Displacement of surrounding rock in a deep circular hole considering double moduli and strength-stiffness degradation

The problem of cavity stability widely exists in deep underground engineering and energy exploitation. First, the stress field of the surrounding rock under the uniform stress field is deduced based on a post-peak strength drop model considering the rock’s characteristics of constant modulus and double moduli. Then, the orthogonal non-associative flow rule is used to establish the displacement of the surrounding rock under constant modulus and double moduli, respectively, considering the stiffness degradation and dilatancy effects in the plastic region and assuming that the elastic strain in the plastic region satisfies the elastic constitutive relationship. Finally, the evolution of the displacement in the surrounding rock is analyzed under the effects of the double moduli characteristics, the strength drop, the stiffness degradation, and the dilatancy. The results show that the displacement solutions of the surrounding rock under constant modulus and double moduli have a unified expression. The coefficients of the expression are related to the stress field of the original rock, the elastic constant of the surrounding rock, the strength parameters, and the dilatancy angle. The strength drop, the stiffness degradation, and the dilatancy effects all have effects on the displacement. The effects can be characterized by quantitative relationships.


Introduction
Circular holes are widely involved in deep underground energy exploitation, nuclear waste storage, and underground space development, such as circular chambers in mines, water conveyance tunnels, mine borehole pressure relief, oil and gas development, and columnar holes in coalbed methane mining [1][2][3][4][5] . In addition to the original rock stress, this type of holes is also subject to different internal pressures. The instability of the chamber is manifested as shrinkage or expansion, which is related to the plastic failure and displacement of the surrounding rock. The instability of the surrounding rock of the hole has brought many challenges to the excavation of deep underground engineering, such as drill hole inclination, hole collapse in soft rock, and jamming of a drilling tool. Therefore, understanding the mechanical response of the surrounding rock in deep circular holes under different combinations of pressures is of great significance for maintaining the stability of deep engineering.
Researchers have done a lot of work in predicting the stress and displacement fields of circular holes. For example, based on the Hoek-Brown failure criterion, Brown et al. [6] analyzed the theoretical solutions for the stress and displacement of circular holes in two models, i.e., elasticplastic and elastic-brittle-plastic. Sharan [7][8] analyzed the displacement of the cavern based on the Hoek-Brown failure criterion in an elastoplastic model. Jiang and Shen [9] studied the elastic, brittle, and plastic expansion of cylindrical holes based on the Drucker-Prager (D-P) criterion and Mohr-Column (M-C) criterion. Wen et al. [10] used the twin-shear strength theory to obtain analytical solutions for the stress and displacement of the failure zone, the plastic zone, and the elastic zone of the circular chamber. For deep well seepage problems, some scholars considered the seepage force as radial force and provided analytical solutions for underwater chambers [11] and porous elastic-brittle-plastic rock [12][13] based on the elastic-brittleplastic model and M-C criterion. The tunnel boring machine (TBM) method or mine blasting method causes initial damage to the surrounding rock of a chamber. Thus, in many reports, the analytical solutions of the stress and displacement of a circular hole were obtained considering the initial damage caused by blasting loads [14][15][16][17] . From the existing results, researchers have established analytical solutions for various circular holes based on different working conditions using different constitutive models. In these solutions, the displacement in the plastic zone was developed based on the following three assumptions. First, the elastic strain in the plastic zone was assumed to be constant and equal to the strain at the elastoplastic interface [18][19][20] . Second, the plastic region was considered as an elastic thick-walled cylinder to obtain the solution of the strain [7] . Third, the elastic displacement of the plastic region was assumed to meet the law of elasticity, and thus the elastic constitutive equation and the stress in the plastic region can be used to obtain the elastic strains [21][22] . Park and Kim [23] compared the calculated displacement in the plastic zone using the above three methods. Based on the comparison results, it was found that the first method would underestimate the displacement value. When the dilatancy angle was zero, the results from the second and third methods were similar, but the difference between the two methods increased as the dilatancy angle increased.
The relevant conclusions in the above reports are of great significance for understanding the instability of deep circular holes. However, rock is an inhomogeneous material with internal cracks. Therefore, it has double moduli in tension and compression. After the chamber was excavated, the surrounding rock in the plastic zone experiences not only the strength drop but also the modulus damage, which was rarely studied in the present achievements. Therefore, this paper focuses on the solution of the displacement field of surrounding rocks under the dualmodulus properties and strength-stiffness degradation, instead of the influence of multi-field coupling or the selection of strength criteria.

