International audienceA correlation algorithm is proposed to measure full three-dimensional displacement fields in a three-dimensional domain. The chosen kinematic basis for this measurement is based on continuous finite-element shape functions. It is furthermore proposed to account for the presence of strong discontinuities, similarly to extended finite element schemes, with a suited enrichment of the kinematics with discontinuities supported by a (crack) surface. An optimization of the surface geometry is proposed based on correlation residuals. The procedure is applied to analyze one loading cycle of a fatigue-cracked nodular graphite cast iron sample by using computed tomography pictures. Subvoxel crack openings are revealed and measured

François Hild

Jean-Philippe Tinnes

Jean-Yves Buffière

Julien Réthoré

Stéphane Roux

Réthoré, Julien

Tinnes, Jean-Philippe

Roux, Stéphane

Buffiere, Jean-Yves

Hild, François

Hal-Diderot

Extended three-dimensional digital image correlation (X3D-DIC)

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 336 (2008) 643–649http://france.elsevier.com/direct/CRAS2B/Extended three-dimensional digital image correlation (X3D-DIC)Julien Réthoré a,1, Jean-Philippe Tinnes b, Stéphane Roux a, Jean-Yves Buffière b,François Hild a,∗a LMT-Cachan, ENS de Cachan / CNRS UMR 8535 / Université Paris 6 / PRES UniverSud Paris, 61 avenue du Président-Wilson,94235 Cachan cedex, Franceb MATEIS, INSA Lyon / CNRS UMR 5510, 7 avenue Jean-Capelle, 69621 Villeurbanne cedex, FranceReceived 31 January 2008; accepted after revision 25 June 2008Presented by Jean-Baptiste LeblondAbstractA correlation algorithm is proposed to measure full three-dimensional displacement fields in a three-dimensional domain. Thechosen kinematic basis for this measurement is based on continuous finite-element shape functions. It is furthermore proposedto account for the presence of strong discontinuities, similarly to extended finite element schemes, with a suited enrichment ofthe kinematics with discontinuities supported by a (crack) surface. An optimization of the surface geometry is proposed based oncorrelation residuals. The procedure is applied to analyze one loading cycle of a fatigue-cracked nodular graphite cast iron sampleby using computed tomography pictures. Subvoxel crack openings are revealed and measured. To cite this article: J. Réthoré etal., C. R. Mecanique 336 (2008).© 2008 Published by Elsevier Masson SAS on behalf of Académie des sciences.RésuméCorrélation d’images numériques tridimensionnelles étendue (CINÉ3D). Un algorithme de corrélation d’images numé-riques est présenté pour mesurer des champs de déplacement tridimensionnel avec des images à trois dimensions. La basecinématique choisie pour cette mesure est constituée de fonctions de forme (continues) d’éléments finis. Pour rendre comptede la présence de discontinuités fortes, à l’instar des schémas par éléments finis étendus, un enrichissement par une cinématiquediscontinue au travers d’une surface (fissurée) est implémenté. La géométrie de cette surface de discontinuité est optimisée enexploitant les résidus de corrélation. Cet algorithme est utilisé pour l’analyse d’un cycle de chargement d’une éprouvette de fonteà graphite sphéroïdal fissurée en fatigue à l’aide d’images tomographiques. Des ouvertures de fissures sensiblement inférieures àla taille du voxel peuvent être détectées et mesurées. Pour citer cet article : J. Réthoré et al., C. R. Mecanique 336 (2008).© 2008 Published by Elsevier Masson SAS on behalf of Académie des sciences.Keywords: Rupture; Discontinuity; PhotomechanicsMots-clés : Rupture ; Discontinuité ; Photomécanique* Corresponding author.E-mail addresses: julien.rethore@insa-lyon.fr (J. Réthoré), stephane.roux@lmt.ens-cachan.fr (S. Roux), jean-yves.buffiere@insa-lyon.fr(J.