The Signature of Deformation Bands in Porous Sandstones

Abstract

Accurate identification and differentiation of various deformation bands in porous sandstone constitute a critical first step towards thorough understanding of these failure patterns. Two conventional ways prevail on band identification for sandstone, one based on a kinematic definition of shear offset (the ratio between shear and compaction displacement S/C) and the other according to a geometric inclination angle between band orientation and the principal stress direction. The two methods are not always consistent with each other, frequently leading to confusions or false identification of band patterns, especially when a deformation pattern is in transition. Employing an advanced multiscale modeling approach, we have reproduced a complete kinematic spectrum of deformation bands in porous sandstone. Enlightened by an exponential relation between \(|\epsilon _v/\epsilon _q|\) and \(|S/C |\) observed in the results, a new, accurate classifier, \(B_i=\epsilon _v/\epsilon _q\), is proposed in this study, where \(\epsilon _v\) and \(\epsilon _q\) denote, respectively, the volumetric and deviatoric strains inside a band. The validity and robustness of \(B_i\) are examined with rigorous mechanical analyses in conjunction with insights drawn from multiscale modeling of localized deformation. Instead of decomposing the displacements, we further propose splitting the deformation gradient into four distinct kinematic components: (1) deviatoric compaction, (2) lateral extension, (3) simple shear, and (4) rigid rotation. As an end member of the kinematic spectrum, a pure dilation band (PDB) is found dominated by lateral extension without apparent compaction, shear, or rotation; a pure compaction band (PCB) is dominated by deviatoric compaction without apparent extension, shear, or rotation; a simple shear band (SSB) distinguishes itself from the previous two with the presence of substantial, proportional compaction, extension, shear, and rotation. As transitional patterns, shear-enhanced dilation band (SEDB), and shear-enhanced compaction band (SECB) are closer to pure volumetric deformation bands (PDB and PCB), while dilatant shear band (DSB) and compactive shear band (CSB) are more similar to simple shear band. In addition, \(B_i\) shows advantages in characterizing spatial variation and temporal transition of band patterns in complex boundary-value problems of sandstone.

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Acknowledgements

The study has been financially supported by the National Natural Science Foundation of China under project 51679207 and the Research Grants Council of Hong Kong through GRF project 16210017. All figures were generated by the open-source data analysis toolkit (Python with the NumPy/SciPy and matplotlib packages, Paraview, and Inkscape). The software code used for modeling and generating the data in this study is based on the open-source parallel hierarchical multiscale modeling code FEM\(\times\)DEM accessible via Yade (https://yade-dem.org/doc/FEMxDEM.html).

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Appendix: Mesh-Dependency Analysis

Appendix: Mesh-Dependency Analysis

The coupled FEM/DEM scheme in this study has been based on a non-regularized FEM formulation which may suffer mesh-dependency and make the post-localization results questionable. Advanced regularization techniques, e.g., Cosserat models and second gradient theory (Desrues et al. 2019; Rattez et al. 2018), are required to resolve this issue. On the other hand, the adoption of high-order elements with reasonable fine mesh has been found helpful to mitigate mesh-dependency (Guo and Zhao 2016b). Two more biaxial compression tests under \(\sigma _0=10\) MPa have been conducted with different meshes (\(6 \times 10\) and \(18 \times 30\)) to analyze the mesh-dependency.

Fig. 11
figure11

Mesh-dependency analysis for the FEM/DEM multiscale modeling. ac Failure mode at the final state in terms of normalized debonding number with two white dots marking the pre-inserted weak points. d The evolution of differential stress (\(\sigma _1-\sigma _0\)) with axial strain. e The evolution of \(\epsilon _v/\epsilon _q\) with axial strain

Figure 11a–c shows the final failure mode of different meshes at \(\epsilon _1=2.0\%\) in terms of normalized debonding number. Despite the different band width and the different in-band damage intensities, the failure mode is similar with the same band angle. The stress–strain relations for the three cases are presented in Fig. 11d. Due to the pre-inserted weak points to trigger the localization, the coarse mesh (\(6 \times 10\)) produces the smallest stress peak and the fine mesh (\(18 \times 30\)) produces the largest. It is worth noting that the difference in peak stress between the medium-fine mesh (\(12 \times 20\)) and the fine mesh is less than 2%. The post-peak behaviors of these two cases are almost the same, as well. Typical in-band RVEs are selected to calculate the evolution of the proposed index \(\epsilon _v/\epsilon _q\) as presented in Fig. 11e. The in-band RVEs in the medium-fine case and the fine case present almost the same evolution of \(\epsilon _v/\epsilon _q\). It is worth noting that \(\epsilon _v/\epsilon _q\) for the coarse mesh tend to converge to the same value despite the difference as the localization occurs due to the stronger effect of the weak points. The results presented in Fig. 11 prove that the adoption of a reasonable fine mesh (\(12 \times 20\) in this study) mitigates the pathological mesh-dependency and justify the post-peak analyses conducted in this study.

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Wu, H., Zhao, J. & Liang, W. The Signature of Deformation Bands in Porous Sandstones. Rock Mech Rock Eng 53, 3133–3147 (2020). https://doi.org/10.1007/s00603-020-02100-8

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Keywords

  • Deformation band
  • Compaction band
  • Kinematic spectrum
  • Classification index
  • Porous sandstone
  • Multiscale modeling