J Réthoré

N Limodin

J-Y Buère

F Hild

W Ludwig

S Roux

Digital volume correlation analyses of synchrotron tomographic images

Keywords: Elastic properties, full-eld measurements, measurement uncertainties, stress intensity factor, X ray tomography

3D images of nodular graphite cast iron samples obtained with synchrotron tomography are analyzed by resorting to digital volume correlation. This technique uses the material microstructure to measure displacement elds within a sample submitted to mechanical loading. Compared to classical 2D digital image correlation, the reconstructed volumes used by DVC tend to increase the uncertainty in the correlation calculations, yet elastic strains of the order of 10 -3 can be measured. Displacement 1 elds around the front of a fatigue crack have also been measured by an extended version of DVC, and are used to extract stress intensity factors (in modes I, II and III) along the crack front. The values are in good agreement with nite element calculations when the experimentally measured displacements prescribed on the sample boundaries are considered as input in the numerical simulations.

List of notations

Among the dierent methods available to the engineer or researcher for measuring displacements in a material under mechanical loading, digital image correlation (DIC), whose early developments date back to the early 1980s [START_REF] Lucas | An Iterative Image Registration Technique with an Application to Stereo Vision[END_REF][START_REF] Burt | Local correlation measures for motion analysis: a comparative study[END_REF][START_REF] Sutton | Determination of Displacements Using an Improved Digital Correlation Method[END_REF],

is gaining growing interest, partly because of the appealing simplicity of the technique, namely, displacement elds are measured by registering two images of the surface of a specimen under dierent loads [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF]. The extension of this technique for measuring three dimensional displacement elds in the bulk of samples under load is relatively recent (see for example 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] for references).

Such measurements are complementary to diraction experiments. First they allow one to relate directly local uctuations in displacements to variations in the microstructure (in a broad sense) avoiding a combined and usually relatively complex use of diraction and imaging [START_REF] Preuss | SiC single bre full-fragmentation during straining in a Ti-6Al-4V matrix studied by synchrotron X-rays[END_REF]. Besides, some information can potentially be obtained with correlation methods for amorphous and/or non diracting materials such as polymeric foams [START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF] or stone wool [START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF].

If the principles underlying image and volume correlations are essentially the same, in practice several distinctions must be pointed out. First, 3D images of a material are not as easily obtained in a non destructive way as 2D images.

Among the few techniques that provide such 3D images, X-ray tomography using either laboratory or synchrotron X ray sources is more and more used.

Tomography images are reconstructed from a set of radiographs, a procedure producing artifacts / noise that must be taken into account when digital volume correlation (DVC) measurements are to be performed [START_REF] Limodin | Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory X-ray source[END_REF].

Second, as explained before, DVC requires the presence of a pattern in the material microstructure. When observing surfaces, this pattern is easy to obtain by spraying for instance black and white paint [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF], in 3D the markers issue is much more complex. Two dierent methods have been employed so far. A rst route is to introduce articially the markers in the studied material [START_REF] Nielsen | Measurements of plastic displacement gradient components in three dimensions using marker particles and synchrotron X-ray absorption microtomography[END_REF][START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF][START_REF] Germaneau | 3D strain eld measurement by correlation of volume images using scattered light: Recording of images and choice of marks[END_REF][START_REF] Germaneau | Comparison between X-ray micro-computed tomography and optical scanning tomography for full 3D strain measurement by digital volume correlation[END_REF][START_REF] Haldrup | A general methodology for fulleld plastic strain measurements using X-ray absorption tomography and internal markers[END_REF], the other one is to directly use the material microstructure [START_REF] Bay | Digital volume correlation: three-dimensional strain mapping using X-ray tomography[END_REF][START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF][START_REF] Toda | A 3D measurement procedure for internal local crack driving forces via synchrotron X-ray microtomography[END_REF][START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF][START_REF] Hild | Three dimensional analysis of a compression test on stone wool[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]. The rst method can in principle be applied to a variety of materials, provided a suitable technique for distributing homogeneously the markers is found. However, the question of how much the presence of those extrinsic features aects the deformation processes remains a critical issue. From that point of view, natural markers appear more attractive, but the number of materials presenting a relevant microstructure is restricted. So far, the deformation of quite dierent naturally marked materials has been studied, namely, trabecular bone [START_REF] Bay | Digital volume correlation: three-dimensional strain mapping using X-ray tomography[END_REF][START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF][START_REF] Benoit | 3D analysis from micro-MRI during in situ compression on cancellous bone[END_REF], polystyrene foam [START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF], stone wool [START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF], wood [START_REF] Forsberg | 3D micro-scale deformations of wood in bending: Synchrotron radiation µCT data analyzed with digital volume correlation[END_REF], cast iron [START_REF] Réthoré | Extended three-dimensional digital image correlation (X3D-DIC)[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] and aluminum alloys [START_REF] Toda | Assessment of the fatigue crack closure phenomenon in damage-tolerant aluminium alloy by in-situ high-resolution synchrotron X-ray microtomography[END_REF][START_REF] Toda | A 3D measurement procedure for internal local crack driving forces via synchrotron X-ray microtomography[END_REF] to name a few. It must be pointed out that in some cases [START_REF] Toda | Assessment of the fatigue crack closure phenomenon in damage-tolerant aluminium alloy by in-situ high-resolution synchrotron X-ray microtomography[END_REF][START_REF] Nielsen | Measurements of plastic displacement gradient components in three dimensions using marker particles and synchrotron X-ray absorption microtomography[END_REF], the method consists in tracking the displacement of the center of mass of the markers (a method called particle tracking) and is therefore dierent from what will be described hereafter.

