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A Peak Dilation Angle Model Considering the Real Contact Area for Rock Joints

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Abstract

The area ratio content of the asperities with different apparent dip angles is studied based on Grasselli’s apparent dip angle distribution function. An approximate expression of the minimum dip angle of the asperities in contact is further obtained according to the contact theory. For regular joints, the peak dilation angle is the average dip angle of all asperities on the contact part. Extending the above idea to the peak dilation angle of rough joints, the peak dilation angle model is derived based on the apparent dip angle distribution of the real contact asperities. In the derivation process, the peak dilation angle of an arbitrary stress state is directly obtained, instead of obtaining the initial dilation angle first and then defining the relation between the peak dilation angle and the initial dilation angle. The more important innovation is that all the parameters in the new model are of physical significance and easy to obtain; furthermore, they are not obtained by fitting test results. Based on 89 sets of test data, the predicted values of the new model are compared with those of the other 4 existing models. The results show that the prediction accuracy of the new model is the best. Besides, the mesh scales of rock joint are discussed, and the size range of the triangle mesh is obtained. It is proposed to remove the isolated points with a large apparent dip angle when processing test data. The shear mechanism of rough joints is further clarified in this study: the dilation is regarded as the average apparent dip angle of the asperities in contact at a certain stress level. This view is easy to understand and follow. It is as simple and beautiful as many natural principles.

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Abbreviations

JRC:

Joint roughness coefficient

JCS:

Joint wall compressive strength (MPa)

JMC:

Joint matching coefficient

3D:

Three-dimensional

Max:

Maximum value

θ :

Dip angle (°)

θ * :

Apparent dip angle (°)

t :

Shear direction

n :

External vector of the triangle element

n 1 :

Projection vector of n

α :

Azimuth

n 0 :

External vector of the shear plane

\(\theta_{ \text{max} }^{*}\) :

Maximum apparent dip angle (°)

A 0 :

Maximum possible contact area ratio

C :

A fitting coefficient to describe the distribution of the apparent dip angles

τ p :

Peak shear strength (MPa)

ϕ b :

Basic friction angle (°)

i p :

Peak dilation angle (°)

σ n :

Normal stress (MPa)

σ c :

Compression strength (MPa)

i 0 :

Dilation angle under zero normal stress (°)

α :

Azimuth (°)

\(A_{{\theta^{*} }}\) :

Ratio between the area of the triangle elements with an effective dip angle greater than θ* and the total area of the joint surface

A 0 :

Ratio between the area of the triangle elements with an effective dip angle greater than 0 and the total area of the joint surface

\(x_{{\theta^{*} }}\) :

Area ratio content of the asperities

\(\theta_{\text{big}}^{*}\) :

Large effective dip angle of triangle elements (°)

\(\theta_{\text{small}}^{*}\) :

Small effective dip angle of triangle elements (°)

x big :

Area ratio content of the asperities when the apparent dip angle is steep

x small :

Area ratio content of the asperities when the apparent dip angle is smooth

C big :

Extreme value of C when the apparent dip angle is steep

C small :

Extreme value of C when the apparent dip angle is smooth

ϕ (z):

Height probability density of the asperities

z :

Peak height of the asperities

m :

Root mean square of the asperities height

E * :

Equivalent elastic modulus (GPa)

R :

Curvature radius of the asperity

θ ave :

Roughness parameter \({{\theta_{\text{max} }^{*} }/ {\left( {C + 1} \right)}}\)

\(\theta_{\text{cr}}^{*}\) :

Minimum apparent dip angle of the asperities in contact at a certain stress level (°)

δ :

Average relative error

i pt :

Test peak dilation angle (°)

i pe :

Estimated peak dilation angle (°)

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Acknowledgements

This study is supported by National Key R&D Programs of China (No. 802015CB575), National Natural Science Foundation of China (Nos. 51478027, 51174012, 41502323) and Beijing Postdoctoral Research Foundation.

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Authors and Affiliations

  1. School of Civil and Transportation Engineering, Beijing Advanced Innovation Center for Future Urban Design, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China

    Liren Ban & Chengzhi Qi

  2. State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084, China

    Weisheng Du

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  1. Liren Ban
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  2. Weisheng Du
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  3. Chengzhi Qi
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Correspondence to Weisheng Du or Chengzhi Qi.

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Ban, L., Du, W. & Qi, C. A Peak Dilation Angle Model Considering the Real Contact Area for Rock Joints. Rock Mech Rock Eng 53, 4909–4923 (2020). https://doi.org/10.1007/s00603-020-02193-1

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