Theory and Numerical Simulation of Deep Rock Mass Based on a Non-Euclidean Model

Bao, Yingji; Jiang, Binsong

doi:https://doi.org/10.1155/2022/4492406

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Volume 2022 | Article ID 4492406 | https://doi.org/10.1155/2022/4492406

Theory and Numerical Simulation of Deep Rock Mass Based on a Non-Euclidean Model

Yingji Bao^1,2and Binsong Jiang¹

Academic Editor: Tongguang Ni

Received27 Sept 2021

Revised10 Nov 2021

Accepted18 Jan 2022

Published14 Feb 2022

Abstract

Rock masses existing in nature may experience closed internal stress under the action of geological structures. This closed internal stress makes the deformation of the rock mass incompatible, so classic continuum theory is not suitable for analyzing the stress and deformation of rock masses. In this study, a non-Euclidean model for rock masses was established based on differential geometry. By choosing the non-Euclidean parameter as the internal variable, a thermodynamic model was constructed. Then, numerical computation based on the non-Euclidean model was applied to a circular tunnel in a deep rock mass. The distribution of internal stress and the effects of rock parameters on the distribution of internal stress were analyzed. Through our research, we conclude that the stress field of the deep rock mass consists of classical stress and internal stress and that the internal stress shows a distinct wavy behavior. The radius of the fractured zone decreases with an increasing Young’s modulus E and Poisson’s ratio , but increases with increasing non-Euclidean parameter ξ.

1. Introduction

In classical continuum theories of rock mechanics, a rock is a continuous, simply connected body that undergoes deformation, and the deformation satisfies the equations of compatibility. The metric tensor for measuring the distances between material particles in the reference configuration is Euclidean. However, rock masses existing in nature may contain closed internal stress. The internal stress makes the deformation of the rock incompatible. Therefore, the classical continuum theory is not suitable for analyzing the incompatible deformation of rock masses [1–5]. It is necessary to introduce non-Euclidean space to describe the incompatible deformation, and differential geometry will be used to describe the closed internal stress.

The relationship between the continuum theory and mathematical theory of differential geometry has been studied by many researchers. Kondo [6] first recognized the relationship between dislocation theory and non-Riemannian theory. Later, Anthony [7] pointed out the relationship of differential geometry to disclinations. Then, Kroner [8] completed the underlying theory for defects and differential geometry, including dislocations and extra matter. These investigations established a relation between the parameters of non-Euclidean geometry and elastic strain, bend twist, and quasiplastic strain of defect theory [9]. The first non-Euclidean continuum model to describe the stress-field distribution around underground working was developed by Myanikov and Guzev [10]. A modification of this model was presented by Guzev and used to describe different rock phenomena, such as zonal disintegration and anomalous deformation of rock samples [11–15]. Investigations were also undertaken by other researchers, and a non-Euclidean continuum model was proposed to investigate the zonal disintegration phenomenon of surrounding rocks by Zhou and Qian [16–19]. These existing models assume that the undeformed body is in three-dimensional Euclidean space. However, there is closed internal stress in the rock mass.

Suppose there is a flat plate in two dimensions. Due to uneven heating and mutual expansion, thermal stress will be generated, and the uneven temperature field causes internal stress in the plate. If we cut the whole plate into tiny elements and release each tiny element into its stress-free natural state, then the flat plate composed of various elements can no longer be in two-dimensional Euclidean space. However, if the flat plate is allowed to warp, it will bulge and become a curved surface. The plate is a two-dimensional Riemannian space immersed in three-dimensional Euclidean space R³. According to this scenario, the undeformed rock mass with closed internal stress should be regarded as a manifold.

In this study, it is assumed that the undeformed rock mass with closed internal stress was a manifold. A non-Euclidean model of rock masses was established based on differential geometry. By choosing the non-Euclidean parameter as the internal variable, a thermodynamic model was constructed. Then, numerical computation based on the non-Euclidean model was applied to a circular tunnel in a deep rock mass. The distribution of internal stress and the effects of rock parameters on the distribution of internal stress were then analyzed.

2. Differential Geometry Method

The initial state of the rock is as a manifold. Moreover, the manifold is an imagined state of undeformed rock that occupies a compact and simply connected region. Consider a material point p in the undeformed body and an m-dimensional manifold M that is referred to as a conﬁguration. We choose the conﬁguration p ∈ U⊂M at time t = t0 as a reference conﬁguration, such that the subsequent deformation and motion of the current conﬁguration are given by q ∈ V⊂N at time t = t with respect to the reference configuration [14] (Figure 1).

