Abstract
Edge-to-edge is the most common form of contact in a two-dimensional joint block system. Discontinuous deformation analysis (DDA) is a powerful tool for handling block systems and treats an edge-to-edge contact as two vertex-to-edge contacts with a couple of normal and shear springs at the extremes of the contact segment. However, this method is unsuitable for evaluating the tensile failure of a joint under asymmetrical tension conditions, even though it can simplify the contact type and increase computational efficiency. This is mainly because the distributed forces at an edge-to-edge contact are simplified as two couples of concentrated forces. To solve this problem, this paper proposes incorporating a multi-spring edge-to-edge contact model with DDA. In the model, several contact pairs with normal and shear springs on an edge-to-edge contact are set to simulate the distributed forces. Several benchmark tests were performed to demonstrate the discrepancy between the original and improved DDAs. The results revealed that the latter more accurately models the stress distribution across the edge-to-edge contact, providing a more reasonable acceleration required to topple a series of blocks resting on an inclined stepped base.
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Abbreviations
- c :
-
Joint cohesion
- dn, ds :
-
Penetration distances in the normal and shear directions
- D i :
-
6 × 1 deformation submatrix of block i
- εx, εy, γxy :
-
x-wise and y-wise strains and the shear strain of a block
- φ :
-
Joint friction angle
- F :
-
Force
- F i :
-
6 × 1 force submatrix of block i
- G :
-
Gravity
- G cr :
-
Critical gravity
- h :
-
Height of block
- i, j, m, n :
-
Indices
- kn, ks :
-
Stiffness of the added normal and shear springs, respectively
- K ij :
-
6 × 6 submatrix of the stiffness matrix of blocks i and j
- L :
-
Contact length
- l :
-
Length of block
- n :
-
Total number of blocks/springs
- R 2 :
-
Correlation coefficient
- σ t :
-
Tensile strength of the joint
- θ :
-
Dip angle of the incline
- σ cr :
-
Critical tensile strength
- T i :
-
Displacement transformation matrix of block i
- u, v :
-
x-wise and y-wise displacements
- u0, v0, r0 :
-
x-wise and y-wise displacements of the point (x0, y0) and rotation angle of the block
- x, y :
-
Coordinates of an arbitrary point of a block
- x0, y0 :
-
Coordinates of the block centroid
- EECi,j–m,n :
-
Edge-to-edge contact with two edges i,j and m,n
- VECi–m,n :
-
Vertex-to-edge contact with vertex i and edge m,n
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Acknowledgements
The authors thank Dr. G. Shi for his significant contribution to DDA development and his diligent work on DDA programming. The program used in this study is an extension of his original code, DDA2012. This study received financial support from the National Natural Science Foundation of China (41977213, 41672286, 51408511, 41761144080), the Science & Technology Department of Sichuan Province (2017JQ0042), and Ministry of Science and Technology of China (KY201801005). The financial support received from these organizations is gratefully acknowledged. The authors wish to thank the editors and reviewers for their time and effort in reviewing our article.
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Zhang, Y., Wang, J., Zhao, J.X. et al. Multi-spring Edge-to-Edge Contact Model for Discontinuous Deformation Analysis and Its Application to the Tensile Failure Behavior of Rock Joints. Rock Mech Rock Eng 53, 1243–1257 (2020). https://doi.org/10.1007/s00603-019-01973-8
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DOI: https://doi.org/10.1007/s00603-019-01973-8