 N load, an excellent match is obtained between the elastic X-FEM computation and the experimental points. Then, at 217 N, the computed curve rises slightly higher as plasticity develops. In Figure 9b, values obtained for K II show fluctuations around 0 through the sample thickness,

i.e. from about 1 to -1 MPa√m with K II values that do not vary significantly with increasing load up to 130 N but do increase at 217 N. K III has an almost zero value that tends to increase slightly with load as shown in Figure 9c. The X-FEM computation for K II and K III (using measured boundary conditions obtained from DIC analyses) shows a good agreement with experimental points. Although the specimen was macroscopically loaded in mode I, small mixed mode displacements occurred.

The position of the crack tip calculated with the SIF extractor is plotted in Figure 10 within the sample thickness. At 82 N and above, the calculated crack tip position in Figure 10 is in agreement with the crack tip position observed in the grey level images although the calculated position is about 80 to 120 µm deeper. In the SIF extractor, the crack length that is calculated corresponds to an equivalent elastic crack. Thus, the larger the plastic zone size at the crack tip, the longer the "equivalent elastic" crack as compared to the "real" or physical crack. Moreover, the manual procedure for estimating the crack front position (Figure 5) in the tomographic image is sensitive to the local crack opening displacement so that it tends to underestimate the crack front position. Both effects contribute to the observed differences.

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To the best of the authors' knowledge, this is the first time that crack opening displacement fields are measured at a local scale by applying DIC on 3D images obtained with a laboratory X-ray tomograph, an equipment which is becoming more and more widespread in materials science laboratories. A problem that is typical of this kind of equipment has arisen. The thermal expansion of the X-ray tube during tomographic data acquisitions results in artificial "dilation" of the sample. For the four studied load levels, spurious strains of the order of 10 -3 , 1.7 × 10 -3 , 1.6 × 10 -3 , 2.7 × 10 -3 , respectively, were measured. This dilation superimposes on the real displacement field inside the sample under tensile loading and biases the interpretation of DIC results. In the present study, this artefact was solved by a correction added to the above-mentioned SIF extraction method. In the near future, other solutions will be tested to solve the problem directly at its source by cooling of the X-ray tube. Provided this dilation artefact is corrected for, the use of a laboratory X-ray source instead of a synchrotron facility facilitates the kind of experiment presented herein.

The DIC method can however only be applied to materials with finely dispersed and resolvable microstructural features. The nodular graphite cast iron used in this study is a widely used material whose microstructure (with natural markers) is perfectly suited for image correlation. However, the present study would not have been possible for example with the 2024 aluminium alloy used by Toda et al. [21; 22] where the natural markers used (pores) have a low volume fraction and a typical size about twice the voxel size used in the present study.

The displacement uncertainty reported in Figure 3 appears to be higher than the levels reported for 2D images [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: application to Portevin-Le Chatelier bands[END_REF]. A possible source of this larger uncertainty lies in the nature of the images used. Tomography images are prone to "noise" because of reconstruction artefacts and improvement in reconstruction of tomographic data may help one to decrease the DIC uncertainty. It is to be noted that, for a given element size, coarsening the image resolution, e.g. from 12 to 36 µm per voxel, was reported to divide the DIC uncertainty by Linking powered by eXtyles more than 3 [START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF]. However, in the present study, a small voxel size (3.5 µm) was preferred in order to have an accurate mapping of the displacement field and of the crack morphology.

An advantage of DIC is also that it allows for 3D rendering of the crack without resorting to threshold tomographic images. Compared to PT [21; 22], DIC is also a more automatic method. Although a robust and automatic particle tracking method is being developed to optimize the tracking efficiency [START_REF] Kobayashi | High-density three-dimensional mapping of internal strain by tracking microstructural features[END_REF], it has still not proved successful for cracked specimens.

Besides, while application of PT to cracked specimens is restricted to the crack tip [START_REF] Toda | A 3D measurement procedure for internal local crack driving forces via synchrotron x-ray microtomography[END_REF], DIC allows for an extended area to be studied, and the full 3D displacement field to be measured so that 3D maps of COD are accessible. Moreover, the exact position of the crack tip is not a pre-requisite to SIFs extraction. As explained in Section 2.5, SIFs are computed with respect to a calculated crack tip position (Figure 10) whose accurate position is not known a priori.

In the present study, different techniques allowed for the detection of crack opening during progressive loading of the specimen. Using direct image analysis to retrieve the crack tip position, an advance of the apparent crack tip position is observed in Figure 5b when the load increases from 44 to 82 N; it is not visible in the residual error field probably because of the medium spatial resolution of the DIC field. This apparent crack advance may be related to PICC that developed during pre-cracking of the larger sample. The larger advance observed on the short crack side would then be consistent with the size of the plastic zone inherited from the larger specimen. At the former free surface of the large sample, the plane stress condition created a plastic zone size that is theoretically about 3 times greater than that in plane strain, i.e. at the long crack side. Consequently, this larger plastic zone exerts its influence on a longer distance away from the crack tip. It is noticeable that the plane stress plastic zone seems to influence a large part of the crack front away from the former free surface. In thin specimens, Chermahini et al. [START_REF] Chermahini | Three-dimensional aspects of plasticity-induced fatigue crack closure[END_REF] also reported that the opening load decreases slightly from the free surface towards the centre of the specimen whereas in thicker specimen, the region of higher opening load was restricted to a relatively narrow region near the free surface as expected. However, in the present study, this result should be considered with caution as the cutting of the small tomography sample might induce stress relaxation so that the stress state that formerly existed inside the larger specimen might not be fully acting in the smaller specimen.

In the 3D images of the crack, a load of 82 N was sufficient to open the crack, and above this load level, the apparent crack tip position (Figure 5) did not vary significantly. However, non negligible values of COD are observed ahead of the crack tip location measured at maximum load (see Figure 8); they might be attributed to the presence of a plastic zone. At 217 N, the mean stress on the uncracked ligament is about 60 % of the monotonic yield strength.

Generalized plasticity is thus not expected. However, due to dissymmetry of the crack and the vicinity of the free surface, the plastic zone size at the crack front is likely to increase rapidly, especially on the short crack side, and the confined plasticity condition may be violated at the crack front. This hypothesis is in agreement with the fact that an elastic computation (see 'E' in Figure 9a) was unable to perfectly match the experimental points at 217 N, while a nearly perfect match was observed at 130 N and below.

