Haizhou Liu

François Hild
email: francois.hild@ens-paris-saclay.fr

Quantifying 3D Crack Propagation in Nodular Graphite Cast Iron Using Advanced Digital Volume Correlation and X-ray Computed Tomography

Keywords: digital volume correlation, 3D crack propagation, 3D crack shape, stress intensity factor, nodular graphite cast iron

Deformation analysis reveals crack tearing, plastic zones, and

Introduction

Nodular graphite cast iron, also known as ductile iron, has been widely recognized and utilized across various industries, including automotive, wind power, and heavy machinery, due to its unique combination of strength, ductility, and fatigue resistance [START_REF] Lacaze | Cast iron: a historical and green material worthy of continuous research[END_REF]. In these applications, the components made of such material are often subjected to cyclic loading, leading to the initiation and propagation of fatigue cracks, which seriously affect the in-service life. The growth of fatigue cracks in materials is often evaluated using the well-known Paris' law [START_REF] Paris | Closure to "Discussions of 'A Critical Analysis of Crack Propagation Laws[END_REF], which establishes a relationship between the stress intensity factor range and the crack growth rate. As a consequence, it is crucial to monitor fatigue crack propagation and measure stress intensity factors (SIFs) for calibrating predictive models.

In the case of two-dimensional (2D) images, digital image correlation (DIC) has gained popularity in experimental and fracture mechanics applications. DIC enables for full-field measurements near crack tips, allowing for the estimation of fracture mechanics parameters such as crack opening displacement (COD) [START_REF] Yates | Quantifying crack tip displacement fields with DIC[END_REF], energy release rate [START_REF] Becker | An approach to calculate the J-integral by digital image correlation displacement field measurement[END_REF], and stress intensity factors (SIFs) [START_REF] Cneill | Estimation of stress intensity factor by digital image correlation[END_REF][START_REF] Gonzáles | A J-integral approach using digital image correlation for evaluating stress intensity factors in fatigue cracks with closure effects[END_REF]. However, when dealing with three-dimensional (3D) crack propagation in thick samples [START_REF] Yusof | Effect of overload on crack closure in thick and thin specimens via digital image correlation[END_REF], relying solely on 2D measurements of CODs and SIFs fails to provide a comprehensive understanding of the bulk behavior.

High-resolution X-ray microcomputed tomography (XCT) enables for the 3D observation of cracks in optically opaque materials such as metals and alloys. To measure 3D displacement fields near crack fronts in loaded specimens, two distinct approaches are employed based on tomography images, namely, 3D particle tracking (PT) [START_REF] Toda | A 3D measurement procedure for internal local crack driving forces via synchrotron X-ray microtomography[END_REF] and digital volume correlation (DVC) [START_REF] Éthoré | Extended three-dimensional digital image correlation (X3D-DIC)[END_REF]. The PT method involves tracking the motions of individual microstructural markers within the material. It is suitable for cases where the images contain discrete features against a homogeneous background (e.g., precipitates within a metallic alloy). Qu et al. [START_REF] Qu | Local crack driving force analysis of a fatigue crack by a microstructural tracking method[END_REF] measured the local crack driving forces in an aluminum alloy fatigue crack, which provided valuable insights into the crack behavior. Toda et al. [START_REF] Toda | In situ observation of ductile fracture using X-ray tomography technique[END_REF] quantified the J-integral and driving force during an in situ tensile fracture test on an aluminum alloy by tracking porosities. Yang et al. [START_REF] Yang | Analysis and esearch on Experimental Process of Water Thermal igration of Freeze-Thaw Cracked ock Based on Particle Tracking ethod and Thermal Imaging Technology[END_REF] performed thermal imaging tests combined with PT to quantify the temporal and spatial changes of water-heat migration both inside and outside rock fractures. Different from PT, DVC focuses on correlating the gray level contrast between reference and deformed volumes.

It has proven to be effective in estimating CODs and SIFs. Furthermore, the measurement uncertainty level of DVC has been demonstrated to be smaller than that of PT [START_REF] Limodina | 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]. éthoré et al. [START_REF] Éthoré | Three-dimensional analysis of fatigue crack propagation using X-ay tomography, digital volume correlation and extended finite element simulations[END_REF] analyzed three-dimensional fatigue crack propagation in the cross-section of a precracked cast iron sample while evaluating SIFs along crack fronts using eXtended DVC.

Lachambre et al. [START_REF] Lachambre | Extraction of stress intensity factors for 3D small fatigue cracks using digital volume correlation and X-ray tomography[END_REF] monitored the growth of a fatigue crack inside a nodular cast iron and computed SIFs along the curved front. Valle et al. [START_REF] Valle | New development of digital volume correlation for the study of fractured materials[END_REF] developed Heaviside based DVC to assess the local COD and SIF under mode I loading conditions in a fractured polymer sample.

Jin et al. [START_REF] Jin | Crack propagation in fine grained graphites under mode I and mixed-mode loading, as observed in situ by microtomography[END_REF][START_REF] Jin | Assessment of the fracture toughness of neutron-irradiated nuclear graphite by 3D analysis of the crack displacement field[END_REF], Shen et al. [START_REF] Shen | Combined evaluation of Young modulus and fracture toughness in small specimens of fine grained nuclear graphite using 3D image analysis[END_REF], and Chen et al. [START_REF] Chen | Fracture toughness evaluation of a nuclear graphite with non-linear elastic properties by 3D imaging and inverse finite element analysis[END_REF] quantified 3D crack propagation of finegrained nuclear graphite under different loading modes using DVC and finite element analyses. Koko et al. [START_REF] Koko | 3-Dimensional analysis of fatigue crack fields and crack growth by in situ synchrotron X-ray tomography[END_REF] characterized the 3D fatigue crack fields of nodular graphite cast iron by combining DVC and finite element simulations. The position of the crack front was determined with three independent techniques and the SIFs were assessed with interaction integrals.

Although DVC is feasible in quantifying damage growth, it still suffers from a big challenge, namely, ill-posedness [START_REF] Bornert | Assessment of digital image correlation measurement errors: methodology and results[END_REF]. An unavoidable compromise between the measurement uncertainty and the spatial resolution has to be found. A global approach to DVC [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: application to Portevin-Le Châtelier bands[END_REF] may overcome this limitation, which generally assumes continuous displacement fields. A straightforward connection between experiments and simulations is obtained when considering the same FE mesh [START_REF] Avril | Overview of identification methods of mechanical parameters based on full-field measurements[END_REF][START_REF] Shakoor | On the choice of boundary conditions for micromechanical simulations based on 3D imaging[END_REF]. However, due to noise sensitivity and limited available information (i.e., intensity levels) within a small spatial resolution (e.g., voxel scale), additional regularization techniques (e.g., Tikhonov regularization [START_REF] Tikhonov | Solutions of ill-posed problems: Vh Winston[END_REF], mechanical regularization [START_REF] Neggers | Complete mechanical regularization applied to digital image and volume correlation[END_REF]) are still desirable to further circumvent the ill-posedness of registration. oreover, a brittle damage law [START_REF] Hild | Damage measurements via DIC[END_REF] and node splitting [START_REF] Vargas | Calibration of cohesive parameters for a castable refractory using 4D tomographic data and realistic crack path from in-situ wedge splitting test[END_REF] may be used to capture displacement discontinuities across cracked surfaces. A mesh refinement scheme [START_REF] Liu | Damage evolution in coal under different loading modes using advanced digital volume correlation based on X-ray computed tomography[END_REF] may also be useful to describe the complex behavior of cracks, especially for the crack surface and its front. Such DVC scheme has demonstrated its efficacy to quantify damage in brittle materials including coal [START_REF] Liu | Damage evolution in coal under different loading modes using advanced digital volume correlation based on X-ray computed tomography[END_REF] and sandstone [START_REF] Liu | ultiscale Damage Analyses of ed Sandstone in niaxial Compression Based on Advanced Digital Volume Correlation[END_REF].