-Y. Buffière), hild@lmt.ens-cachan.fr (F. Hild).1 Now at LaMCoS, INSA Lyon / CNRS UMR 5259, 18-20 rue des Sciences, 69621 Villeurbanne cedex, France.1631-0721/$ – see front matter © 2008 Published by Elsevier Masson SAS on behalf of Académie des sciences.doi:10.1016/j.crme.2008.06.006644 J. Réthoré et al. / C. R. Mecanique 336 (2008) 643–649Version française abrégéeL’étude expérimentale d’échantillons fissurés repose habituellement sur des mesures photomécaniques en sur-face [1]. L’analyse d’images de tomographie aux rayons X constitue une voie intéressante pour caractériser égalementles effets tridimensionnels de la propagation de fissure et sa morphologie. Au-delà de la visualisation de la fissure elle-même et de son exploitation mécanique [2], la corrélation d’images numériques peut être appliquée sur des images3D pour donner accès aux trois composantes du champ de déplacement dans l’ensemble du volume. Afin de résoudrele problème de l’identification d’un champ de déplacement minimisant l’écart entre un volume de référence (Fig. 1a)et le volume déformé (Fig. 1b) corrigé, une formulation globale sur l’ensemble du volume d’intérêt est adoptée.Moyennant l’utilisation d’une cinématique 3D enrichie [3], on accède à la mesure de la discontinuité du champ dedéplacement sur la surface de la fissure. La détection de cette surface et son ajustement est un point important et lecritère utilisé ici est basé sur la carte des résidus de corrélation. Un algorithme utilisant une transformation de Radon3D [4] permet d’optimiser une fonction de niveau dont l’iso-zéro décrit la surface de la fissure.Les performances de l’algorithme de corrélation sont caractérisées pour des images déformées artificiellement.L’enrichissement discontinu qui est implémenté permet d’utiliser des tailles d’élément importantes même dans le casde cinématiques discontinues, et donc de conserver des niveaux d’incertitude faibles, un des avantages des techniquesd’élements finis étendus appliquées à la corrélation d’images (Fig. 2).Un échantillon de fonte à graphite sphéroïdal, préalablement fissuré en fatigue, a été imagé in situ à différents stadesde chargement. Outre l’obtention de cartes de champs de déplacements discontinus tridimensionnels (Fig. 3), il estpossible d’obtenir des résidus de corrélation (Fig. 4) dont les faibles valeurs indiquent un bon niveau de confiance dansles résultats obtenus qui permettent ici de caractériser des ouvertures de fissure résiduelles dues à des phénomènes deplasticité confinée (Fig. 5).1. IntroductionThe experimental analysis of cracked samples or structures usually relies upon surface measurements by variousphotomechanical techniques [1]. With the use of 3D imaging techniques (e.g., computed tomography, magnetic res-onance imaging), it is possible to obtain full 3D pictures of the in situ microstructure. Image analysis may alreadyreveal from single images important morphological information about the 3D crack geometry, which may be coupledto a mechanical modeling [2]. Alternatively, the acquisition of a series of images obtained at different stages of amechanical test opens the way for characterizing not only surface but also bulk phenomena. The resulting full dis-placement field allows for a very sensitive technique to detect crack geometries and opening fields with a resolutionwell below the voxel size.The aim of the present Note is to show that the 3D kinematics introduced in the extended finite element method [3]may be used in a 3D correlation algorithm. When dealing with cracks, there is also a need for describing the cracksurface. This is achieved by optimizing the zero-contour of a level set function thanks to the correlation residuals.Since 3D pictures are used, this technique is referred to as “eXtended 3D Digital Image Correlation” (or X3D-DIC),and corresponds to an extension of the X-DIC technique developed with 2D pictures [5].2. Extended 3D digital image correlation2.1. PicturesX-ray microtomography enables one to get 3D pictures of the local density of solids exploiting the attenuation ofa polychromatic X-ray beam. Image and/or volume correlations require a random texture to measure displacements.With the resolution of the acquisition device available at INSA Lyon, cast-iron samples were selected to provideadequate (random) markers. The graphite nodules, whose typical size is 50 µm, provide a random texture in the images(Fig. 1) at the chosen magnification (i.e., the physical size of one voxel is equal to 13.5 µm). A testing machine insidethe tomograph allows for taking 3D pictures of the specimen during a tensile test. A large sample was first cycled in aconventional servohydraulic testing machine to grow a fatigue crack. It was subsequently machined so that the gaugevolume has a length of 4 mm and a square cross-section of 2 × 2 mm2. The crack is visible on the deformed pictureJ. Réthoré et al. / C. R. Mecanique 336 (2008) 643–649 645(a) (b)Fig. 1. Reference (a, load 20 N) and deformed (b, load 400 N) 3D images of a cracked sample (180 × 120 × 120 voxels, 16-bit digitization). Theload is applied along the longitudinal direction (normal to the crack).Fig. 1. Images de référence (a, 20 N) et déformée (b, 400 N) d’un échantillon fissuré (180 × 120 × 120 voxels, codage sur 16 bits). L’effort estappliqué dans la direction longitudinale (normale à la fissure).of the specimen (Fig. 1). The dynamic range of the reconstructed pictures is equal to 16 bits. Only a part of the picture(180 × 120 × 120 voxels) will be analyzed in the sequel by using a standard personal computer.2.2. Enriched kinematicsTo capture the displacement discontinuity across crack faces, a 3D enriched finite element kinematics is chosen [3]for the measured displacement field uu(x) =∑n∈N ,αaαnNn(x)eα +∑n∈Nenr ,αdαnNn(x)Fn(x)eα (1)where x is the local (voxel) coordinate, Fn the enrichment functions, and Nenr is a sub-set of N in which nodes holdenriched degrees of freedom dαn. In the following, the simplest shape functions Nn are implemented, namely, trilinearpolynomials associated with 8-node cube elements (or C8). The enriched kinematics Fn is introduced by exploitingthe partition of unity [6] property of finite element shape functions, namely, for all x,∑n∈N Nn(x) = 1. The followingdefinition holds for the enrichment functionsFn(x) = H(x) −H(xn) (2)where xn is the position of node n [7] and the function H is equal to 1 above the crack surface and 0 below. Itis conveniently implemented from the level set field φ(x) that gives the signed distance to the crack surface. Thefunction H(x) then reduces to the Heaviside function H of the level set. Only nodes whose support is cut by thesurface hold enriched degrees of freedom. In the present case, the crack front is not detected on the surface andthe support of the discontinuity traverses the entire specimen. By adopting the previous discretization scheme of thedisplacement field, the latter is rewritten for all points x asu(x) = [(x)]{U} (3)where [] is a row vector containing the values of all the interpolation functions m, and {U} the column vector ofthe unknown (standard and enriched) degrees of freedom.646 J. Réthoré et al. / C. R. Mecanique 336 (2008) 643–6492.3. Correlation algorithmBy using the previous displacement basis, the passive advection of the texture between the reference f and thedeformed g images (e.g., see Fig. 1) readsg(x) = f (x + u(x)) (4)From the knowledge of f and g, the measurement problem consists in identifying u as accurately as possible. Toestimate u, the quadratic difference, ϕ2 = [g(x) − f (x + u(x))]2 is integrated over the studied domain ΩΦ2 =∫Ωϕ2 dx (5)This functional is minimized with respect to the set of unknown degrees of freedom (i.e., aαn and dαn). This procedureis performed through successive low-pass filtering of the images (from coarse to fine) and first order expansion ofϕ[δu] = g(x) − f (x + u(n)(x)) − ∇f (x + u(n)(x))δu about already resolved displacements u(n) so that each iterationis a simple linear problem in the incremental correction δu. The method proposed herein is therefore a Galerkinapproach to image correlation. In a matrix-vector setting, the linear system to solve globally over the whole region ofinterest Ω reads[M]i{δU} − {b}i = 0 (6)withMinm =∫ ∫Ω(n(x) · ∇f(x + [(x)]{U}i))(m(x) · ∇f (x + [(x)]{U}i))dx (7)andbin =∫ ∫Ω(n(x) · ∇f(x + [(x)]{U}i))(g(x) − f (x + [(x)]{U}i))dx (8)where {δU} is the incremental correction vector. One key aspect of the displacement field in the presence of cracks isthe shape of the crack surface. The latter is described by the