Using the 3D displacement fields obtained from DIC, the SIF change along the crack front was obtained for the different load levels. K II in Figure 9b varies around a zero value and a small positive value is measured for K III in Figure 9c. Crack geometry and misalignment of the specimen inside the in-situ loading device and / or misalignment of the crack faces after fatigue pre-cracking of the large sample may explain some of this mixed mode displacement during tensile loading. However, mode II and III SIFs remain close to zero, and mode I is predominant, as expected from the prescribed "macroscopic" mechanical testing conditions. K I is observed to increase from the long crack side towards the short crack side in Figure 9a. This is the result of the 3D shape of the crack, namely, a locally lagging crack front induces a higher value of SIF [48; 49].

By combining the results of Figure 7 and 9a it is possible to give an estimate of the value of opening SIF, K op , assuming that mode I is the main opening mode and, also, that the crack actually opens when the measured ΔU x equals 0. [START_REF] Clément | Short crack behaviour in nodular cast iron[END_REF][START_REF] Ogawa | Near-threshold fatigue crack growth and crack closure in a nodular cast iron[END_REF][START_REF] Wasen | Fatigue crack growth thresholds -the influence of Young's modulus and fracture surface roughness[END_REF]. This tends to indicate that DIC coupled with laboratory X-ray tomography can be trusted to assess crack closure in 3D studies of this type of material.

Conclusions

3D-DIC analyses on 3D microtomographic images obtained with a laboratory X-ray source were used to study fatigue crack closure in a nodular graphite cast iron. An in-situ testing machine enabled for progressive loading of the specimen in tension and 3D images acquired at different loading steps allowed for the detection of the opening load. The 3D images were further analyzed using 3D digital image correlation. The nodular graphite cast iron material is well adapted to the correlation technique with graphite nodules that act as natural markers.

Compared to the tracking method used by Toda et al. [21; 22] to analyze crack closure in synchrotron X-ray microtomographic images, DIC is a more automatic method that provides the full displacement field inside the cracked sample. The measured displacement fields were then used to evaluate the COD maps in the specimen cross-section in order to visualize crack opening locally with an increased resolution compared to direct image analysis of the crack. A stress intensity factor extractor adapted for lab-CT, i.e. with correction for the artificial dilation induced by thermal expansion of the X-ray tube, permitted us to compute the stress intensity factor along the crack front from the 3D displacement fields. The confrontation of the COD maps with local SIF allowed for the estimation of a stress intensity factor for opening, K op , equal to 6 MPa√m.

The method described herein could be potentially used to investigate the geometry of fatigue cracks in samples with a larger size (up to 5 mm with the present tomograph) relevant for engineering applications and corresponding to more complex loading cases (e.g. tension / torsion loading). Besides, the tomography technique being non destructive, the same type of study can be performed in situ during fatigue crack propagation. Such experiments have been performed recently and are currently being analysed, they should allow to assess crack closure during mechanical cycling, and to correlate it to local crack growth.

Figure captions

  correspond to crack faces still in contact. At 44 N, contact areas are still observed at the crack front. Only at 82 N and above does the crack fully open on the short crack side and a non-negligible value of ΔU x is observed at the position of the crack tip Linking powered by eXtyles

Figure 1 :

 1 Figure 1 :Fatigue pre-cracking of (a) the large specimen from which a (b) small sample is obtained for (c) microtomography of in-situ loaded specimen.

Figure 2 :

 2 Figure 2 : Extraction of the crack surface from Digital Image Correlation measurements.

Figure 3 :

 3 Figure 3 : Displacement uncertainty and error expressed in voxel as functions of the element size when uniform sub-voxel displacements are prescribed.

Figure 4 :

 4 Figure 4 : 3D rendering of the crack surface at maximum load from (a) the thresholded tomographic image and from (b) the residual error field.

Figure 5 :

 5 Figure 5 : Position of the crack front for different load levels: (a) experimental points and (b) fitting curves.

Figure 6 :

 6 Figure 6 : Displacement (expressed in voxels) fields in the specimen volume under maximum load (macroscopic load applied along x direction). The hatched planes drawn in the U x field indicate the (y, z) planes used for COD mapping in Figure 7.

Figure 7 :

 7 Figure 7 : Maps of ΔU x displacement (COD) in the specimen cross-section with increasing load. The crack front is drawn in white and the dashed-lines along y direction indicate the location of the profiles plotted in Figure 8.

Figure 8 :

 8 Figure 8 : Change with load of the crack opening displacement profile along crack depth (a) on the short and (b) long crack side; the dashed-line corresponds to 3 times the uncertainty in U x and the arrow indicates the position of the crack front plotted in Figure 7. The solid lines correspond to the best fitting curves.

Figure 9 :

 9 Figure 9 : SIF values along the crack front for (a) Mode I, (b) Mode II and (c) Mode III; Measured ('FF') values of SIF (symbols) are compared to computed ('E') values (lines); the hatched area in (a) corresponds to the K op range obtained from literature data.

Figure 10 :

 10 Figure10: Crack tip position of the equivalent "elastic crack" based on the SIF extractor method (symbols); the solid lines correspond to the crack tip positions reported in Figure5b(visual inspection of tomographic images and smoothed by curve fitting).1 Courtesy of W. Ludwig -MATEIS/ESRF

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1 :

Figure(s)Figure 4 :

 4 Figure(s)

Figure 5 :

 5 Figure(s)

Figure

  Figure(s)

Figure 6 :

 6 Figure 6 : Displacement (expressed in voxels) fields in the specimen volume under maximum load (macroscopic load applied along x direction). The hatched planes drawn in the U x field indicate the (y, z) planes used for COD mapping in Figure 7.

  Figure(s)

Figure 7 :Figure 8 :Figure 7 .

 787 Figure 7 : Maps of U x displacement (COD) in the specimen cross-section with increasing load. The crack front is drawn in white and the dashed-lines along y direction indicate the location of the profiles plotted in Figure 8.

Figure 5 )Figure 9 :

 59 Figure(s)

Figure

  Figure(s)

Figure 10 :

 10 Figure 10 : Crack tip position of the equivalent "elastic crack" based on the SIF extractor method (symbols); the solid lines correspond to the crack tip positions reported in Figure 5b (visual inspection of tomographic images and smoothed by curve fitting).

