Joël Lachambre

Julien R Éthor Éa

Arnaud Weck

Jean-Yves Buffiere
email: jean-yves.buffiere@insa-lyon.fr

Extraction of stress intensity factors for 3D small fatigue cracks using digital volume correlation and x-ray tomography

Keywords: X-ray tomography, Digital Volume Correlation, Fatigue crack, In situ, Cast iron, 3D

This paper describes a methodology used to compute Stress Intensity Factor values along the curved front of a fatigue crack inside a nodular cast iron. An artificial defect is introduced at the surface of a small sample. The initiation and growth of a fatigue crack from this defect during constant amplitude cycling is monitored in situ by laboratory x-ray tomography. The method for processing the 3D images in order to compute SIF values is described in detail. The results obtained show variations of the stress intensity factor values along the crack front.

Introduction

In the last decade, in the transport industry, a large research effort has been devoted to the reduction of the structural weight of vehicles in order to reduce fuel consumption and comply with tougher environmental regulation. Globally, this has lead to a more severe use of materials which are submitted to higher mechanical loads than in the past. For theoretical models which aim at predicting the durability of materials this is a challenging evolution. In the field of cyclic mechanical behaviour (fatigue), the Paris law established in the sixties is widely used to model fatigue life of components based on propagation laws.

Such laws are established with standard test procedures which monitor crack propagation rates of a relatively long (generally a few millimetres) through crack which front is assumed to be straight. In practice, however, it might take a very substantial part of the fatigue life of a component for it to contain a fatigue crack comparable in size and shape. While the size issue can be addressed via the similarity principle (up to some extent, see for example (author?) [START_REF] Schijve | Fatigue of Structures and Materials[END_REF] for a discussion on that topic), the effect of the 3D shape of the crack and the variation of stress intensity factor values that it implies is generally neglected.

To establish crack propagation laws of fatigue cracks with a curved front in opaque materials, several techniques have been proposed which are either direct (beach marking) or indirect (compliance measurements) and which all suffer from various drawbacks (see ref [START_REF] Ferrie | Fatigue crack propagation: In situ visualization using X-ray microtomography and 3D simulation using the extended finite element method[END_REF] for a discussion). The availability of powerful x-ray beams which can go through several millimetres of metals, has made it possible to analyse directly the mechanics of crack tips in the interior of an opaque material either by direct imaging or by analysing diffraction patterns or by combining both signals [START_REF] Withers | 3D Crack-tip Microscopy: Illuminating Micro-Scale Effects on Crack-Tip Behavior[END_REF].

The emergence of digital correlation methods, on the other hand, gives access to crack tip displacement fields from which it is possible to extract Stress Intensity Factor (SIF) values. This has been done at the surface [START_REF] Hamam | Stress Intensity Factor Gauging by Digital Image Correlation: Application in Cyclic Fatigue[END_REF][START_REF] Withers | Strain measurement by digital image correlation[END_REF][START_REF] Nowelll | Optical methods for measurement of fatigue crack closure: moiré interferometry and digital image correlation[END_REF][START_REF] Yusof | Effect of overload on crack closure in thick and thin specimens via digital image correlation[END_REF] and also in the bulk [START_REF] Toda | A 3D measurement procedure for internal local crack driving forces via synchrotron X-ray microtomography[END_REF][START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using 3d correlation of laboratory x-ray microtomography images[END_REF][START_REF] Limodin | Influence of closure on the 3d propagation of fatigue cracks in a nodular cast iron investigated by x-ray tomography and 3d volume correlation[END_REF] of cracked samples. In the latter case, the images used are obtained by x-ray tomography and the need for a high spatial resolution restricts the size of the studied samples to a few millimetres. In practice, the samples studied in 3D so far all contain crack tips extracted from larger coupons. The stress state at the tip of those cracks is therefore largely unknown because it is inherited from a larger crack and also because it is likely to have been affected by the machining of the small sample.

This paper describes a methodology used to overcome these issues. An artificial defect (notch) is introduced at the surface of a miniature metallic fatigue sample. The initiation and growth of a fatigue crack from this defect during constant amplitude cycling is monitored in situ by laboratory x-ray tomography. The processing of the 3D images required to compute SIF values from the recorded 3D images is described in detail. The potential of the method for computing SIF values along the front of curved crack fronts is demonstrated.

Materials and experimental methods

Material properties

The material used in this study is a nodular graphite cast iron (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) with a homogeneous dispersion of graphite nodules which are used as natural markers for Digital Volume Correlation (DVC) [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using three-dimensional correlation of laboratory Xray microtomography images[END_REF]. Casting and appropriate heat treatments produce a ferritic matrix microstructure (grain size ∼ 50µm) with a 14% volume fraction of graphite nodules (average diameter 45 µm). The static and cyclic mechanical properties of the material are listed in Table 1 The aim of this work is to characterise cracks initiated and grown in situ. In this material, within smooth samples, fatigue cracks usually initiate on pores which are produced by the solidification process. Such defects are relatively rare. For various reasons [START_REF] Buffiere | Three-dimensional visualisation of fatigue cracks in metals using high resolution synchrotron X-ray micro-tomography[END_REF], the size of the fatigue samples characterised by tomography samples has to be kept small (e.g. section 1.6 × 1.6 mm 2 , gage length ∼ 5mm in this work). The probability to find such a casting defect in a sample extracted randomly from a bar is, hence, very low. An alternative solution has been adopted: cracks are initiated from well controlled artificial sharp notches. In a previous work, FIB machining had been used to produce such notches in an Al sample [START_REF] Ferrie | Fatigue crack propagation: In situ visualization using X-ray microtomography and 3D simulation using the extended finite element method[END_REF]. For the material studied here, however, the notches that could be produced with this technique in a reasonable time (a few hours) were too shallow (maximum depth ∼ 20µm) to act as an efficient initiation site. The notches were then produced by laser machining [START_REF] Weck | Femtosecond laser-based fabrication of a new model material to study fracture[END_REF] using a 1 kHz repetition rate regeneratively amplified Ti:Sapphire laser system employing chirped pulse amplification. A pulse length (full-width half-maximum) of approximately 40 fs is measured using an auto-correlator. A femtosecond laser is chosen as it produces a negligible heat affected zone around the notch due to the high intensity of the laser pulse which directly vaporizes the material. For all experiments presented here, rotating polarization obtained with a spinning zero order half wave plate was used. The beam is focused with a 10x microscope objective using an energy per pulse of 50 µJ. Cast iron samples are placed on a computer controlled x-y translation stage for precise sample positioning under the laser beam. The notches are produced by translating the stage back and forth under the laser beam until the desired depth is obtained.

The results shown in this paper have been obtained with a notch located at the centre of one of the lateral surface of the sample (surface crack) 2 . The notch is perpendicular to the applied cyclic stress and has approximately a square shape 180 × 180 µm 2 (see figure 1) with an initial opening of 15µm and a tip radius (both measured by optical inspection at the sample surface) of about 7µm.

X-ray tomography

A laboratory tomograph has been used. The X-ray source is operated at an accelerating voltage of 85 kV with a transmission tungsten target. The minimum source/detector distance imposed by the machine (see below) is 15 mm resulting in a voxel size of 2.5 µm. A current of 174 µA and a mean of 3 radiographs with an exposure time of 1 second have been used; 720 angular positions have been taken which corresponded to a scan duration of 45 minutes. The reconstruction time (weighted filtered back projection, Feldkamp algorithm) on a Desktop computer with a bi Intel xeon quad core processor, 3 GHz CPU, 64 GBytes RAM, takes approximately 15 minutes for a 560 × 560 × 900 voxels 3D image.

