Abstract
As part of the rock mass, both the macroscopic flaws such as joints and the mesoscopic flaws such as microcracks affect the strength and the deformational behavior of rock mass. Existing models can either handle any one of them alone, and a model which can consider the co-effect of these two kinds of flaws on rock mass mechanical behavior is not yet available. This study focusses on rock mass with nonpersistently closed joints and establishes a new damage constitutive model for it. Firstly, the damage model for the intact rock which contains only the mesoscopic flaws is introduced. Second, the expression of the macroscopic damage variable (tensor) which can consider the joint geometrical and mechanical properties at the same time is obtained based on the energy principle and fracture theory. Third, the damage variable based on coupling the macroscopic and mesoscopic flaws is deduced based on Lemaitre strain equivalence hypothesis, and then the corresponding damage constitutive model for rock mass with nonpersistently closed joints under uniaxial compression is set up. Finally, the test data for the intact rock under uniaxial compression are adopted to validate the proposed model. A series of calculation examples verify that the proposed model is capable of presenting the effect of joint geometrical and mechanical properties on the rock mass mechanical behavior.
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Abbreviations
- SIF:
-
Stress intensity factor
- CDM:
-
Continuum damage mechanics
- \({\alpha}\) :
-
Joint dip angle (°)
- \({\varepsilon}\) :
-
Strain
- \({{\bf {\Omega}}}\) :
-
The second-order damage tensor
- \({{\bf {\Omega}}^{{\bf k}}}\) :
-
The damage tensor for the kth set of joints
- Ω 12 :
-
The coupled damage tensor
- \({\tau}\) :
-
Shear stress (MPa)
- \({\varphi}\) :
-
Joint internal friction angle (°)
- \({\mu}\) :
-
The friction coefficient of the joint face
- τeff :
-
The slide force along the joint face (MPa)
- \({\theta}\) :
-
The propagation angle of the wing crack at the joint tip (°)
- \({\phi}\) :
-
The persistent ratio of the nonpersistent joints
- \({\sigma}\) :
-
Normal stress (MPa)
- \({\tilde {\sigma }}\) :
-
The effective stress (MPa)
- \({\varepsilon _{12}, \varepsilon _1, \varepsilon _2, \varepsilon _0}\) :
-
The strain of the rock mass samples with both macroscopic and mesoscopic flaws, with only macroscopic flaws, with only mesoscopic flaws and without any flaws, respectively
- a :
-
The joint half length (cm)
- a k :
-
The size of the jointed area (m2)
- A :
-
The joint area (m2)
- B :
-
The joint depth (cm)
- D :
-
Damage
- D 0 :
-
The damage along the loading direction
- D 1, D 2 :
-
Macroscopic and mesoscopic damages, respectively
- D 12 :
-
The coupled damage variable
- E :
-
Young’s modulus (MPa)
- \({\tilde {E}_{12}, \tilde {E}_1, \tilde {E}_2, E_{0}}\) :
-
The elastic modulus of the rock mass samples with both macroscopic and mesoscopic flaws, with only macroscopic flaws, with only mesoscopic flaws and without any flaws, respectively (MPa)
- [E 0 ]:
-
The second-order elastic tensor of intact rock (MPa)
- [E]:
-
The second-order elastic tensor jointed rock mass (MPa)
- F :
-
An elemental strength parameter or stress level
- G :
-
Young’s and shear moduli of the intact rock (MPa)
- I :
-
The second-order unit tensor
- K I and K II :
-
The first and second stress intensity factors (SIF) of the joint tip, respectively (\({{\rm MPa} \sqrt{m}}\))
- K I0, K II0 :
-
SIF of one single I and II types of joints, respectively (\({{\rm MPa} \sqrt{m}}\))
- l * :
-
The length variable (cm)
- l :
-
The propagation length of the wing cracks (cm)
- l 0 :
-
The average space between two neighboring joints (m)
- m, F 0 :
-
The distribution parameters
- n :
-
The number of all failed ones under a certain load
- n k :
-
The orientation vector of the joint
- N 0 :
-
The number of all mesoscopic elements
- N :
-
The joint number
- \({P( \varepsilon)}\) :
-
The percentage of damaged ones out of the total number of the microcells in the rock
- U E :
-
The unit volume elastic strain energy (MN m)
- V:
-
The volume of rock mass (m3)
- Y:
-
The emission of damaged strain energy (MN m)
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Liu, H., Zhang, L. A Damage Constitutive Model for Rock Mass with Nonpersistently Closed Joints Under Uniaxial Compression. Arab J Sci Eng 40, 3107–3117 (2015). https://doi.org/10.1007/s13369-015-1777-8
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DOI: https://doi.org/10.1007/s13369-015-1777-8