An enhanced equivalent continuum model for layered rock mass incorporating bedding structure and stress dependence
Introduction
Layered rock mass is one of the rock masses which possess clear and definite bedding structure.1 Rock layers with different thicknesses and bedding planes, cemented or separated, comprise the whole rock mass. From engineering point of view, the mechanical properties of layered rock mass, such as its stiffness directionality2 and mixed failure modes,3 have closer relationship with instability problems encountered in underground excavations. It is well recognized that the relative orientation of rock layers with respect to underground openings has prominent effect on the deformation and stability of surrounding rocks.4, 5 Besides, layer thickness also influences the size of failure zone as well as the equivalent rock mass stiffness.6, 7 In order to better predict the mechanical responses of layered rock mass, a suitable constitutive model which is defined based on laboratory tests and monitoring information8 is usually needed. Such constitutive model specialized for layered rock mass should reflect fundamental properties of this rock mass, such as the orientation and layer thickness, meanwhile incorporates the essential features exhibited under different stress states, such as the anisotropy in both stiffness and strength, and their evolution with loading history. Engineering tasks involving the analysis of excavation and support design in layered rock mass could thus be more conveniently realized using models of this kind. Dynamic adjustment of excavation sequence and support optimization could also be more scientific-oriented.
The mechanical response of layered rock mass has so far been characterized by various numerical methods. According to the differences among the assumptions and algorithms, layered rock mass is represented either as discrete body or equivalent continuum. The discrete element method (DEM)9 treats rock mass as an assemblage of blocks or particles. In block-based codes, bedding planes are defined as boundaries of rock layers according to their orientation and layer thickness,10 while in particle-based codes, bedding planes are defined either as boundaries of particles or as a group of particles within a band.11 Therefore by using DEM bedding planes should be first explicitly built up, and then assigned independent constitutive relations.12 The advantage of this method is that it can more realistically capture the large deformation and failure mechanism along bedding plane under structurally-controlled conditions. Using particle-based model, the micro-crack initiation process within layers could be simulated13 simultaneously. Computational efficiency of this method is obviously influenced by the minimum thickness of layers. In fact, thickness less than 10 cm could hardly be handled by DEM when dealing with engineering-scale problems. Another limitation is that acquisition of the microscopic parameters used in particle codes usually needs tedious calibration. Recent development in hybrid continuum-discontinuum (HCD) method14 provides more insight into the failure mechanism of rock masses. The initiation and propagation of cracks could be reproduced by special element or separation/slip along element boundaries,15 and governed by fracture mechanics principles, for example, the combined finite-discrete element method (FDEM),16 the numerical manifold method17 or extended FEM.18 Layered rock mass is treated in a similar way by HCD19 as by DEM, namely independently defining rock layers and bedding planes. Computational cost is still heavy using HCD because the spontaneous fracturing process and the subsequent block movements require more variables and computer memories, despite the fact that failure phenomenon may be more realistic. Besides, parameters describing the fracture process would not be readily available.
Considering the computational expense of large-scale excavation simulation, the equivalent continuum method20 is still the most suitable way at present. In this method, bedding planes are not necessarily defined explicitly, although there are various joint elements aimed at reproducing the discontinuous distributions of displacement or stress field. The main goal of the equivalent continuum is to render results comparable to DEM results while maintaining relatively high performance in large-scale engineering computations. According to the difference of representing bedding planes, the equivalent continuum method could be further divided into the following three subcategories:
Completely equivalent model: Effects of bedding planes are smeared into the continuum description of rock mass, including the classical elasto-plastic theory,21 coupled elasto-plastic damage theory,22 as well as the micropolar theory23 which introducing additional degrees of freedom and characteristic length.
Explicit joint element: Bedding planes are represented by various joint elements, such as Goodman element,24 Desai element,25 or the special interface in FLAC3D26 commercial code. Strictly speaking, using joint element usually obtains a result analogous to DEM. However, the fundamental ideas behind the two methods are different. Therefore joint element is still regarded as a continuum method.