In situ cycling

Cyclic loading of the sample is performed in situ using a miniature hydraulic fatigue machine [START_REF] Lachambre | 3D Analysis of a Fatigue Crack in Cast Iron Using Digital Volume Correlation of X-ray Tomographic Images[END_REF]. In order to decrease the voxel size in the reconstructed 3D images, the sample must be as close as possible to the laboratory X-ray source (divergent beam configuration), thus the sample is placed into a long slender aluminium tube which diameter and thickness have to be kept as small as possible to be transparent to X-rays. In this work the diameter of the tube (Al 2024 alloy) is 16 mm and the thickness 1 mm [START_REF] Buffiere | In Situ Experiments with X ray Tomography: an Attractive Tool for Experimental Mechanics[END_REF].

As the contrast of the crack on the radiographs is blurred by the presence of numerous graphite nodules, crack growth in the interior of the sample is assessed by recording a tomographic scan. The crack shape evolution is studied through the residual of the DVC which correspond to the difference between the reference and the deformed and corrected volumes as described in [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using three-dimensional correlation of laboratory Xray microtomography images[END_REF].

Once the sample is set up in the loading stage, two scans are recorded at low (40 MPa) and high stress (290 MPa) to characterise the initial state. The sample is then cycled in situ at a constant stress amplitude of 300 MPa (R=0.1). One hundred thousand cycles are applied prior to the first recorded scan. Then 2 scans are recorded every 15 kcycles, one at 40 MPa and another at 290 MPa, to monitor the crack evolution as described in [START_REF] Limodin | Influence of closure on the 3D propagation of fatigue cracks in a nodular cast iron investigated by X-ray tomography and 3D volume correlation[END_REF]. Because fatigue cycling is performed at constant stress amplitude, the crack accelerates during propagation; near the end of the experiment, smaller steps are used between two scans according to the approximate crack speed da dN observed on the residuals.

Digital volume correlation

Optical flow equation and resolution

The optical flow equation assumes that the grey level variations are due to the displacement of the material points only. Its generic form is written as

f (X) = g(X + u(X)). (1)
In this equation, X is the vector denoting the position of the current point with respect to the image or volume coordinate system and u the unknown displacement vector field. f and g, are the reference and deformed grey level functions representing the image or volume of the sample. Once a basis functions has been chosen to describe the variations of the displacement field in space

u(X) = i∈[1,N ] φ i u i = [φ]{U}, (2)
a non-linear least-squares resolution strategy is adopted. The unknow vector U is then obtained as a minimizer of

η 2 = ROI (f (X) -g(X + [φ]{U}))dX. ( 3
)
The minimization of η 2 results in the resolution of a series of linear systems until convergence is reached. More details on the resolution strategy can be found in the references listed in the next sections.

Displacement measurements

For the estimation of the displacement field, a finite element basis function is adopted as proposed in (author?) [START_REF] Besnard | finite-element' displacement fields analysis from digital images: Application to portevin-le châtelier bands[END_REF] for the 2D case and in (author?) [START_REF] Roux | Three-dimensional image correlation from x-ray computed tomography of solid foam[END_REF] for the 3D case. This basis function is enriched in the spirit of the eXtended Finite Element Method [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF] by using discontinuous enrichment functions. Discontinuous enrichment functions of the Heaviside type are adopted to measure the displacement discontinuity across the crack faces as in (author?) [START_REF] Réthoré | Extended digital image correlation with crack shape optimization[END_REF] which is dedicated to 2D measurements. In the present paper, this strategy is extended in 3D.

Due to the noise in the data acquisition chain (X-ray source, scintillator, CCD) and to the ill-posed nature of the problem (a vector field is searched for whereas only a scalar equation can be written), resolution/uncertainty compromise must be solved. Previous studies have shown that the uncertainty which alter the measured displacements increases when the element size decreases [START_REF] Besnard | finite-element' displacement fields analysis from digital images: Application to portevin-le châtelier bands[END_REF][START_REF] Roux | Three-dimensional image correlation from x-ray computed tomography of solid foam[END_REF][START_REF] Réthoré | Extended digital image correlation with crack shape optimization[END_REF][START_REF] Limodin | Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory x-ray source[END_REF]. Thus, when trying to capture crack tip field, which exhibits strong variations i.e. , requires high spatial resolution, small element size is needed. Without any further improvement, the required element size would lead to very high noise level. To circumvent this drawback, a regularization is added to the DVC equation. For the analysis of a fatigue cycle, the small scale yielding assumption is fulfilled and an elastic behaviour of the material can be adopted within a loading cycle. Based on the work by (author?) [START_REF] Réthoré | An extended and integrated digital image correlation technique applied to the analysis of fractured samples[END_REF] in 2D, an elastic regularization is added to the minimization problem. In the formalism then introduced by (author?) [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Leclerc | Digital volume correlation: what are the limits to the spatial resolution?[END_REF], a cut-off wave length c below which the variations of the displacement field are governed by linear elasticity, is introduced. This regularization uses the equilibrium gap method and it enforces that the linear elastic balance of momentum equations are satisfied. Using this strategy, and contrary for example to particle tracking techniques, there is no need to have a sufficiently high marker density. The weakness of the image texture is compensated for by the mechanical regularization that enforces smoothness and elastic balance of momentum constraint. The regularization includes the traction free boundary condition on the crack faces and also on the sample surface. This kind of boundary conditions holds meaningful mechanical information and it is thus important for the finite element mesh to incorporate both an accurate description of the crack faces and of the sample surface. For these purposes, the DVC formalism is also used to extract an accurate geometrical description of the sample boundaries and of the crack surface.

Contour and crack detection

From the acquired grey level volumes, the optical flow equation is used to compute the 3D displacement field between two scans. As the sub-surface depth of the crack analysed in this work remains small (∼ 500µm), displacement fields must be computed very close to the sample surface, specially in the first stages of growth. Because an elastic reularization is used, the description of stress free surfaces must also be accurate. This is achieved by using the optical flow equation as explained in the following paragraphs. A similar strategy is also used for determining the crack surface.

The detection of these surface relies on the same problem which is to extract the silhouette of an object in a 3D image. Due to the graphite nodules, a direct thresholding of the x-ray scans is neither adapted for extracting the crack surface nor for meshing the sample surface. For the crack surface, the correlation residual (a voxel reconstruction of the optical flow residual) can be used as proposed in (author?) [START_REF] Réthoré | Extended digital image correlation with crack shape optimization[END_REF] or (author?) [START_REF] Rannou | Three dimensional experimental and numerical analysis of a fatigue crack[END_REF]. However due to the small voxel size used, direct thresholding of this residual volume is again not accurate enough. A methodology based on the optical flow equation is used as follows. Inspired by (author?) [START_REF] François | Identification of the shape of curvilinear beams and fibers[END_REF] and (author?) [START_REF] Réthoré | Curve and boundaries measurement using bsplines and virtual images[END_REF], the conservation of grey levels is written between a virtual scan f * and the reference one f .

f * (X) = f (X + s(X)) + b(X). ( 4
)
f * is a virtual volume that describes a silhouette of simple geometrical shape with an ideal grey level transition between black and white. b is a grey level function used to account for the lower spectrum of the actual scan. s is the displacement of the virtual object surface so that the actual reference scans f at the modified surface location matches the virtual scan. For the sample surface detection, the virtual volume shape is a cylinder and for the crack surface, a virtual plane is used. The variations of s are described using B-splines functions and the same non-linear least-squares minimization as for the displacement measurement is used.