In many cases, the evaluation of 3D displacement fields near the cracked surface has been limited to small cracks, typically with CODs less than 4 voxels (abbreviated as vx hereafter) [START_REF] Éthoré | Three-dimensional analysis of fatigue crack propagation using X-ay tomography, digital volume correlation and extended finite element simulations[END_REF]. The mesh size in the vicinity of the crack front is generally larger than 8 vx [START_REF] Lachambre | Extraction of stress intensity factors for 3D small fatigue cracks using digital volume correlation and X-ray tomography[END_REF]. Yet a precise estimation of COD and SIF relies on accurate descriptions of the crack surface. Thus, voxel-scale resolutions are essential for capturing the details of the crack opening behavior. Furthermore, the whole process of gradually cracking from small to large cracks (i.e., COD > 10 vx) is also of significance to understand the fracture behavior. Therefore, it is crucial to further improve DVC techniques to handle both small and large crack openings for a better description of crack propagation.

In this study, a voxel-scale digital volume correlation (DVC) approach utilizing 8-noded cube elements is employed to quantify the 3D crack propagation and key parameters (i.e., displacement and strain fields, CODs, and SIFs) of a nodular graphite cast iron under tensile (in situ) loading. The aim is to show that the DVC framework enabled very large CODs to be measured. The first part introduces the material, in-situ experimental setup, and reconstructed volumes to provide an initial understanding of the 3D crack shape. The next section describes advanced DVC and SIF extraction algorithms. A performance assessment is first conducted using a virtual ode-I crack case. Last, DVC results of the real test case are presented and discussed.

Material and experiment

1 Material

This study focuses on the analysis of a ferritic cast iron specimen. The composition of the material consists of 3.4 wt.% C, 2.6 wt.% Si, 0.05 wt.% g, 0.19 wt.% n, 0.005 wt.% S and 0.01 wt.% P. The ferritic matrix was obtained through casting and subsequent heat treatment, involving ferritization at 880°C followed by air cooling. The volume fraction of graphite nodules was approximately 14%. The carbon and iron matrix exhibit significant differences in X-ray attenuation, thereby resulting in high contrast for DVC registration purposes [START_REF] Limodina | 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 material properties include a Young's modulus of 175 GPa, a yield stress of 315 Pa, and a Poisson's ratio of 0.27 [START_REF] Limodina | 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]. These properties correspond to the macroscopic response of the material with no distinction between the ferritic matrix and the graphite nodules. A horizontal pre-crack was introduced through fatigue loading on a larger specimen, from which the analyzed sample was subsequently extracted.

2 In situ tensile test

A region of interest ( OI) of 220×270×940 vx 3 (i.e., 1.4×1.7×5.9 mm 3 ) was imaged using the micro-CT scanner (North Star Imaging X50+) of L PS with a voxel resolution of 7.3 𝜇m.

A total of nine tomographic scans was acquired at different stages of tensile loading. The displacement was kept constant during each scan. The reconstructed volumes are denoted as 𝑉 0 , 𝑉 1 , 𝑉 2 , 𝑉 3 , 𝑉 4 , 𝑉 5 , 𝑉 6 , 𝑉 7 , and 𝑉 8 , as depicted in Fig. 1. The initial four scans (𝑉 0 , 𝑉 1 , 𝑉 2 , and 𝑉 3 ) revealed minimal crack reopening, which refers to precrack reopening under the influence of a new tensile load. In contrast, scans 𝑉 4 , 𝑉 5 , and 𝑉 6 highlighted more crack openings and propagation. Last, scans 𝑉 7 and 𝑉 8 indicated ductile fracture with significantly larger crack openings associated with ductile tearing. It is worth noting that with the selected testing machine, the sample is not visible when mounted.

for which 𝛟 𝑖 (𝐱) is the shape function associated with the degrees of freedom 𝜐 𝑖 . A Gauss Newton iterative procedure is then applied, leading to linear systems [START_REF] Hild | Toward 4D mechanical correlation[END_REF]. echanical regularization related to the local equilibrium gap [START_REF] Claire | A finite element formulation to identify damage fields: the equilibrium gap method[END_REF] is further considered to render the solution less affected by acquisition noise and reconstruction artifacts

Φ 𝑚 2 = {d𝛖} T [𝐊 𝒎 ] T [𝐊 𝒎 ]{d𝛖} (3)
where [𝐊 𝒎 ] is the rectangular stiffness matrix of the regularized nodes, and {d𝐮} the displacement increment between two analyzed volumes. Such regularization is based on the assumption of Hencky elasticity at the local level [START_REF] Hild | ultiscale DIC applied to pantographic structures[END_REF], which is beneficial to prevent overfiltering, especially in plastic regions. Therefore, the total cost function reads

Φ 𝑡 2 = Φ 𝑐 2 + 𝜔 𝑚 Φ 𝑚 2 (4)
The weight 𝜔 𝑚 is proportional to a length ℓ 𝑟𝑒𝑔 raised to the power 4 [START_REF] Hild | An extended and integrated digital image correlation technique applied to the analysis of fractured samples: The equilibrium gap method as a mechanical filter[END_REF], namely, the socalled regularization length. In addition, a master-slave [START_REF] Shariati | Probing Constitutive odels of Bohus Granite with In Situ Spherical Indentation and Digital Volume Correlation[END_REF] regularization was used to mitigate displacement fluctuations on the Dirichlet boundaries. In such DVC scheme, the spatial resolution can be lowered to the voxel-scale [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF]. The damaged elements traversed by the cracked surface are removed to capture a large-scale crack opening. ore details on such schemes are found in ef. [START_REF] Liu | Damage evolution in coal under different loading modes using advanced digital volume correlation based on X-ray computed tomography[END_REF].

Stress intensity factor extraction

In 2D analyses [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF], Williams' series [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF] have been employed to determine SIFs for nearly straight crack paths [START_REF] Oux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Vargas | Comparison of two full-field identification methods for the wedge splitting test on a refractory[END_REF]. However, there have been limited applications in the 3D context.

Notably, Limodin et al. [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 Lachambre et al. [START_REF] Lachambre | Extraction of stress intensity factors for 3D small fatigue cracks using digital volume correlation and X-ray tomography[END_REF] calculated SIFs by performing leastsquares projection of displacement fields measured via DVC. In this study, SIFs are determined from displacement fields obtained through an advanced DVC approach. Initially, the implementation is presented using a virtual cracking test case, and then it applied to an in situ tensile test conducted on nodular graphite cast iron.

Williams' series

It is assumed that each nodal layer orthogonal to the mean direction of the crack front can be treated separately. For each nodal layer, the reference 3D displacement field reads

𝐮(𝑟, 𝜃) = ∑ ∑ ω 𝑛 𝑗 𝑝 𝑛 𝑛=𝑝 𝑖 𝐼𝐼𝐼 𝑗=𝐼 𝚿 𝑛 𝑗 (𝑟, 𝜃) (5)
where the vector fields are defined in the polar coordinate system (𝑟, 𝜃) centered about the crack front position, 𝜔 𝑛 𝑗 the amplitudes to be determined, n the index of Williams' series (in this work, n = -1, 0, …,7), and j = I is related to the mode I (opening) regime

𝚿 𝑛 𝐼 = 𝑟 𝑛/2 2𝜇√2𝜋 [𝜅exp(𝑖 𝑛 2 𝜃) - 𝑛 2 exp ( 𝑖𝜃 -𝑖 𝑛 2 𝜃) + {(-) 𝑛 + 𝑛 2 )exp(-𝑖 𝑛 2 𝜃}] (6
𝚿 𝑛 𝐼𝐼𝐼 = 𝑟 𝑛/2 2𝜇√2𝜋 cos( 𝑛(𝜃-𝜋) 2 ) (8)
with 𝜅 = 3 -4𝜈 for plane strain conditions, and 𝜈 the Poisson's ratio. The amplitudes 𝜔 0

𝑗

give access to the rigid body translations, amplitudes 𝜔 1 𝑗 to SIFs, 𝜔 2 𝐼 and 𝜔 2 𝐼𝐼𝐼 provide the T-stress, 𝜔 2 𝐼𝐼 offers one rigid body rotation. Higher-order (sub-singular) fields explain deviations from the assumption of infinite medium theory [START_REF] Affagard | Identification method for the mixed mode interlaminar behavior of a thermoset composite using displacement field measurements and load data[END_REF]. In addition, the first mode I super-singular field (n = -1) enables the crack front position to be determined by canceling out the corresponding amplitude [START_REF] Oux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF]. Assuming that there is a small offset 𝑑 of the crack front position along the propagation path, it is expressed as [START_REF] Hild | Digital image correlation and fracture: an advanced technique for estimating stress intensity factors of 2D and 3D cracks[END_REF]

𝑑 = 2𝜔 -1 𝐼 𝜔 1 𝐼 (9)

Displacement projection

Considering a cracked surface with a roughly estimated crack front (Fig. 2), DVC measured displacements are projected onto Williams' series over circular domains (i.e., pacman planes). The Williams' matrix [𝐒 𝝎 ] collecting all unitary fields 𝚿 𝑛 𝑗 for each pacman plane is constructed, and the least squares minimization leads to a linear system

[𝐒 𝝎 ]{𝝎} = {𝐮 𝐩 } (10)
in which {𝝎} gathers all unknown amplitudes of Williams' series, and {𝐮 𝐩 } the column vector of measured nodal displacements. A least squares step is run to estimate all unknown amplitudes, and a new crack front position is estimated by using the update to the crack offset.

