Preamble

Study on the Plastic Zone in Layered Surrounding Rock Based on the Modified Hoek–Brown Criterion

Ma Chang1,2, Ding Wenqi1,2, Zhang Qingzhao1,2

1 Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China
2 Key Laboratory of Geotechnical and Underground Engineering, Ministry of Education, Shanghai 200092, China

Abstract

To investigate the distribution characteristics and evolution of the plastic zone in layered rock, this study derived the plastic zone boundaries under hydrostatic and non-hydrostatic stress fields based on the modified Hoek–Brown criterion for anisotropic rock. The influences of bedding inclination, degree of anisotropy, and lateral pressure coefficient on the plastic zone morphology were systematically analyzed. The results indicate that rock mass anisotropy has a significant influence on the morphology of the plastic zone. The expansion of the plastic zone predominantly occurs within 30° of the normal direction to the bedding planes. As the degree of anisotropy increases, the shape of the plastic zone evolves from an approximately circular form to a distinctive butterfly pattern. Furthermore, under high lateral pressure coefficients, the combined effect of rock mass anisotropy and lateral stress significantly amplifies plastic zone development around the tunnel shoulders.

Keywords: modified Hoek–Brown criterion; layered rock; rock anisotropy; plastic zone

The stress field in the surrounding rock is illustrated in Figure 2 [FIGURE:2], where P0 represents the vertical in-situ stress, λ is the lateral pressure coefficient (with λP0 being the horizontal stress), and R0 denotes the tunnel radius. The polar coordinate system is defined by angle θ and radius r, with σr, σθ, and τrθ representing the radial, tangential, and shear stresses at any point in the rock mass, respectively. The bedding plane inclination is designated as α.

Tunnel excavation induces significant changes in the stress state of surrounding rock, transforming the rock mass around the opening from a triaxial to a biaxial stress state and triggering stress redistribution. When the induced secondary stress exceeds the yield limit of the rock mass, an elastoplastic zone forms. Rock within the plastic zone undergoes plastic flow and potential failure, making the determination of plastic zone extent critical for stability control and support design in tunnel engineering [1]. Under hydrostatic pressure conditions, analytical solutions for stress and deformation can be obtained through elastoplastic analysis to delineate the plastic zone boundary. However, under non-hydrostatic stress fields, the irregular shape of the plastic zone boundary precludes straightforward analytical solutions. Existing research primarily employs the approximate Kastner method, which bypasses detailed plastic zone analysis by directly substituting the Kirsch stress solution for a circular tunnel under biaxial unequal pressure into the rock strength criterion to obtain an approximate plastic zone boundary equation [2-3]. The selection of an appropriate strength criterion is paramount for accurate elastoplastic analysis of surrounding rock. Commonly used criteria include the Mohr-Coulomb criterion [4] and the Hoek-Brown criterion.

Layered rock masses, frequently encountered in tunnel and underground engineering projects, exhibit pronounced anisotropic mechanical properties that complicate deformation and strength characteristics of the surrounding rock, leading to significant variations in failure modes and deformation patterns at different locations [6]. Current research on plastic zones primarily focuses on isotropic rock masses, while studies on anisotropic rock masses such as layered rock are mainly conducted through numerical simulation methods [7], with theoretical investigations remaining scarce. Regarding anisotropic strength criteria for layered rock masses, H. Saroglou [8] modified the Hoek-Brown criterion by introducing a coefficient kβ to account for anisotropic effects on the parameter mi, proposing that both kβ and the uniaxial compressive strength σcβ vary with the angle β between the maximum principal stress direction and the rock bedding planes.

This study investigates the distribution characteristics of plastic zone boundaries in layered rock masses under both hydrostatic and non-hydrostatic stress fields based on the modified Hoek-Brown criterion, and examines the influences of bedding inclination, rock mass properties, and lateral pressure coefficient on the plastic zone extent.

2. Analytical Derivation of Plastic Zone Boundary

2.1 Basic Assumptions

To derive concise analytical solutions and practical analysis methods for tunnels, the following fundamental assumptions are adopted:

(1) The tunnel cross-section is circular with an axial length far exceeding its cross-sectional dimensions, allowing the excavation problem to be simplified as a plane strain condition.