Stress intensity factors calculation

In previous 2D analyses [START_REF] Mcneill | Estimation of stress intensity factor by digital image correlation[END_REF][START_REF] Abanto-Bueno | Investigation of crack growth in functionally graded materials using digital image correlation[END_REF][START_REF] Yoneyama | Automatic evaluation of mixedmode stress intensity factors utilizing digital image correlation[END_REF][START_REF] Hammam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF], Williams' series [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF] were extensively used to extract stress intensity factors and crack tip position directly from digital images [START_REF] Roux | Stress intensity factor measurement from digital image correlation: post-processing and integrated approaches[END_REF] or by least-squares projection [START_REF] Réthoré | Optimal and noise-robust extraction of fracture mechanics parameters from kinematic measurements[END_REF] of the displacement field. This strategy has also been used in 3D for through cracks with nearly straight crack fronts by (author?) [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using 3d correlation of laboratory x-ray microtomography images[END_REF][START_REF] Limodin | Influence of closure on the 3d propagation of fatigue cracks in a nodular cast iron investigated by x-ray tomography and 3d volume correlation[END_REF]. In the present experiment where curvilinear crack fronts are studied, the original methodology needs some improvements.

Williams' series

Considering an infinite homogeneous elastic solid with isotropic behaviour, Williams [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF] has obtained the in-plane and out-of-plane solution for the stress distribution produced by a semi-infinite crack. The solution is decomposed over a series of displacement functions,indexed by n for each mode m, u n m and v n m , along the crack direction and orthogonal to the crack direction respectively. These in-plane fields for mode I and mode II are written in a polar coordinate system (r, θ) as

u n I (r, θ) = r n/2 κ cos nθ 2 - n 2 cos (4 -n)θ 2 + n 2 + (-1) n cos nθ 2 (5)
v n I (r, θ) = r n/2 κ sin nθ 2 - n 2 sin (4 -n)θ 2 - n 2 + (-1) n sin nθ 2 (6)
and

u n II (r, θ) = -r n/2 κ sin nθ 2 + n 2 sin (4 -n)θ 2 + n 2 -(-1) n sin nθ 2 (7) v n II (r, θ) = r n/2 κ cos nθ 2 + n 2 cos (4 -n)θ 2 - n 2 -(-1) n cos nθ 2 . (8)
For mode III, only the out-of-plane displacement w n III is described:

w n III (r, θ) = r n/2 cos(n(θ -π)/2). (9)
Based on a linearization of the perturbation due to a mispositioning of the crack tip, it was shown by (author?) [START_REF] Roux | Stress intensity factor measurement from digital image correlation: post-processing and integrated approaches[END_REF] that the crack tip position can be estimated by using the first mode I super-singular field (n = -1). An iterative search is performed so that the position of the equivalent elastic crack tip, i.e. corresponding to a vanishing amplitude of the super-singular term, is progressively reached. This iterative search is initialized with a semi-elliptical crack front extracted from the DVC residuals as shown in Figure 4d. Then at each iteration of the procedure, the displacement interpolated on circular projection domains is projected onto a basis containing displacement functions for n varying from -3 to 9 for the 3 modes. As long as the equivalent elastic crack tip position is not reached, for a given position along the crack front, the first mode I super singular amplitude is not vanishing and the stress intensity factor estimation given by n = 1 amplitudes is not reliable. Once a sufficiently small shift (typically less than a tenth of voxel) to the equivalent elastic crack tip is estimated, then the procedure is stopped and the stress intensity factors values and crack tip positions are stored for this position along the initial estimate of the crack front.

Adaptation to 3D crack front

First, the projection onto the Williams series is performed over circular domains which orientation depends on the local normal to the crack surface and to the crack front. As illustrated on Figure 2, the projection domain is adapted to the local crack geometry in order to reduce the approximation due to a misalignement of the local crack tip coordinate system with the projection plane. Note that during the iterative search of the crack tip position for a given position along the front, the projection domain is moved along the local normal to the initial semi-elliptical crack front. The second improvement consists in using the modification suggested by (author?) [START_REF] Leblond | The stress field near the front of an arbitrarily shaped crack in a three-dimensional elastic body[END_REF] to account for crack curvature. It is shown in [START_REF] Leblond | The stress field near the front of an arbitrarily shaped crack in a three-dimensional elastic body[END_REF] that the expression of the stress field in the crack front vicinity is modified when considering the perturbation due to locally curved crack morphology. From these corrected expressions of the stress field, a corrected displacement basis is obtained. This corrected basis is then used to extract the stress intensity factors and crack tip position from the measured displacement field by a least-square projection. The method has been validated by tracking curved crack fronts within displacement fields obtained from linear elastic finite element simulations. Robust and accurate detection is obtained when such noise free displacement fields are used.

Results and analysis

In this section, a shallow surface crack of maximum depth around 500 µm is analysed for three different numbers of fatigue cycles (120, 140 and 160 kcycles).

Figure 3 shows a displacement field obtained using the proposed methodology. Note that in order to avoid the elastic regularization prescribing the crack front position, a small cylindrical region over the crack is front is excluded from this regularization for which the cut-off wave length was set to 256 voxels. The mesh size in the vicinity of the crack front is 8 voxels.

As mentioned before, elastic regularization of the DVC problem requires an accurate description of the crack and sample surfaces; Figure 4 shows the results obtained using the method described in section 3.3 for the crack surface after 160 000 fatigue cycles. It can be seen on this figure that the method used gives a good description of the slightly curved sample surface and of the crack surface topography. About the latter, the method used to extract the crack surface tends to smooth the crack deflections which are usually observed in this type of material when graphite nodules are encountered by the crack tip [START_REF] Cocco | Graphite nodules and fatigue crack propagation micromechanisms in a ferritic ductile cast iron[END_REF]. However as shown on Fig. 4c, some roughness of the crack surface with a typical range of ±12 voxel i.e. ±30 µm is detected; this value is close to the graphite nodules average size.

The detection of the crack front using the method described in section 4.1 is illustrated in figure 5 for the crack observed after 160 000 fatigue cycles. The corresponding distribution of K I values along the crack front is shown in figure 6 for three different crack positions corresponding to different numbers of fatigue cycles. These results have been obtained by projecting the displacement field measured between two scans acquired at the minimum and maximum loads of the same loading cycle. The extraction is performed using discs of 100 voxels in diameter and Williams functions for n from -3 to 9. The region in the vicinity of the crack front which is not submitted to elastic regularization is not used for the extraction as well as a band of 40 voxels along the crack faces.

To the author's knowledge, this is the first time that K values are extracted by from experimentally measured displacements along the front of semi elliptical low fatigue cracks initiated and propagating in situ from a notch in an optically opaque material. Compared to previous results published in the literature, in the same material [START_REF] Limodin | Analysis and Artifact Correction for Volume Correlation Measurements Using Tomographic Images from a Laboratory X-ray Source[END_REF][START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using three-dimensional correlation of laboratory Xray microtomography images[END_REF] or in Al alloys [START_REF] Toda | A 3D measurement procedure for internal local crack driving forces via synchrotron X-ray microtomography[END_REF][START_REF] Zhang | Three-dimensional fatigue crack growth behavior in an aluminum alloy investigated with in situ high-resolution synchrotron X-ray microtomography[END_REF] studied by particle tracking, the cracks studied here have built their own plastic zone. This one has not been altered by the preparation of the sample from a larger specimen. If the displacement fields studied in this work are therefore more realistic, the orders of magnitude of the crack openings are however at the limit of what can be achieved by the correlation method (see for example the much larger values of crack opening values in [START_REF] Limodin | Influence of closure on the 3D propagation of fatigue cracks in a nodular cast iron investigated by X-ray tomography and 3D volume correlation[END_REF] for a comparison) which has to be "assisted" by the introduction of an elastic regularization. At the intersection between the crack front and the specimen surface, the results given by the method might be considered with some care due to lower robustness of the crack front detection. This can be due to the use of the Williams' series that may not be appropriate for describing the mechanical state in the vicinity of the crack vertex.

