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An Analytical Model for Two-Order Asperity Degradation of Rock Joints Under Constant Normal Stiffness Conditions

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Abstract

Jointed rock masses during underground excavation are commonly located under the constant normal stiffness (CNS) condition. This paper presents an analytical formulation to predict the shear behaviour of rough rock joints under the CNS condition. The dilatancy and deterioration of two-order asperities are quantified by considering the variation of normal stress. We separately consider the dilation angles of waviness and unevenness, which decrease to zero as the normal stress approaches the transitional stress. The sinusoidal function naturally yields the decay of dilation angle as a function of relative normal stress. We assume that the magnitude of transitional stress is proportionate to the square root of asperity geometric area. The comparison between the analytical prediction and experimental data shows the reliability of the analytical model. All the parameters involved in the analytical model possess explicit physical meanings and are measurable from laboratory tests. The proposed model is potentially practicable for assessing the stability of underground structures at various field scales.

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Abbreviations

CNL:

Constant normal load

CNS:

Constant normal stiffness

\(A_{\rm u}\) :

Amplitude of the critical unevenness

\(A_{\rm w}\) :

Amplitude of the critical waviness

c :

Cohesion

\(c_{\rm u}\) :

Wear constant of the critical unevenness

\(c_{\rm w}\) :

Wear constant of the critical waviness

\(d_{\rm n}\) :

Dilation angle

\(d^{\rm u}_{\rm n}\) :

Dilation angle of the critical unevenness

\(d^{\rm w}_{\rm n}\) :

Dilation angle of the critical waviness

E :

Young’s modulus

F :

Shear stiffness reduction factor

\(i_0\) :

Initial asperity angle of the critical waviness

\(i_{\mathrm{mob}}\) :

Mobilisable asperity angle of the critical waviness

\(k_{\rm n}\) :

Normal stiffness of surrounding rock

\(k_{\rm s}\) :

Elastic shear stiffness

n :

Maximum loop number

\(S_{\rm u}\) :

Geometric area of the critical unevenness

\(S_{\rm w}\) :

Geometric area of the critical waviness

\(\alpha _0\) :

Initial asperity angle of the critical unevenness

\(\alpha _{\mathrm{mob}}\) :

Mobilisable asperity angle of the critical unevenness

\(\delta _{\rm n}\) :

Joint dilation

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

Maximum elastic shear displacement

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

Plastic shear displacement

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

Wavelength of the critical unevenness

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

Wavelength of the critical waviness

\(\nu\) :

Poisson’s ratio

\(\rho\) :

Rock density

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

Uniaxial compressive strength of rock

\(\sigma _{\rm n}\) :

Instant normal stress

\(\sigma _{\rm n0}\) :

Initial normal stress

\(\sigma _{\rm n}\;(i)\) :

Normal stress at iterative step i

\(\sigma _{\rm n}\;(i-1)\) :

Normal stress at iterative step \(i-1\)

\(\sigma _{\rm T}\) :

Transitional stress

\(\sigma ^{\rm u}_{\rm T}\) :

Transitional stress of the critical unevenness

\(\sigma ^{\rm w}_{\rm T}\) :

Transitional stress of the critical waviness

\(\tau\) :

Shear stress

\(\tau _{\rm mob}\) :

Mobilisable or bounding shear stress

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

Basic friction angle

\(\Delta \delta\) :

Increment of shear displacement

\(\Delta \delta _{\rm n}\) :

Dilation increment

\(\Delta \delta ^{\rm e}_{\rm s}\) :

Increment of elastic shear displacement

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

Increment of plastic shear displacement

\(\Delta \sigma _{\rm n}\) :

Increment of normal stress

\(\Delta \tau\) :

Increment of shear stress

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Acknowledgements

Yingchun Li thanks the support from the Fundamental Research Funds for the Central University (Grant No. DUT17RC(3)032) and the Open Laboratory for Deep Mine Construction, Henan Polytechnic University, China (Grant No. 233 2015KF-02). Wei Wu gratefully acknowledges the support of the Start-Up Grant from Nanyang Technological University, Singapore.

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Authors and Affiliations

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

    Yingchun Li

  2. The Open Laboratory for Deep Mine Construction, Henan Polytechnic University, Jiaozuo, China

    Yingchun Li

  3. School of Civil and Environmental Engineering, Nanyang Technological University, Singapore, Singapore

    Wei Wu

  4. School of Civil Engineering, Shaoxing University, Shaoxing, China

    Bo Li

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  1. Yingchun Li
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Correspondence to Wei Wu.

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Li, Y., Wu, W. & Li, B. An Analytical Model for Two-Order Asperity Degradation of Rock Joints Under Constant Normal Stiffness Conditions. Rock Mech Rock Eng 51, 1431–1445 (2018). https://doi.org/10.1007/s00603-018-1405-5

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