For any p ∈ M, there exists a neighborhood U of p, such that U is homeomorphic to an open set in R^m, i.e., the homeomorphism is φ_U: U⟶φ_U (U).where Xⁱ is the local coordinate of the point p ∈ U.

For any q ∈ N, there exists a neighborhood V of q, such that V is homeomorphic to an open set in Rⁿ, i.e., the homeomorphism is ψ_V: ⟶ψ_V (V).where xⁱ is the local coordinate of the point q ∈ V.

The deformation of the body is a continuous map from the initial configuration smooth manifold M to the final configuration smooth manifold N. F: M⟶N. Compatible coordinate charts (U, φ_U) exist at points p ∈ M and (V, ψ_V) at f(p) ∈N, such that the mapis C^∞ at the point φ_U (p).

Moreover, smooth maps F between M and N induce linear maps between tangent spaces and between cotangent spaces. F: M⟶N, p ∈ M, and q = F(p) ∈V. is a linear map of the tangent space induced by F. We define the map : T_pM⟶T_qN as follows (Figure 2):

The adjoint map of is : M⟶N which is called the differential map induced by F.

Xⁱ and xⁱ are the local coordinates of points p and q. The deformation map F can be expressed near p by the functions

The increment vector dx of point q ∈ N in deformed body N belongs to cotangent space T_qN, and the increment dX of the corresponding point X ∈ M in undeformed body M belongs to tangent space T_pM. They are related by the differential map .

Thus, the action of on the natural basis {dx^α, 1 ≤ α ≤ n} is given by

The matrix representation of in the natural bases dx and dX is exactly the Jacobian matrix (∂F^α∕∂Xⁱ)_p. This relates the infinitesimal line segment dX ∈ M to the corresponding segment dx ∈ N.

Map plays a role in the coordinate transformation from T_qN to T_pM. Since no material point vanishes, there is an inverse relation for :where ⁻¹ is the deformation gradient.

A volume element dV in undeformed body M is related to the volume element dv in deformed body N through determinant J of the deformation gradient F by dv = JdV.where J is referred to as the Jacobian.

The deformed body N is a manifold with a topological structure and smooth structure. During deformation, some vectors act on it, such as body force and external force. They are vector bundles on N (Figure 3).

N is called the base space. π: E⟶N is a smooth surjective map called a bundle projection, and V = R^s is an s-dimensional vector space.

If there is no mass ﬂux, the total mass of undeformed body M is conserved in deformed body N:where ρ₀ and ρ are the mass densities before and after deformation, respectively. The time differential form using Reynold’s transport theorem gives the following mass conservation law:where is a vector of the tangent bundle and ∇ is the gradient operator. The divergence makes a linear map div: TM⟶C^∞M.

Newton’s second law states that in an inertial frame, the rate of linear momentum is equal to the applied force. Here, by applying the second law to a continuum region, the linear momentum of the deformed body N is given bywhere t is an external force per unit area called the stress vector, which acts on the boundary ∂Ω. It is a tensor of the tensor bundle T₀¹(N), and the body force per unit volume b acting in volume N is a tensor of tensor bundle TN.

Substituting (12) into (13) yields the following:where σ is called the Cauchy stress, which gives a transformation law that maps the unit outward n to the traction t acting on the surface by t = σ^Tn. σ is a tensor of tensor bundle T₀²(N), and n is a tensor of tensor bundle T₀¹(N).

Now, we need to measure the extent of deformation of an elemental length located at a material point. To do so, we should introduce a metric on the smooth manifold. If the metric is a smooth, nondegenerate positive symmetric second-order covariant tensor field, then the manifold is called a Riemannian manifold, and the metric is a Riemannian metric. We compare length |dx| with its original length |dX| by comparing the diﬀerence of both lengths as a squared measure:where is the metric tensor of manifold N, and G_αβ is the metric tensor of manifold M.

Substituting the deformation gradient yields

Then, the deformation measure can be written as

This tensor ε is referred to as a strain tensor.

