Abstract
We investigate the deformation localization at brittle–ductile transition in axisymmetric compression tests of rock analogue material GRAM1 made of bonded rigid particles. This transition in rocks (as well as in GRAM1) is recognized by the formation of a network of conjugate deformation localization bands in the postmortem rock samples and by a shallow stress reduction followed by a stress plateau in stress–strain curves. GRAM1 is much weaker than hard rocks. Therefore, brittle–ductile transition occurs in GRAM1 at a low confining pressure, 0.3 MPa, which is much smaller than that for real rocks. This allows using a transparent pressure cell and applying the digital image correlation technique to visualize the deformation evolution. Taking advantage of this technique and of the detailed characterization of GRAM1’s constitutive properties in the previous studies, we show that the initiation of deformation localization bands occurs in the dilatant (positive dilatancy factor \(\beta\)) and strain-softening (negative hardening modulus \(h\)) deformation regime. During the band evolution, the deformation within it becomes compactive (\(\beta\) < 0) and is accompanied by the material hardening (\(h\) > 0), which causes the band to widen and new bands to form successively resulting in a progressively densified band network. The formation of a conjugate band network at brittle–ductile transition is thus due to the transition from dilatancy to compaction and from softening to hardening during the inelastic straining.
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Abbreviations
- \({\sigma _{\text{m}}}\) :
-
Mean stress
- \(q={\sigma _1} - {P_{\text{c}}}\) :
-
Differential stress
- \({P_{\text{c}}}\) :
-
Confining pressure
- \({\sigma _1},{\sigma _2},{\sigma _3}\) :
-
Principal stress
- \({\sigma _{{\text{bdt}}}}\) :
-
Stress at brittle–ductile transition
- \(\overline {\tau }\) :
-
Von Mises stress
- \(\theta\) :
-
Lode angle
- \(h\) :
-
Hardening modulus
- \(E\) :
-
Young’s modulus
- \(\nu\) :
-
Poisson’s ratio
- \({V_{\text{c}}}\) :
-
Sample shortening rate
- \(C\) :
-
Correlation coefficient
- D :
-
Correlation window area
- \({X_i},\;{x_i}={\phi _0}\left( {{X_i}} \right)\) :
-
Coordinates (in pixels) of homologous points in the reference and deformed images
- \({\phi _0}\) :
-
Transformation function
- \(f\left( {{X_i}} \right),\;g\left( {{x_i}} \right)\) :
-
Grey levels of the point i in both the reference and deformed images
- \(\overline {f} ,\;\overline {g}\) :
-
Averages of the grey levels over all the pixels in a subset in the reference and deformed images
- \({P_{{\text{bdt}}}}\) :
-
Confining pressure at brittle–ductile transition
- \(\psi\) :
-
Angle between deformation bands and sample axis
- \({\widetilde {\varepsilon }_{{\text{ax}}}}\) :
-
Nominal axial strain
- \({\varepsilon _{{\text{ax}}}}\) :
-
Axial strain
- \({\gamma _{\text{m}}}\) :
-
Maximum shear strain
- \({\varepsilon _{\text{c}}}\) :
-
Circumferential/horizontal strain
- \({\varepsilon _{\text{r}}}\) :
-
Radial strain
- \({\varepsilon _1},~{\varepsilon _2}\) :
-
Principal strains
- \(\varepsilon\) :
-
Volume strain
- \({\bar {\gamma }^{\text{p}}}\) :
-
Accumulated inelastic equivalent shear strain
- \({\varepsilon ^{\text{p}}}\) :
-
Accumulated inelastic volume strain
- \(\beta\) :
-
Dilatancy factor
- \({V_{\text{b}}}\) :
-
Band propagation rate
- G :
-
Shear modulus
- BDT:
-
Brittle–ductile transition
- AE:
-
Axisymmetric extension
- AC:
-
Axisymmetric compression
- DIC:
-
Digital image correlation
- GRAM1:
-
Granular rock analogue material 1
- \(\Delta L\) :
-
Side length of the square correlation window
- \(\Delta l\) :
-
Maximum deviation distance between the positions of grid element centers in the procedure of computing the displacement field
- \({d_{\text{R}}}\) :
-
Band thickness measured on the deformed GRAM1 specimen
- \({d_D}\) :
-
Band thickness defined from the DIC data
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Acknowledgements
This work was supported by the Côte d’Azur Observatory, the Region Provence Alpes Côte d’Azur and GeoFracNet Consortium.
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Tran, TPH., Bouissou, S., Chemenda, A. et al. Initiation and Evolution of a Network of Deformation Bands in a Rock Analogue Material at Brittle–Ductile Transition. Rock Mech Rock Eng 52, 737–752 (2019). https://doi.org/10.1007/s00603-018-1641-8
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DOI: https://doi.org/10.1007/s00603-018-1641-8