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Multiscale Damage Analyses of Red Sandstone in Uniaxial Compression Based on Advanced Digital Volume Correlation

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Abstract

X-ray computed tomography (XCT) combined with digital volume correlation (DVC) has proven to be a powerful tool for bulk deformation measurements of rocks subjected to in-situ experiments. Traditional DVC (i.e., local/global approaches) is generally applied to roughly characterize damage growth by mapping strain localization. However, due to the brittleness of sandstone, damage detection and quantification are very challenging for small spatial resolutions, especially at the microscale (i.e., voxel levels). In this paper, an advanced global approach (i.e., multimesh DVC) was developed, in which mechanical regularization, brittle damage law, and mesh refinement were considered. Such DVC scheme provides an adapted mesh based on damage activity to measure crack opening displacements at the mesoscale and eventually at the voxel scale. An in-situ uniaxial compression test applied to red sandstone was carried out. Kinematic fields and damage development were analyzed at different scales via multimesh DVC. Macroscale (i.e., specimen-scale) analyses showed the overall deformation characteristics of the specimen by mean strain curves. Mesoscale (i.e., element-scale) results displayed the crack opening displacement fields at sub-voxel resolution. Microscale (i.e., voxel-scale) studies focused on local damage growth using extremely small spatial resolutions (i.e., one voxel). All these investigations quantitatively revealed microcrack initiation, propagation, and coalescence to form the final macrocrack, providing a powerful proof for understanding damage mechanisms in rocks.

Highlights

  • Macroscale, mesoscale, and microscale damage characterization and quantification in uniaxial compression of red sandstone.

  • Multimesh DVC considering mechanical regularization, damage, and multimesh refinement to measure full-field deformation.

  • Very fine (sub-voxel) crack opening displacement fields via voxel-scale multimesh DVC to reveal damage initiation and growth.

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Availability of Data and Materials

The raw/processed data required to reproduce these findings will be available upon request.

Code Availability

The code will be available upon request.

Notes

  1. The following convention is used. Positive strains refer to expansion, and negative strains refer to contraction.

Abbreviations

\(b\) :

Exponent of power-law interpolation

\(D\) :

Damage variable

f :

Volume in the reference configuration

\(\left\{\mathbf{f}\right\}\) :

Nodal force vector

g :

Volume in the deformed configuration

k :

Weight factor for uncertainty assessment

\(\left[\mathbf{K}\right]\) :

Stiffness matrix

\(\left[{\mathbf{K}}_{D}^{\mathbf{e}}\right]\) :

Damaged elementary stiffness matrix

\({\ell}\) :

Mesh size

\({{\ell}}_{\mathrm{reg}}\) :

Regularization length

\({\varvec{u}}\) :

Sought displacement field

\({v}_{i}\) :

Nodal displacement

\(\left\{{\varvec{v}}\right\}\) :

Displacement column vector

\(w\) :

Mechanical weight

\({\rho }_{\mathrm{dvc}}\) :

Dimensionless correlation residual

\({\rho }_{\mathrm{mec}}\) :

Mechanical residual

\({\rho }_{r}\) :

Correlation residual

\({S}_{\mathrm{COD}}\) :

Standard deviation of CODs

\({\epsilon }_{1}\) :

Maximum principal strain

\({\epsilon }_{3}\) :

Minimum principal strain

\({\epsilon }_{v}\) :

Volume strain

\(\sigma\) :

Standard deviation

\({\sigma }_{z}\) :

Applied force

\({\sigma }_{\mathrm{COD}}^{\mathrm{std}}\) :

Standard COD uncertainty

\({\sigma }_{\mathrm{COD}}\) :

COD uncertainty

\({\mu }_{\mathrm{COD}}\) :

Mean value of CODs

\({V}_{D}\) :

Damaged volume

\({V}_{\mathrm{COD}}\) :

Fraction of damaged volume

\({\sigma }_{{\mathrm{u}}_{x}}\), \({\sigma }_{{\mathrm{u}}_{y}}\), \({\sigma }_{{\mathrm{u}}_{z}}\) :

Standard uncertainties of nodal displacements

\({\sigma }_{{\epsilon }_{1}}\), \({\sigma }_{{\epsilon }_{2}}\), \({\sigma }_{{\epsilon }_{3}}\) :

Standard uncertainties of element-wise principal strains

\({\Phi }_{c}^{2}\) :

L2-Norm of correlation residual

\({\Phi }_{m}^{2}\) :

L2-Norm of force residual

\({{\varvec{\Psi}}}_{i}^{C8}\) :

Shape function of C8 elements

AE:

Acoustic emission

COD:

Crack opening displacement

DVC:

Digital volume correlation

FE:

Finite element

FWHM:

Full width at half maximum

GLR:

Gray-level residual

PMMA:

Polymethyl methacrylate

ROI:

Region of interest

RMS:

Root mean square

VOI:

Volume of interest

μXCT:

X-ray computed microtomography

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Funding

This work was financially supported by the National Key Research and Development Program of China (2022YFC2904102), Science Fund for Creative Research Groups of the National Natural Science Foundation of China (52121003), the Open-fund of State Key Laboratory for Fine Exploration and Intelligent Development of Coal Resources, and the China Scholarship Council.

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Authors and Affiliations

  1. School of Mechanics and Civil Engineering, China University of Mining and Technology, Beijing, 100083, China

    Haizhou Liu & Lingtao Mao

  2. CentraleSupélec, ENS Paris-Saclay, CNRS LMPS-Laboratoire de Mécanique Paris-Saclay, Université Paris-Saclay, 91190, Gif-sur-Yvette, France

    Haizhou Liu, Xuyang Chang & François Hild

  3. State Key Laboratory for Fine Exploration and Intelligent Development of Coal Resources, China University of Mining and Technology, Beijing, 100083, China

    Lingtao Mao

Authors
  1. Haizhou Liu
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  2. Lingtao Mao
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  3. Xuyang Chang
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  4. François Hild
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Contributions

HL conceptualization, methodology, investigation, and writing—original draft. LM resources, formal analysis, validation, supervision, and writing—review; XC resources and supervision; FH conceptualization, formal analysis, validation, supervision, and writing—review.

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Correspondence to Lingtao Mao.

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Liu, H., Mao, L., Chang, X. et al. Multiscale Damage Analyses of Red Sandstone in Uniaxial Compression Based on Advanced Digital Volume Correlation. Rock Mech Rock Eng 56, 8623–8641 (2023). https://doi.org/10.1007/s00603-023-03504-y

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  • DOI: https://doi.org/10.1007/s00603-023-03504-y

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