Mathematical cover refinement of the numerical manifold method for the stability analysis of a soil-rock-mixture slope
Introduction
To investigate the stability of slopes, a lot of methods including the limit analysis methods [1], the LEMs (limit equilibrium methods) [2], [3], [57], [58], [59] and the numerical methods [4] have been proposed. However, compared to the limit analysis methods, the LEMs and the numerical methods are more frequently used. The theory foundations of the LEMs are very simple. In addition, the computational cost of the LEMs is very cheap. However, in the LEMs the sliding body of the slope is assumed as a rigid body, and the deformation of the slope cannot be considered. Furthermore, the distribution mode of internal forces between different contact slices has to be assumed [5]. Hence, the numerical methods which can overcome the defects of the LEMs are more attractive than the LEMs.
The FEM [4], [34], as a representative of the numerical methods, was initially proposed for structure mechanic problems. The deformation of the slope can be simulated with the FEM. In addition, the assumption about the distribution mode of internal forces of the slope that made in the LEMs is not needed in the FEM. Due to the attractive advantage of the FEM, it has been adopted by many geotechnical engineers to investigate the stability of slopes [6], [7]. Nevertheless, the FE meshes should coincide to the physical meshes (such as fracture faces, joints, material interfaces, and so on). This defect of the FEM hinders the further application of the FEM for problems with complex geometry, such as the SRM(soil-rock-mixture) problems (Fig. 1) presented in [8], [9], [10]. To improve the performance of FEM, many methods have been proposed, see for example in [11], [12], [13], [14], [15], [16], [17], [31], [32].
In other front, the numerical manifold method (NMM) [18] which can also be considered as an improvement of the FEM has been proposed. In the NMM, the mathematical meshes can be easily generated, since the mathematical meshes don't have to coincide to the physical meshes. Additionally, the contact technique in the NMM is mature. Since the advent of the NMM, it has been adopted by a lot of researchers to solve various types of problems, see for example in [19], [20], [21], [22], [23], [24], [25], [33], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56].
Recently, the NMM has been applied for SRM problems [26]. In the SRMs (Fig. 1), the mechanical property of rock blocks is much better than that of the soil matrix. The deformation and the failure of the SRM slopes are mainly due to the failure and deformation of the rock-soil interfaces and the soil matrix. In the TNMM (Traditional NMM), mathematical meshes are usually uniformly constructed to discretize the domain of the SRM slope. However, from the viewpoint of computational efficiency, this scheme is not economical.
In the present work, to improve the computational efficiency of the TNMM, an improved NMM with mathematical cover refinement (RNMM) is proposed for SRM slopes. In this RNMM, refined mathematical meshes are deployed at specified domains including the rock-soil interfaces and the slope toe where stress concentration may occur. Compared with the FEM, the generation of refined mathematical meshes in the RNMM is much easier, since refined mathematical meshes in the RNMM don't have to coincide to the physical meshed. In addition, the recently proposed scheme called MLC (mathematical cover systems with multiple layers) [26] is adopted to further improve the computational efficiency of the RNMM. In the scheme of MLC, a mathematical mesh (a triangular mesh) is adopted for a rock block to form its corresponding mathematical cover system, while uniform mathematical meshes are used to form the corresponding mathematical cover system for the soil matrix. To be more specifically, the deformations of the rock blocks in the SRMs are simulated with constant strain mode.
The strength reduction method [27], [28] is further incorporated into the RNMM and the RNMM + MLC (RNMM with MLC) to evaluate the factors of safety of the slopes. With the proposed numerical models, stability analyses about two typical slopes including a soil slope and a SRM slope are investigated.
Section snippets
The governing equation and the boundary conditions
The governing equations in regarding to a solid problem can be expressed as [26]in which gi, σij, Ω and ρ are the gravity acceleration, stress tensor, problem domain and density of the medium, respectively.
The traction boundary condition and the displacement boundary condition for the solid problem are expressed as:in which the displacement boundary is denoted as Γu, while the traction boundary is denoted as Γt.
Basic concepts in the TNMM
In the NMM, two types of CSs (cover systems) are adopted, which are the mathematical CS and the physical CS [29]. The mathematical CS is formed through mathematical meshes. A MP (mathematical patch) is a domain formed by a series of grids having a communal node. Cutting all the MPs with the physical meshes forms all the physical patches (PPs). The union of all the PPs forms the physical CS. In the NMM, the ME (manifold element) is the basic element for integration. Each ME is the communal part
Numerical examples
For the purpose of comparison, the following methods are adopted, which are
- (1)
TNMM: Traditional NMM. In the TNMM, mathematical meshes which are uniformly distributed are adopted to form the mathematical cover system.
- (2)
TNMM+MLC: TNMM enriched using the mathematical cover systems with multiple layers. In the TNMM + MLC, a mathematical mesh (a triangular mesh) is adopted for a rock block to form its corresponding mathematical cover system, while uniform mathematical meshes are used to form the
Conclusions
To improve the computational efficiency of the TNMM (traditional NMM), a NMM with mathematical cover refinement (RNMM) is adopted. Additionally, the recently proposed scheme called MLC (mathematical cover systems with multiple layers) is adopted to further improve the computational efficiency of the RNMM. The strength reduction method is incorporated into the proposed numerical models to get the factors of safety (FOSs). With the RNMM+MLC (RNMM with MLC), two examples about slope stability
Acknowledgements
The two anonymous reviewers are gratefully acknowledged for their helpful and constructive comments and suggestions. This study is supported by the Youth Innovation Promotion Association CAS, under the grant no. 2020327; and the National Natural Science Foundation of China, under the grant nos. 51609240, 11572009 and 51538001.
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