zero-contour of a level set. To position the latter, a firstcorrelation evaluation is performed without enriched kinematics (dαn = 0), referred to as C8. The residual map ϕ,which is evaluated for each voxel, is used to detect the crack face. By considering small volumes (e.g., 43 voxels)containing the zone where the residuals are the highest (i.e., where an enrichment is needed), the correspondingelementary residual is Radon transformed to determine the local tangent plane to the crack face. When the crack facelevel set is initialized, an enriched kinematics is sought (i.e., dαn = 0), which is referred to as X-C8.It is worth noting that this detection algorithm does not require the crack opening to be greater than one voxel.In this sense, it is much better suited for our purpose than other segmentation techniques which could have beenconsidered (see e.g., Ref. [8]). Moreover it allows for iterative adjustments of the surface geometry within the sameframework.2.4. PerformancesBefore performing a correlation on the two pictures of Fig. 1, it is worth analyzing the performance of the correla-tion algorithm. A common baseline analysis consists in prescribing known displacements to the reference picture andevaluate the displacement uncertainty. Two cases are considered. First, a uniform displacement ranging from 0 to 1pixel is prescribed. This allows one to estimate both the systematic and fluctuating parts of the error in the measureddisplacement. The fluctuation part is typically much larger than the bias. Thus, only the latter is documented hereinand quantified through the standard deviation of the displacement field. It is shown for both C8 and X-C8 algorithmsin Fig. 2a. Even though the kinematic basis of the extended algorithm is richer than in the C8 version, the overall per-formance is conserved. It levels off similarly to the C8 algorithm, yet at the (low) value of about 2 × 10−2 voxel. Tocapture without enrichment a rapid variation of the displacement such as a discontinuity, a small element size wouldbe needed, thus implying a rather large uncertainty. Conversely, with enrichments, the full benefit of the very smallJ. Réthoré et al. / C. R. Mecanique 336 (2008) 643–649 647(a) (b) (c)Fig. 2. Displacement field uncertainties as functions of the element size for C8 and extended (X-C8) algorithms when a uniform displacement isprescribed (a), or a discontinuous displacement (b). (c) Displacement jump uncertainty when a discontinuous displacement is prescribed.Fig. 2. Incertitudes en déplacement en fonction de la taille des éléments pour un algorithme C8 et X-C8 lorsqu’un champ constant (a) ou discontinu(b) est imposé. (c) Incertitude en saut de déplacement lorsqu’un champ discontinu est imposé.(a) (b)Fig. 3. Displacement field (a) and displacement jump (b) along the loading axis (normal to the crack orientation) expressed in voxels(1 voxel = 13.5 µm) when the two 3D pictures of Fig. 1 are considered.Fig. 3. Champs de déplacement (a) et de saut de déplacement (normal à la fissure) exprimés en voxels (1 voxel = 13.5 µm) pour l’analyse des deuximages de la Fig. 1.uncertainty attached to large element sizes is preserved without any prejudice to the description of the discontinuity.This result is shown in Fig. 2b for the displacement field, and for the displacement jump in Fig. 2c. This analysisshows the benefit of resorting to enriched kinematics when one wants to capture displacement discontinuities on acoarse mesh (i.e., the basic idea of extended finite element procedures).3. ExampleThe displacement field obtained in the direction of the loading axis is shown in Fig. 3. Elements of size 16 × 16 ×16 voxels are considered. The displacement map over the entire volume of interest is shown together with the valuesof the displacement jump over the crack faces on the detected crack geometry. Displacement jumps of the order of1.5 voxel (1 voxel = 13.5 µm) are obtained at one of the corners of the crack, while the sample face average crackopening (i.e., discontinuity of the displacement component normal to the crack surface) is of the order of one voxel.Note however, that as one progresses along the crack towards its front, an opening of 0.1 voxel is resolved.648 J. Réthoré et al. / C. R. Mecanique 336 (2008) 643–649(a) (b)Fig. 4. Thresholded correlation residuals prior to (a) and after (b) enriching the measurement basis.Fig. 4. Résidus de corrélation seuillés avant (a) et après (b) enrichissement de la base de mesure.