(2) The surrounding rock is a homogeneous, continuous, ideal elastoplastic material, with stresses in the plastic zone satisfying the modified Hoek-Brown criterion. For computational simplicity, the tangential stress σθ direction is approximated as the maximum principal stress direction, where β represents the angle between σθ and the rock bedding planes, calculated using Equation (2). Due to symmetry of the surrounding rock, the plastic zone exhibits central symmetry, and analysis is therefore restricted to the range θ = 0 to π.

(3) The rock mass is subjected to vertical and horizontal pressures with a lateral pressure coefficient λ, while rock self-weight is neglected.

(4) No support is installed, resulting in zero support pressure at the tunnel periphery.

2.2 Strength Criterion and Mechanical Model

H. Saroglou [8] proposed a modified Hoek-Brown criterion for layered rock masses, expressed as Equation (1). In this formulation, σ1 and σ3 represent the maximum and minimum principal stresses, respectively; mi denotes the strength parameter; while σcβ and kβ are correction coefficients for the uniaxial compressive strength and the parameter mi. Both coefficients vary with the angle β between the maximum principal stress direction and the rock bedding planes, as illustrated in Figure 1 [FIGURE:1].

2.3 Plastic Zone Boundary for λ = 1

Under hydrostatic conditions (λ = 1), the stress components in the tunnel surrounding rock depend only on the radial coordinate r. The equilibrium and geometric equations simplify to:

[Equations]

According to the modified Hoek-Brown strength criterion:

[Equation]

Considering axisymmetric conditions and substituting the strength criterion (Equation (1)) into the equilibrium equation (Equation (4)):

[Equation]

Solving this differential equation yields the expression for radial stress σr:

[Equation] where C is an integration constant. With no support pressure (σr = 0 at r = R0), the constant C can be determined as:

[Equation]

Substituting this result into Equation (9) gives:

[Equation] and subsequently into Equation (7) yields the tangential stress σθ:

[Equation]

At the elastoplastic interface (r = Rp), the continuity condition requires:

[Equation]

From elastic theory, the relationship between principal stresses and stress components in plane strain conditions is:

[Equation]

Substituting the strength criterion (Equation (1)) into Equation (16) provides:

[Equation] which, when solved simultaneously with the previous equations, yields the plastic zone radius Rp.

2.4 Plastic Zone Boundary for λ ≠ 1

When the lateral pressure coefficient λ ≠ 1, the plastic zone boundary of the surrounding rock assumes an irregular shape, making analytical derivation difficult. The conventional approach involves first determining the elastic stress field in the rock mass and then substituting this elastic solution into the rock plastic strength criterion to obtain an approximate failure zone.

The far-field in-situ stress can be decomposed into a biaxial equal-pressure component and a component with compression vertically and tension horizontally. Applying the principle of superposition yields the Kirsch stress solution for a circular tunnel under biaxial unequal pressure. For an unsupported tunnel, the elastic stress field is given by:

[Equations for stress components]

The principal stresses can be expressed in terms of stress components as:

[Equation]

Substituting the strength criterion (Equation (1)) into Equation (17) yields the plastic zone boundary equation. Table 2 [TABLE:2] presents a comparison between analytical and approximate solutions.

3. Case Study

Numerical examples are presented using different rock types from reference [8]. The rock mass parameters are summarized in Table 1 [TABLE:1], with a tunnel radius R0 = 6 m, in-situ stress P0 = 50 MPa, and mi = 6. The degree of anisotropy is characterized by RC = σci(90)/σci(min), where σci(90) is the uniaxial compressive strength perpendicular to bedding (β = 90°) and σci(min) is the minimum uniaxial compressive strength across all bedding orientations. A higher RC value indicates more pronounced anisotropy.

Figure 3 [FIGURE:3] illustrates the plastic zone boundaries under hydrostatic pressure (λ = 1) for various bedding inclinations α. The results demonstrate that anisotropic characteristics significantly influence the yield and failure patterns of the surrounding rock. Across all inclination angles, plastic zone expansion concentrates within 30° of the bedding plane normal. This occurs because the rock mass strength parameters σcβ and kβ attain minimum values when the angle β between the maximum principal stress σθ and bedding planes is 30°, rendering the rock more susceptible to yielding under identical stress conditions.

Figure 4 [FIGURE:4] presents plastic zone morphologies for different rock types at α = 0°. For rock masses with low anisotropy (small RC), the plastic zone boundary approximates a circular shape. As the degree of anisotropy increases, the variation in plastic zone radius across different θ angles becomes more pronounced, exhibiting a distinctive butterfly pattern. The plastic zones in schist and marble are substantially smaller than those in gneiss A and B, primarily due to the significantly higher strength parameters σcβ and kβ in the gneiss materials.

