Elliptical Hertz-Based General Closure Model for Rock Joints

Appendices

Appendix 1: Composite Morphology

As shown in Fig. 8a, contacts are not only formed near the tips of the asperities on single surface, but also between any two points on the surfaces. If the two surfaces are forced to be closer together, new contacts will appear due to the deformation of the existing contacts. Under the un-deformed state, we define two parallel reference planes, which are separated at a distance d₀. From these planes we measure the height of each surface z₁ and z₂. Summing the heights at each point along the surface, we obtain the function $z_{\text{c}} (x) = z_{\text{upper}} (x) + z_{\text{lower}} (x)$ as shown in Fig. 8b, where, $z(x)$ is the xth measured topographical height, $z_{\text{c}} (x)$ is the xth topographical height of composite topography, $z_{\text{upper}} (x)$ and $z_{\text{lower}} (x)$ are the topographical heights of the upper and lower surfaces of a joint, respectively.

Fig. 8

Sketch of two rough surface profiles of a joint and the derived composite topography (Cook 1992)

Appendix 2: Asperity Spatial Features

The asperity peaks randomly distributed on a rock joint should be first identified when adopting Hertz-based theoretical models to predict the closure process. For three-dimensional rock joint, the peak is defined as a point with height greater than its four closest points (Greenwood 1984), as shown in Fig. 9. Equation (13) is the criterion to identify the asperity peaks on a rough rock joint. The equivalent radius of each identified peak can then be calculated based on the five points using the least-square method. The mean peak density is determined by the ratio of the number of asperity peaks to the nominal area of rock joint; the mean peak radius is determined by the ratio of the sum of all the peaks’ equivalent radii to the peak number; and the mean peak height is determined by the ratio of the sum of all the peaks’ height to the peak number:

$$ Z_{i,j} > Z_{i - 1,j} , Z_{i + 1,j} , Z_{i,j - 1} , Z_{i,j + 1} $$

(13)

where, $Z_{i,j}$ is the asperity peak height at the point (i, j) in 3D domain; $i$ and $j$ are integers.

Fig. 9

Schematic diagram for identifying asperity peaks on rock joint (Greenwood 1984)