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Analytical Prediction of the Shear Behaviour of Rock Joints with Quantified Waviness and Unevenness Through Wavelet Analysis

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Abstract

We present an analytical model for the shear behaviour of a rock joint with waviness and unevenness. The waviness and unevenness of a natural joint profile are quantitatively separated through wavelet analysis. The critical waviness and critical unevenness of a joint profile are subsequently determined. The degradation process of each-order asperity is predicted by considering the role of plastic tangential work in shear, by which the sheared-off asperity area and the dilation angle are quantified. Both the dilation angles of critical waviness and critical unevenness decay, as plastic tangential work accumulates. The analytical predictions are compared with the experimental data from direct shear tests on both regular- and irregular-shaped joints. Good agreement between analytical predictions and laboratory-measured curves demonstrates the capability of the developed model. Therefore, the model is capable of assessing the stability of rock-engineering structures with ubiquitous joints.

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Abbreviations

\(A_\text {u}\) :

Amplitude of critical unevenness

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

Amplitude of critical waviness

\(A^0_\text {u}\) :

Initial amplitude of critical unevenness

\(A^0_{\text {w}}\) :

Initial amplitude of critical waviness

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

Sheared area ratio

\(a^{\text {s}}_\text {u}\) :

Sheared area ratio of critical unevenness

\(a^{\text {s}}_{\text {w}}\) :

Sheared area ratio of critical waviness

\(c_\text {u}\) :

Degradation coefficient of critical unevenness

\(c_{\text {w}}\) :

Degradation coefficient of critical waviness

d\(S^{\text {s}}_\text {u}\) :

Increment of the sheared area of critical unevenness

d\(W^{\text {p}}_{\text {s}}\) :

Increment of plastic tangential work

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

Increment of shear stress

F :

Reduction factor

\(i_0\) :

Initial inclination angle of critical waviness

\(i_{\text {d}}\) :

Dilation angle of critical waviness

\(i^{\text {m}}_{\text {d}}\) :

Mobilisable dilation angle of critical waviness

K :

Dimensionless coefficient

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

Joint shear stiffness

\(S^0_\text {u}\) :

Initial area of critical unevenness

\(S^0_{\text {w}}\) :

Initial area of critical waviness

\(S_\text {u}\) :

Unsheared area of critical unevenness

\(S^{\text {b}}_\text {u}\) :

Area of critical unevenness base

\(S^{\text {s}}_\text {u}\) :

Sheared area of critical unevenness

\(S_{\text {w}}\) :

Unsheared area of critical waviness

\(S^{\text {b}}_{\text {w}}\) :

Area of critical waviness base

\(S^{\text {s}}_{\text {w}}\) :

Sheared area of critical waviness

\(W^{\text {s}}_{\text {e}}\) :

Elastic tangential energy

\(W^{\text {p}}_{\text {s}}\) :

Plastic tangential work

\(\alpha _0\) :

Initial inclination angle of critical unevenness

\(\alpha ^{\text {d}}_{\text {m}}\) :

Mobilisable dilation angle of critical unevenness

\(\delta _{\text {ave}}\) :

Average percent error

\(\delta _{\text {ave}} (\delta _n)\) :

Average percent error of dilation

\(\delta _{\text {ave}} (\tau )\) :

Average percent error of shear stress

\(\delta _n^{\text {pre}}\) :

Predicted dilation

\(\delta _n^{\text {exp}}\) :

Experimental dilation

\(\delta ^{\text {e}}_{\text {ms}}\) :

Maximum elastic shear displacement

\(\delta ^{\text {e}}_{\text {s}}\) :

Elastic shear displacement

\(\delta ^{\text {p}}_{\text {s}}\) :

Plastic shear displacement

\(\delta _n\) :

Joint dilation

\(\Delta \delta _n\) :

Incremental dilation

\(\Delta \delta ^{\text {p}}_{\text {s}}\) :

Plastic shear displacement increment

\(\lambda _\text {u}\) :

Wavelength of critical unevenness

\(\lambda ^0_\text {u}\) :

Initial wavelength of critical unevenness

\(\lambda _{\text {w}}\) :

Wavelength of critical waviness

\(\lambda ^0_{\text {w}}\) :

Initial wavelength of critical waviness

\(\sigma _{\text {c}}\) :

Uniaxial compressive strength of rock

\(\sigma _n\) :

Normal stress

\(\sigma ^{\text {n}}_{\text {T}}\) :

Transitional normal stress

\(\sigma ^{\text {s}}_{\text {T}}\) :

Transitional shear stress

\(\tau \) :

Shear stress

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

Basic frictional strength

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

Mobilisable shear strength

\(\tau ^{\text {exp}}\) :

Experimental shear stress

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

Predicted shear stress

\(\phi _{\text {b}}\) :

Basic friction angle

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Acknowledgements

Yingchun Li thanks the financial supports from the National Natural Science Foundation (Grant No. 51809033), the China Postdoctoral Science Foundation (Grant No. 2018M631789), the National Key Research and Development Plan (Grant No. 2018YFC1505301), and the Fundamental Research Funds for the Central Universities (Grant No. DUT17RC(3)032).

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  1. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China

    Yingchun Li, Shengyue Sun & Chun’an Tang

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  1. Yingchun Li
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Li, Y., Sun, S. & Tang, C. Analytical Prediction of the Shear Behaviour of Rock Joints with Quantified Waviness and Unevenness Through Wavelet Analysis. Rock Mech Rock Eng 52, 3645–3657 (2019). https://doi.org/10.1007/s00603-019-01817-5

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