Figure(s)

  Figure(s)

  27 voxel. Around 82 N, the measured values of K I vary from 7.5 to 3.5 MPa√m from the short crack side (fully open) to the long crack side (crack locally closed). At 130N, K I varies from 13 to 6 MPa√m and both sides of the crack are fully open. The value of K op can thus be estimated close to 6 MPa√m. This level is in very good agreement with values reported in the literature by different authors on the same material but using mechanical compliance techniques

Table 1

 1

	Reference	X-Ray source	Crack	Voxel size in µm	Method	Material	Displacement uncertainty (in voxels)	Section of analysis (mm) voxels] [subvolume in	Deformation range (%)
	[19]	laboratory	no	35	DIC	Trabecular bone	3.5  10 -2	15	0.4 -1.8
	[22]				Manual tracking	Al2024-T351 with	5  10 -1
		synchrotron yes	0.7		800 pores 7.2 µm		0.8  0.9	Elastic
	[21]				Tracking	in 	10 -1 (on COD)
						Aluminium with
	[23]	synchrotron	no	1.5	Tracking	added (5000) W markers 10 µm in	2  10 -1	0.45  0.45	2.7 -9.5 (plastic)
						
	[20]	laboratory	no	36 (20,12)	DIC	Open-cell aluminium foam	5  10 -2	5  5 [28  16  26]	No loading
	[24]	synchrotron	no	1.5	Tracking	Al/AlO 2 with added (~38000) W	5  10 -1 (on markers'	1 [67 3 ]	10 (plastic)
						markers	position)
	[25]	synchrotron	no	0.474	More automatic tracking	Copper alloy with 188 artificial pores 15 µm in 	1 (on markers' position)	0.25  0.3	3.5 -22 (plastic)
	[18]	synchrotron	no	4.91	DIC	Polypropylene solid foam	3  10 -1 -3  10 -2	15 [400 3 ]	10 -30

Table ( s)

 (

Acknowledgments

This work was funded by the CETIM Foundation grant entitled PROPAVANFIS: 'Advanced methods for the experimental and numerical analyses of crack propagations under complex loadings.' The authors are also grateful to Dr. W. Ludwig for the use of his in-situ testing machine.


Yevgeny Tsodikovich

Xavier Venel
email: xvenel@luiss.it

Anna Zseleva
email: anna.zseleva@maastrichtuniversity.nl

E Solan

D Lagziel

J Flesch

G Ashkenazi

The Regularity of the Value Function of Repeated Games with Switching Costs *

Keywords: Journal classification numbers: C72, C73 Switching Costs, Repeated Games, Stochastic Games, Zero-sum games

We study repeated zero-sum games where one of the players pays a certain cost each time he changes his action. We derive the properties of the value and optimal strategies as a function of the ratio between the switching costs and the stage payoffs. In particular, the strategies exhibit a robustness property and typically do not change with a small perturbation of this ratio. Our analysis extends partially to the case where the players are limited to simpler strategies that are history-independent, namely static strategies. In this case, we also characterize the (minimax) value and the strategies for obtaining it.

Introduction

We consider a repeated normal-form zero-sum game where at each time step, the minimizing player pays the maximizer both the "standard" outcome of the game and an additional fine if he switched his previous action. This model, namely the traveling inspector model, was presented in [START_REF] Filar | The traveling inspector model[END_REF] and is used to study different scenarios with switching costs [START_REF] Xu | Optimal patrol planning for green security games with black-box attackers[END_REF], [START_REF] Yavuz | An analysis and solution of the sensor scheduling problem[END_REF], [START_REF] Darlington | A stochastic game framework for patrolling a border[END_REF] to mention a few).

A game with switching costs is equivalent to a stochastic game where the states correspond to the previous action taken by the minimizing player [START_REF] Filar | The traveling inspector model[END_REF]. Since only the minimizing player controls the states, this is a single-controller stochastic game and there exists an optimal strategy that is stationary (depends solely on the state) and can be computed using one of several standard tools, such as Raghavan and Syed [2002] and Raghavan [2003]. We characterize completely the value and the optimal stationary strategies, and show that they belong to a finite set that depends only on the underlying one-shot game, not on the switching costs. Our main finding is that the value function is piece-wise linear in the weight between the switching costs and the "standard" stage payoff, c. Consequently, the optimal strategy depends only on the segment in which c is situated, not its value, and the minimizer can play optimally without knowing the exact weight (as in [START_REF] Rass | Numerical computation of multi-goal security strategies[END_REF]).

In some applications there is an additional requirement to use only history-and timeindependent strategies (denoted static strategies), i.e., to use the same mixed action in every stage [START_REF] Rass | Defending against advanced persistent threats using gametheory[END_REF][START_REF] Schoenmakers | Repeated games with bonuses[END_REF]. We repeat our analysis with this constraint, and study the properties of the optimal payoff1 in static strategies and the corresponding optimal strategies. In general, the optimal payoff is not piece-wise linear but the previous result for stationary strategies carry on in the special case of switching costs that are independent of the actions. In this case, our results theoretically explain why in empirical works, such as [START_REF] Liuzzi | Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems[END_REF], the optimal static strategies only change slightly (if at all) in response to a small change in the ratio between the switching costs and the stage payoffs.

The Switching Costs Model

A zero-sum game with switching costs is a tuple Γ pA, S, cq where A pa ij q is an m ¢ n matrix, S ps ij q is an n ¢ n matrix with non-negative entries, and c ¥ 0. At time step t, Player 1 (the maximizer) chooses an integer iptq in the set rms t1, . . . , mu and Player 2 (the minimizer) chooses jptq rns. The stage payoff that Player 2 pays Player 1 is a iptqjptq cs jpt¡1qjptq , where cs jpt¡1qjptq represents the cost of switching. 2 We assume that keeping the same action is costless (s jj 0). The extension to mixed strategies is standard, but it should be noted that jpt ¡ 1q is known at time t, even if it was determined by a mixed action.

