Mechanisms of Anisotropy in Salt Rock Upon Microcrack Propagation

Abstract

Salt rock is a polycrystalline material of interest for geostorage because of its low permeability and potential to self-heal by pressure solution at favorable stress and temperature conditions. It is often assumed that microcrack propagation and healing lead to isotropic stiffness changes. The goal of this study is to check this assumption and to gain a fundamental understanding of the mechanisms that control the accumulation of damage and irreversible deformation. Cyclic axial loading tests are performed under a confining pressure of 1 MPa on synthetic salt rock generated by thermal consolidation. The stress–strain curves and the microstructure images taken at key stages of the cycles reveal the formation of a complex system of sliding and wing microcracks, the orientation of which is loading dependent. We interpret the mechanisms that control the coupled evolution of crack families by a discrete wing crack elastoplastic damage (DWCPD) model. Crack propagation is controlled by Mode I and Mode II fracture mechanics criteria. Sliding “main” cracks grow if a cohesive frictional criterion is met, while the wing cracks propagate in tension. Displacement jumps at crack faces are related to the deformation of the rock representative elementary volume (REV). The DWCPD model can capture the nonlinear stress–strain relationship and the degradation of stiffness during the loading cycles. Simulations show that microcracks occur following two stages: (1) wing cracks initiate and main cracks do not propagate; (2) wing cracks and main cracks then propagate simultaneously. Higher friction at the crack faces leads to higher strength. With a larger cohesion, salt rock strength increases, damage development is delayed and exhibits a stick-slip evolution. At higher confinement, the initiation of wing cracks is delayed, which results in an increase of strength. The damage rate is higher in specimens that are damaged prior to compression than in the ones that are not. The proposed DWCPD model can be extended to any polycrystalline semi-brittle material, and can be applied to understand the formation of crack patterns in geostorage facilities.

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Abbreviations

\(\varPsi ^*_{\text {s}}\) :

Helmholtz free energy of the REV

\(G^*\) :

Gibbs energy of the REV

\(\overrightarrow{n},\overrightarrow{l}\) :

Direction normal to a main crack plane and a wing crack plane

\(\varvec{\sigma }, \varvec{\epsilon }\) :

Microscopic stress and strain tensors of a representative elementary volume (REV)

\(\varvec{\sigma }^m, \varvec{\sigma }^w\) :

Stress fields that are applied at main crack faces and wing crack faces

\(\sigma ^m_{\text {n}},\varvec{\sigma }^m_{\text {t}}\) :

Normal stress and the tensor of tangential stress that apply on the faces of the main crack

\(\sigma ^m_{\text {l}}\) :

Net tangential stress that applies on the faces of the main crack in the direction l

\(\sigma ^w_{\text {n}}\) :

Normal stress that applies on the faces of the main crack

\(\varvec{\epsilon }^m, \varvec{\epsilon }^w\) :

Strain fields on main cracks and wing cracks

\(\varvec{\epsilon }^e\) :

Elastic strain of the matrix

\(\varvec{\epsilon }^{{\text {ed}}}\) :

Recoverable strain induced by the loss of stiffness

\(\varvec{\epsilon }^E, \varvec{\epsilon }^p\) :

Elastic strain and plastic strain of the REV

\({\varvec{t}}^m\) :

Traction on a main crack plane

\(\mu , c\) :

friction coefficient and cohesion of main cracks

\(N^m, B^m\) :

Normal and frictional indexes of a main crack

\(\beta ^m, \varvec{\gamma }^m\) :

Volume fraction of the normal displacement jumps and shear displacement jumps of main cracks

\(s_0, s_1\) :

Normal and shear elastic compliance of cracks

\(\overrightarrow{T}\) :

Shear force applies at the faces of the main crack

\(V_{{\text {REV}}}\) :

Actual volume of the REV

\(M_{i}\) :

Number of cracks in family i

\(a^m, a^w\) :

Crack lengths of main cracks and wing cracks

\(\rho ^m, \rho ^w\) :

Crack densities of main cracks and wing cracks

\(\beta ^w\) :

Volume fraction of the normal displacement jumps of wing cracks

\(\varvec{C}_{\text {o}}\) :

Elastic stiffness of the matrix

Q :

Number of main crack families

\({\mathbb {N}}_{ijkl}, {\mathbb {T}}_{ijkl}\) :

Fourth-order tensor operators

\(f_{\text {I}}, f_{{\text {II}}}\) :

Crack propagation criteria for Mode I and Mode II

\(K_{{\text {Ic}}}, K_{{\text {IIc}}}\) :

Crack toughness for Mode I and Mode II

\(K_{\text {o}}, \sigma _{\text {c}}\) :

Constitutive parameters for toughness

\(\varvec{\varOmega }\) :

Macroscopic damage variable of the REV

\(\varvec{\varOmega}_{\text {m}}, \varvec{\varOmega}_{\text {w}}\) :

Macroscopic damage variable of main cracks and wing cracks

d :

Trace of Macroscopic damage variable of the REV

\(f_{\text {p}}\) :

Plastic yield surface function

g :

Plastic potential function

\(q, p, \theta\) :

Deviatoric stress, mean stress, and Lode’s angle

\(J_2, J_3\) :

The second and third stress invariants

e :

Cohesion constant of the rock

\(\alpha _{\text {p}}\) :

Plastic hardening function

\(m_\theta\) :

The parameter controlling the effect of Lode’s angle

\(\chi\) :

The parameter controlling the effect of damage

\(\eta\) :

The parameter controlling the boundary of the compressive dilation zone

R :

The parameter controlling plastic hardening rate

\(\lambda , \omega\) :

Plastic multiplier and the plastic hardening variable

\(\alpha _{\text {p}}^o, \alpha _{\text {p}}^m\) :

The plastic yielding threshold and the maximum of the hardening function

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Acknowledgements

This research was supported by the U.S. National Science Foundation under Grants CMMI-1362004/1361996 (“Collaborative research: Linking Salt Rock Deformation Regimes to Microstructure Organization”) and under Grant CMMI-1552368 (“CAREER: Multiphysics Damage and Healing of Rocks for Performance Enhancement of Geo-Storage Systems—A Bottom-Up Research and Education Approach”).

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Correspondence to Xianda Shen.

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Shen, X., Arson, C., Ding, J. et al. Mechanisms of Anisotropy in Salt Rock Upon Microcrack Propagation. Rock Mech Rock Eng 53, 3185–3205 (2020). https://doi.org/10.1007/s00603-020-02096-1

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Keywords

  • Salt rock
  • Cyclic loading
  • Micro-mechanics
  • Wing cracks
  • Elastoplactic damage model