The previous step is repeated until all the absolute values of the shift 𝑑 are all less than one voxel. Last, the corresponding SIFs are extracted from the converged solution.

Results and discussions

Virtual crack testing case

Generating cracked volume

A virtual crack testing case is presented to assess the performance of voxel-scale DVC. x y z 5% of the dynamic range) to simulate acquisition noise and reconstruction artifacts. Such noise level was estimated from the standard deviation of gray level residuals between the first two scans (i.e., scans 𝑉 0 and 𝑉 1 ), which is rather high in order to be challenging.

Voxel-scale DVC results

The registration between the above reference and deformed volumes was performed with advanced DVC. An adaptive pyramidal mesh (i.e., ℓ = -8-4-2 vx) was used to account for voxel-scale damage while ensuring computational efficiency. The regularization length and the master/slave boundary regularization mesh size are equal to 80 vx and 48 vx, respectively.

SIFs extraction results

The crack front and SIF were determined using the procedure outlined in Section 3.2. In the lower-left corner of Fig. 5(a), a pacman-like mesh is displayed along the crack front. Here, the mesh size in the xOz plane ranged from coarse (6 vx) to fine (3 vx), while the mesh size along the y-direction was set to 6 vx. The removed inner and outer circles have a radius of 10 vx and 150 vx, respectively. As observed in Fig. 5(a), both measured crack fronts obtained via DVC gray level residuals (i.e., crack defined as gray level residuals exceed the noise level threshold of 10 GL) and Williams' series were in strong agreement with the prescribed one.

The corresponding absolute errors in the measurement of the crack front along the sample thickness are almost within 1 vx. The estimated mode I SIF 𝐾 𝐼 (left y-axis in Fig. 5(b)) follows an identical trend with the exact solution but exhibits a slight decrease, possibly due to the influence of enforced elastic regularization. Despite this decrease, the relative error (Fig. 5 (b)) remains within the range of 1% to 1.5%, indicating a good performance of the proposed identification scheme.

DVC results and discussions of an in-situ tensile test

1 Standard uncertainty

A cubic OI with a size of 200×200×200 vx 3 extracted from the reference volume (i.e., scan 𝑉 0 ) was used to evaluate the standard measurement uncertainty of the proposed DVC framework. The equivalent speckle size, determined by autocorrelation [START_REF] Inguet | On the validation of a priori estimates of standard displacement uncertainties in T3-stereocorrelation[END_REF] was found to be 2.4 vx. The root-mean-square ( S) image gradients along the three directions, denoted as S(∇𝑓 𝑥 ) = 13.1, S(∇𝑓 𝑦 ) = 12.4, and S(∇𝑓 𝑧 ) = 12.3 GL/vx, respectively, indicated that the image contrast was almost identical along the three directions. To assess the standard displacement and maximum principal strain (i.e., the strain tensor within one element is solved by finite difference of nodal displacements) uncertainties, white Gaussian noise with a standard deviation of 10 gray levels was added. The standard deviations of displacement and maximum principal strain were approximately 0.1 vx and 0.001, respectively, as shown in Fig. 6. Without regularization, increasing the mesh size resulted in reduced standard uncertainty because more voxels were used to evaluate each nodal displacement.

Introducing regularization significantly reduced the standard uncertainty for a given mesh size following a power law trend observed in all three fitted curves. This regularization allowed for measuring displacements at very small spatial resolutions (e.g., 10 voxels in this study)

while maintaining low uncertainty. However, excessive regularization may lead to oversmoothing, since elasticity is enforced in areas where it is not applicable, such as in the cracked zone. Consequently, a compromise between mechanical regularization and image correlation residuals has to be found for each scan.

2 Mesoscale damage analyses

A mesh with split nodes on the whole OI was produced (Fig. 7 The thresholded gray level residual fields (Fig. 8) for the different scans reveal distinct characteristics of the crack. Each scan exhibits two vertically distributed bands due to imperfect volume reconstruction. Furthermore, regions with high residuals indicate the presence of the crack, enabling for the visualization of its shape and propagation. Notably, the crack surface appears to be nearly horizontal and progressively propagated inward with the applied loading. Threshold = 50 gray levels for scans 𝑉 3 , 𝑉 4 , and 𝑉 5 , or 80 GL for scans 𝑉 6 , 𝑉 7 , and 𝑉 8 .

Voxel-scale damage analysis

Crack growth updating strategy

To investigate the crack development at the voxel-scale, a crack growth updating strategy based on advanced DVC was performed on a small OI containing the crack, as shown in Fig. 9. There are two cycles in the whole process, namely, the first cycle for crack quantification for the current scan (orange arrow in Fig. 9(a-d)), and the second cycle for crack identification in the next scan (blue arrow in Fig. 9(e-g)). The proposed methodology consists of the following steps:

1) The reference volume 𝑉 0 and the deformed one 𝑉 𝑛 (i.e., the n-th scan, here n = 4) are taken as initial input to estimate the crack path with the registration residuals (Fig. 9(a)), which are extracted from the thresholded gray level residual field (i.e., scan 𝑉 4 in Fig. 8);

2) The corresponding crack surface (Fig. 9(b)) is fitted based on the residual points. A coarsegraining approach is used (Fig. 10(a-d)) in which the residual points far from the fitted surface are removed in each fitting step to suppress outliers. The crack front is estimated by fitting and eventually checked by the uncertainty of CODs;

3) An adaptive multiscale mesh (i.e., ℓ = 24-8-4-2 vx) for advanced DVC calculations is generated based on the fitted crack surface (Fig. 9(c)). The damaged elements are removed to describe discontinuity. The voxel-scale mesh (i.e., 2 vx) surrounding the crack surface allows for a precise description of the crack morphology. The master mesh size for boundary regularization is equal to 48 vx;

3D crack shape and propagation

From the above-mentioned procedure, the 3D growing crack surface in each scan was determined based on the gray level residual fields. Furthermore, considering the uncertainty of COD, particularly near the crack front, allowed for the characterization of the 3D crack shape and its propagation as illustrated in Fig. 11(a). The crack exhibits a tendency to bypass or traverse the graphite nodules, thereby resulting in a wavy crack surface. In the early loading stages (scans 𝑉 1 to 𝑉 3 ), the crack reopened slightly with a very small COD below the CT resolution. In the medium loading stage (scans 𝑉 4 to 𝑉 5 ), the crack front penetrated deeper into the sample leading to more crack opening (above the CT resolution). However, there was only a small and stable growth of the crack front observed between scans 𝑉 4 and 𝑉 5 . In the final loading stages (from scans 𝑉 6 to 𝑉 8 ), the crack front experienced significant inward propagation, accompanied by changes in the 3D crack shape. These findings provide insights into the progressive behavior of the crack under (re)loading conditions.