Figure 5 [FIGURE:5] shows plastic zone shapes under various lateral pressure coefficients λ. When λ < 1, the plastic zone extends more extensively at the tunnel sidewalls. As λ increases, the butterfly characteristic becomes increasingly evident. For bedding inclinations of 0° and 60°, plastic zone development at the tunnel shoulders is particularly pronounced, attributed to the superposition of anisotropic effects and elevated lateral pressure. The influence of the lateral pressure coefficient on the plastic zone becomes more significant for highly anisotropic rock masses.

To evaluate the error introduced by the approximate method for non-hydrostatic conditions, analytical and approximate solutions for the plastic zone radius of gneiss A at λ = 1 were compared across various θ angles, as presented in Table 2 [TABLE:2]. The small discrepancies between the two solutions demonstrate that the approximate method adequately captures the plastic zone morphology.

4. Conclusions

Previous studies [9] have shown that when λ < 1, the plastic zone assumes a transverse elliptical shape with greater extent at the tunnel sidewalls than at the crown and invert, whereas high λ values produce a butterfly-shaped plastic zone with significantly larger radii at the tunnel shoulders.

Based on the modified Hoek-Brown criterion for anisotropic rock masses, this study derived plastic zone boundaries for layered rock under both hydrostatic and non-hydrostatic stress fields, and analyzed the distribution characteristics and evolution patterns under varying bedding inclinations, rock mass properties, and lateral pressure coefficients. The main conclusions are:

(1) Rock mass anisotropy exerts a significant influence on plastic zone distribution. Plastic zone expansion predominantly occurs within 30° of the bedding plane normal because rock strength parameters decrease substantially when the angle between the maximum principal stress and bedding planes is 30°, facilitating rock yield.

(2) For weakly anisotropic rock masses, the plastic zone boundary approximates a circular shape. As anisotropy increases, pronounced differences in plastic zone radius develop around the tunnel, exhibiting a characteristic butterfly distribution.

(3) Rock mass anisotropy amplifies the effect of lateral pressure on plastic zone morphology. Under high lateral pressure coefficients, the superposition of anisotropic behavior and lateral stress leads to more significant plastic zone development at the tunnel shoulders.

References

[1] Wang Weijun, Fan Lei, Zhao Zhiqiang, et al. Research progress on roadway surrounding rock support theory and technology based on plastic zone control[J]. Journal of China Coal Society, 2024, 49(01): 320-336.

[2] Chen Liwei, Peng Jianbing, Fan Wen, et al. Analysis of plastic zone in circular roadway surrounding rock under non-uniform stress field based on unified strength theory[J]. Journal of China Coal Society, 2007, (01): 20-23.

[3] Zhang Fei, Liu Defeng, Zhang Xiaobo, et al. Engineering geological analysis of plastic zone distribution in roadway surrounding rock under non-uniform in-situ stress field[J]. Coal Mining Technology, 2013, 18(05): 56-59.

[4] Hu Z, An X, Li F, et al. The shape characteristics of circular tunnel surrounding rock plastic zone in the complex stress field[J]. Arabian Journal of Geosciences, 2022, 15(2):

[5] Pan Yang, Zhao Guangming, Meng Xiangrui. Elastoplastic study of roadway surrounding rock based on Hoek-Brown strength criterion[J]. Journal of Engineering Geology, 2011, 19(05): 637-

[6] Xia Binwei, Chen Guo, Kang Yong, et al. Deformation and failure characteristics and stability evaluation of layered rock mass surrounding rock[J]. Hydrogeology & Engineering Geology, 2010, 37(04): 48-52.

[7] Zhang Yujun, Zhang Siyuan. Application of modified Hoek-Brown criterion to layered rock mass[J]. Chinese Journal of Rock Mechanics and Engineering, 2017, 36(S1): 3307-3313.

[8] Saroglou H, Tsiambaos G. A modified Hoek–Brown failure criterion for anisotropic intact rock[J]. International Journal of Rock Mechanics and Mining Sciences, 2008, 45(2): 223-234.

[9] Li Tao, Zhao Guangming, Meng Xiangrui, et al. Analysis of plastic zone in circular roadway surrounding rock under non-uniform stress field considering intermediate principal stress[J]. Coal Engineering, 2014, 46(03): 68-71.