The process repeats indefinitely, and the payoff is the undiscounted average. More precisely, let pσ, τ q be a pair of strategies in the repeated game and denote by iptq, jptq the chosen actions at time t according to σ, τ (given the history). We set jp0q 1 (an assumption that has no effect on the payoff) and define the payoff to be γpσ, τ q lim inf

T ÑV E σ,τ £ 1 T T ţ1 a iptqjptq cs jpt¡1qjptq
.

(

) 1
This game is equivalent to a stochastic game where each state represents the previous action of Player 2. The sets of actions in all states are rms and rns, the payoff in state k when Player 1 plays i and Player 2 plays j is a ij cs kj , and the next state is j. The class of games where only one player controls the state transitions was studied by [START_REF] Ja Filar | Algorithms for solving some undiscounted stochastic games[END_REF], who showed that the value exists and obtained in strategies that depend solely on the current state, not on the history, namely stationary strategies. It follows that the value of Γ exists, and it is obtained in strategies that depend solely on the previous action of Player 2.

Definition 1. A stationary strategy is a strategy that depends in each t on the previous action of Player 2, jpt ¡ 1q, but not on the rest of the history or t itself. Hence, a stationary strategy is a vector of n mixed actions, one to follow each possible pure action of Player 2.

Definition 1 concerns both players but in any case the dependence is on the previous action of Player 2, as he is the one paying switching costs and controlling the states.

Following the literature [START_REF] Schoenmakers | Repeated games with bonuses[END_REF][START_REF] Rass | Numerical computation of multi-goal security strategies[END_REF][START_REF] Liuzzi | Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems[END_REF], we also consider strategies independent of the time and of the history.

Definition 2. A static strategy is a strategy that plays the same mixed action in each stage, regardless of the history. Hence, a static strategy is one mixed action played repeatedly.

Suppose Player 1 uses a static strategy x and Player 2 uses the static strategy y. We obtain a closed form formula for the payoff in matrix notation:

gpcqpx, yq x T Ay cy T Sy.

(2)

Typically, the value may not exist in static strategies. Instead, the figure of merit studied in the literature is the minimax defined as ṽpcq min y max x gpcqpx, yq. For simplicity, we refer to it as the value in static strategies, but it should be understood only as the minimax value.

Results

A Useful Lemma for Parametric One-Shot Games

We study the value of a parametric one-shot game, where the payoff is linear in each parameter in a way that is independent of the actions of Player 1. The Lemma stands alone, as it might be of general interest. Formally, let A be an m ¢ n zero-sum game, and b 1 , . . . , b n be nonnegative constants. For each x px 1 , . . . , x n q, the game Γpxq is defined to be the one-shot zero-sum game whose payoff matrix is a ij b j x j , and its value is denoted by vpxq.

Lemma 1. The value function vpxq : R n Ñ R is continuous, increasing in every parameter, concave and piece-wise linear in every direction.

Proof Let I 1 , I 2 be some subsets of the rows and columns (resp.) of the one-shot game A. We check if Player 2 can make Player 1 indifferent among all the actions in I 1 , using a completely mixed action over I 2 , and does Player 1 prefer them over the actions not in I 1 .

Fix k I 1 . We look for a vector y R n of Player 2, such that the support of y is exactly I 2 ( °yj 1 with dj I 2 : y j ¡ 0, and dj I 2 : y j 0), Player 1 is indifferent among the actions in I 1 and prefers actions in I 1 over other actions:

6 9 9 8 9 9 7 °jI 2 y j a kj °jI 2 y j a lj dl I 1 ztku, °jI 2 y j a kj ¥ °jI 2 y j a lj dl I 1 . (3)
If there exists solutions to this system of equations, we denote one of them by ypI 1 , I 2 q. Suppose Player 2 uses ypI 1 , I 2 q in Γpxq for some x. Player 1 is indifferent among the actions in I 1 and prefers them over other actions if Eq. (3) holds with pa ¤j b j x j q instead of a ¤j . This is indeed the case, as the b j x j terms cancel out. Therefore, when Player 1 best responds he chooses an action in ∆pI 1 q, and the variable x j appears in the payoff linearly with the slope y j b j . Denote by lpI 1 , I 2 qpxq the payoff function, which is linear in x.

Fix x and a pair of optimal strategies pp ¦ , q ¦ q. This profile has a corresponding support pair pI 1 pxq, I 2 pxqq. By definition, ypI 1 pxq, I 2 pxqq makes Player 1 indifferent between the actions he chooses with non-zero probability according to p ¦ , hence the pair pp ¦ , ypI 1 pxq, I 2 pxqqq is optimal and so does ypI 1 pxq, I 2 pxqq. It follows that vpxq lpI 1 pxq, I 2 pxqqpxq. Moreover, the value function vpxq is continuous in each x i (this is a polynomial game, [START_REF] Shapley | Basic solutions of discrete games[END_REF]), hence the support pairs can only change when the payoffs they induce are equal. Since there is a finite number of supports, there is a partition of R n into finitely many disjoint convex and closed sets (one for each pair of supports) such that the restriction of vpxq to each of them is the restriction of some linear map to this set. We conclude that vpxq is piece-wise linear in any direction of R n , and since the slopes are convex combinations of the non-negative coefficients b i , vpxq is also increasing in each variable.

Finally, vpxq is concave. Let c 1 , c 2 , c 3 R n be three points on the same line such that vpxq is linear on the segments rc 1 , c 2 s and rc 2 , c 3 s. For c ¦ pc 1 , c 2 q, let ypc ¦ q be the optimal action of Player 2 chosen using the above method. If Player 2 plays ypc ¦ q regardless of c, the payoff is a linear function of c that coincides with the value on rc 1 , c 2 s. In the region rc 1 , c 2 s, the value function must be below this line, so the slope of c must decrease. To conclude, on this line, vpxq is a piece-wise linear function with decreasing slope, i.e. concave.

The Main Results

The Value Function in Stationary Strategies

Theorem 1. For every c ¥ 0, the game has a value in stationary strategies denoted by vpcq.

This function is continuous, increasing, concave, piece-wise linear, and eventually constant.

If vpcq is linear on rc, cs, then Player 2 has a strategy which is optimal for all c rc, cs.