The positions of the crack front along the sample thickness for different scans quantitatively reveal the crack shape and its propagation (Fig. 11(b)). The crack front shape had large fluctuations in scans 𝑉 1 and 𝑉 2 with a mean crack length of approximately 200 μ𝑚 (or 30 vx). For scan 𝑉 3 , the crack front exhibited an inclined profile with deeper propagation on the left along the sample thickness than on the right. The mean crack length is approximately 400 μ𝑚 (or 60 vx), which is much smaller than that in scan 𝑉 4 (i.e., 600 μ𝑚 or 90 vx). A smaller increment of crack length (i.e., 80 μ𝑚 or 90 vx) is observed between scans 𝑉 4 and 𝑉 5 , which proves that the opening was gradual. In addition, the crack length gradually increased from the left to the right along the sample thickness indicating that crack opening was not uniform. Larger increments in crack length for the last three scans indicate ductile tearing and the crack front shape was almost arch-like due the presence of plasticity.

Displacement fields in the bulk of the sample

The displacement fields in the reference and deformed configuration of the last scan (𝑉 8 ) are displayed in Fig. 12, with the top and bottom rows representing each configuration, respectively. The crack introduced a significant and noticeable discontinuity in the 𝑢 𝑥 and 𝑢 𝑧

Von Mises strain fields in the bulk of the sample

Based on the previous analysis, it was determined that the sample underwent plastic deformation. To assess the shape and size of the plastic zone in different scans, von ises equivalent strain fields near the crack front were computed. According to a linear constitutive model, the yield strain (𝜖 𝑦 ) was roughly estimated via Hooke's law given the yield stress (𝜎 𝑦 ) and Young's modulus (E) at the macroscopic level (i.e., 𝜖 𝑦 = 𝜎 𝑦 /E = . × -3 ).

Considering the numerous sources of uncertainty, such as acquisition noise and the presence of complex microstructure and deformation, the plastic zone was defined as the region where the von ises strain exceeded nine times the standard strain uncertainty or five times the estimated yield strain (i.e., the strain offset was equal to 9 × -3 ). The crack did not open and propagate significantly before scan 𝑉 5 . Therefore, it was reasonable to assume the absence of additional plastic deformation from scans 𝑉 1 to 𝑉 4 . With the selected offset strain, the plastic zone remained very small in the first four scans. Figure 13 illustrates the thresholded von ises strain fields from scans 𝑉 5 to 𝑉 8 . The different viewing angles presented in the top and bottom rows enhance the understanding of the distribution characteristics on the front and rear faces of the sample. Interestingly, the thresholded von ises strain field was only observed at the

x y z crack front in scan 𝑉 4 , thereby indicating that the current deformation is still within the elasticity stage. Based on the spatial patterns and variations of the von ises strain fields, it is observed that the plastic zone gradually increased with the applied loading. Specifically, the average plastic zone radius (i.e., the distance between the crack front and the profile in each yOz plane of plastic zone) for scans 𝑉 5 , 𝑉 6 , 𝑉 7 , and 𝑉 8 are approximately 300 μ𝑚, 450 μ𝑚, 660 μ𝑚 , and 815 μ𝑚 , respectively. The localized regions with a gradual lighter color surrounding the crack tip are indicative of the areas experiencing significant deformation and plastic flow. In particular, the presence of high strain concentrations in the immediate vicinity of the crack front suggests large deformation and stress concentrations. Furthermore, a roughly symmetrical pattern is observed around the crack tip in all four scans, indicating a nearly aligning tensile loading condition. Slight asymmetry suggests that crack propagation was not uniform, potentially leading to variations in crack growth behavior.

Crack opening displacements

Thanks to voxel-scale measured displacement fields, CODs along three directions (as described in Fig. 18 in Appendix A) for different scans were estimated by calculating the displacement jumps between the upper and lower surfaces of the crack mouth. The norm of these displacement jumps was considered as the average COD (Fig. 14) for which the detection level (i.e., 0.08 vx or 0.60 μm) was defined as 4 times the standard uncertainty based on the first two scans. The COD decayed closer to the crack front. In addition, the COD distribution along the sample thickness (i.e., x) direction was not uniform, namely, the COD level on the The solid and dashed lines represent the best fitting curves and uncertainty level, respectively.

SIFs estimation

To quantify the level of stress intensification at the crack front, the SIFs (i.e., 𝐾 𝐼 , 𝐾 𝐼𝐼 , and 𝐾 𝐼𝐼𝐼 in Fig. 16) were estimated using the post-processing method from scans 𝑉 1 to 𝑉 5 . The reference volume (i.e., scan 𝑉 0 ) and its deformed state (i.e., corrupted by white Gaussian noise with a standard deviation of 10 GL) were used to evaluate the measurement uncertainty of SIFs.

The standard uncertainties of 𝐾 𝐼 , 𝐾 𝐼𝐼 , and 𝐾 𝐼𝐼𝐼 were . , . a d . MPa√m, respectively.

The mode I SIF 𝐾 𝐼 shown in Fig. 16(a) is almost identical along the crack front in each scan meaning a symmetric distribution of mode I stress, and it suggests that crack propagation is more likely to occur in a uniform manner. Similar observations were reported for graphite [START_REF] Shen | Combined evaluation of Young modulus and fracture toughness in small specimens of fine grained nuclear graphite using 3D image analysis[END_REF].

Sub-voxel crack opening in the first three scans (i.e., 𝑉 1 to 𝑉 3 ) were quantified accurately.

Small but clear differences are observed among them. For scans 𝑉 4 and 𝑉 5 , the crack opening is larger than one voxel. It was thus expected that mode I SIF 𝐾 𝐼 would increase in a more pronounced way. The increase of 𝐾 𝐼 with loading (especially for scans 𝑉 4 and 𝑉 5 ) illustrates higher levels of opening stress intensity near the crack front.

In Fig. 16

Correction of the crack front by Williams' series

The crack fronts in each scan were identified using gray level residuals from DVC and Williams' series, as shown in Fig. 17. A good agreement is observed between the two methods from scans 𝑉 1 to 𝑉 5 , thus indicating the effectiveness of the proposed voxel-scale DVC scheme in locating the crack front when plastic deformation was limited. However, a significant bias occurred in the last three scans (i.e., 𝑉 6 -𝑉 8 ) due to the increased presence of plasticity.

This bias is attributed to two main factors. First, the presence of more complex deformations

Comparison of RMS residuals between Williams' series and DVC measurements

The nodal displacements in each pacman-like plane (as described in Fig. 20 to 𝑉 3 , which was expected because of the presence of the crack with sub-voxel opening. For scans 𝑉 4 and 𝑉 5 , displacement residuals are a bit higher than the uncertainty level but still within a low range (i.e., less than 0.07 vx), which indicates that the SIFs evaluation was trustworthy. However, higher displacement residuals with a few tenths of voxel are observed from 𝑉 6 to 𝑉 8 proving gradual dominance of plasticity near the crack front.

The comparison of gray level residuals (as shown in Fig. 18

Conclusion

An advanced DVC scheme combined with XCT was employed to investigate 3D crack propagation in nodular graphite cast iron under in situ tensile loading. The following main conclusions were drawn from this study:

• Voxel-scale crack analyses based on advanced DVC is a comprehensive methodology for accurately quantifying crack growth. By utilizing a crack growth updating strategy, the crack path and surface were precisely captured even for very large openings. This technique provided valuable insights into the characteristics of 3D crack shape and its propagation, including tendencies to pass around or traverse graphite nodules resulting in a rough cracked surface. Three distinct crack growth stages were identified under dominant mode I loading, namely, crack reopening, crack propagation, and ductile tearing.

•

The observed discontinuities in the displacement fields near the crack front indicated significant crack opening and tearing, which were due to ductile fracture of the sample. oreover, the von ises strain fields illustrated the plastic zone surrounding the crack front and its gradual increase with the applied load. The presence of high strain intensifications near the crack tip suggested large deformations and localized stress intensifications.

• CODs extracted from the measured displacement fields provided quantitative insights into crack propagation. The non-uniform distribution of CODs along the sample thickness with a decay toward the crack front, highlighted variations in opening regime. In addition, the entire cracking stages from reopening to ductile tearing was clearly revealed by CODs, thus indicating the efficiency of the proposed framework. However, the presence of substantial plastic deformation near the crack front was challenging in accurately pinpointing the crack front, thereby indicating the need for further advancements in capturing deep cracked regions.

•

The estimation of SIFs offered valuable information about stress intensification at the crack front. The dominance of mode I, with limited influences from modes II and III, showed that crack reopening and early propagation occurred under essentially uniform levels along the crack front. Large SIFs in scans with significant crack propagation indicated the gradually more pronounced plasticity. Variations in shear and tearing deformation characteristics along the sample thickness further highlighted the complex distribution during crack growth.