Description of problem
As shown in Fig. 1(a), the deep circular chamber with a radius a is subject to the hydrostatic pressure p 0 . The inner wall of the chamber is subject to an internal pressure p i . At this time, three stresses of the surrounding rock σ r , σ θ , and σ z are all principal stresses. p i can be regarded as the virtual internal supporting force provided by the tunnel face during the excavation of the roadway. As the tunnel face moves away from the section, p i gradually decreases. When it decreases to a certain value, the surrounding rock close to the free face of the roadway enters the plastic zone, and then strength drop and modulus damage occur. At this time, the three principal stresses are all compressive stresses. In addition, p i can also be regarded as the internal pressure, the drilling pressure, or the high internal water pressure of the water and oil pipelines. When the force is greater than a certain value, the surrounding rock of the roadway also undergoes the plastic deformation, and the strength and modulus of the plastic zone drop. At this time, σ r and σ z are still compressive stresses, while the tangential stress σ θ changes from the tensile force to the compressive force from the inner boundary to the outer boundary of the hole.
Let the compressive stress be positive and the tensile stress be negative. The dimensionless radial coordinates are defined as t = r/a (t 1), the boundary of the plastic zone of the surrounding rock is defined as t = t p , and the position for the tangential stress to change sign is t = t s . The deformation of the surrounding rock can be divided into the following zones (see Figs. 1(b) and 1(c)).
, the elastic tension and compression zone (ETCZ). Ω 3 Π zone: t t s , σ r > 0, σ θ > 0, σ z > 0, the ECZ. According to the theory of different elastic moduli in tension and compression [24] , Ω 1 Π zone and Ω 2 Π zone should be analyzed by the dual-modulus theory, and the other zones can be analyzed by the constant modulus.

Elastic constitutive and deterministic equations of rock considering double moduli
Assume that the elastic moduli in compression and tension of the surrounding rock are E + and E − , respectively, and the transverse deformation coefficients are μ + and μ − , respectively. According to the theory of different elastic moduli in tension and compression [24][25] , the plane strain elasticity constitutive is where b 11 = a 11 a 33 − a 13 a 31 a 33 , b 12 = a 12 a 33 − a 13 a 32 a 33 , For deep surrounding rocks, σ r > 0, and σ z > 0. Therefore, a 11 = a 33 = 1/E + , and a 21 = a 31 = a 13 = a 23 = −μ + /E + . a 12 , a 22 , and a 32 are set based on the compressive and tensile signs of the tangential stress σ θ . The plane differential equation for axisymmetric problems can be simplified as The geometric equation is Let σ r = Φ/r, and σ θ = dΦ/dr. The deterministic control equations of stress can be obtained by combining Eqs. (2) and (3), , which is related to the double moduli of the rock.
The general solution of elastic stress can be obtained using Eq. (4), where C 1 and C 2 can be determined by the stress boundary conditions.

Yield and damage behaviors of surrounding rock
The M-C criterion is used to describe the strength characteristics of the surrounding rocks. Set the initial yield criterion to After the strength drop, the subsequent yield criterion is where in which c and ϕ are the initial cohesion and internal friction angle of the surrounding rock, respectively, and c r and ϕ r are the cohesion and internal friction angle of the surrounding rock in the residual stage after the strength drop, respectively. Assume that the elastic strain is smaller than the plastic strain and that the strain in the plastic region obeys the non-associated flow rule, as shown in Fig. 2. Then, the following equation can be satisfied: where the superscript p represents the plastic region, and the subscript p represents the plastic strain. Considering the modulus damage of the surrounding rock in the plastic zone, and assuming that the elastic strain in the plastic zone satisfies the law of elasticity, the elastic strain in the plastic zone can satisfy the following equation: where the superscript r indicates the modulus after damage.  2) and (7) and considering the boundary conditions of t = 1 and σ r = p i , the stress solutions of the plastic zone can be obtained as follows: The elastoplastic interface stress satisfies σ r +σ θ = 2p 0 . Substituting this equation into Eq. (6), the radial stress at the elastoplastic interface can be obtained as follows: Based on σ p r (t p ) = σ tp r , the radius of the plastic zone can be obtained as follows: Using Eq. (5) and the boundary conditions of t = t p , σ e r (t p ) = σ tp r and t → ∞, σ e r = p 0 , the stress solutions of the elastic zone can be obtained as follows: 3.1.2 Displacement solutions Both elastic and plastic deformations exist in the plastic zone, and the total strains can be expressed as follows: According to Eq. (8), the following equation can be obtained for this condition: where Θ = (1 + sin ψ)/(1 − sin ψ), in which ψ is the dilatancy angle of the surrounding rock.
If only the displacement caused by excavation is considered, the deformation caused by the original rock stress before excavation should be removed, and the elastic strains should meet the following equation: For plane axisymmetric problems, the surrounding rock has only the radial displacement u. Combining Eqs. (3), (15), and (16), the differential equations for the displacement of the plastic zone can be obtained as follows [23] : where f (t) = (b r 11 + Θb r 21 )σ p r + (b r 12 + Θb r 22 )σ p θ − (b r 11 + b r 12 + Θb r 21 + Θb r 22 )p 0 . The displacement at the elastoplastic interface is Combine Eqs. (17) and (18). Then, the displacement of the plastic zone can be obtained as follows: The displacement of the plastic zone can be obtained by integrating the above equation, where Combining Eqs. (1) and (13), the displacement of the elastic zone can be obtained as follows:

Analytical solutions in Case B
3.2.1 Stress solutions (i) Stress in Ω 1 Π zone Since σ r > 0, σ θ < 0, σ 1 = σ r , and σ 3 = σ θ , combining Eqs. (2) and (6), the stress solutions of the plastic zone can be obtained, The stress at the elastoplastic interface is The radius of the plastic zone can be obtained by σ p r (t p ) = σ tp r and Eq. (12), (ii) Stress in Ω 2 Π zone The boundary conditions of the Ω 2 Π zone are σ r = σ tp r when t = t p , and σ θ = 0 when t = t s . The stress solutions can be obtained by Eq. (5), (iii) Stress in Ω 3 Π zone In this zone, a 11 = a 22 , and b 11 = b 22 . Thus, λ = 1, and the boundary conditions are σ θ = 0 when t = t s , and σ r = p 0 when t → ∞.
The stress solutions can be obtained by Eq. (5), According to the stress continuity condition (σ r ) t=s − = (σ r ) t=s + , the transcendental equation of s can be obtained, 3.2.2 Displacement solutions (i) Displacement in Ω 1 Π zone In the plastic zone, the following equation is obtained: The elastic strain still satisfies Eq. (9), and the differential equation for displacement control is rewritten as where g(t) = (b r 21 +Θb r 11 )σ p r + (b r 22 +Θb r 12 )σ p θ − (b r 21 + b r 22 + Θb r 11 + Θb r 12 )p 0 . The displacement at the elastoplastic interface can still be obtained by Eq. (19), The displacement of the plastic zone can be obtained by integrating the above equation, where (ii) Displacement in Ω 2 Π zone The displacement in this zone can be obtained by combining Eqs. (1) and (2), (iii) Displacement in Ω 3 Π zone The displacement in this zone can be obtained by combining Eqs. (1) and (26),

Unified solution of displacement in two cases
From the above analysis, the displacement of the plastic zone can be written in a unified form for both cases, The expression of each parameter is summarized in Table 1.

Analysis and comparison of solution
The tensile and compressive elastic modulus ratio is defined as E = E + /E − , the ratio of Poisson's ratios is defined as μ = μ + /μ − , and the residual strength coefficient in the plastic strength drop zone is defined as λ c = c r /c and λ ϕ = ϕ r /ϕ. Considering the modulus damage in the strength drop zone, the residual modulus coefficient is set to be D E = E + r /E + = E − r /E − ; that is, the tensile and compressive moduli are considered to be degraded at the same level, and the transverse deformation coefficient damage is not considered. The values of the parameter in the two cases in Eq. (34) are shown in Table 2. Table 1 Values of variables and constants in unified solution of displacement The values of the basic calculation parameters in both cases are shown in Table 3.  Figure 3 shows the influence of compressive and tensile dual-modulus characteristics on the displacement of the surrounding rock. In Case A, the elastic constants in tension and compression of the surrounding rock are the same; that is, the tensile and compressive elastic modulus ratio E = 1.0, and the ratio of Poisson's ratios μ = 1.0. The displacement of the surrounding rock is not affected by the difference of modulus. In Case B, the shear stresses, radial stresses, and axial stresses in Ω 1 Π and Ω 2 Π have opposite signs. Therefore, the difference between compressive and tensile moduli has a great impact on the displacement. Figure 3(a) shows the evolution law of the displacement of the surrounding rock when the tensile and compressive elastic modulus ratio E is different. At smaller E , the displacement of the surrounding rock in the plastic zone is larger. As E increases, the displacement of the plastic zone decreases. When E < 1.0, the effect of the difference between tensile and compressive elastic moduli on the displacement of the plastic zone is more significant than that at E > 1.0. The influence of E on the displacement in the Ω 3 Π elastic zone exhibits the same trend as that in the Ω 1 Π zone. In comparison, the effect of E on the displacement in the Ω 3 Π elastic zone is less significant. In addition, E has no effect on the relative radius t p of the plastic zone, but it has a greater effect on the relative radius t s of the Ω 2 Π zone. As E increases, t s also continues to expand outward. From Fig. 3(b), the influence of μ on the displacement of the surrounding rock is just the opposite to the influence of E . With the increase in μ , the displacements of the surrounding rock in Ω 1 Π and Ω 2 Π zones increase continuously and have the same trend. μ has no effect either on the relative radius of the plastic zone t p . The relative radius of the Ω 2 Π zone decreases with the increase in μ , but the change is not significant. when λ ϕ drops to 10% of its original value, the maximum displacement of the surrounding rock is increased by only 32.3%.