When dealing with 3D images (or volumes), the pattern that is used for measuring displacements elds corresponds to a volume within the material, the term Digital Volume Correlation (DVC) has been chosen and will be used hereafter. Historically, DVC has been rst used to characterize relatively large displacement / strain levels such as those obtained during plastic deformation of metallic materials [START_REF] Nielsen | Measurements of plastic displacement gradient components in three dimensions using marker particles and synchrotron X-ray absorption microtomography[END_REF] or elastic strains in low moduli materials [START_REF] Bay | Digital volume correlation: three-dimensional strain mapping using X-ray tomography[END_REF][START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF][START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF]. The elastic strains in metallic materials being quite small (typically less than 10 -3 ), they are quite challenging to be measured by DVC as the uncertainty levels encountered with this technique tend to be larger than those observed in 2D-DIC [START_REF] Limodin | Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory X-ray source[END_REF]. An explanation for these higher uncertainty levels is discussed herein.

It is also shown that DVC can be used to evaluate accurately elastic strains in metallic samples under load using synchrotron X ray tomographic images.

Finally a variant of DVC, which is called eXtended DVC (or X-DVC [START_REF] Réthoré | Extended three-dimensional digital image correlation (X3D-DIC)[END_REF][START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF]), is also described and used for measuring displacement elds in the bulk of a material containing a crack. This technique gives a direct evaluation of crack opening maps and allows for the evaluation of the local crack driving force via the extraction of stress intensity factor proles [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF].

Material

The studied material is a ferritic cast iron (3.65 wt%C, 3.2 wt%Si, 0.04 wt%Mg, < 0.1 wt%Mn, <0.005 wt%S, 0.02 wt%P) whose microstructure consists of a ferritic matrix containing a homogeneous distribution of nearly spherical graphite nodules (volume fraction: 13 %, average inter-nodule spacing: 50 µm, average nodule diameter: 50 µm). Those nodules have a lower X ray attenuation than the (ferritic) iron matrix, and are therefore easily imaged by tomography; they allow for an easy registration of the reconstructed images. [Figure 1 about here.]

3D imaging and mechanical tests

The synchrotron tomography experiments were performed at the European Synchrotron Radiation Facility (ESRF) in Grenoble (France) on beamline ID19.

The details of the tomography experimental set up of ID19 can be found in

Ref. [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]. A set of 600 radiographs (scan) was recorded during a 180-degree rotation on a Charge Coupled Device (CCD) camera with a square array of 2048 × 2048 pixels (exposure time: 3 s). This detector was coupled with a uorescent screen via optical lenses. The white beam coming from the synchrotron ring was rendered monochromatic by a multilayered monochromator. The energy of the beam was set to 60 keV. The monochromatic beam was parallel so that no geometric magnication was possible. Instead, the voxel size was the result of the camera optics used.

