Abstract
A numerical model for thermo-hydro-mechanical strong couplings in an elasto-plastic Cosserat continuum is developed to explore the influence of frictional heating and thermal pore fluid pressurization on the strain localization phenomenon. This model allows specifically to study the complete stress–strain response of a rock specimen, as well as the size of the strain localization zone for a rock taking into account its microstructure. The numerical implementation in a finite element code is presented, matching adequately analytical solutions or results from other simulations found in the literature. Two different applications of the numerical model are also presented to highlight its capabilities. The first one is a biaxial test on a saturated weak sandstone, for which the influence on the stress–strain response of the characteristic size of the microstructure and of thermal pressurization is investigated. The second one is the rapid shearing of a mature fault zone in the brittle part of the lithosphere. In this example, the evolution of the thickness of the localized zone and the influence of the permeability change on the stress–strain response are studied.
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Abbreviations
- c :
-
Cohesion
- C:
-
Heat capacity of the porous material
- \(c_{\text {hy}}\) :
-
Hydraulic diffusivity
- \(C^e_{ijkl}\) :
-
Elastic stiffness tensor
- \(c_{\text {th}}\) :
-
Thermal diffusivity
- \(e_{ijk}\) :
-
Levi–Civita symbol
- F :
-
Yield surface
- G :
-
Shear modulus
- \(G_{\text {c}}\) :
-
Cosserat shear modulus
- \(g_i\) :
-
Coefficients in the generalized plastic strain invariant
- \(h_i\) :
-
Coefficients in the generalized stress invariant
- K :
-
Bulk modulus
- L :
-
First modulus of the elastic flexural bending rigidity tensor
- M :
-
Second modulus of the elastic flexural bending rigidity tensor
- \(M_c\) :
-
Third modulus of the elastic flexural bending rigidity tensor
- \(M^e_{ijkl}\) :
-
Elastic flexural bending rigidity tensor
- n :
-
Eulerian porosity
- p :
-
Pore pressure
- Q :
-
Plastic potential
- q :
-
Hardening variable
- R :
-
Cosserat internal length
- S :
-
External surface of the specimen
- T :
-
Temperature
- \(u_i\) :
-
Displacements
- \(\alpha\) :
-
Coefficient of thermal expansion of the porous medium
- \(\beta\) :
-
Dilatancy
- \(\beta ^f\) :
-
Compressibility of the fluid phase per unit volume
- \(\beta ^s\) :
-
Compressibility of the solid phase per unit volume
- \(\beta ^*\) :
-
Mixture compressibility per unit volume
- \(\chi\) :
-
Permeability of the porous medium
- \(\delta _{ij}\) :
-
Kronecker symbol
- \(\eta ^f\) :
-
Viscosity of the liquid phase
- \(\gamma _{ij}\) :
-
Strain tensor
- \(\gamma _{[ij]}\) :
-
Antisymmetric part of the strain tensor
- \(\gamma ^e_{ij}\) :
-
Elastic strain tensor
- \(\gamma ^p\) :
-
Generalized plastic strain invariant
- \(\gamma ^p_{ij}\) :
-
Plastic strain tensor
- \(\kappa _{ij}\) :
-
Curvature tensor
- \(\kappa _{[ij]}\) :
-
Antisymmetric part of the curvature tensor
- \(\kappa ^e_{ij}\) :
-
Elastic part of the curvature tensor
- \(\kappa ^p_{ij}\) :
-
Plastic part of the curvature tensor
- \(\kappa _{(ij)}\) :
-
Symmetric part of the curvature tensor
- \(\lambda ^f\) :
-
Coefficient of thermal expansion of the fluid phase per unit volume
- \(\lambda ^s\) :
-
Coefficient of thermal expansion of the solid phase per unit volume
- \(\lambda ^*\) :
-
Coefficient of thermal expansion of the soil–water mixture per unit volume
- \(\mu\) :
-
Friction coefficient
- \(\mu _{ij}\) :
-
Couple-stress tensor
- \(\varOmega\) :
-
Volume of the specimen
- \(\omega ^c_i\) :
-
Cosserat rotations
- \(\omega ^c_{ij}\) :
-
Tensor of rotations of the microstructure
- \(\varOmega _{ij}\) :
-
Macroscopic rotation tensor
- \(\psi _i\) :
-
Test functions
- \(\rho\) :
-
Density of the porous material
- \(\sigma '\) :
-
Effective mean stress
- \(\sigma _{ij}\) :
-
Symmetric part of the total stress tensor
- \(\tau _{ij}\) :
-
Total stress tensor
- \(\tau _{[ij]}\) :
-
Antisymmetric part of the total stress tensor
- \(\tau\) :
-
Generalized deviatoric stress invariant for Cosserat continua
- \(\tau '_{ij}\) :
-
Effective stress tensor
- \(\varepsilon _{ij}\) :
-
Symmetric part of the strain tensor
- \(\xi\) :
-
Parameter that enables to switch between different hardening
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Acknowledgements
The second author, I.S., would like to acknowledge support of the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant agreement no. 757848 CoQuake).
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Rattez, H., Stefanou, I., Sulem, J. et al. Numerical Analysis of Strain Localization in Rocks with Thermo-hydro-mechanical Couplings Using Cosserat Continuum. Rock Mech Rock Eng 51, 3295–3311 (2018). https://doi.org/10.1007/s00603-018-1529-7
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DOI: https://doi.org/10.1007/s00603-018-1529-7