Influence of dual-modulus characteristics on displacement
In the case of double moduli (Case B), with the drop of cohesion and friction angle, the displacement of the surrounding rock increases significantly, especially in the plastic zone. Unlike Case A, the radius of the surrounding plastic zone increases with the drop of the strength parameter, but the displacement at the elastoplastic interface is not changed. In addition, the friction angle drop has a more significant effect on the displacement of the surrounding rock than the cohesive force drop. When λ ϕ is dropped to 10% of the original value, the maximum displacement of the surrounding rock increases by 148%. With the same drop of cohesion, the maximum displacement of the surrounding rock increases by only 63%.

Influence of dilatancy effect
The dilatancy effect has great influence on the displacement of the surrounding rock in the plastic zone as shown in Fig. 6. Under different dilatancy angles, the displacements of the surrounding rock in Case A and Case B show different characteristics. In Case A, the displacement of the surrounding rock shrinks inward. As the dilatancy angle increases, the convergent displacement of the surrounding rock increases significantly, showing a dilatation effect. In Case B, the displacement of the surrounding rock expands outward. As the dilatancy angle increases, the displacement of the surrounding rock tends to decrease, showing a compacting effect on the plastic zone due to the large pressure in the roadway. In comparison, the dilatancy effect has more significant influence in Case A.

Influence of stiffness degradation on displacement
After excavation of the chamber, in addition to the strength drop of the surrounding rock in the plastic zone, the stiffness is also significantly degraded. Figure 7 shows the effect of the residual modulus coefficient on the displacement of the surrounding rock in both cases. For the convenience of analysis, it is assumed that the elastic moduli of tension and compression are attenuated by the same degree. Under the conditions of constant modulus and double moduli, the displacement of the surrounding rock increases obviously with the deterioration of stiffness. As the modulus gets more degraded, the displacement of the surrounding rock in the plastic zone is increased faster. The above analysis considers the influence of the tension-compression double-modulus characteristics of the surrounding rock, the strength of the plastic zone, the stiffness degradation, and the dilatancy effect on the displacement of the surrounding rock. From the perspective of the displacement of the surrounding rock, Park and Kim [23] analyzed in detail the validity of three different calculation methods of elastic strain in the plastic zone and the equivalent replacement of M-C and Hoek-Brown parameters. If the double-modulus characteristics of the rock and the stiffness degradation of the plastic zone are not considered, the solution in the study is consistent with the solution proposed by Park and Kim.

Conclusions
In this paper, a unified analytical solution for the displacement of the surrounding rock is obtained for the problem of circular holes in deep underground engineering, considering different characteristics of the tensile and compressive elastic moduli of the surrounding rock, the strength-stiffness degradation, and the dilatancy behavior in the plastic zone. In addition, the effects of the tensile and compressive elastic modulus ratio, the residual strength, the residual stiffness, and the dilatancy angle on the displacement of the surrounding rock are analyzed. The main conclusions are as follows: (i) According to the sign change of tangential stress, the surrounding rocks under a constant modulus can be divided into two zones, i.e., the PDZ and the ECZ. On the other hand, the surrounding rocks under double moduli can be divided into three zones, i.e., the PDZ, the ETCZ, and the ECZ.
(ii) Considering the strength drop, the stiffness degradation, and the dilatancy effect, the analytical solutions of displacement of the surrounding rock in the plastic zone under the conditions of constant modulus and dual moduli using non-associated flow laws and the M-C yield criterion can adopt a unified expression.
(iii) Under the dual moduli, the tensile and compressive elastic constant ratio has a significant effect on the displacement of the surrounding rock. With the increase in the tensile and compressive elastic modulus ratio E , the displacement of the plastic zone is decreased. However, with the continuous increase in the ratio of Poisson's ratio μ , the displacement of the surrounding rock appears to continuously increase. In comparison, the impact of E is more significant. Although E has no effect on the relative radius of the plastic zone, it has a great effect on the boundary between the two zones of the ETCZ and the ECZ.
(iv) The strength drop, the stiffness degradation, and the dilatancy effect in the plastic zone affect the displacement of the surrounding rock in both cases. As the strength drop of the surrounding rock is greater and the stiffness degradation is more severe, the displacement in the plastic zone is greater. However, the dilatancy characteristics have a different effect. The dilatancy angles have opposite influence on the displacement of the surrounding rock under the constant modulus and double moduli.