Reconstruction of the tomographic data was performed with a ltered backprojection algorithm developed at ESRF that provided 8-bit gray scale 3D images with an isotropic voxel size. The nodule size is large enough to allow for a relatively low voxel size to be used for tomography. This, in turn, enables for the observation of millimetric samples, the only limitation for the sample size being the overall attenuation of the material that has to allow for at least 10 % transmission of the incoming X ray beam. In practice a voxel size of 5.1 µm was chosen so that each nite element in the DVC analysis (see below) could contain at least one nodule (the mean distance between nodules is of the order of 50 µm, or 10 voxels). Therefore analyses dealing with elements whose size is greater than 9 voxels does not need any regularization. For smaller lengths, it is possible to perform successfully a registration (even at the voxel level). This procedure, called voxel-scale DVC (or V-DVC), calls for a mechanical regularization [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF].

The rst experiment described in this paper consists of a tensile test on a smooth sample (i.e., with no crack) that was loaded to a maximum level of 273 N, and then unloaded. Its cross section is equal to 1.6 × 0.8 mm 2 . Five tomographic scans were acquired during which the load level was equal to 22, 150, 200, 250 and 273 N. The standard uncertainty of the load cell is equal to 5 N. In the second experiment, a specimen containing a crack has been studied during an in situ cyclic test thanks to a dedicated testing machine. This millimetric sample (cross section: 1.6 × 1.6 mm 2 ) has been cut out of a larger fatigue coupon [START_REF] Limodin | Inuence 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 fatigue machine was directly installed on the rotation stage of the micro-tomography setup. The sample has been cycled in air with a constant stress amplitude (load ratio: R = 0.1) and a frequency of 40 Hz. At the beginning of the experiment, scans were acquired at intermediate loading steps from the maximum to the minimum load of the rst unloading / loading cycle (Figure 2). Then during the in situ fatigue experiment, scans were recorded at dierent time intervals with the specimen held under maximum load. When crack growth was detected in these images, another complete loading / unloading sequence was recorded. The fatigue crack interactions with the microstructure gives a roughness of the crack path of the order of the grain/nodule size, which is small enough to ease further modeling. In the present case, one result will be shown when 45,000 cycles were applied to the sample. The reference conguration corresponds to the acquisition at minimum load, and the deformed conguration to a scan at maximum load.

[Figure 2 about here.]

4 Digital Volume Correlation (DVC)

Volume correlation consists in registering two volumes with the help of a displacement eld (to be determined). To estimate the unknown displacement eld u, the quadratic dierence

φ 2 = [f (x) -g(x + u(x))] 2 is integrated over the studied domain Ω Φ 2 = ∫ Ω φ 2 (x)dx (1)
and minimized with respect to the degrees of freedom u i of the sought displace-

ment eld u(x) = ∑ j u j n j (x) (2)
where n j (x) are the components of the chosen kinematic basis. In the present case, a 3D nite element kinematics is chosen [START_REF] Zienkievicz | The Finite Element Method[END_REF] for the sought elds, so that n j correspond to shape functions, here selected as trilinear polynomials associated with 8-node cube elements (or C8-DVC [START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF]). The method used herein is therefore a Galerkin approach to digital volume correlation. Other approaches can be found, namely, as in 2D applications, the most commonly used correlation algorithms consist in registering locally small zones of interest in a sequence of pictures to determine local displacement components [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF]. The same type of hypotheses are made in three-dimensional algorithms [START_REF] Bay | Digital volume correlation: three-dimensional strain mapping using X-ray tomography[END_REF][START_REF] Smith | Digital volume correlation including rotational degrees of freedom during minimization[END_REF][START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF][START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF]. With C8-DVC there is only one correlation parameter to be chosen, namely, the size ℓ of the element edge.

The second kinematic basis that will be used herein is enriched as in X-FEM simulations [START_REF] Moës | A nite element method for crack growth without remeshing[END_REF]. The partition of unity property [START_REF] Babuska | The partition of unity method[END_REF] of the shape functions n e is used ∑ e n e (x) = 1

(3)

to construct the enriched approximation of each component u of the displacement eld

u(x) = ∑ j∈N u j n j (x) + ∑ j∈Ncut d j h j (x)n j (x) (4)
where u j are the degrees of freedom associated with the standard (i.e., continuous) shape functions, and N cut the set of nodes that hold additional degrees of freedom d j associated with the enrichment function h j (x) dened in [START_REF] Zi | New crack-tip elements for XFEM and applications to cohesive cracks[END_REF] h

j (x) = H(x) -H(x j ) ( 5
)
where H is the Heaviside step function whose value is 0 below the crack surface and 1 above, and x j gives the position of node j. When associated with a C8 shape functions, it is referred to as XC8-DVC [START_REF] Réthoré | Extended three-dimensional digital image correlation (X3D-DIC)[END_REF][START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF].

