Abstract
To describe the mechanical behavior of interfaces existing in rocks, a three-dimensional contact element model with a certain thickness was proposed that could simultaneously consider the heterogeneity, damage evolution and state variation of interfaces. The Weibull distribution function was employed to describe the rock heterogeneity. The damage evolution of the contact element was analyzed by using the equivalent damage method. Two criteria—the Mohr–Coulomb criterion and the maximum tensile stress criterion—were employed to assess the damage occurrence. Once the element was identified as being completely damaged, the state variation of the contact element was determined. Three types of element states—contact without sliding, contact with sliding, and separation—were taken into consideration. The model was then implemented and programmed into the finite element method and validated with experimental measurements and theoretical solutions for three classical rock mechanical problems, including one static problem (i.e., a direct shear test) and two dynamic problems (the seismic responses of rock joints and split-Hopkinson pressure bar tests), in an attempt to examine the applicability of the proposed contact element model. The results demonstrated that the contact element model can effectively model the rock interfacial problems and can therefore help solve problems in discontinuous rock masses.
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Abbreviations
- l, w, t :
-
Length, width and thickness of the contact element
- σcr, σtr :
-
Residual compressive and tensile strength
- σ1, σ3 :
-
Maximum principal stress and minimum principal stress
- σc, σt :
-
Uniaxial compressive and tensile strength
- \(\bar{\varepsilon }\) :
-
Equivalent principal strain
- D :
-
Damage variable
- εc0, εt0 :
-
Strains at σc and σt
- εcu, εtu :
-
Ultimate compressive and tensile strain
- f c :
-
Element shear stress
- Φ :
-
Frictional angle
- λc, λt :
-
Ratio of σc to σcr and ratio of σt to σtr
- \(\left\{ \delta \right\}^{\text{e}}\) :
-
Element displacement
- u, v, w :
-
Nodal displacement in the x, y and z directions
- Δu, Δv, Δw :
-
Displacement difference between the upper and lower surfaces of contact element in the x, y and z directions
- N1, N2, N3, N4 :
-
Shape functions
- γxz, γxz, εn :
-
Shear strains and normal strain
- {σ0}:
-
Initial stress
- G, E :
-
Shear modulus and elastic modulus
- [K]e, [K]:
-
Element stiffness matrix and total stiffness matrix
- λs, λn :
-
Shear and normal stiffness
- u, f :
-
Nodal displacement vector and nodal loading vector
- σ 0 :
-
Adhesive strength
- fs, fd :
-
Static and dynamic friction coefficient
- σn, τs :
-
Normal stress and shear stress
- α, α0 :
-
Parameters in the Weibull distribution function
- m :
-
Heterogeneity index
- [S]:
-
Elastic matrix
- T p :
-
Transmission coefficient
- Z :
-
Seismic impedance
- Ω :
-
Angular frequency
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Acknowledgements
We thank L. Zhang from Dalian University of Technology for providing laboratory measurements. This research was financially supported by the Department of Science and Technology of Guangdong Province (No. 2019ZT08G315) and the National Natural Science Foundation of China (Nos. 51627804 and 51904189).
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Liao, Z., Ren, M., Tang, C. et al. A three-dimensional damage-based contact element model for simulating the interfacial behaviors of rocks and its validation and applications. Geomech. Geophys. Geo-energ. Geo-resour. 6, 45 (2020). https://doi.org/10.1007/s40948-020-00171-z
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DOI: https://doi.org/10.1007/s40948-020-00171-z