Proof This game is equivalent to a stochastic game where the current state corresponds to the pure action chosen by Player 2 in the previous time period, so the set of states is also rns. In this stochastic game, only one player controls the transitions, so existence follows from [START_REF] Ja Filar | Algorithms for solving some undiscounted stochastic games[END_REF]. Moreover, according to [START_REF] Filar | A matrix game solution of the single-controller stochastic game[END_REF], the value of the game is the same as the value of a one-shot normal form game, whose pure actions are the pure stationary actions in the stochastic game (each pure action in the one-shot game is a vector of size n dictating which pure action to choose at each state). Given a pure strategy of Player 1 and a pure strategy of Player 2, the payoff is linear in c with coefficients depending on the strategies. Moreover, the coefficient of c is determined only by the strategy of Player 2 since only he bears switching costs. Hence, there exists a ij and b j such that the payoff can be denoted by a ij b j c, with b j ¥ 0 since it is a convex combination of the elements of S.

Lemma 1 can be applied to this one-shot game by setting for all i : x i c, so vpcq vpc, . . . , cq and it is continuous, concave, increasing, and piece-wise linear. Since vpcq is bounded by the pure minimax of A, it cannot strictly increase for all c and is eventually constant.

Suppose vpcq is linear on rc, cs and fix c ¦ pc, cq. Let σ c ¦ be an optimal strategy for Player 2 in the corresponding stochastic game and suppose Player 2 plays σ c ¦ regardless of c whereas Player 1 best responds (as a function of c). When Player 2 plays σ c ¦ , he fixes the transition probabilities between states, and these are now independent of c. Hence, the fraction of time spent in each state is constant and independent of c. In state k, the payoff matrix is of the form a ij s kj c, so the part corresponding to c in each column is identical and linear with the same slope, regardless of the row. When mixing the columns in state k according to σ c ¦ , the expected payoff of the rows are parallel lines. Player 1 best responds in each state, but since the lines are parallel, his choice does not depend on c. Thus, the payoff in each state is a linear function of c, and the total payoff, which is a weighted average of these functions with the percentage of time spent in each state as weights, is a linear function of c, denoted by f c ¦ pcq. Since vpcq is the value, vpcq ¤ f c ¦ pcq, with equality at c ¦ . These two lines that intersect once must coincide and σ c ¦ obtains the value in the entire segment.

Optimal strategies are therefore robust to small changes of c: knowing the exact c is not necessary to play optimally, and it is almost universally unnecessary to adjust the strategy as c changes. Moreover, if Player 2 wishes to minimize separately his stage payoff in the repeated game and his switching costs, an alternative approach is to consider a game where the payoff is their convex combination, as in our model [START_REF] Rass | Numerical computation of multi-goal security strategies[END_REF]. Theorem 1

shows that the exact weights of the two goal functions are of small significance for Player 2, whose optimal strategy comes from a finite set that depends solely on A. Interestingly, this is not true for Player 1, and his optimal strategy typically depends on the exact c.

The concavity of vpcq comes from a compromise between the stage payoff and the switching costs. The higher the c, the more costly it is to switch so it is better to play a strategy that rarely changes actions, in expense of some loss in the stage game A.

To conclude, the family of functions described in Theorem 1 is the widest possible. Any function that holds these properties corresponds to a zero-sum game with switching costs, hence our theorem provides a complete characterization of the value function of such games.

Remark. Let vpcq : r0, Vq Ñ R be a continuous, increasing, concave, piece-wise linear function and eventually constant. Then, there exists a one-shot game A and a switching costs matrix S such that vpcq is the value of the game pA, S, cq.

Sketch of the proof. The value function of the game

A ¡ 2b 1 0 2b 2 0 v 0 2b 1 0 2b 2 v © with S ¤ ¥ 0 2β 1 M M M 2β 1 0 M M M M M 0 2β 2 M M M 2β 2 0 M M M M M 0 , where M 4 β i is vpcq 6 9 9 8 9 9 7 b 1 β 1 c if c r0, c 1 s, b 2 β 2 c if c rc 1 , c 2 s, v if c rc 2 , Vq.
The generalization to vpcq with more than 3 segments is straightforward.

The Value Function in Static Strategies

The minimax value in static strategies exhibits similar properties to vpcq, except piece-wise linearity. Piece-wise linearity can be obtained by adding the common assumption that the switching costs are independent of the switched actions, i.e. s ij 1 for i $ j. This case arises when the costs stem from the act of "switching" itself and do not depend on the actions being switched [START_REF] Bl Lipman | Switching costs in frequently repeated games[END_REF][START_REF] Bl Lipman | Switching costs in infinitely repeated games[END_REF][START_REF] Schoenmakers | Repeated games with bonuses[END_REF].

Theorem 2. The minimax value ṽpcq in static strategies is a continuous, increasing, and concave semi-algebraic function.

If, in addition, the switching costs are uniform (s ij 1 for all i $ j), then ṽpcq is piecewise linear and for each rc, cs where ṽpcq is linear, there exists a static strategy for Player 2 which is optimal for all c rc, cs.

Proof Consider gpcqpx, yq from Eq. ( 2) and recall that by definition, ṽpcq min (4)

Note that arg max

x gpcqpx, yq depends solely on y and not on c. Since gpcqpx, yq is continuous, so is ṽpcq. Moreover, ṽpcq is semi-algebraic since gpcqpx, yq is a polynomial in each variable.

The function ṽpcq is increasing: Let 0 ¤ c 1 c 2 , and y c 2 be the arg min from Eq. (4). ṽpc 2 q max x∆prmsq tgpc 2 qpx, y c 2 qu ¥ max x I ∆prmsq tgpc 1 qpx I , y c 2 qu ¥ ṽpc 1 q, where the first inequality follows from c 1 c 2 and S ¥ 0 (the maximizing x is the same as it depends solely on y c 2 ). The last inequality follows from the definition of ṽpc 1 q.

The function ṽpcq is concave: Let β p0, 1q. Then

ṽpβc 1 p1 ¡ βqc 2 q min y 3 max x tx T Ayu pβc 1 p1 ¡ βqc 2 qy T Sy A ¥ min y 3 β max x gpc 1 qpx, yq A min y 3 p1 ¡ βq max x gpc 2 qpx, yq A βṽpc 1 q p1 ¡ βqṽpc 2 q.
The inequality is obtained since term-by-term minimization yields a smaller result then minimizing the entire sum. This completes the main part of the proof.