Between the above two schemes: No bedding planes are explicitly defined, while the effects of bedding planes are equivalently reflected by constitutive equations. This idea first comes from the microplane model27 widely used in concrete analysis, and another similar model called the multilaminate model.28 Both models are proposed to describe the equivalent macroscopic behavior of concrete or rock from a microscopic perspective. The contacts among aggregates are abstracted as fictitious planes, or microplanes. Each plane has a unique local stress-strain relation. By integrating over the entire orientation domain, the macroscopic stress-strain relation is acquired. The ubiquitous-joint model and bilinear strain-hardening/softening ubiquitous-joint model developed in FLAC3D29 is just a simplified version of the microplane model because only one set of planes is considered. However, the ubiquitous-joint model further defines the elasto-plastic calculations for rock and joint independently. Compared with the completely equivalent model, the displacement and stress distributions computed by these implicit models can better reflect the effect of bedding planes. However, if these models are applied to evaluate the mechanical responses of layered rock mass, several deficiencies must be acknowledged. The first limitation is that the layer thickness is not explicitly expressed in the formulation. In fact the layer thickness could substantially affect the magnitude and distribution of both the displacement and failure zone depth. Another drawback is that the anisotropy of stiffness and strength of intact bedded rocks are ignored. In addition, engineering problems require that the appropriate model is capable of predicting reasonable response of rock mass induced by stress state change.
In order to solve engineering problems involving excavations in layered rock mass in a continuum framework and to overcome the above mentioned deficiencies, in this paper we first present a brief summary of the mechanical properties of bedded rock and layered rock mass. On the basis of the ubiquitous-joint model, we propose an enhanced model which takes into account important features of layered rock mass including the stiffness and strength directionality, the stress dependence of mechanical parameters, and the effect of layer thickness. The formulations and numerical implementation are explained in detail. A series of numerical tests are then performed to validate the proposed model by comparing with the laboratory tests and with theoretical solutions.
Section snippets
Bedded rocks
Several important mechanical properties of bedded rocks have been revealed by various laboratory tests, among which the anisotropy of stiffness and strength has been reported by different researchers.30, 31 It is therefore necessary to consider this anisotropy for single layer comprised by bedded rock even if bedding planes are thought to be the major source of rock mass anisotropy. Besides anisotropy, results of triaxial tests30 have proved that the apparent moduli of bedded rocks increase
Basic assumptions
According to the above discussions, several basic assumptions for the enhanced model are: (a) Rock layers and bedding planes comprise the layered rock mass. Bedding planes are the unique discontinuities. Rock layers extend ad infinitum in space. (b) Bedding planes are perfectly cemented before failure. The rock mass is a transversely isotropic body before failure takes place and remain to be transversely isotropic after failure. (c) Micro-cracks within or through the layer will change the
Uniaxial compression: apparent Young's modulus and UCS
A cylindrical body discretized with hexahedron elements is built up to perform numerical uniaxial compression test. Diameter and height of this body is 50 and 100 mm, respectively. A total of 35,700 nodes and 33,000 elements are used. By controlling the velocity of the grid points (≈ 2.5 × 10−8 m/s) at both ends of the sample, we are able to simulate uniaxial and triaxial compressions, and by using FISH functions we achieve servo-controlled tests via adjusting the loading rate according to the
Effect of layer thickness
As is illustrated in Fig. 12, convergence in the case of 0.1 m thickness is more pronounced than that in the case of 1 m, which is a noticeable feature of the enhanced model and can be anticipated according to the model formulations. A somewhat similar example is proposed to further investigate this effect. A 20 m × 20 m plate with a horse-shoe-shaped hole in center (Fig. 13(a)) is used together with the parameters listed in Table 7. The initial vertical stress is 10 MPa and the
Conclusions
This paper mainly deals with the problem of modeling the mechanical behaviors of layered rock mass in the perspective of equivalent continuum. Based on the ubiquitous-joint model, essential features of layered rock mass are incorporated and formulated, thus forming an enhanced model.
Major modifications to the original model include: (a) the intrinsic directionality in both stiffness and strength for bedded rocks are characterized by the transversely isotropic elastic constitutive law and
Acknowledgements
This study is financially supported by the Wudongde Project Construction Department, China Three Gorges Projects Development Co., Ltd. and by Chinese Natural Science Foundation with the Grant nos. 11232024 and 41320104005.
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2022, Tunnelling and Underground Space TechnologyCitation Excerpt :The presence of bedding planes was even ignored in some researches (Li et al., 2014; Xu et al., 2015, 2017; Xiao et al., 2018; Jiang et al., 2021). Conversely, at shallow depth, where stresses were low to moderate, the deformation was usually controlled by the mechanical responses of bedding planes (Hoek, 2007; Fekete and Diederichs, 2013; Zhou et al., 2016a, 2017, 2019; Ding et al., 2019). Detailed investigations of development of the deformations inside the rock medium can provide insight valuable to geotechnical engineers.