As mentioned before there is no equivalent experimental data available to compare with our results. X-ray diffraction which has been used by several authors for studying crack tip displacements [START_REF] Steuwer | In situ analysis of cracks in structural materials using synchrotron X-ray tomography and diffraction[END_REF][START_REF] Croft | Fatigue history and in-situ loading studies of the overload effect using high resolution X-ray strain profiling[END_REF] in the bulk of metals for through cracks does not have the necessary spatial resolution for the curved crack fronts of the short cracks studied here. One can however compare the experimental SIF curves of figure 6 with calculations either analytical or making use of the finite element (FE) method. We have shown elsewhere [START_REF] Lachambre | Développement d'une méthode de caractérisation 3d des fissures de fatigue à l'aide de la corrélation d'images numériques obtenues par tomographie x[END_REF] that the Raju and Newman analytical approximation [START_REF] Raju | Stress intensity factors for a wide range of semielliptical surface cracks in finite-thickness plates[END_REF] gives SIF values which are very close to FE calculations (less than 5% discrepancy) even in the case of a curved crack front showing some irregularities (elastic calculation), therefore in this paper our experimental results are only compared with the analytical calculation (figure 6).

This comparison is qualitatively satisfying: the maximum (resp. minimum) SIF values are measured at the surface (resp. in the bulk) of the sample (140 and 160 kcycles curves) in agreement with the trend observed on the analytical curves. The latter being a good average of the measured ones for the three crack fronts investigated. When looking at the variation of the SIF values along the crack fronts, it can be seen that the range of variation between the maximum (surface) and minimum (bulk) values is much larger for the experimental curves (∼ 9 MPa √ m) than for the analytical ones (∼ 1 MPa √ m). The size of the cracks detected by the Williams series at the surface (2c) is smaller than the size of the crack detected in the bulk (a), this gives the characteristics U shape of the K I curve shown on figure 6. In other words, the crack front is lagging behind at the surface. This so-called tunnelling effect which is well known for fatigue cracks in macroscopic CT specimens cycled with a load ratio close to 0.1 is usually attributed to a larger closure effect at the surface (plane stress -larger plastic zone) [START_REF] Schijve | Fatigue of Structures and Materials[END_REF]. It has also been observed for elliptical cracks [START_REF] Schijve | Fatigue of Structures and Materials[END_REF] and corner cracks [START_REF] Tong | T-stress and its implications for crack growth[END_REF] and even in the case of small (millimetric) tomographic samples [START_REF] Ferrie | Fatigue crack propagation: In situ visualization using X-ray microtomography and 3D simulation using the extended finite element method[END_REF].

The values of K shown on figure 6 are close to the typical value of K op which can be found in the literature for the cast iron studied (between 4 and 6 MPa √ m [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using three-dimensional correlation of laboratory Xray microtomography images[END_REF]) and therefore such closure effects are likely to happen. The fact that, in nodular cast iron, closure plays a role during the propagation of such very small cracks had already been suggested by Clément et al. [START_REF] Clement | Short crack behaviour in nodular cast iron[END_REF]. Our results tend to show, in addition, that the level of closure might vary along the front of the crack.

Finally, figure 7 shows the comparison the experimental crack growth rates measured in this work with some literature data on the same material [START_REF] Nadot | A study of natural cracks initiated on casting defects by crack front marking[END_REF]. The points obtained at the surface and in the bulk in this work fall in the characteristic short crack zone of the Paris curve; the size of the crack at 160 000 cycles being probably at the transition between the short and long crack regime. This behaviour is again similar to that observed by by Clément et al. [START_REF] Clement | Short crack behaviour in nodular cast iron[END_REF] for artificial cracks of comparable sizes. Besides, the points measured in this work at the surface tend to fall closer to the nominal curve while bulk measurements are closer to the effective curve. This is in agreement with the above mentioned argument of a lower level of closure in the bulk.

Compared to previously published results [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using three-dimensional correlation of laboratory Xray microtomography images[END_REF][START_REF] Limodin | Influence of closure on the 3D propagation of fatigue cracks in a nodular cast iron investigated by X-ray tomography and 3D volume correlation[END_REF], the experimental setup has been improved in order to avoid the effect of an initial unknown plastic state. One drawback, however, is that x-ray tomography limits the size of the sample. Therefore the physical size of the crack is obviously limited. As a result, graphite nodules that are used for supporting the DVC analysis introduce strong heterogeneities in the local material properties resulting in strong interactions between crack propagation and the material microstructure. This can be seen for example on figure 5 where the irregularities of the crack front detected by the Williams series can be correlated with the presence of a porosity and /or nodules. As discussed above, the SIF values which were extracted from the measured displacement fields are nevertheless coherent with simulations and results on large samples. It can be expected, however, that a material with smaller microstructural markers (but large enough to be imaged by the tomography) would increase the quality of the results obtained.

Conclusions

Laboratory X-ray tomography has been used to monitor the initiation and growth of a fatigue crack in a nodular cast iron sample. SIF values along the curved crack front were extracted from displacement fields by DVC. Compared to previous work where the tips of through cracks coming from larger coupons were examined, this method has the advantage producing a crack with its "own" plastic zone/stress field . A laser machined notch is used to initiate in situ the small fatigue crack (physical size of the order of 500 µm). Some tools have been developed in 3D to map with a reasonable accuracy the sample and crack surfaces automatically. The small size of the studied crack (< 500µm) and the low values of the crack opening which had to be measured are at the limit of of what can be detected with classical correlation methods for describing crack tip displacement fields with a high enough spatial resolution. This problem was solved by assisting correlation with a mechanical regularisation. The SIF values obtained are in agreement with analytical calculations and reveal a typical short crack behaviour. Larger values of SIF were found close to the sample surface; this was attributed to a higher level of closure compared to the bulk of the specimen. The method presented here is limited by the relatively large size of the nodules, which act as microstructural markers for the correlation, compared to the physically small size of the crack.

Figure 1 :

 1 Figure 1: a) Schematic drawing of the fatigue sample showing the dog bone shape of the sample used and the location of the notch introduced in the square cross-section b) SEM observation of the fracture surface; the white striped area underlines the approximately square shape of the artificial defect introduced by laser machining.

Figure 2 :

 2 Figure 2: Schematic 3D rendering of the planes used for the projection of the displacement onto the Williams' series. Red lines (perpendicular to the crack surface) and blue lines (perpendicular to the crack front) show the local coordinate frame along the crack front.

Figure 3 :

 3 Figure 3: a)Example of a displacement field for a surface crack after 160 000 fatigue cycles (Mode I opening -stress direction along z). The shape of the crack in the bulk of the sample is also shown. The "smooth" part of the field corresponds to the part of the sample which is regularized whereas displacements within the "speckled" regions close to the crack tips at the sample surface (arrows) are only driven by DVC. b) Map of the crack opening.

Figure 4 :Figure 5 :

 45 Figure 4: 3D renderings of the crack surface extracted from the experimental data. a) and b) show lateral views of the surface while c) is a top view with a color scale (one voxel = 2.5 µm) giving a quantitative idea of the topography. d) shows a top view of the correlation residuals which is a very good approximation of the crack size. On the sub-figures c) and d) the dark elliptical line shows the shape of the crack which is used for initiating the iterative crack tip detection while the nearly horizontal line marks the sample surface. (see the text for details)

22

 22

Figure 6 :

 6 Figure 6: Evolution of the mode I stress intensity factor along the 3D crack front shown in figure 5.

  or dc/dN (mm/c)

Figure 7 :

 7 Figure 7: Paris curve of the studied material (triangles) [45] with the experimental points (circles and squares) obtained in this work .

Table 1 :

 1 . Mechanical properties of the studied material

	Young's Yield Poisson's Fracture	∆K th
	modulus stress (GPa) (MPa)	ratio -	toughness (Nominal, R=0.1) (MPa √ m) (MPa √ m)
	170	300	0.27	80	8

Corner notches have also been produced; the results obtained are similar but not shown here

Acknowledgements

The measurements described here were carried out under ANR funding granted through ANR-09-BLAN-0009-01 (RUPXCUBE Project). The authors would like to acknowledge Dr. M.C.Baïetto, Prof. A.Gravouil, Dr. F.Hild, Mr. C.Roux and Dr. S.Roux for fruitful discussions during the PhD work of Joël Lachambre.