In summary, voxel-scale crack analyses using advanced DVC in conjunction with XCT, provided valuable insights into crack propagation, 3D crack shape, displacement and strain fields, CODs and SIFs. These findings contribute to a better understanding of the crack behavior and mechanical response under in situ loading conditions. The reported results also showed deviations from the purely elastic framework based on Williams' series when extensive plastic zones and ductile tearing were observed at the end of the experiment. ore advanced models are needed to describe and quantify such phenomena (e.g., via finite element simulations).

Engineering fracture mechanics. 1987;28:101-12. https://doi.org/10.1016/0013-7944(87)90124-X.

Fig. 1

 1 Fig. 1 3D rendered volumes for different scans for increasing applied displacements (the corresponding forces are indicated in parentheses)

  ) j = II to mode II (in-plane shear) to mode III (tearing)

Fig. 2

 2 Fig. 2 Schematic 3D rendering of 2D pacman planes centered about the crack front used for the projection of the displacement onto Williams' series. The black arrows define the crack frame in polar coordinate systems (𝑟, 𝜃) and the purple plane highlights the cracked surface.

Figure 3 1 )

 31 Figure3illustrates the process for generating deformed volumes, which involves the following steps:1) Consider a reference volume 𝑉 0 (𝐱 𝐯 ) (Fig.3(a)) whose size is 210 × 210 × 350 vx3 extracted from the original scan 𝑉 0 , in which 𝐱 𝐯 are integer-valued voxel positions;2) A voxel-wise displacement field 𝐮(𝐱 𝐯 ) (Fig.3(b)) is calculated according to the pure mode I crack solution in Williams' series with a sloping straight crack front;3) Find non-integer positions 𝐱 𝟎 in 𝑉 0 corresponding to integer positions 𝐱 ̅ 𝐯 in the deformed volume 𝑉(𝐱 ̅ 𝐯 ) based on the inverse deformation mapping 𝐱 𝟎 = 𝛃 -1 (𝐱 ̅ 𝐯 ), in which 𝜷(𝐱 𝟎 ) = 𝐱 𝟎 + 𝐮(𝐱 𝟎 ) = 𝐱 ̅ 𝐯 . In the present implementation, a Newton iterative scheme was used to determine 𝐱 𝟎 ; 4) Generate 𝑉(𝐱 ̅ 𝐯 ) = 𝑉 0 (𝐱 𝟎 ) (Fig.3(c)) using spline interpolation of the gray levels. A flat crack surface is observed in the middle of the z-direction with a straight crack front (in blue). Last, Gaussian white noise is added with a standard deviation of 10 gray levels (i.e.,

Fig. 3

 3 Fig. 3 Schematic views of generating a virtually cracked volume. (a) eference volume. (b) Voxel-wise displacement field 𝑢 𝑧 . (c) Deformed volume for which white Gaussian noise was added.

Figure 4 (

 4 Figure 4(a) displays the measured displacement field 𝑢 𝑧 with the overlaid multiscale mesh. The prescribed (Fig. 3(b)) and measured (Fig. 4(a)) displacement fields show strong agreement, thereby indicating that the proposed DVC framework can efficiently capture displacement discontinuities with voxel-scale meshes around the cracked surface. Furthermore, the histogram of nodal displacement residuals (defined as the difference between measured and prescribed quantities) is reported in Fig. 4(b) to assess measurement errors. The presence of a Gaussian-like distribution within the range of -0.1 to 0.1 vx indicates high quality performance even in such challenging condition. Higher displacement residuals are mainly observed at the boundary and near the crack front due to the limitations of the master-slave and mechanical

Fig. 4

 4 Fig. 4 (a) easured displacement field 𝑢 𝑧 expressed in voxels. (b) Histogram of nodal displacement residuals.

Fig. 5 (

 5 Fig. 5 (a) Crack front determined by different methods. The used pacman-like mesh is shown in the lower-left corner. (b) Identified mode I SIF and corresponding relative error.

Fig. 6

 6 Fig. 6 Standard measurement uncertainties as functions of regularization length for (a) nodal displacements and (b) maximum principal strain

Fig. 7 (

 7 Fig. 7 (a) DVC mesh with split nodes (in red) laid over the reference rendering volume. The mesh size is equal to 10 vx. (b) echanical vs. DVC residuals for different regularization lengths.

Fig. 8

 8 Fig. 8 Thresholded gray level residual fields (unit: GL) for different stages of loading.

x y z 4 ) 5 )Fig. 9

 459 Fig. 9(g); 7) The 6 steps are repeated until all scans are studied.

Fig. 11

 11 Fig. 11 Crack characteristics for the different scans. (a) 3D renderings of the cracked surface. (b) Corresponding crack front positions along the sample thickness.

  displacement components, which correspond to modes III and I, respectively. This discontinuity enables for the evaluation of crack tearing and opening at a local scale. On the other hand, only minor discontinuities are observed in the mode II (i.e., 𝑢 𝑦 ) displacement field, indicating limited shearing. Furthermore, arc-shaped displacement distributions are observed near the crack front and along the thickness direction of the sample. The displacement jumps in terms of 𝑢 𝑥 , 𝑢 𝑦 , and 𝑢 𝑧 at the crack mouth were approximately 40 μ𝑚 (or 5.4 vx), 20 μ𝑚 (or 2.7 vx), and 300 μ𝑚 (or 41.1 vx), respectively. Significant crack opening and tearing indicate the occurrence of ductile fracture within the bulk of the sample. This fracture resulted in a strong alteration in the geometry before and after deformation.

20 Fig. 12

 2012 Fig. 12 Bulk displacement fields (unit: μ𝑚) in the reference (top row) and deformed (bottom row) configuration of the last scan (𝑉 8 ). (a,d) 𝑢 𝑥 , (b,e) 𝑢 𝑦 , (c,f) 𝑢 𝑧 components.

Fig. 13

 13 Fig. 13 Thresholded von ises strain fields under two different viewing angles. (a,e) Scan 𝑉 5 . (b,f) Scan 𝑉 6 . (c,g) Scan 𝑉 7 . (d,h) Scan 𝑉 8 .

  Fig. 14 COD fields (unit: μm) for different scans. (a) 𝑉 1 , (b) 𝑉 2 , (c) 𝑉 3 , (d) 𝑉 4 , (e) 𝑉 5 , (f) 𝑉 6 , (g) 𝑉 7 and (h) 𝑉 8 .

Fig. 15

 15 Fig. 15 COD profiles for three different sections. (a) Front line. (b) iddle line. (c) ear line.

  Fig. 16 (a) 𝐾 𝐼 , (b) 𝐾 𝐼𝐼 and (c) 𝐾 𝐼𝐼𝐼 SIF profiles along the crack front for the first five scans.

  Fig. 17 Crack front for different scans determined by DVC residuals and Williams' series.

  in Appendix B) was obtained based on the Williams' series and DVC measurements, respectively. The corresponding S displacement residuals (i.e., 𝑟𝑚𝑠(𝒖 𝑊 -𝒖 𝑑𝑣𝑐 ) , where 𝒖 𝑊 is the Williams' series calculated nodal displacements and 𝒖 𝑑𝑣𝑐 the DVC measured one ) for each scan are plotted in Fig. 18 (a). The uncertainty level (i.e. 0.028 vx as depicted in Fig. 18 (a) in gray line) is defined as 5 times the standard deviation of displacement assessed from the first two scans. Displacement residuals less than the uncertainty level are observed from scans 𝑉 1

  (b) and Fig.21in Appendix C) between the Williams' series (i.e., gray level residual is calculated by the difference between the reference volume and deformed volume corrected by the nodal displacements evaluated by Williams' series) and DVC measurements exhibits a consistent trend as analyzed in displacement residuals. Specifically, the difference in S gray level residuals remains very small from scans 𝑉 1 to 𝑉 3 , then experiences a slight increase from scans 𝑉 4 to 𝑉 5 , and undergoes a substantial surge from scans 𝑉 6 to 𝑉 8 . The examination of both types of residuals is correlated with plastic activity from scans 𝑉 6 to 𝑉 8 , which explains why only the SIFs up to volume 𝑉 5 were reported.