To evaluate the quality of the following correlation results, the normalized correlation residual is considered

η(x) = |φ(x)| max Ω (f ) -min Ω (f ) (6)
and its mean value ⟨η⟩ is computed over the whole correlation volume Ω. When the measured displacements are unknown, this quantity is the only estimator available to assess the quality of the registration (Section 5.1), and the adequacy of the kinematic basis to the analyzed experiment (Section 5.2).

It is to be stressed that the quality of the enriched kinematics relies on the ability to determine precisely the support of the discontinuity, i.e., the crack surface. To reach this goal, it revealed convenient to exploit the residual map obtained with a continuous kinematics [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF]. In that case, mismatch errors accumulate on the surface of discontinuity as it violates the kinematic assumption (i.e., continuity of the displacement eld). An automatic procedure can then be designed to nd a surface which maximizes its intersection with the residual eld, thereby providing a good approximation of the crack surface in the reference image. [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] Results and analyses

Tensile test on smooth sample

To illustrate the usefulness of the correlation residuals, Figure 3 shows the latter ones at dierent steps of the registration procedure when the rst load level is analyzed. In Figure 3(a) the initial dierence |f -g| is given before the correlation procedure has started. The microstructure is clearly visible, and ⟨η⟩ = 11.6 %. A rst step consists in evaluating the mean rigid body translation.

The result of this rst correction is shown in Figure 3(b). The mean correlation residual ⟨η⟩ = 5.4 % is signicantly reduced. However, there are zones for which the registration has still to be improved. Figure 3(c) shows the nal dierence |f -g| at convergence. The microstructure cannot be recognized, and ⟨η⟩ = 2.9 %. This low value gives condence in the measured displacement eld.

It is conrmed when the histogram of the gray level residuals (Figure 3(d)) is compared with that of the reference volume (Figure 1(b)).

[Figure 3

about here.]

To evaluate the performance of the correlation code, a reference volume is considered. A constant sub-voxel displacement is applied to create an articial volume that is registered with the reference one. The mean value is very close to the prescribed one (i.e., a dierence less than one thousandth of one voxel). However, there exists a scatter characterized by a standard deviation σ u = 0.044 voxel (for 16-voxel elements) whose level is more than one order of magnitude higher than the bias. The value of σ u is not as low as what is observed in 2D-DIC [START_REF] Besnard | Finite-element displacement elds analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF], but classical in DVC. Higher values are generally observed in DVC due to reconstruction artifacts associated with tomography [START_REF] Davis | Artefacts in X-ray microtomography of materials[END_REF][START_REF] Ketcham | New algorithms for ring artefact removal[END_REF][START_REF] Limodin | Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory X-ray source[END_REF]].

An additional cause is related to the fact that the analyzed 16-Mvoxel volume is such that its external surface to volume ratio is of the order of ≈ 1/39 voxel -1 as compared with a ratio of external perimeter to surface ratio generally less than ≈ 1/256 pixel -1 for 1-Mpixel pictures. Therefore, the weight of the degrees of freedom belonging to the external faces is more important in the present case than for regular pictures. Remembering that the degrees of freedom belonging to an external surface have a higher uncertainty level (half of the elements are missing for a node belonging to a face, 3/4 for an edge and 7/8 for a corner), higher surface to volume ratios will lead to higher values of uncertainty levels [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF].

Figure 4 illustrates this eect, namely, when only inner nodes are considered (inner ROI), the uncertainty levels are less than when all nodes are considered.