Furthermore, M and N are two smooth manifolds, and f: M⟶N is a smooth map. If f is the immersion and h is a Riemannian metric, thenwhere is the Riemannian metric of M. (17) indicates that the metric of manifold M can be induced by the differential map and the metric of manifold of N.

The Riemann tensor R of the initial configuration is a function:

The Riemann tensor R of the current configuration is a function.

Substituting (18) and (16) into (19), equation (19) can be rewritten as

During excavation in a deep rock mass, a circular tunnel is a typical plane strain problem. Thus, the Riemann tensor R can be obtained in a two-dimensional manifold as

The other component of R_ijkl is equal to 0. On the right of (21) is the compatible condition of classical mechanics. Due to the existence of closed internal stress, the deformation of rock masses is not compatible. Then, the tensor contraction of R_ijkl is a second-order tensor R_ij, and the components arewhere R is the tensor contraction of R_ij, and R = 2R₁₂₁₂.

3. The Thermodynamics Model and Discussion

Classical thermodynamics provide valuable information that can assist in the solving of practical problems in many ﬁelds of science and engineering. The physical state of the system can be described by variables such as internal energy, entropy energy, and free energy. They are decided by independent parameters; these parameters are the internal variables. The choice of the internal variables is not unique. It can be decided by the level of the model, the nature of the thermodynamic system, and the accuracy of the system state. The independent parameters consist of external variables (the strain tensor, deformation gradient, and temperature) that can be observed and internal variables that relate to the material properties of the system.

In geotechnical engineering, the strain tensor ε, temperature T, and Riemann tensor R_ij can be chosen as external and internal variables. Now, we consider the deformation from the initial configuration to the final configuration, and this deformation is elastic. Thus, the Helmholtz free energy can be rewritten as

According to the second law of thermodynamics, the Clausius–Duhem inequality will bewhere ρ is the density of the body, s is the entropy, q is the heat flux, and σ_ij is the Cauchy stress tensor. The inequality above will be satisfied for any T, Q, h, and ε. Then, the following thermodynamic restrictions hold true:

Equation (25) indicates that the stress tensor is equal to the derived free energy, in accordance with classical elasticity theory.

Supposing that the rock material is isotropic at a constant temperature, we can extend the free energy to second order withwhere λ and μ are Lame’s constants, and ν and ξ are the parameters related to the internal structure of the rock. Substituting equation (28) into (25) gives

The underground rock experiences the effects of construction and heat, and the deformation will be incompatible and lead to a nonuniform stress field. A part of the stress will be lost by the tunnel excavation, but the residual will remain in the rock. In equation (29), for the initial undeformed body, the strain tensor is equal to zero. Then, the rock experiences an initial stress ξR_ij, which is the self-balancing stress closed in the rock. This value is related to the metamorphic and tectonic movement of rock. From a physical point of view, the existence of nonzero self-equilibrated stress ﬁelds in a continuous medium is due to the presence of structural defects in the material.

The stress can be regarded as the sum of classical stress Σ_ij and t_ij:

According to the principles of thermodynamics with internal state variables, we define the force Q aswhere Q is a generalized force conjugated to R, which is an internal variable that relates to the internal stress of the rock mass and contributes to the dissipation of energy.

Recall that equation (29) can be written as

For the deformation from the initial configuration to the final configuration, the equilibrium and boundary conditions are

Substituting (32) and (33) into (21), we obtain

Because the stress of the rock mass is self-balanced, Δσ_kk = 0. Equation (34) can be rewritten aswhere . Equation (44) is Poisson’s equation, and the solution iswhere J₀ and Y₀ are the Bessel function and Neumann function, respectively. A and B are the parameters related to the structural defects of the material.

Considering a circular tunnel in the plane strain state, the total stress can be written as

The rock mass experiences gravitational stress and internal stress before excavation. The construction of underground engineering projects breaks the stress balance, and so, the stress will redistribute. In the plane strain state, the shear stress of the internal stress dissipates with the deformation, but normal stress still exists in the rock mass and shows a periodic change with radius. If this stress reaches a critical value, the rock will break.