Peter Görner

Xavier Simon

Alexis Boivin

Sébastien Bau

Sampling Efficiency and Performance of Selected Thoracic Aerosol Samplers

Keywords: thoracic aerosol, sampling efficiency, sampler performance, exposure measurement

Measurement of worker exposure to a thoracic health related aerosol fraction is necessary in a number of occupational situations. This is the case of workplaces with atmospheres polluted by fibrous particles, such as cotton dust or asbestos, and by particles inducing irritation or bronchoconstriction such as acid mists or flour dust. Three personal and two static thoracic aerosol samplers were tested under laboratory conditions. Sampling efficiency with respect to particle aerodynamic diameter was measured in a horizontal low wind tunnel and in a vertical calm air chamber. Sampling performance was evaluated against conventional thoracic penetration.

Three of the tested samplers performed well, when sampling the thoracic aerosol at nominal flow rate and two others performed well at optimized flow rate. The limit of flow rate optimization was found, when using cyclone samplers.

INTRODUCTION

Workplace exposure to atmospheric aerosols is usually measured in one of three conventional health-related aerosol fractions (CEN, 1993;[START_REF] Iso | Directive 161of European Commission from17 th December 2009 establishing the third list of[END_REF]ACGIH, 1995). Measurement of the inhalable, thoracic or respirable fraction is necessary depending on the definition of Threshold Limit Value (TLV) for different particulate substances. While inhalable and respirable aerosol measurement is common practice, the thoracic fraction is only rarely measured. The thoracic aerosol fraction was first measured in the US cotton industry to assess the risk of byssinosis [START_REF] Corn | Methods to assess airborne concentrations of cotton dust[END_REF] by using a vertical elutriator operated at 7.4 L.min -1 [START_REF] Lynch | Air sampling for cotton dust[END_REF]. The Workshop on Fiber Toxicology Research Need, held in 1989, recommended "the use of size selective inlets such as thoracic particulate matter for fibrous aerosol sampling" [START_REF] Dement | Overview: Workshop on Fiber toxicology research needs[END_REF]. This recommendation continues to be applied nowadays especially when sampling of asbestos.

The aerosol thoracic fraction is also relevant in the case of substances having a local effect on the conductive respiratory airways, more specifically irritation or bronchoconstriction [START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF], for example acid mists and flour dust [START_REF] Laurière | Physical and biochemical properties of airborne flour particles involved in occupational asthma[END_REF]. In the wake of a number of epidemiological studies [START_REF] Soskolne | Laryngeal cancer and occupational exposure to sulfuric acid[END_REF][START_REF] Soskolne | A populationbased case-control study of occupational exposure to acids and the risk of lung cancer: Evidence of specificity of association[END_REF], it was suggested that the conventional thoracic fraction would best indicate exposure of employees to sulfuric acid [START_REF] Lippmann | Basis for a particle size-selective TLV for sulfuric acid aerosol[END_REF]. In 2009, EU Directive 2009/161/EU defined the indicative Occupational Exposure Limit Value (OELV) for sulfuric acid mist as 0.05 mg.m -3 in the thoracic fraction.

A suitable thoracic aerosol sampler with a sampling efficiency that meets standardized sampling criteria (CEN, 2014) is necessary in the workplace exposure measurement. In the last 20 years, a number of thoracic aerosol samplers have been developed and their efficiencies have been tested using isometric or fibrous aerosols [START_REF] Kauffer | The use of a new static device based on the collection of the thoracic fraction for the assessment of the airborne concentration of asbestos fibres by transmission electron microscopy[END_REF][START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF][START_REF] Maynard | Measurement of aerosol penetration through six personal thoracic samplers under calm air conditions[END_REF][START_REF] Maynard | Thoracic size-selection of fibres: Dependence of penetration on fibre length for five thoracic sampler types[END_REF][START_REF] Jones | Thoracic sizeselective sampling of fibres: Performance of four types of thoracic sampler in laboratory tests[END_REF], while other thoracic aerosol samplers have been recently developed. The aim of this study was to measure in the laboratory sampling efficiency of these samplers and to compare it with conventional thoracic penetration criteria. Bias maps were plotted for a series of particle size distributions likely to occur in the workplace [START_REF] Görner | Industrial aerosol measurement according to the new sampling conventions[END_REF] to evaluate their performance.

MATERIALS AND METHODS

Selected thoracic aerosol samplers

Table 1 lists and details the thoracic aerosol samplers studied along with their respective flow rates. Sampler choice was guided by availability, flow rate, expected efficiency and experience in their use and handling.

Although worker exposure is measured using personal samplers, use of static devices may be preferable in specific cases. Static sampling generally allows implementation of higher sampling flow rates and collection of more material for more accurate analysis. Static sampling is also recommended for pollution mapping studies on industrial sites.

Photograph

In this study, the CATHIA-T is considered as the reference sampler what is based on its known sampling efficiency [START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF][START_REF] Maynard | Measurement of aerosol penetration through six personal thoracic samplers under calm air conditions[END_REF][START_REF] Maynard | Thoracic size-selection of fibres: Dependence of penetration on fibre length for five thoracic sampler types[END_REF][START_REF] Jones | Thoracic sizeselective sampling of fibres: Performance of four types of thoracic sampler in laboratory tests[END_REF]. This enables the reliability of our experimental methodology to be checked.

Experimental methodology

Experimental aerosol sampling efficiency as a function of aerodynamic diameter is usually measured in dust chambers or wind tunnels, to which monodisperse or polydisperse aerosol generators are fitted. Airborne particles are sampled by a reference probe and a candidate sampler. For a given particle size D ae , the ratio of concentrations sampled by the candidate C c (D ae ) to those sampled by reference probe C r (D ae ) provides the aerosol sampling efficiency E(D ae ). Thus

E(D ae ) = C c (D ae ) / C r (D ae )
The reference probe sampling efficiency is assumed to be 100% within the range of particle diameters investigated. The experimental efficiency of the candidate sampler is compared with the conventional thoracic efficiency curve (CEN, 1993;[START_REF] Iso | Directive 161of European Commission from17 th December 2009 establishing the third list of[END_REF]ACGIH, 1995) defined by the probability of particle penetration beyond the larynx. This probability is expressed by mathematical relation provided by [START_REF] Soderholm | Proposed international conventions for particle size-selective sampling[END_REF] and commented by [START_REF] Görner | Industrial aerosol measurement according to the new sampling conventions[END_REF].

The experimental method used in this study has been described in detail and qualified in our earlier publications. We used two experimental set-ups: a horizontal low velocity (0.15 m.s -1 ) wind tunnel [START_REF] Fabriès | Realization of a dust tunnel -Response of some air sampling instruments used in industrial hygiene[END_REF][START_REF] Fabriès | Equipment for the study of sampling instruments with real time measurement of the aerosol concentration[END_REF][START_REF] Görner | Study of fifteen respirable aerosol samplers used in occupational hygiene[END_REF] and a vertical dust chamber [START_REF] Görner | Aerosol sampling by annular aspiration slots[END_REF][START_REF] Görner | Laboratory study of selected personal inhalable aerosol samplers[END_REF]. The horizontal wind tunnel is approximately 10 m in length. Within this tunnel, the sampler is placed in the test zone corresponding to a 1 m  1 m crosssectioned area located around the middle of the tunnel. The flow is stabilized between the aerosol generation and the test area by means of a 6-mm honeycomb mesh completed by a 0.75-mm grid mesh at its outlet. Besides, the calm air tunnel provides a downward supply of aerosol particles to its measuring zone. Only the particle generator and small dilution air streams enter the cylindrical section with 0.4 m in diameter, making the air flow very close to calm air (0.027 m.s -1 ).