Fig. 18

 18 Fig. 18 esiduals comparison for each scan between Williams' series and DVC measurements.

  along sample thickness. (b) Corresponding difference in S gray level residuals.

Table 1

 1

	. DVC hardware parameters
	Tomograph	North Star Imaging X50+
	X-ray source	X ayWorX XWT-240-CT
	Target/Anode	W
	Filter	none
	Voltage	140 kV
	Current	80 𝜇A
	Tube to detector	473.5 mm
	Tube to object	20.0 mm
	Detector	Dexela 2923
	Definition	1536 × 1944 pixels
	Number of projections	1000
	Angular amplitude	360°
	Frame average	3
	Frame rate	11 fps
	Acquisition duration	25 min
	econstruction algorithm	filtered back-projection
	Gray levels amplitude	8 bits
	Volume size	220 × 270 × 940 voxels (after crop)
	Field of view	1.6 × 2.0 × 6.9 mm 3 (after crop)
	Image scale	7.4 𝜇m/voxel

3. Methods 3.1 Digital volume correlation

	𝐮(𝐱, {𝝊}) = ∑ 𝜐 𝑖 𝛟 𝑖 (𝐱) 𝑖
	In global DVC, the displacement fields u(x, {𝜐}) between the reference volume 𝑉 0 and
	deformed volume 𝑉 𝑛 is measured by minimizing the L -norm of the gray level residuals [32]
	Φ 𝑐 2 = ∑ [𝑉 0 (𝐱) -𝑉 𝑛 (𝐱 + 𝐮(𝐱, {𝝊})] 2 ROI	(1)
	Where the sought displacement field 𝐮(𝐱, {𝝊}) is parameterized in an FE sense (e.g., 8-noded
	cube elements)

Ackno ledgements

Dr. N. Limodin and Prof. J.-Y. Buffière kindly provided the precracked sample. Dr. Amine Bouterf performed the in situ test. This work has benefited from the support of the French "Agence Nationale de la echerche" through the "Investissements d'avenir" Program under the reference AN -10-EQPX-37 AT ECA.


Y Borjon-Privé

A Dalles

R Hamayon

C Jacquemoud

J-L Lambert

V Vaté

CHAMANISMES ET CHRISTIANISMES EN SIBERIE

Depuis 2002, le GSRL accueille un groupe de chercheurs, anthropologues, spécialisé sur la Sibérie. Celui-ci s'est étoffé avec les années jusqu'à constituer un des plus importants groupes de recherche sur cette aire en France. Ces chercheurs inscrivent leur activité de manière complémentaire dans le cadre du Centre des études mongoles et sibériennes et s'investissent dans la revue Études mongoles et sibériennes, centrasiatiques et tibétaines qui lui est associée 1 . C'est en tant que « groupe », avec une perspective comparative et en faisant dialoguer nos terrains et nos recherches que nous avons choisi, dans cette étude, d'aborder le religieux en Sibérie. Notre comparaison se fonde sur une diversité d'approches, d'expériences, mais aussi de contextes temporels, puisque nous représentons trois générations de chercheurs, ayant effectué des recherches de terrain à des moments différents, ajoutant ainsi une perspective diachronique. Notre comparaison se fonde également sur une variété de régions 2 , habitées par différents peuples. Au sein de ces espaces, nos recherches ont porté essentiellement sur les peuples autochtones. Ces peuples, minoritaires au sein d'un territoire qui représente 80% de l'espace russe, parlent des langues non-slaves appartenant à différentes familles linguistiques. Pour des raisons notamment historiques, géographiques, démographiques, et politiques, ces peuples présentent des situations contrastées. Tout d'abord du point de vue démographique, d'après les chiffres officiels accessibles, qui montrent [p. 504] essentiellement des tendances : les Bouriates sont aujourd'hui environ 461 000 (dans toute la Russie, mais seulement 290 000 en Bouriatie), les Altaïens 77 000, les Khantys 30 900, les Mansis 12 300, les Tchouktches autour

de 16 000, les Dolganes 7800, les Nanaïs 12 000 (aussi appelés Hezhe en Chine, où ils sont environ 5000), et les Nganassanes 1 200. Cet aspect démographique a des incidences sur le degré de préservation des langues vernaculaires et sur la situation politique de ces peuples, notamment en termes de représentation dans les instances au pouvoir, dans un contexte où la plupart sont extrêmement minoritaires. Tous ces peuples sont traditionnellement chasseurs, pêcheurs et/ou éleveurs, même si un grand nombre d'autochtones exercent aujourd'hui des professions de toutes natures, comme comptable ou instituteur, au village ou en ville. [START_REF] Hamayon | Chamanismes, fondements et pratiques d'une forme religieuse d'hier et d'aujourd[END_REF][START_REF] Hamayon | Une figure pour le Ciel ? ou De la difficulté de construire une 'religion nationale[END_REF].

Parmi cette diversité, le christianisme ou, pourrait-on dire, les christianismes occupent une place importante et ce sur l'ensemble des terrains qui nous intéressent, au contraire du bouddhisme qui n'a eu un impact qu'au sud de la Sibérie. Dès le XVIIIe siècle, la plupart des Sibériens sont en contact avec le christianisme orthodoxe. Cette rencontre avec l'orthodoxie a lieu par deux voies principales : l'oeuvre des missionnaires sur le terrain liée à la politique impériale et la diffusion d'une orthodoxie populaire par les colons slaves (Znamenski 1999b[START_REF] Kan | Russian Orthodox Missionaries at Home and Abroad: The Case of Siberian and Alaskan Indigenous Peoples[END_REF].

Après la période soviétique, pendant laquelle l'ensemble des pratiques religieuses a subi les conséquences de la politique athéiste, mais à des niveaux distincts3 , différentes dénominations protestantes, dont certaines appartiennent au protestantisme évangélique, se sont implantées en Russie à partir des années 1990, avec plus ou moins de succès selon les régions et selon les dénominations. Quelle que soit l'incidence de ces dénominations protestantes quant au nombre de convertis « parler d'interaction, c'est en effet d'abord marquer une rupture avec une lecture univoque du colonialisme comme processus de domination économico-politique, comme mouvement commandé par des forces ou des structures ou dirigé par les agents de l'État […], ou encore comme un système de conversion unilatérale de sujets subordonnés, avec son corollaire idéologique la résistance ou le refus des dominés. L'idée est que la simple équation de la domination et de la résistance ne permet pas de comprendre la complexité de la situation coloniale. L'interaction désigne un processus d'influences réciproques entre les partenaires en présence et d'articulation entre les dimensions (politiques, économiques, culturelles) en jeu ».

Cette exploration impose d'accorder une attention particulière aux pratiques ou au discours sur ces pratiques. La question des interactions religieuses constitue un axe de recherche peu développé dans l'espace post-soviétique. En effet, après avoir suscité un intérêt certain après la fin de l'Union Soviétique, aujourd'hui le religieux n'est plus autant au coeur des agendas de recherches. Ce point peut être mis en perspective avec les difficultés liées au terrain où le religieux, en particulier la situation de certains mouvements protestants parfois non enregistrés, peut constituer un sujet délicat à aborder.