Suppose in addition that s ij 1 for i $ j. Under this assumption, y t Sy 1 ¡ ||y|| 2 , and ṽpcq c min . For each i rms, let I i be the subset of ∆prnsq such that i is a best response for Player 1 in the one-shot game A to all y I i . It is well-known that I i is a compact, convex polyhedron. Now, suppose Player 2 is restricted to playing only in I i . The best response for Player 1 is i and the payoff is c min

yI i 2 A i y ¡ c||y|| 2 @
, where A i is the ith row of A. This is a minimization of a concave function over a polyhedron, so the minimum is attained at one of its vertices. There are m polyhedrons and each one has finitely many vertices, so the set of candidate optimal strategies is finite. Following a similar argument as in Theorem 1, ṽpcq is piece-wise linear and on each segment where it is linear, there exists a static strategy that obtains the value for all cs in the segment.

The requirement for uniform switching costs is essential for the piece-wise linearity of ṽpcq. For example, it is easy to verify for the game A ¡200 1 200 200 1 ¡200 ¨with the switching costs matrix S ¡ 0 1 100 1 0 1 100 1 0 © , the optimal static strategy for Player 2 strongly depends on c, and in the domain c

1 98 , 1 2 $ , the value is ṽpcq 1 ¡ p1¡2cq 2 192c .
With the additional assumption that the switching costs are uniform, the optimal static strategy comes from a finite set and is robust to the exact value of c. This time it is also true for the best responses of Player 1, since he best-responds purely. This provides a theoretical explanation for the numerical results of [START_REF] Rass | Numerical computation of multi-goal security strategies[END_REF] and [START_REF] Liuzzi | Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems[END_REF], which indicate that slightly changing c has a small effect, if any, on the optimal strategies.

The above result can be used to search more efficiently for the optimal strategy within the set, by eliminating from consideration strategies with too-high expected per-stage switching cost. The idea is that if a particular strategy y c ¦ is optimal for some c ¦ , the concavity dictates that y is not optimal for c ¡ c ¦ if y T Sy ¡ y T c ¦ Sy c ¦ . A similar approach can be used for the optimal stationary strategy as a consequence of Theorem 1, although calculating the slope for a given stationary strategy is more complicated.

2 ¢ 2 Games

We finalize by showing that in 2 ¢ 2 games, the value and the minimax value in static strategies coincide. This generalizes Theorems 4.1, 4.2, and Corollary 4.1 in [START_REF] Schoenmakers | Repeated games with bonuses[END_REF] for the case that s 12 $ s 21 . The main idea is that there are only two candidates for being the optimal stationary strategy: the optimal strategy of A and the pure minimax of A, both are also static strategies.

Proposition 1. Let A ¡ α β γ δ
© be a zero-sum game, and S 0 s 12 s 21 0 ¨the switching costs of Player 2. The value of pA, S, cq can be obtained by a static strategy, i.e., ṽpcq vpcq.

Proof If the value of A without switching costs can be obtained in pure actions, then vpcq ṽpcq v for all c. Otherwise, w.l.o.g., α ¡ β, γ and δ ¡ β, γ. Let p p0, 1q be the unique probability of playing the left column (action L) in A that achieves the value.

In each state, in a similar way to Eq. ( 5), Player 1 is indifferent between his two actions if and only if Player 2 plays p, otherwise Player 1 plays purely one of them, regardless of the switching costs. Thus, in each state either Player 2 plays purely too or he plays the mixed action p. There remains only 4 types of possible stationary strategies to consider: (i) Play p in both states; (ii) Play in a way that never reaches one of the states (e.g play L after s L );

(iii) Play purely but always visit all the states (Play L in state s R and R in s L ); and (iv)

Play purely in only one state and p in the other (e.g. R in state S L ).

Options (i) and (ii) correspond to static strategies whereas option (iii) is clearly suboptimal. We show by contradiction that a strategy of type (iv) is not optimal. Let κ be the continuation payoff after playing R (the continuation payoff after playing L is normalized to 0). When we add the switching costs and the continuation payoffs to the games, we obtain two one-shot games:

©

To be optimal in the stochastic game, the profile must be optimal in each of these oneshot games. Consider state s L . Since it is optimal to play purely R, we necessarily have α ¥ β κ cs 12 or γ ¥ δ κ cs 12 . If only the first equation is true, the equilibrium is mixed and R is not optimal. If only the second equation is true, then there is a contradiction: α ¡ γ ¥ δ κ cs ¡ β κ cs ¡ α. It follows that both equations are true, i.e. the right column dominates the left, and in particular γ ¥ δ cs 12 κ.

In state s R , however, the optimal strategy is mixed and therefore such dominance is impossible. If the direction of the original inequalities (on α, β, γ, δ) remains correct, then δ κ ¥ γ cs 21 and s 12 s 21 ¤ 0, which is a contradiction. In the other case that both inequalities change direction, the contradiction is constructed using the first row.

  Ayu cy T Sy B .

Typically the game has no value in static strategies. Hereafter, when discussing the value in such strategies, we are referring to the minimax value, and when discussing optimal strategies, the minimax strategies.

We emphasize that the game remains a zero-sum game, as the switching costs are transferred to the adversary, Player 1. If part of the switching costs dissipate, the game is no longer a zero-sum game.

anonymous referee for their highly valuable comments. YT acknowledges the support of the French National Research Agency Grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University -A*MIDEX. XV acknowledges the financial support by the National Agency for Research, Project CIGNE (ANR-15-CE38-0007-01) and is member of Gnampa-Indam (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni).


Romain Pascaud
email: romain.pascaud@isae.fr

Francisco Pizarro

Thierry Callegari

Laurent Liard

Olivier Pascal

Olivier Pigaglio

Micro-décharge plasma comme élément limiteur de puissance dans des dispositifs micro-ondes en technologie circuit imprimé

L'utilisation d'une micro-décharge plasma comme élément limiteur de puissance est proposée pour la protection des récepteurs micro-ondes. Plusieurs circuits planaires intégrant des micro-décharges ont été caractérisés expérimentalement afin d'évaluer leur performance.