(a) (b)(c) (d)Fig. 5. Crack opening displacements along the loading axis (normal to the crack orientation) expressed in voxel (1 voxel = 13.5 µm) for fourdifferent load levels: 100 N (a), 200 N (b), 400 N (c) and 20 N (d).Fig. 5. Champs d’ouverture normale à la fissure exprimés en voxel (1 voxel = 13.5 µm) pour différents niveaux de chargement : 100 N (a), 200 N (b),400 N (c) et 20 N (d).J. Réthoré et al. / C. R. Mecanique 336 (2008) 643–649 649Fig. 4 illustrates the enrichment procedure on the analyzed pictures. First, a calculation with no enrichment isperformed and yields the correlation residuals ϕ that are computed for each voxel x (Fig. 4a). For illustration purposes,the residuals are thresholded. When the proposed optimization procedure is followed, the corresponding residuals areshown in Fig. 4b on the crack surface. The average value of ϕ over the entire volume of interest is 1.8% of the dynamicrange of the reference image, except on the crack faces themselves where the average residual is 2.6%. The residualsare very small, thereby proving that the displacement estimate is likely to be trustworthy.Last, Fig. 5 shows the crack opening fields (i.e., the crack opening value supported by the crack surface) forfour different applied load levels. First, when the applied load is equal to 100 N, no opening is measured since thedisplacement jump levels are very close to the uncertainties (Fig. 2c). Second for a load of 200 N, there is a significantopening that becomes even more important when the maximum load level (400 N) is reached. When the sample issubsequently unloaded (20 N), a residual crack opening is observed.4. Summary and perspectivesThe present technique of 3D image correlation allows for the measurement of 3D displacement fields in the pres-ence of cracks. It opens the way to a study of fatigue crack propagation based on an in-depth determination of thecrack geometry. Moreover, stress intensity factors along the crack front may be evaluated by post-processing the mea-sured data, provided the crack front is localized (by another level set). Under cyclic loadings, such a tool may clarifythe role of partial crack closure induced by crack tip plasticity, not only from a global examination, or free surfaceinvestigations, but in the entire sample volume.AcknowledgementsThis work is part of a project (PROPAVANFIS) funded by the CETIM foundation.References[1] P.K. Rastogi (Ed.), Photomechanics, Topics in Applied Physics, vol. 77, Springer, Berlin, Germany, 2000.[2] E. Ferrié, J.-Y. Buffière, W. Ludwig, A. Gravouil, L. Edwards, Fatigue crack propagation: In situ visualization using X-ray microtomographyand 3D simulation using the extended finite element method, Acta Mat. 54 (4) (2006) 1111–1122.[3] N. Moës, A. Gravouil, T. Belytschko, Non-planar 3D crack growth by the extended finite element and level sets. Part I: Mechanical model, Int.J. Numer. Meth. Eng. 53 (11) (2002) 2549–2568.[4] H. Barrett, The Radon transform and its applications, in: Progress in Optics, North-Holland, Amsterdam, The Netherlands, 1984, pp. 217–286.[5] J. Réthoré, S. Roux, F. Hild, From pictures to extended finite elements: Extended digital image correlation (X-DIC), C. R. Mecanique 335(2007) 131–137.[6] I. Babuska, J.M. Melenk, The partition of unity method, Int. J. Numer. Meth. Eng. 40 (4) (1997) 727–758.[7] G. Zi, T. Belytschko, New crack-tip elements for XFEM and applications to cohesive cracks, Int. J. Numer. Meth. Eng. 57 (15) (2003) 2221–2240.[8] B. Bourdin, A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford–Shah functional, Numer. Math. 85(2000) 609–646.

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Extended three-dimensional digital image correlation (X3D-DIC)

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