Interactions entre orthodoxie et chamanismes à partir du XVIIIe siècle

Lors de l'introduction de l'orthodoxie, l'interpénétration du chamanisme et du christianisme est mutuelle. Du point de vue autochtone, il n'y a pas vraiment d'incompatibilité formelle. En tant que système de croyances, le chamanisme ne comporte (d'une manière générale) que des entités spirituelles qu'il peut concevoir comme homologues des humains parce qu'elles doivent être « nourries », selon deux oppositions fondées essentiellement sur les dichotomies humain/animal, vivant/mort. Ainsi, les saints chrétiens sont parfois réinterprétés à partir du modèle des âmes de morts malheureux à qui il faut rendre un culte pour qu'elles ne puissent pas nuire pas (Hamayon 1990, p. 683-697 ;[START_REF] Hamayon | A Challenging Technique Involving Imaginary Figures of Power among the Pre-Soviet West-Buryats[END_REF]. Tout comme les [p. 508] Bouriates ont plus tard intégré dans leurs représentations les âmes des Communards parisiens de 1871, en affirmant que celles-ci ont fini par se noyer dans le lac Baïkal où elles sont devenues des esprits donneurs de poisson (Hamayon 1990, p. 582), certains peuples chamanistes, par exemple les Dolganes, ont auparavant adopté Saint Nicolas et Saint Élie et ont nourri les icônes avec du sang, comme on nourrit les autres supports d'esprits (Popov 2003, p. 160-161, p. 213-214, p.250 ;Vasil'ev 1909, p. 271 ;Borjon-Privé, terrains 2014[START_REF] Hamayon | Une figure pour le Ciel ? ou De la difficulté de construire une 'religion nationale[END_REF]. Dans les représentations comme dans la pratique rituelle, c'est Dieu que le chamane craint le plus : seuls les spécialistes les plus forts peuvent se rendre chez lui pour des requêtes ou recevoir des conseils et bénédictions. C'est dans sa direction que le chamane conduit, ou fait mener par ses esprits auxiliaires, l'âme des personnes en danger pour les sauver de la mort (Vasil'ev 1909, p. 281). Au moment des funérailles, c'est encore vers le ciel et Dieu que sont envoyés les défunts, baptisés ou non. Chez les Dolganes comme chez les Russes, le cercueil est enterré. La croix orthodoxe, systématiquement présente, y compris pour les sépultures de chamanes, comporte des représentations d'oiseau en bois. En effet, l'oiseau est considéré comme la meilleure figuration possible de l'âme afin que celle-ci puisse rejoindre le paradis céleste. Ensuite, cette âme pourra revenir dans le monde des hommes, dans la même famille, et le prénom du nouveau-né sera celui du défunt (Popov 2003, p. 126 ;Borjon-Privé, terrains 2013-2017). Nommé bourkhanisme, du nom de la divinité nouvellement vénérée Ak-Bourkhan 6 , le 6 Ak-Bourkhan signifie « Bouddha blanc » ou « divinité blanche » en mongol. Bourkhanisme est le nom russe donné au mouvement, tandis qu'il est nommé Ak-T'aŋ (« Foi/Manière de faire blanche ») ou Altaj-T'aŋ (« Foi/Manière de faire altaïenne ») chez les autochtones. Notons à ce sujet que les Altaïens, à la différence de leurs voisins Mongols ou Touvas, n'ont pas été intensément bouddhisés au cours de l'Histoire. C'est la raison mouvement a évincé les chamanes du devant de la scène religieuse et les a remplacés, notamment, par les bardes. Le mouvement bourkhaniste comporte également une grande part de nationalisme. Ainsi, dans un chant de louange dédié à Ojrot-Khan 7 , l'envoyé d'Ak-Bourkhan, nous découvrons comment le messager est présenté comme un héros qui viendra combattre les Russes, rendra l'Altaï aux Altaïens et les libérera du joug des colons (Danilin 1993, p. 95).

Par ailleurs, dans certains endroits, des Russes de Sibérie délaissent l'orthodoxie pour se tourner vers le chamanisme. Comme le montre Znamenski pour le nord-est sibérien (1999a, p. 23), « afin de survivre dans l'environnement local difficile, [les Russes] devaient suivre les manières de faire des autochtones [native ways] ». En Sibérie, différents peuples ont été conduits par les Russes, à partir de la fin du [p. 510] XVIIe siècle, à prêter serment sur l'ours, et non sur les Évangiles ou sur la croix comme les chrétiens, ce qui a modifié le statut de l'ours dans les représentations autochtones (Lambert 2007(Lambert -2008)). Mais on trouve aussi des occurrences où, dans l'ouest sibérien, des Russes se mettent à jurer sur l'ours comme les chamanistes ! De la même façon, à proximité de l'embouchure du fleuve Amour, des Russes accrochaient encore récemment des supports d'esprits empruntés aux Nivkhs au-dessus de la porte d'entrée pour protéger leur maison (A. Dalles, terrain 2013 à Khabarovsk).

Si le rapport du chamanisme à l'orthodoxie n'est sans doute pas le même qu'au protestantisme, des interactions se produisent également dans des contextes de conversion à différentes formes du protestantisme évangélique contemporain.

Interactions entre chamanismes et différentes formes du protestantisme

Comme a pu l'observer Clément Jacquemoud dans l'Altaï aujourd'hui, le chamanisme, même s'il a perdu de sa vigueur avec la période soviétique, reste présent dans l'imaginaire et est perçu par les Altaïens comme la « forme religieuse traditionnelle ». Les discours des autochtones qui adoptent le christianisme, en particulier évangélique, montrent que les convertis se sentent obligés de se positionner par rapport au chamanisme et à ses acteurs. C'est souvent après leur pour laquelle l'idée de Bourkhan n'est associée en Altaï qu'au mouvement religieux bourkhaniste (Jacquemoud 2015, p. 68-69). 7 Ojrot-Khan est une figure légendaire qui personnifie les khans de l'« empire » oïrate (ou Khanat dzoungar, 1616-1756), et dont les hauts faits ont été transmis de génération en génération (Znamenski 2005, p. 31). Potapov (1991, p. 26, 69, 180) fait du principe héréditaire l'un des « dogmes » fondamentaux du chamanisme des Turcs de Sibérie. Ainsi, concernant les Altaïens, on sait que les esprits auxiliaires du chamane qui l'élisent sont les âmes tös des défunts de sa lignée paternelle ou maternelle (Anohin 1924, p. 23-30 ;Potapov 1991, p. 67 ;[START_REF] Verbickij | Altajskie inorodcy. Sbornik ètnografičeskih statej i issledovanij[END_REF]Verbickij [ ] 1993, p. 63), p. 63). Les bardes sont eux aussi concernés par ce principe (Jacquemoud 2015, p. 91 ;Surazakov 1973, p. 439). Beaucoup plus au sud, dans la Sibérie extrême orientale, chez les Nanaïs, les derniers chamanes ont disparu à la fin des années 1990 [START_REF] Bulgakova | Nanai Shamanic Culture in Indigenous Discourse, Fürstenberg/Havel: Kulturstiftung Sibirien[END_REF]. Même si les rituels et les pratiques chamaniques sont aujourd'hui peu connus des Nanaïs, cette méconnaissance est perçue par les intéressés comme dangereuse : en effet, on considère que les héritiers potentiels -ceux qui devraient devenir chamane mais aujourd'hui ne sont pas en mesure d'identifier leur ascendance chamanique -sont alors hantés par des esprits sans maîtres [START_REF] Bulgakova | Nanaï Shamans under Double Oppression. Was the Persecution by Soviet Power Stronger than the Power of Shamanistic Spirits?[END_REF]. Il semble alors, selon les données recueillies par Anne Dalles, que la possibilité de sortir de cet héritage dangereux justifie que certains se tournent vers les Églises protestantes. Dieu est supposé pouvoir mettre fin à un héritage chamanique qu'il n'est plus possible d'assumer.

Un autre problème concerne la question des âmes des morts dont on n'est plus en mesure d'assurer le bon traitement sans chamane. En effet, lors d'importants rituels, le grand chamane nanaï emmenait les âmes des morts dans l'au-delà. Or, le dernier de ces grands rituels a eu lieu dans les années 1970 [START_REF] Smoljak | Šaman: ličnost', funkcii, mirovozzrenie (narody Nižnego Amura)[END_REF]. En l'absence de chamane, ces âmes devenues errantes ne peuvent pas atteindre le monde des morts et, dit-on, hantent les vivants. Elles sont considérées comme la cause des dépressions, de l'alcoolisme et d'un taux de suicide très élevé.

Figure 1 -

 1 Figure 1 -Terrains en Sibérie (carte ©Yann Borjon-Privé)

  Chez les Dolganes, le Dieu chrétien est bien intégré dans le système religieux autochtone depuis le XIXe siècle : il est parfois créateur omnipotent, parfois seulement organisateur d'un monde préexistant et sans humains. Il est un référent invisible dans le ciel, dominant à distance le monde des hommes et des esprits sur terre. On lui attribue différentes dénominations en fonction de ce qui est attendu de lui. Dans ces représentations, les saints jouent le rôle d'assistants de Dieu, parachevant ou réaffirmant son intervention. Quant à la Mère de Dieu, la Vierge Marie, son rôle est essentiellement lié à la fécondité des femmes ; elle veille aussi à la survie des enfants et favorise la naissance des héros (Tret'jakov 1869, p. 420-423).