In both cases, a solid polydisperse aerosol was produced by a fluidized-bed aerosol generator [START_REF] Guichard | Aerosol generation using fluidised beds[END_REF]. This unit was complemented by a corona discharger to ensure the electrical neutrality of the generated particles. The experimental aerosol was composed of glass microspheres (BL 0-50, Verre Industrie  , France) with a particle density of 2.46 g.cm -3 . The experimental aerosol size distribution should be within the thoracic penetration interval below about 30 µm). In the horizontal wind tunnel, the log-normal particle size distribution of the generated aerosol was characterized by a mass median aerodynamic diameter MMAD of 10.30 µm and a geometric standard deviation GSD of 1.98. The aerosol size distribution in the horizontal wind tunnel was shifted towards smaller particles because of coarse particle sedimentation in the horizontal low air stream. In the vertical dust chamber, MMAD was 15.06 µm and GSD was 2.04.

Aerodynamic Particle Sizer (APS 3321, TSI Inc.) was used to measure particle sizeresolved aerosol concentration [START_REF] Lidén | Comparison of measured respirable dust sampler penetration curves with sampling conventions[END_REF][START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF]. APS measurements are based on the time-of-flight measurement of accelerated particles, which is directly related to their aerodynamic diameter. The APS aerodynamic diameter range was within the 0.5 to approximately 20 µ interval.

The nominal aerosol flow rate of the APS was 5 L.min -1 . When the tested sampler operated at a different flow rate, an in-house isokinetic isoaxial aerosol diluter [START_REF] Görner | Study of fifteen respirable aerosol samplers used in occupational hygiene[END_REF] was integrated in two different ways based on whether the sampling flow rate was higher or lower than 5 L.min -1 .

Different reference probes were used, depending on the sampler under study. The flow rate of the reference probe was invariably the same as the flow rate of the tested sampler. The connection tube to the aerosol diluter was also the same to ensure identical transmission efficiency in both cases. The reference probes were designed based on the models provided by [START_REF] Hangal | Overall efficiency of tubular inlets sampling at 0-90 degrees from horizontal aerosol flows[END_REF] and [START_REF] Grinshpun | A general equation for aerosol aspiration by thin-walled sampling probes in calm and moving air[END_REF] for calculating aspiration efficiency with respect to flow rate, inlet diameter and wind direction. Reference probe efficiency was close to 1 with minimum of 0.98 at 20 µm within the measured particle size interval.

Reference and candidate sampling can be performed simultaneously or consecutively. Simultaneous sampling requires perfect spatial homogeneity of the aerosol and two APSs. We therefore opted for consecutive sampling. Reference and candidate sampling was performed on the tunnel axis in the following order: referencecandidatereference. Aerosol stability in time was continuously checked by an optical particle counter (Grimm 1.108, Grimm Industry GmbH, Germany). The experimental efficiency data were accepted only when the aerosol number concentration varied by no more than ±5% in each channel during the entire sampling sequence (referencecandidatereference).

To ensure the reliability of our experimental results, we used both experimental setups -the horizontal low velocity wind tunnel and the vertical dust chamber -to measure the sampling efficiency of the reference sampler CATHIA-T. Except CATHIA-T, which inlet is omnidirectional, all other samplers have "oriented" sampling orifices. When comparing the results with various sampler orientations in the horizontal tunnel, slight differences were observed. We therefore used the horizontal low wind speed tunnel which allows orientationaveraged results to be provided for all personal samplers in this study. Furthermore, the wind speed in this tunnel (0.15 m.s -1 ) is close to the wind speeds currently encountered in the workplace. Indeed, [START_REF] Baldwin | A survey of wind speeds in indoor workplaces[END_REF] reported that 85% of workplace wind speeds are lower than 0.3 m.s -1 and lower than 0.1 m.s -1 in most cases). Thus, the methodology adopted in this work leads to results as close as possible to the real sampling situation in slightly moving air and with random oriented workers wearing a sampler.

Efficiency data treatment and sampler performance calculation

Each data point and error bar represents the mean value and one standard deviation for 3 to 10 experimental runs. Experimental efficiency data were modeled by a suitable analytical function using a weighted least-squares method [START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF]. A decreasing cumulative log-normal law was used to model cyclone efficiency because cyclone separators exhibit naturally similarly shaped efficiency curves. For other samplers, the decreasing cumulative log-normal law was multiplied by the [START_REF] Vincent | On the quantitative definition of the inhalability of airborne dust[END_REF] inhalability function (noted V.A. in the figures). It is indeed this type of combined law that effectively defines the conventional thoracic and respirable penetrations [START_REF] Soderholm | Proposed international conventions for particle size-selective sampling[END_REF].

The decreasing cumulative log-normal law can be written as:

E(D ae ) = [1-F(x)],
where F(x) is the cumulative probability function of a standardized normal variable x:

x = ln(D ae / D 50 ) / ln(GSD), where D 50 is the particle diameter, for which the penetration efficiency is 50%. GSD is the Geometric Standard Deviation. D 50 and GSD are parameters to be optimized to fit the experimental efficiency data.

The [START_REF] Vincent | On the quantitative definition of the inhalability of airborne dust[END_REF] inhalable efficiency function is expressed as: E(D ae ) = P 1 ×(1+exp(-P 2 ×D ae )),

where P 1 and P 2 are the parameters to be adjusted (they are 0.5 and 0.06 respectively in the conventional definition of inhalability).

The model enabled us to compare experimental sampling efficiency with the conventional thoracic penetration curve and to compute a bias map of aerosol concentration measurements for a set of simulated aerosol size distributions. The bias calculation method has been described elsewhere [START_REF] Lidén | The performance of respirable dust samplers: Sampler bias, precision and inaccuracy[END_REF][START_REF] Görner | Industrial aerosol measurement according to the new sampling conventions[END_REF][START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF]CEN, 2014). The Bias Performance Criterion (BPC) [START_REF] Görner | Study of fifteen respirable aerosol samplers used in occupational hygiene[END_REF] was then calculated to provide a simple expression of sampler performance. BPC corresponds to the proportion of simulated aerosols, for which the mass concentration measurement bias is within ±10%. The simulated aerosols comprised a set of log-normal distributions with 1 < MMAD < 25 µm in 1 µm increments and 2 < GSD < 3.5 in 0.25 increments.

RESULTS AND DISCUSSION

Validation of methodology by experiments on CATHIA-T

The CATHIA-T (Tecora, France) sampler was designed at INRS and is protected by a French patent [START_REF] Fabriès | Capteur individuel portatif pour recueillir une fraction déterminée des suspensions solides d'une atmosphère[END_REF]. CATHIA-T's high flow rate (7 L.min -1 ) imposes a high volume pump, so this sampler can only be used for static sampling [START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF][START_REF] Kauffer | The use of a new static device based on the collection of the thoracic fraction for the assessment of the airborne concentration of asbestos fibres by transmission electron microscopy[END_REF]. The instrument has been previously investigated [START_REF] Maynard | Measurement of aerosol penetration through six personal thoracic samplers under calm air conditions[END_REF][START_REF] Maynard | Thoracic size-selection of fibres: Dependence of penetration on fibre length for five thoracic sampler types[END_REF][START_REF] Jones | Thoracic sizeselective sampling of fibres: Performance of four types of thoracic sampler in laboratory tests[END_REF] and is currently used mainly for asbestos sampling in France.

The CATHIA-T sampler was included in this study in order to check the reliability of the experimental methods used. The experimental data were fitted by multiplying the decreasing cumulative log-normal law by the [START_REF] Vincent | On the quantitative definition of the inhalability of airborne dust[END_REF] inhalability function as explained in previous section. The minimum value of least-squares function χ 2 [START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF] is reached, when parameter P 2 is zero. This means that the best fit corresponds to the log-normal law multiplied by a constant 2P 1 .