The larger the element, the smaller the volume / surface ratio in terms of degrees of freedom. For instance for 8-voxel elements, there are 81% inner nodes. This value falls down to 14% for 64-voxel elements when a 144 × 288 × 400-voxel ROI is considered. This property explains the reason for the dierence in terms of measurement uncertainty when only inner nodes are considered or all nodes of the ROI. Figure 4 shows that the larger the element, the smaller the uncertainty. This result is typical of the compromise between measurement uncertainty and spatial resolution. A power law between the displacement uncertainty σ u and the element size ℓ

σ u = a α+1 ℓ α (7)
where a is a constant expressed in voxels, interpolates very well the results of the uncertainty analysis. The power α is equal to 1.8 when only inner nodes are considered, and 1.1 for all measured nodes. Parameter a is of the order of one voxel in both cases. In the following analyses, 16-voxel C8 elements are used (i.e., ℓ = 16 voxels).

[Figure 4 about here.]

The raw information provided by DVC is displacement elds. Figure 5 shows 3D renderings of displacement elds in the three directions. It is observed that the main contribution is given by rigid body motions (translation and rotation).

However, the latter ones do not reveal the mechanical properties of the material.

A post-processing of the displacement eld is generally required to study, say, cracked samples [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 | Inuence 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][START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF] or strain elds [START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF][START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF][START_REF] Benoit | 3D analysis from micro-MRI during in situ compression on cancellous bone[END_REF]. Local strains computed from the gradient of the measured displacement show a large level of uctuations that may be ltered or smoothed at the expense of faithfulness to the measurement.

Global (mean) strains, at the level of the ROI itself, may provide an estimate of, say, the elastic moduli. Mean strains can however be computed in dierent ways leading to more or less secure determinations.

[Figure 5 about here.]

A rst way of evaluating a mean longitudinal strain is to compute the mean displacements of the upper ⟨u u ⟩ and lower ⟨u l ⟩ faces of the ROI. Let n x , n y and n z denote the number of elements in the x, y, and z directions, respectively.

The mean longitudinal strain is dened as

ϵ l = ⟨u u ⟩ -⟨u l ⟩ n z ℓ (8)
In the present case, n x = 9, n y = 18, and n z = 25. The total number of measured degrees of freedom is therefore equal to 3(n x +1)(n y +1)(n z +1) = 14, 820.

In the following uncertainty analysis, it is assumed that each measured degree of freedom is uncorrelated with any other one. This is only a rst approximation.

Exact results can be found in Ref. [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF]. The standard uncertainty σ ⟨u⟩ of ⟨u u ⟩ and ⟨u l ⟩ is therefore equal to

σ ⟨u⟩ = σ u √ (n x + 1)(n y + 1) (9)
so that the corresponding standard uncertainty σ ϵ of the mean longitudinal strain ϵ l reads

σ ϵ = √ 2σ u ℓn z √ (n x + 1)(n y + 1) (10)
In the present case, σ ⟨u⟩ = 0.003 voxel, and σ ϵ = 1.1 • 10 -5 . Figure 6 shows the stress / strain curve deduced from the analysis of the ve scans. The identied value of Young's modulus is E = 187 GPa.

[Figure 6 about here.]

A second way of evaluating the mean strain is to use all measured points.

The corresponding displacement eld is linearly interpolated and its rst order gradient is then used to estimate the nominal strain tensor [START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF]. For the mean longitudinal strain ε, its standard uncertainty becomes

σ ε = σ u ℓ √ (n x + 1)(n y + 1) nz 6 ( nz 2 + 1 ) (n z + 1) (11)
so that the two standard strain uncertainties are related by

σ ε = σ ϵ √ 3n z ( nz 2 + 1 ) (n z + 1) (12)
Equation [START_REF] Germaneau | 3D strain eld measurement by correlation of volume images using scattered light: Recording of images and choice of marks[END_REF] shows that for values of n z greater than 2, σ ε is less than σ ϵ . When n z ≫ 1, the ratio of the two uncertainties is √ 6/n z . If the level of uncertainty is an issue, it is more desirable to use the interpolation of the whole measured eld.

However, it has to be checked that the residuals are suciently low to make sure that the interpolation is meaningful. In the present case, σ ε = 5 • 10 -6 . Figure 6 shows the stress / strain curve deduced from the analysis of the ve scans. The identied value of Young's modulus is E = 156 GPa. This result is in good agreement with classical values of nodular cast iron (160 ± 3 GPa [START_REF] Dierickx | Etude de la microstructure et des mécanismes d'endommagement de fontes G.S. ductiles : inuence des traitements thermiques de ferritisation[END_REF][START_REF] Limodin | Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory X-ray source[END_REF]).

