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A simple way to use X-ray micro-tomography to infer elastic properties of heterogeneous materials: application to sedimentary rocks


Macroscopic mechanical properties of materials depend directly on their microstructure. Microscopy, and more specifically tomography, is a key method for studying microstructures. Here, we propose a simple way to use an X-ray tomogram to infer local elastic properties. We distinguish between two scenarios of microstructure images. In the first scenario, the material is composed by very apparent phases so the image can be easily segmented into a set of subspaces with homogenous properties. In the second scenario, the image, as that of sedimentary rocks, contains poorly contrasted phases, including strong intra-phase heterogeneities. For this case, we propose an alternative to segmentation techniques in order to factor in material heterogeneities. To do this, we use the local X-ray attenuation to define elastic moduli. Then, we compute up-scaled elastic moduli by solving the mechanical equilibrium. Finally, we confirm our method by comparing the up-scaled elastic moduli to indentation experiments performed at the same scale.

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  1. 1

    Haralick RM, Shapiro LG (1985) Image segmentation techniques. Comput Gr Image Process 29(1):100–132

  2. 2

    Arbelaez P, Maire M, Fowlkes C (2011) Contour detection and hierarchical image segmentation. J Malik IEEE Trans PAMI 33(5):898–916

  3. 3

    Lalaouia L, Mohamadib T, Djaalab A (2015) New method for image segmentation. Proc Soc Behav Sci 195:1971–1980

  4. 4

    Bossa N, Chaurand P, Vicente J, Borschneck D, Levard C, Aguerre-Chariol O, Rose J (2015) Micro- and nano-x-ray computed-tomography: a step forward in the characterization of the pore network of a leached cement paste. Cem Concr Res 67:138–147

  5. 5

    Hubler MH, Gelb J, Ulm F-J (2017) Microtexture analysis of gas shale by XRM imaging. J Nanomech Micromech 7(3):04017005

  6. 6

    Houben ME, Desbois G, Urai JL (2014) A comparative study of representative 2D microstructures in shaly and sandy facies of opalinus clay (Mont Terri, Switzerland) inferred form BIB-SEM and MIP methods. Mar Pet Geol 49:143–161

  7. 7

    Saif T, Lin Q, Bijeljic B, Blunt MJ (2017) Microstructural imaging and characterization of oil shale before and after pyrolysis. Fuel 197:562–574

  8. 8

    Hackley PC, Cardott BJ (2016) Application of organic petrography in North American shale petroleum systems: a review. Int J Coal Geol 163:8–51

  9. 9

    Agrawal A (2009) A technical and economic study of completion techniques in five emerging U.S. gas shale plays. Thesis, Texas A&M University

  10. 10

    Phillips DH, Lannutti JJ (1997) Measuring physical density with x-ray computed tomography. NDT E Int 30(6):339–350

  11. 11

    Kucuk N, Tumsavas Z, Cakir M (2013) Determining photon energy absorption parameters for different soil samples. J Radiat Res 54(3):578–586

  12. 12

    Obliger A, Valdenaire P-L, Capit N, Ulm FJ, Pellenq RJ-M, Leyssale J-M (2018) Poroelasticity of methane-loaded mature and immature kerogen from molecular simulations. Langmuir 34(45):13766–13780

  13. 13

    Pellenq RJ-M, Lequeux N, van Damme H (2008) Engineering the bonding scheme in C–S–H: the iono-covalent framework. Cem Concr Res 38:159–174

  14. 14

    Mavko G, Mukerji T, Dvorkin J (2009) The rock physics handbook, 2nd edn. Cambridge University Press, Cambridge

  15. 15

    Sato H, Ono K, Johnston CT, Yamagishi A (2005) First-principles studies on the elastic constants of a 1:1 layered kaolinite mineral. Am Mineral 90:1824–1826

  16. 16

    Ashby M, Shercliff H, Cebon D (2013) Materials: engineering, science, processing and design, 3rd edn. University of Cambridge, Cambridge

  17. 17

    Farag MM (2014) Materials and process selection for engineering design, 3rd edn. Taylor & Francis, London

  18. 18

    Zargari S, Prasad M, Mba KC, Mattson ED (2013) Organic maturity elastic properties, and textural characteristics of self resourcing reservoirs. Geophysics 78(4):D223–D235

  19. 19

    Moulinec H, Suquet P (1995) A fft-based numerical method for computing the mechanical properties of composites from images of their microstructures. IUTAM Symp Microstruct Prop Interact Compos Mater 37:235–246

  20. 20

    Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157(1–2):69–94

  21. 21

    Valdenaire P-L (2016) Crystal plasticity - transport equation and dislocation density. Ph.D. Thesis, Mines ParisTech

  22. 22

    Schneider M, Ospald F, Kabel M (2016) Computational homogenization of elasticity on a staggered grid. J Mech Phys Solids 105(9):693–720

  23. 23

    Galin LA (1961) Contact problems in theory of elasticity. In: Sneddon IN (ed) Translated by Moss H, North Carolina State College

  24. 24

    Sneddon IN (1965) The relation between load and penetration in the axi-symmetric Boussinesq problem for a punch of arbitrary profile. Int J Eng Sci 3:47–57

  25. 25

    Laubie H, Monfareda S, Radjaï F, Pellenq RJ-M, Ulm F-J (2017) Disorder-induced stiffness degradation of highly disordered porous materials. J Mech Phys Solids 106:207–228

  26. 26

    Laubie H, Radjai F, Pellenq R, Ulm F-J (2017) Stress transmission and failure in disordered porous media. Phys Rev Lett 119:075501

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Financial support was provided by foundation AMU. The authors also thank Carl Zeiss X-ray microscopy for generously conducting the imaging work at their facilities. P.-L. Valdenaire thanks Shell Game Changer program for support.

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Correspondence to Pierre-Louis Valdenaire.

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Valdenaire, P., Perrin, J., Grauby, O. et al. A simple way to use X-ray micro-tomography to infer elastic properties of heterogeneous materials: application to sedimentary rocks. J Mater Sci 55, 3347–3353 (2020). https://doi.org/10.1007/s10853-019-04241-w

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