The CATHIA-T sampling performance provided by the bias map (Fig. 1b) mirrors previous results [START_REF] Fabriès | Personal thoracic CIP 10-T sampler and its static version CATHIA-T[END_REF] obtained under different methodological conditions with different experimental set ups, test aerosols (mineral dusts) and data fits. This confirms the equivalence of the experimental methods, especially in the case of sampler with omnidirectional inlet. Indeed, we prefer showing in this paper the result obtained in calm air chamber, not published yet, instead of similar results from horizontal wind tunnel. Note that all further results concerning personal samplers are issued from the horizontal wind tunnel for the reason given at the end of the section "Experimental methodology".

Cyclone GK 2.69

The BGI GK 2.69 cyclone (Mesa Laboratories, USA) was designed by [START_REF] Kenny | Characterization and modeling of a family of cyclone aerosol preseparators[END_REF] and is mainly used for respirable aerosol sampling at a flow rate of 4.2 L.min -1 [START_REF] Stacey | Collection efficiencies of high flow rate personal respirable samplers when measuring Arizona road dust and analysis of quartz by X-ray diffraction[END_REF]. For thoracic aerosol sampling, the flow rate must be reduced to 1.6 L.min -1 [START_REF] Maynard | Measurement of aerosol penetration through six personal thoracic samplers under calm air conditions[END_REF].

Experimental sampling efficiency was measured in the horizontal low velocity wind tunnel. The sampler was asymmetrical so it was tested in four orientations: forward, backward, left or right with respect to the air flow. Three runs were performed in each position. The backward orientation is highly improbable in personal sampling and the operator is moving so an average of three orientations (forward, left and right) was considered to represent the sampling efficiency (Fig. 2a). The experimental standard deviation SD is mainly caused by sampler orientation (contribution of 60-80% to overall SD) rather than by run reproducibility. Experimental efficiency points were modeled by a decreasing cumulative log-normal law, also reported in Fig. 2a. The above parameters and those of conventional penetration (CEN, 1993) allow us to simulate numerically thoracic aerosol concentrations for a set of different polydisperse aerosols and this led to the bias map illustrated in Fig. 2b. This bias map shows that the GK 2.69 sampler oversamples slightly (up to 12%) the aerosol thoracic fraction. The BPC indicates that the bias remain below 10% for 93% of the aerosol distributions considered. The GK 2.69 cyclone was previously studied for the respirable (4.2 L.min -1 ) and thoracic (1.6 L.min -1 ) fractions [START_REF] Kenny | Characterization and modeling of a family of cyclone aerosol preseparators[END_REF][START_REF] Maynard | Measurement of aerosol penetration through six personal thoracic samplers under calm air conditions[END_REF][START_REF] Lee | Performance of high flow rate samplers for respirable particle collection[END_REF]. In Fig. 3, these earlier results are compared with this study by means of a 'D 50 vs. flow rate' data plot. The ASTEX experimental points reflect the D 50 values previously obtained in the INRS laboratory for the thoracic flow rate of the GK 2.69 based on two different methods (API and Coulter) applied in an international round robin test (the Aerosol Sampler Testing Exchange) [START_REF] Kenny | Aerosol Sampler Testing Exchange (ASTEX)[END_REF][START_REF] Lidén | Presentation of the results from round 1 of ASTEX[END_REF][START_REF] Görner | Experimental method to determine the efficiency of aerosol samplers using the Coulter Counter[END_REF]. Fig. 3 also includes the INRS D 50 value for the 4.2 L.min -1 respirable flow rate provided by the present study. All experimental results closely follow the cyclone design model (http://bgi.mesalabs.com/cyclonefractionators/) at higher flow rates (≈4 L.min -1 ). At lower flow rates (<2 L.min -1 ), the "D 50flow rate" function is very steep and a large D 50 shift can be observed with a small flow rate variation. This dictates meticulous flow rate control in real field sampling. The Rascal cyclone (Mesa Laboratories, USA) is of similar construction to the GK 2.69 cyclone. It operates at a higher flow rate (8.5-9.5 L.min -1 ), when sampling the respirable aerosol fraction. The manufacturer does not indicate the flow rate for sampling the thoracic fraction. [START_REF] Thorpe | Evaluation of the penetration characteristics of a high flow rate personal cyclone sampler for NIOSH[END_REF] measured the variation in D 50 with respect to flow rate within the 7-10 L.min -1 range. At 7 L.min -1 , the D 50 was 4.9 µm. Using these data, [START_REF] Thorpe | Evaluation of the penetration characteristics of a high flow rate personal cyclone sampler for NIOSH[END_REF] proposed a D 50 -flow rate equation for the cyclone, specifically D 50 =10.826 e -0.1138Q (D 50 is expressed in µm and Q is the flow rate expressed in L.min -1 ). It is obvious that the flow rate must be much lower than 7 L.min -1 in order to shift D 50 up to 10 µm, the aerodynamic diameter that corresponds to the cut point of the thoracic conventional curve. We experienced flow rates lower than 7 L.min -1 . Fig. 4a illustrates our results along with a new data fit. Fairly close agreement of these data between 10 and approximately 3.5 L.min -1 can be observed. The experimental D 50 started to deviate from the fitted curve at lower flow rates. This was confirmed by new HSL data of Thorpe published in [START_REF] Lee | Calibration of high flow rate thoracic-size selective samplers[END_REF]. Fig. 4b shows that the efficiency curves for flow rates below 3 L.min -1 start to be irregular and at 2.5 L.min -1 , particle selection is entirely uncertain. In this condition, the flow velocity is probably too small to maintain the cyclone vortex. It follows that the 3 L.min -1 flow rate is the Rascal sampler's lowest limit of usage but it is otherwise very suitable for thoracic aerosol sampling. Fig. 5a illustrates experimental efficiency points obtained in the horizontal low velocity wind tunnel with the sampler air flow rate of 3 L.min -1 . These data are mean values of readings for three orientations (forward, left and right, 3 runs each) as used for the GK 2.69 cyclone. The solid line is the log normal fitting curve for the efficiency data. The bias map (Fig. 5b) was computed using the following optimized parameters:

D 50 =9.40 µm GSD=1.53
The bias map (Fig. 5b) shows that the GK 4.162 cyclone (Rascal-T) measures the particle mass concentration with an acceptable bias of ±10% for 98% of aerosol distributions used for the BPC calculation.

In a recent study [START_REF] Lee | Calibration of high flow rate thoracic-size selective samplers[END_REF], Lee and Thorpe re-measured the sampling efficiency of this cyclone using two methods: polydisperse aerosol and APS for Thorpe and monodisperse aerosol and Scanning Electron Microscopy (SEM) for Lee. While the former established an optimal flow rate of between 2.7 and 3.3 L.min -1 for thoracic aerosol sampling, the latter reported the optimal flow rate to be 3.5 L.min -1 . In conclusion, the authors stated, "Therefore, it is recommended that the GK 4.162 should be operated at a flow rate of 3.4 L.min -1 pending information in other laboratories". In our study, the best BPC was found for a flow rate of 3.0 L.min -1 , a value lying in the middle of Thorpe's interval.

Cyclone IFA FSP10

German occupational health and safety institute IFA (Institut für Arbeitsschutz) uses an FSP10 (Fein Staub Probe) high volume aerosol sampler for respirable particles in the workplace. This is a modified [START_REF] Cossey | A higher-flow rate cyclone for determination of respirable dust[END_REF] cyclone operated at 10 L.min -1 designed by Riediger (1998-not published) at the BIA (Berufsgenossenschaftliches Institut für Arbeitssicherheit). Based on the original [START_REF] Cossey | A higher-flow rate cyclone for determination of respirable dust[END_REF] experimental 'D 50 vs. flow rate' function (D 50 =68.7 Q -1.15 , where Q is the flow rate in L.min -1 ) a new, stainless steel version of this cyclone was designed with intention to sample the thoracic fraction of the aerosol at a flow rate of 5.34 L.min -1 [START_REF] Breuer | Sulfuric acid at workplacesapplicability of the new Indicative Occupational Exposure Limit Value (IOELV) to thoracic particles[END_REF].