Fatigue test on pre-cracked sample

Figure 7 shows a 3D rendering of the three components of the displacement eld of the cracked sample under load obtained by X-FEM after 45,000 fatigue cycles when measured displacements (using X-DVC) are prescribed as boundary conditions on the top and bottom faces of the mesh.

[Figure 7 about here.]

As expected, the main opening regime of the crack is in mode I, while some lower amplitude displacements are also recorded in mode II and III. The enriched degrees of freedom in the X-DVC approximation are used to extract 2D maps of the crack opening displacement (COD) for the three modes across the cracked surface at maximum load in Figure 8. It is worth noting that the whole surface was not enriched but only the cracked part. For mode I, the COD changes with the applied load can be found in Ref. [START_REF] Limodin | Inuence 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 details of the COD elds are to be interpreted with caution as measurement uncertainties tend to be larger over the enriched degrees of freedom of the chosen kinematics (see maps of mode II and III CODs of Figure 8). With this point in mind, it can be concluded that the crack is mostly open in mode I and that the enrichment was chosen on a suciently large area. Moreover, an approximate determination of the crack geometry (supporting the enrichment) may also induce spurious bias in the evaluation of the opening. Consequently, a quantitative analysis using the raw COD maps is likely to yield poor results in terms of, say, stress intensity factors.

[Figure 8 about here.]

Another way of interpreting the displacements elds measured in the bulk of the specimen is to use them to extract the values of Stress Intensity Factors (SIF) along the crack front. First, least squares techniques were implemented by using 2D solutions [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] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Yoneyama | Evaluating mixed-modestress intensity factors from full-eld displacement eldsobtained by optical methods[END_REF].

An alternative technique to determine

SIFs is to resort to interaction integrals [START_REF] Réthoré | Estimation of mixed-mode stress intensity factors using digital image correlation and an interaction integral[END_REF][START_REF] Réthoré | Noise-robust Stress Intensity Factor Determination from Kinematic Field Measurements[END_REF][START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF]. It was utilized to analyze the results presented in Ref. [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF], but 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 can be treated separately. For each of these planes (nearly orthogonal to the crack front), the 3D displacement eld 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 v(r, θ) = ∑ pmin≤p≤pmax [a p ϕ ϕ ϕ p I (r, θ) + b p ϕ ϕ ϕ p II (r, θ)] ( 13
)
where a p and b p are real numbers. The reference elds for mode I are

ϕ ϕ ϕ p I (r, θ) = r p/2 2µ √ 2π [ κe ipθ/2 - p 2 e i(4-p)θ/2 + ( p 2 + (-1) p )e -ipθ/2 ] (14)
and for mode II

ϕ ϕ ϕ p II (r, θ) = ir p/2 2µ √ 2π [ κe ipθ/2 + p 2 e i(4-p)θ/2 -( p 2 -(-1) p )e -ipθ/2 ] ( 15
)
where r and θ denote the in-plane coordinates associated with the usual crack tip frame, µ Lamé's shear modulus, and κ a dimensionless elastic (or Kolossov) coecient related to Poisson's ratio ν equal to (3 -ν)/(1 + ν) for plane stress conditions, or 3 -4ν for plane strain conditions (used in the sequel). The factor √ 2π is introduced to match the usual denitions of K I and K II SIFs [43], which are thus equal respectively to amplitudes a 1 and b 1 associated with elds ϕ ϕ ϕ 1

I and ϕ ϕ ϕ 1 II .
Let us rst note that the status of elds ϕ ϕ ϕ p I and ϕ ϕ ϕ p II is very dierent for p less than or greater than 1. For p > 1, the elds are called subsingular. They have no impact on the crack tip. Conversely, the attached stress elds 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 boundary conditions. For p < 0, the elds are referred to as supersingular. Traditionally, the supersingular elds are ignored because their asymptotic behavior 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, these solutions are not rejected [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF][START_REF] Roux | Digital Image Correlation and Fracture: An Advanced Technique for Estimating Stress Intensity Factors of 2D and 3D Cracks[END_REF]. However, it makes no sense to extrapolate them inside the process zone. They rather characterize the (vanishing) inuence of nonlinearities present in the process-zone in the outer elastic domain. Out-of-plane motions w induced by mode III displacements whose canonical eld ϕ p III reads