4. Analysis Based on Numerical Computations

4.1. The Distribution of Internal Stress around a Deep Tunnel

The model for numerical computation [20–23] was from Tangkou mine coal, Shandong Province, China. During the excavation of the tunnel, the completed roadway was damaged successively, and the surrounding rock was fractured violently. The material parameters used in the simulations were from a depth of 1028 m, including a uniaxial compressive strength of rock σ_c = 35 MPa, Young’s modulus E = 30 GPa, and Poisson’s ratio = 0.25. The cross-section of the tunnel is approximately a circle, and the radius is r₀ = 3 m. The stress of the surrounding rock is equivalent in vertical and horizontal orientations, and the stress p₀ = 23.4 MPa.

Substituting these parameters into equations (37) and (38), we can obtain

The parameter ξ is related to the internal stress of the rock mass and can be chosen by the radius of the fracture zone. According to geological observations, the fracture zone radius r_c = 1.58 r₀; thus, we have the following boundary conditions for the function R:

Substituting (36) into (41) gives

By solving equations (42) and (43), ω = 7.18 and ξ = 5.57 GPa can be obtained.

Meanwhile, fracture appears when the stresses in the rock mass reach a certain critical value. From a physical viewpoint, this means that it is necessary to use a force criterion, whose fulfillment in a selected region corresponds to the fracture zone. In this study, the Mises criterion is applied.

Substituting (39) and (40) into (44), we obtain A = 5.05 × 10⁻³ and B = 3.57 × 10⁻³. Thus, the relation between the stress components and radius is shown in Figure 4. The distribution of hoop stress and radial stress shows a distinct wavy behavior. The range of fluctuation reflects the magnitude of internal stress and is related to in situ stress and rock behavior.

According to the prediction of the numerical computation, the second crest of the stress is located at 2.36 r₀, and the observation value is 1.58 r₀, as shown in Figure 5. The prediction value is 31% greater than the observation value because the fracture of the rock mass at r = r_c is a region failure.

The fluctuation location and magnitude of the internal stress were influenced by the parameters of the rock mass, such as Young’s modulus E, Poisson’s ratio , and non-Euclidean parameter ξ, which affected the fracture zone of the surrounding rock of the tunnel, and the location of the crest stress is given in Table 1.

Figures 6 and 7 show the effects of Young’s modulus E on the distribution of the internal stress crest in the surrounding rock mass around a deep circle tunnel. It can be observed that the radius of the internal stress crest decreases with an increase in Young’s modulus E. These results indicate that the magnitude of internal stress and energy existing in the rock mass increases with Young’s modulus. The larger Young’s modulus is, the closer the fracture location is to the tunnel.

(a)

(b)

(c)

Figures 8 and 9 show the effects of Poisson’s ratio on the distribution of the internal stress crest in the surrounding rock mass around a deep circle tunnel. The radius of fractured zones decreases with an increase in Poisson’s ratio .

(a)

(b)

(c)

Figures 10 and 11 show the effects of the non-Euclidean parameter ξ on the distribution of the internal stress crest in the surrounding rock mass around a deep circle tunnel. It can be observed that the radius of fractured zones increases with an increase in the non-Euclidean parameter ξ.

(a)

(b)

(c)

5. Conclusions

(1)Because there is closed internal stress in the rock mass, the deformation of the rock mass is incompatible. The undeformed body is a manifold immersed in three-dimensional Euclidean space. The metric tensor of the space is the Riemannian metric instead of the Euclidean metric δ_ij. The deformation of the rock mass is a continuous map from the undeformed manifold M to the deformed manifold N. The incompatible deformation of the rock mass is related to the Riemannian curvature tensor R_ijkl of the manifold.(2)The rock mass experiences gravitational stress and internal stress before excavation. The construction of underground engineering projects breaks the balance, and the stress will redistribute. In the plane strain state, the shear stress of the internal stress dissipates with the deformation, but normal stress still exists in the rock mass and shows a periodic change with radius. If this component reaches a critical value, the rock will break.(3)For the rock mass with internal stress, with the increase of elastic modulus E, the peak value of hoop stress decreases, and the position of peak stress is closer to the tunnel. It shows that the harder the rock mass is, the more severe the damage near the tunnel surface is. With the increase of Poisson’s ratio , the hoop stress of rock mass decreases, which indicates that the larger Poisson’s ratio is, the larger the deformation of rock mass under the same confining pressure, and more internal stress will be released. With an increase of non-Euclidean parameter ξ, the radius of the fractured zone decreases.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was financially supported by General Project of the National Natural Science Foundation of China (51174196).

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Copyright © 2022 Yingji Bao and Binsong Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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