  Cette relation entre chamanisme et christianisme ne doit pas être analysée comme une influence du christianisme sur le chamanisme. Il s'agit bien d'un dialogue ou d'une « conversation » pour reprendre le terme utilisé par les Comaroff dans le contexte sud-africain pour rendre [p. 509] compte de la relation et de la conversion au christianisme (Comaroff et Comaroff 1991, 1997). En effet, certains peuples chamanistes sont allés « piocher » dans le christianisme ce qui pouvait leur servir. Par exemple, certains ont même pu s'approprier le rituel du baptême et s'en servir dans leurs pratiques animistes. C'est notamment le cas des Ougriens de l'Ob qui, à partir du début du XIXe siècle, convertissent rituellement l'ours grâce à des aspersions d'eau. Par cette opération, la bête tuée en forêt devient l'enfant vivant envoyé aux hommes par le Père, le Dieu du ciel, et c'est alors à ce titre que l'ours est honoré lors d'une importante fête collective qui intègre de multiples éléments venant du monde russe (Lambert 2015). D'autres peuples sont même allés plus loin. Par exemple, les Altaïens, en réaction à l'arrivée massive des colons russes, ont développé au début du XXe siècle un mouvement millénariste autochtone de grande ampleur s'inspirant notamment de l'eschatologie chrétienne.

  conversion que certains commencent à s'intéresser au chamanisme, dont ils n'avaient précédemment qu'une connaissance superficielle. Dans cette approche, le chamanisme est perçu comme un repoussoir fascinant : il se présente comme référent d'une pratique à proscrire, tout en constituant une source d'inspiration continue. Certains s'approprient par exemple des capacités rituelles généralement associées au chamanisme, telles que la guérison. Le parler en langues est également comparé au langage chamanique. Dans le nouveau contexte, ces aptitudes ne sont plus considérées comme le fruit d'une élection ou d'une transmission par les esprits, mais comme résultant de la volonté directe de Dieu. On assiste à une réinterprétation, par les convertis, du discours des spécialistes religieux du chamanisme, affirmant que ce ne peuvent être les esprits qui transmettent des capacités extraordinaires (car ils sont perçus comme des émanations du diable) mais Dieu lui-même. L'élection devient alors [p. 511] singulière, indépendante du voeu des esprits des ancêtres chamanes, qui liaient auparavant le spécialiste religieux à une lignée 8 . Certaines formes d'expression « traditionnelles » sont également employées pour exprimer sa nouvelle foi. C'est, par exemple, le cas d'un chanteur d'épopée altaïen, d'un barde, converti au pentecôtisme, qui a créé une épopée retraçant la vie de Jésus, s'affirmant inspiré par le Saint-Esprit. Par cette création, ce barde fait dialoguer les conceptions évangéliques avec les conceptions sibériennes -pour lesquelles la récitation de l'épopée est un rituel en tant que tel dont on attend une efficacité comparable à celle d'un rituel proprement chamanique (Jacquemoud 2017a, 2017b). Dans le Nord-est sibérien, en Tchoukotka, conversations et réinterprétations sont également à l'oeuvre en contexte de conversion au protestantisme évangélique. Certaines de ces réinterprétations servent à justifier les pratiques du passé relevant du chamanisme tout en invitant à les abandonner. Par exemple, un parallèle est établi entre la fête d'élevage réalisée à l'automne pour propitier la période de reproduction des rennes (Ŋènrir''un), et un passage du Lévitique qui décrit le rite de purification d'un lépreux (chapitre XIV, versets 10 à 14). Dans ce passage du Lévitique, il est écrit que, d'un agneau sacrificiel égorgé, « le prêtre prend du sang […] [et] le met sur le lobe de l'oreille droite de celui qui se purifie, sur le pouce de sa 8 Leonid Pavlovič

  main droite et sur le pouce de son pied droit ». Cette onction est mise en relation par certains convertis avec un moment du rituel d'élevage qui consiste à effectuer avec le sang d'un renne abattu rituellement des dessins sur le visage et certaines parties du corps des membres d'une même famille. Chaque famille possède et réalise des dessins qui lui sont spécifiques (pour plus d'information sur ce rituel cf. Vaté 2011 ; 2013). Cette mise en parallèle par les convertis eux-mêmes de ces deux types de pratiques, l'une observable sur le terrain et l'autre décrite dans la Bible, permet aux acteurs concernés d'affirmer que les Tchouktches sont bien présents dans la Bible, puisque leurs rituels y sont décrits. L'idée mise en avant est la suivante : si les ancêtres ne vivaient pas tout à fait [p. 512] comme des chrétiens, ils ne vivaient pas non plus sans Dieu, puisque leurs pratiques sont issues de la Bible. Le message implicite semble être qu'il ne faut pas condamner trop facilement et trop radicalement les pratiques tchouktches du passé. Mais, maintenant que « Jésus est venu », les Tchouktches doivent suivre le Nouveau Testament et non plus vivre en fonction de l'Ancien Testament (à ce sujet, cf. aussi Vaté 2009).

C

  'est auprès d'un prêtre orthodoxe, le Père Nikanor, installé dans la région depuis 2011, que certains Nanaïs de Najhin ont trouvé la solution. Si les âmes des morts ne peuvent plus être conduites dans un au-delà lointain, celles-ci sont maintenant amadouées par des offrandes sur les tombes dans l'espoir qu'elles ne nuisent pas aux vivants. Le Père Nikanor permet aussi aux Nanaïs d'allumer des feux pour les enterrements où l'on brûle traditionnellement les objets et les offrandes destinés au défunt. Pour satisfaire les âmes des morts, on réutilise par ailleurs les commémorations orthodoxes habituellement réalisées pour les morts, comme celle effectuée six jours après Pâques (Radonica). [p. 513] Conclusion Au travers d'une pluralité d'exemples sibériens provenant tant d'époques que de régions différentes, on voit que le chamanisme en tant que tel ne peut pas légitimement être étudié en dehors de son contexte et sans tenir compte des systèmes religieux, essentiellement chrétiens et bouddhistes, avec lesquels il a nécessairement interagi depuis plusieurs siècles. De par leur forme, ces interactions sont extrêmement variées et les circulations sont multiples. On peut parler également de l'importance du chamanisme pour les populations slaves traditionnellement orthodoxes établies en Sibérie. On peut s'interroger sur l'existence de modalités d'interaction avec le chamanisme plus spécifiques à l'orthodoxie ou aux différentes formes de protestantismes présentes en Sibérie. La comparaison des terrains montre la complexité des situations et l'importance des contextes et, par là-même, la difficulté de faire émerger ces spécificités. Dans la façon dont s'établissent ces interactions entre chamanismes et christianismes, les personnalités et les interprétations des représentants religieux, et plus largement des acteurs, jouent un rôle central. Cette réflexion mérite cependant des approfondissements. Les contextes de la Russie impériale, soviétique et contemporaine engendrent des modalités de « conversation religieuse » différentes. L'Union Soviétique était, on le sait, une société extrêmement ritualiste. Certaines sociétés sibériennes avaient bien compris qu'elles pouvaient récupérer la forme de certains de ces rituels soviétiques locaux pour les investir de leurs propres valeurs et représentations chamaniques. Dans le contexte post-soviétique actuel, pour des sociétés profondément marquées par des décennies d'idéologie athéiste voulant promouvoir un « homme nouveau », le recours au religieux n'est plus nécessairement un

En même temps, malgré la politique athéiste, un certain nombre de pratiques, essentiellement privées, se sont parfois maintenues. Celles-ci ont pu servir de base, en certaines circonstances, à des démarches de réinventions contemporaines.

On peut discuter de la définition de ce qu'est un converti. Nous employons ici ce terme pour désigner les individus qui ont choisi d'adopter un champ de référence, pour leurs pratiques et valeurs, relevant d'un domaine religieux autre que celui dans lequel ils ont été élevés/ont grandi (ici en l'occurrence, le christianisme), sans présumer de la durée que revêtira cette adoption.

Les traductions des citations de l'anglais au français sont de Virginie Vaté.