ϕ p III (r, θ) = r p/2 cos[p(θ -π)/2] ( 16
)
allow one to determine K III . In practice, in-plane solutions are considered for p ranging between -3 and 5. The value p = -1 captures the eect of a shift of the crack tip position in the plane with respect to the tip of the equivalent elastic crack [START_REF] Roux | Digital Image Correlation and Fracture: An Advanced Technique for Estimating Stress Intensity Factors of 2D and 3D Cracks[END_REF]. By canceling out its contribution, the equivalent crack tip position is estimated. 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 a -3 is related to a 1 = K I . As all odd p < 0 functions lead to singular displacements at the crack tip, the displacement data at nodes whose distance to the crack tip or to the crack mouth is less than 20 voxels are not considered in the analysis. The result of such an extraction is illustrated in Figure 9, which also shows for comparison purposes the results of a numerical extended nite element (X-FEM) calculation based on the real geometry of the crack.

[Figure 9 about here.]

From the results of Figures 7 and8 it appears that the sample mechanical loading does not correspond to a pure mode I opening of the crack, probably because of slight misalignments between the loading stage and the sample. To overcome this problem, the measured displacement elds were prescribed on the top and bottom surfaces of the analyzed volume. This analysis departs from that proposed in Ref. [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF] in which a uniform pressure was considered to match the applied load. Further, the procedure detailed above is used to extract stress intensity factors from the X-FEM displacement eld whereas the interaction integral was used in Ref. [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF]. When this procedure is carried out, a good agreement between the K I values extracted from DVC results and those resulting from the X-FEM calculation are obtained (Figure 9). A better agreement is obtained in terms of SIF proles with the present procedure when compared with that proposed in Ref. [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF]. The level of uctuations in terms of stress intensity factors is similar when X-DVC and X-FEM simulations are compared. This result shows that the uctuations are not caused by measurement uncertainties alone, but also by the approximate determination of the crack geometry that was the same in X-DVC and X-FEM calculations. Another source of uctuation might be related to the scale at which the analyses were performed and the corresponding material microstructure, which was modeled as a homogeneous and elastic medium.

By comparing crack opening opening displacement maps and the K I values obtained by DVC, it has been possible to show that the crack opens at a value of K I close to 6.5 MPa √ m. This value, which has been consistently found for other samples investigated with the same procedure [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 | Inuence 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], is in very good agreement with the range of values that are given in the literature for nodular cast iron [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]. It was also possible to clearly correlate crack growth (resp. arrest) to crack opening (resp. closure) [START_REF] Limodin | Inuence 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]. These eects are not discussed herein.

Conclusion

Digital volume correlation (DVC) was used to measure displacement elds from 3D synchrotron X-ray tomographic images of cast iron specimens under mechanical loading. The uniform distribution of graphite nodules that are easily imaged by tomography provides a natural pattern for performing correlation avoiding to resort to articial markers, which might have altered the mechanical behavior of the material. Any other material presenting similar microstructural features could in principle be studied by the same technique. In practice, those features should present an attenuation contrast with the surrounding matrix large enough to be imaged by X rays, and their typical size in the reconstructed images should correspond to a few voxels .

For the volumes used analyzed with DVC, the ratio of external surface to volume is larger than that of external perimeter to surface of classical pictures used in 2D correlation. This was shown to be responsible for an increase in uncertainty levels obtained with DVC compared to 2D image correlation. However, it was possible to estimate Young's modulus of the studied material by using DVC (i.e., elastic strains of the order of 10 -3 ) with an accuracy that increased when the whole measured eld was used instead of the that of external boundaries to estimate mean strains.

In the case of cracked samples, the use of X-DVC enabled for the measurement of displacement elds even close to the crack surface. Those displacement elds were used to extract stress intensity factors (modes I, II and III) along the crack front. The values obtained were found in good agreement with nite el-

In practice the actual size of the features will depend on the voxel size i.e., on the resolution used, which in turn is given by the dimension of the specimen whose projections should t on the detector used for tomography [START_REF] Buère | In Situ Experiments with X ray Tomography: an Attractive Tool for Experimental Mechanics[END_REF].