Introduction

Le couplage direct au travers des antennes rend les récepteurs micro-ondes très vulnérables vis-à-vis des agressions micro-ondes de forte puissance du type HPM (High-Power Microwave). Si les agressions en dehors de la bande-passante du récepteur peuvent être contrées par l'utilisation de filtres sélectifs en fréquence, la protection de l'électronique contre des signaux HPM dans la bande-passante requiert l'insertion de dispositifs limiteurs de puissance dans la chaîne de réception [START_REF] Bilotta | Receiver protectors : A technology update[END_REF]. Cependant, l'utilisation d'un limiteur de puissance en amont des composants sensibles, en particulier des amplificateurs à faible bruit, se traduit aussi par une augmentation du facteur de bruit et donc une réduction de la sensibilité du récepteur.

Il existe différentes topologies de limiteurs de puissance micro-onde basées sur l'utilisation de diodes Schottky ou PIN [START_REF] Bahl | 10W CW broadband balanced limiter/LNA fabricated using MSAG MESFET process[END_REF][START_REF] Yang | A novel analysis of a Ku-band planar p-in diode limiter[END_REF], de couches minces de matériaux supra-conducteurs [START_REF] Booth | A selfattenuating superconducting transmission line for use as a microwave power limiter[END_REF] ou de dioxide de vanadium [START_REF] Givernaud | Microwave power limiting devices based on the semiconductor-metal transition in vanadium-dioxide thin films[END_REF] ou encore de décharges plasmas [START_REF] Bäckström | Can intentional electrical discharges be used for HPM protection[END_REF][START_REF] Cross | Theory and demonstration of narrowband bent hairpin filters integrated with AC-coupled plasma limiter elements[END_REF]. Parmi ces solutions, des implémentations spécifiques ont été proposées pour fournir des solutions compactes à faible pertes d'insertion en intégrant directement la fonction de limitation de puissance dans des circuits du récepteur comme un amplificateur à faibles bruit [START_REF] Bahl | 10W CW broadband balanced limiter/LNA fabricated using MSAG MESFET process[END_REF], une antenne [START_REF] Bäckström | Can intentional electrical discharges be used for HPM protection[END_REF] ou un filtre [START_REF] Cross | Theory and demonstration of narrowband bent hairpin filters integrated with AC-coupled plasma limiter elements[END_REF].

Dans cette communication, nous proposons une approche qui consiste à intégrer une micro-décharge plasma de type MCSD (Micro-hollow Cathode Sustained Discharge) au sein de circuits micro-ondes imprimés afin d'obtenir la fonction de limitation de puissance.

Micro-décharge plasma de type MCSD

comme élément limiteur de puissance

2.a. Description de la MCSD

Le dispositif MCSD permet d'obtenir un plasma très localisé et stable pour une large gamme de pression de gaz [8]. Comme on peut le voir sur la Figure 1, il est constitué d'une cathode, d'une feuille diélectrique et de deux anodes. Un trou de diamètre Φ MHCD est percé dans la structure composée de la cathode, de la feuille diélectrique et de l'anode 1. L'anode 2 est placée au-dessus de l'anode 1 à une distance d. A noter qu'en l'absence d'anode 2, on obtient la configuration connue sous le nom de MHCD (Micro-Hollow Cathode Discharge) [START_REF] Schoenbach | High-pressure hollow cathode discharges[END_REF].

Lorsqu'une tension DC de quelques centaines de volts est appliquée entre la cathode et l'anode 1 du dispositif, une micro-décharge plasma de type MHCD est créée à l'intérieur du trou de diamètre Φ MHCD . La présence de la feuille diélectrique permet d'éviter l'apparition de cette micro-décharge ailleurs que dans le trou. Son épaisseur h MHCD ainsi que le diamètre du trou Φ MHCD (généralement Φ MHCD = h MHCD ) sont choisis en accord avec la loi de Paschen qui définit une valeur optimale pour le produit entre la pression du gaz P et l'épaisseur h MHCD afin de minimiser la tension de claquage V b pour un gaz donné [10]. Une fois la micro-décharge MHCD allumée, on obtient la transition vers le mode MCSD en étirant la micro-décharge initiale dans l'espace inter-anodes par l'application d'une tension modeste (quelques dizaines de volts) entre l'anode 1 et l'anode 2. La tension requise pour obtenir la transition entre les modes MHCD et MCSD est considérablement réduite vis-à-vis de la tension de claquage nécessaire pour générer une décharge plasma directement entre la cathode et l'anode 2 (la distance h MHCD + d n'est pas optimale au sens de Paschen).

Caractérisation expérimentale de limiteurs de puissance utilisant la MCSD

Plusieurs circuits imprimés micro-ondes ont été caractérisés pour évaluer les performances en termes de protection d'une micro-décharge plasma de type MCSD.

3.a. Dispositif de caractérisation expérimentale

Un dispositif spécifique a été mis en oeuvre pour la caractérisation expérimentale des différents circuits. Comme le montre la Figure 4, le circuit sous test est placé dans une enceinte en Pyrex connectée à trois unités différentes.

La première unité concerne le contrôle dans l'enceinte du type de gaz et de sa pression. Une pompe est tout d'abord utilisée pour faire le vide dans l'enceinte avant l'injection du gaz étudié à une certaine pression P .

La deuxième unité permet la régulation de la puissance électrique utilisée pour pré-ioniser le limiteur de puissance. Une source de tension continue délivre ainsi une tension V DC entre l'anode 1 et la cathode pour allumer la micro-décharge de pré-ionisation. La résistance R = 10 kΩ sert à limiter le courant I DC injecté à la décharge de sorte à éviter la transition du plasma vers l'arc.

Enfin, la troisième unité contient l'instrumentation micro-onde. Une source micro-onde continue, à puissance et fréquence ajustable, est connectée à l'entrée d'un coupleur directionnel, utilisé pour mesurer les puissances incidente et réfléchie au niveau du circuit sous test. La puissance de sortie du circuit sous test est évaluée à partir d'une sonde de mesure de puissance connectée à la sortie du circuit au travers d'un second coupleur directionnel.

En résumé, ce dispositif expérimental permet le contrôle du type de gaz, de sa pression et de la puissance injectée à la décharge de pré-ionisation, tout en effectuant des mesures de puissance micro-onde.

