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A One-Dimensional Elastoplastic Model for Capturing the Nonlinear Shear Behaviour of Joints with Triangular Asperities Based on Direct Shear Tests

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Abstract

In most cases, the behaviour of rock masses is dominated by weak planes such as joints, along which shear is more likely to occur and deformation is usually nonlinear. Thus, it is of great significance to understand the shear process and mechanisms of joints. We investigated the shear process and mechanics of joints with regular and irregular triangular asperities through a series of direct shear tests, where a high-speed camera was used to record the deformation and failure of asperities. It was found that, for joints with regular asperities, the nonlinear shear behaviour is mainly related to shear failure of asperities. For joints with irregular asperities, this nonlinearity results from both shear failure of asperities and the interactions between them. Based on the findings, it was assumed that the shear stiffness/strength of a joint is the sum of shear stiffness/strength of individual asperities. By this assumption, estimation methods were developed to predict shear strength, shear displacement, and shear stiffness of joints with irregular asperities. A one-dimensional elastoplastic model was then put forward to describe the nonlinear shear behaviour of the irregular joints. In this model, a spring element is connected in series with a modified Saint Venant element. Different from the traditional Saint Venant element, this modified one incorporates the concept of alterable yield limit and thus can reflect the evolution of shear stiffness during shearing. Eventually, with these estimation methods and this new elastoplastic model, the nonlinear shear process of the joints could be expressed mathematically. The applicability of this model was verified through the comparison between fitting results and laboratory measurements. Our findings could provide insights into the evolution of shear stiffness and help to gain a better understanding on the shear process of rough joints.

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Abbreviations

\(\tau _{{{\text{initial}}}}^{{\text{*}}}\) :

Initial yield limit on shear stress–displacement curves

\({u_{\text{T}}}\) :

Shear displacement of joint

\({u_i}\) :

Shear displacement of each asperity

\({k_i}\) :

Shear stiffness of each asperity

\({k_{{\text{joint}}}}\) :

Shear stiffness of joint

\(M\) :

The number of asperities on joint

\({k_{{\text{lin}}}}\) :

Linear shear stiffness on shear stress–displacement curves

\({u_{{\text{yield}}}}\) :

Shear displacement at initial yielding strength

\({u_{{\text{peak}}}}\) :

Shear displacement at peak strength

\({\tau _{{\text{peak~}}}}\) :

Peak strength of joint

\({\tau _{{\text{residual}}}}\) :

Residual shear strength of joint

\(\sigma\) :

Normal stress

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

Residual frictional coefficient

\(C\) :

Apparent cohesion of joint

\({\text{d}}{\tau _{\text{H}}}\) :

Stress increment in Hooke’s element

\({\text{d}}{u_{\text{H}}}\) :

Displacement increment in Hooke’s element

\({k_1}\) :

Spring stiffness in Hooke’s element

\({k_2}\) :

Spring stiffness in St.V or M_St.V element

\({\tau ^*}\) :

Yield limit of St.V model or M_St.V element

\(\Delta l\) :

Pre-deformation of a spring in St.V or M_St.V element

\(\psi\) :

Friction angle in St.V or M_St.V element

\({\text{d}}{\tau _{{\text{M}}\_{\text{S}}}}\) :

Stress increment in M_St.V element

\({\text{d}}{u_{{\text{M}}\_{\text{S}}}}\) :

Displacement increment in M_St.V element

\(\theta\) :

Dip angle of frictional surfaces in M_St.V element

\({\text{d}}{\tau _{\text{T}}}\) :

Total stress increment in H-M_St.V model.

\({\text{d}}{u_{\text{T}}}\) :

Total displacement increment in H-M_St.V model

\({k_{\text{e}}}\) :

Elastic stiffness in the H-M_St.V model

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

“Plastic stiffness” in the H-M_St.V model

\({k_{\text{I}}}\) :

Total stiffness of the H-M_St.V model-I

\({k_{{\text{II}}}}\) :

Total stiffness of the H-M_St.V model-II

\({a_{\text{h}}}\) :

The projected length of shear stress–displacement curve in hardening phase

\({a_{\text{s}}}\) :

The projected length of shear stress–displacement curve in softening phase

\({a_{{\text{l-e}}}}\) :

The extended shear displacement in the linear elastic phase

\({A_{\text{h}}}\) :

The empirical constant in the hardening phase

\({A_{\text{s}}}\) :

The empirical constant in the softening phase

\(\Delta {u_{\text{h}}}\) :

Relative shear displacement in hardening phase

\(\Delta {u_{\text{s}}}\) :

Relative shear displacement in softening phase

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Acknowledgements

This research is financially supported by National Key R&D Program of China (nos. 2018YFC0407002 and 2016YFC0600702), the National Natural Science Foundation of China (no. 41402241), and the Hong Kong Jockey Club.

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Authors and Affiliations

  1. State Key Laboratory of Hydraulic Engineering Simulation and Safety, School of Civil Engineering, Tianjin University, Tianjin, China

    J. B. Zhu

  2. Sichuan University-The Hong Kong Polytechnic University Institute for Disaster Management and Reconstruction, Sichuan University, Chengdu, China

    H. Li

  3. Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China

    H. Li

  4. State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, China

    J. H. Deng

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  1. J. B. Zhu
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  2. H. Li
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Correspondence to J. B. Zhu.

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Zhu, J.B., Li, H. & Deng, J.H. A One-Dimensional Elastoplastic Model for Capturing the Nonlinear Shear Behaviour of Joints with Triangular Asperities Based on Direct Shear Tests. Rock Mech Rock Eng 52, 1671–1687 (2019). https://doi.org/10.1007/s00603-018-1674-z

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