K

  I , K II , K III mode I, II, and III stress intensity factors ℓ element size n e nite element shape function n j component of kinematic basis n x , n y , n z number of nodes in the x, y and z directions N total set of nodes N cut set of nodes that hold additional degrees of freedom p

Figure 1 (

 1 Figure 1(b).

26 2 27 3 28 4 30 6 32 8 33 9

 262728303233 ement calculations when the experimentally measured displacements were used as boundary conditions.List of Figures1 ROI (a) of the analyzed volume in Section 5.1 (144×288×400 voxels, 1 voxel ↔ 5.1 µm, 8-bit digitization) and corresponding gray level histogram (b). . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic illustration of the tests performed (a). A red point corresponds to the acquisition of one tomographic scan. Several scans at dierent load levels are recorded to perform DVC measurements during the fatigue test. Picture of the in situ testing machine mounted on the tomographic set up (b). . . . . . . . . . Initial (a), with rigid body translation correction (b), and at convergence (c) gray level residuals |f -g| when the rst load level is analyzed. The nal histogram of gray level residuals (d) can be compared with that of the reference volume shown in Figure 1(b). Standard displacement uncertainty σ u as a function of the element size ℓ for C8-DVC applied to the reference volume shown in Figure 1 when shifted by 0.5 voxel in all three directions. The dashed lines correspond to a power law interpolation [Equation (7)]. 29 5 3D rendering of transverse (a,b) and longitudinal (c) displacement elds expressed in voxels (1 voxel ↔ 5.1 µm). . . . . . . . Stress / strain plot for the two ways of evaluating the mean strain. The dashed lines show a linear interpolation used to estimate the macroscopic Young's modulus. . . . . . . . . . . . . . . . . . . . 31 7 3D rendering of the displacement elds calculated by X-FEM in the cracked sample under maximum load after 45,000 fatigue cycles: longitudinal (a), in-plane (b), and out-of-plane (c) directions. The scale bar is expressed in micrometers, the cube edge is 288-voxel long (i.e., 1.46 mm). The boundary conditions used for the calculation correspond to the experimental displacements obtained by X-DVC. . . . . . . . . . . . . . . . . . . . . . . . . . Maps of crack opening displacement for modes I (a), II (b), and III (c) at maximum load after 45,000 cycles. The displacements are expressed in voxels (1 voxel ↔ 5.1 µm). The crack front is close to the lower edge. . . . . . . . . . . . . . . . . . . . . . . . . Proles of K I , K II and K III along the crack front at the maximum load of the fatigue cycle after 45,000 cycles determined experimentally by post-processing X-DVC results, and numerically by resorting to X-FEM analyses. . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: ROI (a) of the analyzed volume in Section 5.1 (144 × 288 × 400 voxels, 1 voxel ↔ 5.1 µm, 8-bit digitization) and corresponding gray level histogram (b).

Figure 2 :

 2 Figure 2: Schematic illustration of the tests performed (a). A red point corresponds to the acquisition of one tomographic scan. Several scans at dierent load levels are recorded to perform DVC measurements during the fatigue test. Picture of the in situ testing machine mounted on the tomographic set up (b).

Figure 3 :

 3 Figure 3: Initial (a), with rigid body translation correction (b), and at convergence (c) gray level residuals |f -g| when the rst load level is analyzed. The nal histogram of gray level residuals (d) can be compared with that of the reference volume shown in Figure 1(b).

Figure 4 :

 4 Figure 4: Standard displacement uncertainty σ u as a function of the element size ℓ for C8-DVC applied to the reference volume shown in Figure 1 when shifted by 0.5 voxel in all three directions. The dashed lines correspond to a power law interpolation [Equation (7)].

Figure 5 :

 5 Figure 5: 3D rendering of transverse (a,b) and longitudinal (c) displacement elds expressed in voxels (1 voxel ↔ 5.1 µm).

Figure 6 :Figure 7 :

 67 Figure 6: Stress / strain plot for the two ways of evaluating the mean strain.The dashed lines show a linear interpolation used to estimate the macroscopic Young's modulus.

Acknowledgements

This work was funded by Agence Nationale de la Recherche under the grant ANR-09-BLAN-0009-01 (RUPXCUBE Project).