3.b. Ligne de transmission microruban

Le premier circuit étudié est une ligne de transmission en technologie microruban d'impédance caractéristique Z 0 = 50 Ω. Cette ligne de transmission de largeur W microruban = 1, 8 mm est imprimée sur un substrat de constante diélectrique ǫ r = 3, 55 et d'épaisseur h microruban = 0, 813 mm. Le trou cylindrique est placé au centre de la ligne avec comme dimensions Φ microruban = 1 mm, h MHCD = 1, 52 mm et Φ MHCD = 1, 5 mm. La caractérisation expérimentale de la loi de Paschen pour cette géométrie donne une pression optimale dans l'Argon de 10 torr, soit 1,33 kPa. La tension de claquage V b pour la micro-décharge de pré-ionisation est alors minimale et égale à 335 V.

Cette ligne de transmission a été caractérisée à l'aide du dispositif expérimental de la Figure 4. Un courant de pré-ionisation I DC de 2,5 mA a été mis en place avec de l'Argon à 10 torr. Une fois allumée, la micro-décharge de pré-ionisation requiert une tension de 254 V entre la cathode et l'anode 1 pour être maintenue. La consommation de puissance est donc de 635 mW pour la micro-décharge et 62,5 mW pour la résistance R du circuit de commande. La Figure 5 présente les puissances transmise et réfléchie en fonction de la puissance incidente pour cette ligne de transmission dans de l'Argon à 10 torr. Sans préionisation, et pour cette gamme de puissance, on observe un comportement linéaire entre les puissances transmise et réfléchie et la puissance incidente. En revanche, lorsqu'un courant de pré-ionisation I DC de 2,5 mA est considéré, on note un comportement non-linéaire du dispositif lorsque la puissance incidente dépasse la valeur seuil de 29 dBm. Dans ce cas, la micro-décharge plasma de préionisation est tirée dans le trou de la section microruban et le limiteur est à l'état "on". Pour cet exemple, l'excursion en puissance est réduite car une décharge plasma est créée au niveau du connecteur SMA de la ligne de transmission dès que la puissance incidente dépasse 31 dBm.

3.c. Filtre passe-bande microruban

Comme le monte la Figure 6, le second circuit imprimé étudié est un filtre passe-bande imprimé sur un substrat d'épaisseur h microruban = 0, 813 mm. Ce filtre consiste Les paramètres S simulés et mesurés de ce filtre sont présentés sur la Figure 7. La fréquence de résonance mesurée est égale à 2,415 GHz. Tant que le courant de préionisation reste inférieur à 2,5 mA, on ne note pas de changement sur les paramètres S du filtre lorsque la microdécharge de pré-ionisation est allumée. Les pertes d'insertion apportées par le dispositif sont alors strictement nulles. Ceci reste vrai tant que la puissance micro-onde pour la caractérisation des paramètres S reste faible de sorte à ne pas enclencher le mode limiteur de puissance.

Ce filtre a été caractérisé à l'aide du dispositif expérimental de la Figure 4 à la fréquence de 2,415 GHz et dans les mêmes conditions que celles utilisées pour l'exemple de la Section 3.b. On observe sur la Figure 8 une limitation de la puissance transmise lorsque la puissance incidente dépasse 21 dBm dans le cas d'un courant de pré-ionisation de 2,5 mA. Cette valeur seuil est largement réduite par rapport au cas de la ligne de transmission du fait de l'intensification du champ électrique dans le trou à la résonance. Ce limiteur de puissance se comporte alors comme un limiteur réflectif qui limite la puissance transmise en sortie à 18 dBm lorsque la puissance incidente dépasse 21 dBm.

Conclusion

La faisabilité de circuits imprimés intégrant une microdécharge plasma comme élément limiteur de puissance a été démontrée expérimentalement. Ce nouveau concept de limiteur actif sans pertes d'insertion pourrait permettre la protection des récepteurs micro-ondes contre des agressions HPM sans dégradation de leurs performances.

Figure 1 .

 1 Figure 1. Configuration générale d'une MCSD.

Figure 2 .

 2 Figure 2. Schéma d'un circuit imprimé intégrant une MCSD : (a) géométrie 3D et (b) vue de coupe.

  2.b. Intégration de la MCSD dans un circuit impriméComme le montre la Figure2, un circuit imprimé micro-onde classique nécessite peu de modifications pour accueillir une MCSD. Ainsi, une cathode et une feuille diélectrique sont ajoutées sous le plan de masse du circuit imprimé, et un trou cylindrique est percé sur toute la hauteur de l'empilement final avec un diamètre qui peut différer selon la couche considérée. Ce trou cylindrique a pour vocation à accueillir la micro-décharge plasma. Sa position correspond à une zone du circuit imprimé où le champ électrique micro-onde généré est maximal afin de minimiser au final le seuil de protection du limiteur. Le diamètre Φ microruban est petit devant la longueur d'onde de sorte que sa présence n'affecte pas le comportement nominal du circuit imprimé. Les dimensions de la partie inférieure Φ MHCD et h MHCD sont optimisées pour minimiser V b pour une pression de gaz donnée. Cette configuration peut être vue comme une MCSD où le plan de masse et le conducteur principal du circuit imprimé font respectivement office d'anodes 1 et 2.2.c. Fonctionnement du limiteur de puissancePour obtenir le comportement de limiteur de puissance du dispositif, une phase de pré-ionisation est requise. Elle consiste à allumer une micro-décharge MHCD à l'intérieur du trou de diamètre Φ MHCD sous le plan de masse du circuit imprimé. Pour cela, on applique entre la cathode et le plan de masse du circuit imprimé une tension continue supérieure à la tension de claquage V b du gaz. Pour des courants I DC faibles, la MHCD reste localisée sous le plan de masse du circuit imprimé, limitant ainsi son interaction avec le circuit.

Figure 3 .

 3 Figure 3. Fonctionnement d'un circuit imprimé intégrant une MCSD comme limiteur de puissance : (a) vue de coupe du limiteur à l'état "off" et (b) vue de coupe du limiteur à l'état "on".

Figure 4 .

 4 Figure 4. Dispositif expérimental.

Figure 5 .

 5 Figure 5. Bilan de puissance mesuré à 2.45 GHz pour la ligne de transmission sans et avec micro-décharge de pré-ionisation dans de l'Argon à 10 torr.

Figure 6 .

 6 Figure 6. Géométrie du filtre passe-bande microruban.

Figure 7 .

 7 Figure 7. Paramètres S simulés et mesurés pour le filtre passe-bande microruban.