We measured the efficiency of the German respirable cyclone FSP10 in late 1990s for former BIA institute at a flow rate of 10 L.min -1 . Laboratory testing gave a steep penetration curve with D 50 =4.37 µm, which agrees with the expected respirable fraction of the aerosol. The morphology of today's IFA cyclone is totally different; we therefore checked the respirable sampling efficiency of this cyclone prior to measuring its efficiency at 5.34 L.min -1 . The results obtained at 10 L.min -1 yield a cutoff diameter D 50 close to 4 µm (Fig. 6a), in accordance with our earlier measurement. But, at a flow rate of 5.34 L.min -1 we found a cut point D 50 =7.2 µm, which is far from the required 10 µm for thoracic efficiency. The measured sampling efficiency of the cyclone for a series of flow rates ranging from 3 to 10 L.min -1 yields a D 50 =f(Q) function (Fig. 6b) different from that of [START_REF] Cossey | A higher-flow rate cyclone for determination of respirable dust[END_REF]. The required 10 µm cut point is approached between 3 and 3.5 L.min -1 . Precise determination of the optimum thoracic flow rate is rather difficult because the cyclone begins to operate irregularly in this range of flow rates and the vortex collapses at 2.5 L.min -1 (Fig. 6a). [START_REF] Vaughan | Construction and testing of an axial flow cyclone preseparator[END_REF] was the first to describe this phenomenon, stating, "There is a marked kink in the penetration curve for the lowest flow rate". Fig. 7a illustrates the mean experimental data at 3.5 L.min -1 for nine efficiency measurements; three per forward, left and right position with respect to the horizontal low windspeed tunnel air flow. Results of sampling efficiency (Fig. 7a) and performance of the IFA FSP10 cyclone (Fig. 7b) indicate that this is a good thoracic aerosol sampler at an optimum flow rate of 3.5 L.min -1 . [START_REF] Lee | Calibration of high flow rate thoracic-size selective samplers[END_REF] came close to the same result at 4 L.min -1 using a different method using six monodisperse aerosols.

Although the flow rate of 3.5 L.min -1 is more convenient for personal sampling than the former higher flow rate, the weight of the IFA FSP10 thoracic sampler makes it more suitable for static sampling.

Parallel Particle Impactor -PPI-T

The thoracic PPI-T (PPI, SKC, USA) was designed to sample the thoracic health-related aerosol fraction for assessment of worker personal exposure to airborne particles. It operates at a flow rate of 2 L.min -1 . The thoracic aerosol fraction is selected by simultaneous implementation of four parallel impactor stages and a final collection filter. The thoracic sample consists of the particles collected on the final filter.

Fig. 8a illustrates the sampling efficiency for the PPI, measured in the horizontal low wind speed tunnel, along with data provided by [START_REF] Trakumas | Parallel particle impactornovel size-selective particle sampler for accurate fractioning of inhalable particles[END_REF]. Both experimental data sets are very close and were fitted by above mentioned combined law. The modeled sampling efficiency curve (Fig. 8a) was compared to the conventional definition of thoracic penetration. Fig. 8b illustrates the bias map calculated for the thoracic PPI sampler. The PPI-T parallel impactor offers fair thoracic aerosol sampling performance for most particle size distributions present in the workplace (Fig. 8b). It will sample 71% of aerosols from the bias map with a positive bias smaler than 10%.

CONCLUSION

Five thoracic aerosol sampling devices were tested under laboratory conditions using a spherical polydisperse aerosol and the Aerodynamic Particle Sizer. The experimental method was checked by measuring the efficiency of known static thoracic sampler CATHIA-T. The sampling efficiency of the CATHIA-T turned out to be close to that found in previous studies. This proves the reliability of the experimental method implemented in this study. While it is not a personal sampler (due to size and high flow rate of 7 L.min -1 ), the CATHIA-T sampler has the advantage of collecting large quantity of particles for subsequent analysis.

Our results show that both BGI cyclones sample correctly the thoracic fraction of an ambient aerosol: the GK 2.69 at its nominal flow rate 1.6 L.min -1 and the GK 4.162 at a flow rate of 3.0 L.min -1 . Similar in performance to the GK 2.69 sampler, the GK 4.162 has the advantage of collecting almost twice the particulate matter thanks to its higher flow rate.

IFA cyclone FSP10 with its suggested 5.34 L.min -1 nominal flow rate did not assume conventional thoracic penetration. Our results show that cyclone samples the thoracic aerosol fairly well at the optimized flow rate of 3.5 L.min -1 .

The IFA cyclone is much larger and heavier than the other cyclones, and is therefore less suited to personal exposure measurement. The optimized low flow rates of both the IFA and GK 4.162 cyclones are close to the operating limit because the cyclonic vortex tends to collapse at approximately 2.5 L.min -1 .

The Parallel Particle Impactor PPI-T was found to be a suitable thoracic aerosol sampler at its nominal flow rate of 2.0 L.min -1 . It is small and lightweight, and is suitable for personal exposure measurement.

Aerosol sampling efficiency was measured using solid aerosol particles (glass beads) in this study. It was observed that sampling efficiency could vary depending on the particle phase. In theory, particle motion in air depends only on the aerodynamic size and shape of the particles. In practice, particles entering a sampler can come into contact with its walls and, in this case, solid particles will tend to rebound, while liquid particles will more easily adhere to the internal surfaces of the sampler. Sampler efficiency could therefore be slightly modified, in the case of liquid aerosol sampling in the workplace (e.g. sulfuric acid mist (EU 2009)). Prior to using aerosol samplers for liquid aerosol sampling, we recommend checking their sampler efficiency with at least one monodisperse liquid aerosol of approximately 10 µm particle size, i.e. close to the expected D 50 cut point.

A field study is planned to be carried out in different industrial situations to compare the behavior of thoracic samplers in workplace conditions.

Fig. 1 .

 1 Fig. 1. Sampling efficiency (a) and bias map (b) for CATHIA-T aerosol sampler at 7 L.min -1

Fig. 2 .

 2 Fig. 2. Sampling efficiency (a) and bias map (b) of Cyclone GK 2.69 aerosol sampler operated at 1.6 L.min -1

Fig. 3 .

 3 Fig. 3. "D 50flow rate" function for cyclone GK 2.69

Fig. 4 .

 4 Fig. 4. "D 50 vs. flow rate" function (a) and sampling efficiencies at different flow rates (b) of the Cyclone GK 4.162 (Rascal)

Fig. 5 .

 5 Fig. 5. Sampling efficiency (a) and bias map (b) of cyclone GK 4.162 operated at 3.0 L.min -1 .

Fig. 6 .

 6 Fig. 6. Sampling efficiencies at different flow rates (a) and 'D 50 vs. flow rate' function (b) for the IFA FSP10 cyclone.

Fig. 7 .

 7 Fig. 7. Sampling efficiency (a) and bias map (b) of the IFA FSP10 cyclone at 3.5 L.min -1

Fig. 8 .

 8 Fig. 8. Sampling efficiency (a) and bias map (b) of PPI thoracic aerosol sampler at 2.0 L.min -1

Table 1 .

 1 Thoracic aerosol samplers under study

	Sampler	CATHIA-T	Cyclone GK 2.69 SS-T	Cyclone GK 4.162 Rascal-T	Cyclone IFA FSP10	Parallel Impactor PPI-T
	Provider	Tecora (F)