Research Article | Open Access

Xuyang Shi, Wei Zhou, Liang Chen, Qingxiang Cai, Ming Li, Zhaolin Li, Boyu Luan, "A New Unified Solution for Circular Opening considering Different Strength Criteria and the Postpeak Elastic Strain Form", *Advances in Civil Engineering*, vol. 2020, Article ID 8872201, 21 pages, 2020. https://doi.org/10.1155/2020/8872201

# A New Unified Solution for Circular Opening considering Different Strength Criteria and the Postpeak Elastic Strain Form

**Academic Editor:**Zhiqiang Yin

#### Abstract

The strength criterion is an extremely important basis for evaluating the stability of surrounding rock and optimizing the support pressure design. In this paper, nine different strength criteria are summarized and simplified based on the reasonable assumption. Then, a new unified criterion equation is established, which includes all strength theories proposed by this paper. Meanwhile, a new unified closed-form solution for circular opening based on the newly proposed unified criterion equation is deduced with the infinite and finite external boundary combining with the nonassociative flow rule under plane strain conditions. In the plastic zone, four different elastic strain assumptions are applied to solving the plastic zone deformation considering the effect of rock mass damage. The solution’s validity is also verified by comparison with the traditional solution. Finally, the influences of strength criteria, dilation coefficient, elastic strain form of plastic zone, and rock mass damage on the mechanical response of surrounding rock are discussed in detail. The research result shows that TR and VM criteria give the largest plastic zone radius, followed by IDP, MC, and MDP criteria, and seem to underestimate the self-strength of rock mass; The CDP criterion gives the smallest plastic zone radius and may overestimate the self-strength of rock mass; UST0.5, GSMP, GMC, and GLD criteria that reasonably consider the effect of internal principal stresses give an intermediate range and can be strongly recommended for evaluating the mechanics and deformation behavior of surrounding rock; as the dilation coefficient gradually increases, the dimensionless surface displacement presents the nonlinear increase characteristics; the deformation of plastic zone and the ground response curve, which are closely related to the strength criteria, are also greatly influenced by the elastic strain assumption in the plastic zone and rock mass damage degree. The assumption that the elastic strain satisfies Hook’s law (Case 3) may be more reasonable compared with the continuous elastic strain (Case 1) and thick-walled cylinders (Case 2) assumptions; in addition, the Young’s modulus power function damage model seems to give more reasonable solution for the deformation of plastic zone and is suggested to be a preferred method for solving plastic displacement. The research results can provide very important theoretical bases for evaluating the tunnel stability and support design reliability of different lithology rock masses in underground engineering.

#### 1. Introduction

Accurate prediction for the stress and displacement distribution of circular opening plays a crucial role in evaluating the mechanics and deformation behavior of rock mass in civil, mine, and oil engineering and natural gas development engineering. The circular opening may include tunnel, vertical shaft, boreholes, and pile foundation holes. In the early stage, many closed-form solutions of circular opening are obtained by regarding the outer boundary as the infinite boundary with association or nonassociative flow rules [1–9]. However, the semianalytical solutions considering the finite boundary are rarely deduced because of the complexity of the calculation process [6, 10–13]. In practice, it is possible to calculate the outer boundary as an infinite boundary only if it is much larger than the diameter of circular opening. Therefore, this simplified calculation method has certain limitations in many geological engineering, such as shallow-buried tunnel.

As for most of the analyses reported in the past, the solutions were given considering different yield criteria, such as linear Mohr–Coulomb (MC), Tresca (TR), and nonlinear Hoek–Brown (HB) criteria [2, 4–9, 14–17]. Nevertheless, the above studies did not consider the effect of intermediate principal stress on the mechanical response of surrounding rock. Many research results have shown that the intermediate principal stress exerts significant influence on the failure behavior of rock mass [6]. In addition, for rock mass as a natural geological material, the yield failure criterion is more complicated under the influence of internal crack and joint. Therefore, it is extremely difficult to reconcile the calculation results with the field measured results if only one or two yield criteria are used to predict the stresses and displacement behavior of surrounding rock. In this paper, as shown in Table 1, different yield criteria, such as Mogi–Coulomb (GMC) [9, 10], Drucker–Prager (CDP, MDP, and IDP) [3, 4, 14, 28–30], Generalized Lade–Duncan (GLD) [31, 35], Generalized SMP (GSMP) [18, 27, 31], and von Mises (VM) criteria [15, 26], will be summarized and simplified and then used to study the mechanical response of rock mass.

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The compressive stress is positive and the tensile stress is negative ; b is the intermediate principal stress coefficient; ; ; . |

For an elastic-brittle plastic rock mass, the postpeak deformation is closely related to the assumed form of elastic strain in the plastic zone. Brown et al. researched the ground response curve for the rock tunnel by assuming that the elastic strain in the plastic region was equal to that on the elastic-plastic interface [36]. Sharan presented a series of new closed-form solutions for the prediction of displacements around circular openings in a brittle rock mass with nonlinear Hoek–Brown criterion by regarding the elastic strain of plastic zone as the thick-walled cylinder [16, 37]. Yu and Zhang and Reed et al. held that the elastic strain in the plastic zone satisfied the generalized Hooke’s theorem and then derived the deformation of plastic zone with nonassociative flow laws [9, 38–40]. In addition, Park summarized the above three different definitions for elastic strains in the plastic zone and analyzed the deformation law of plastic zone with Mohr–Coulomb and Hoek–Brown criteria under three different case conditions [9]. However, the postpeak Young’s modulus attenuation along the radii direction is ignored in this study.

In this paper, different yield criteria of rock mass are firstly summarized and then a unified yield criterion form is derived by simplifying the above criteria. Next, a new closed-form solution for stresses and deformation distribution around a circular opening subjected to the hydrostatic pressure at the finite or infinite outer boundary is also obtained with the new proposed unified yield criterion. In the plastic zone, five different definitions for elastic strains in the plastic zone and nonassociated flow rule are adopted to establish the radii displacement solution. The correctness of the solution is also verified by comparison with a series of traditional solution and numerical simulation results. Finally, the influences of strength theory and elastic strain definitions of plastic zone on the mechanical response of surrounding rock are discussed in detail.

#### 2. Brief Description of Yield Criterion

##### 2.1. Mohr–Coulomb Criterion

The linear Mohr–Coulomb criterion is widely used in geotechnical engineering because of its simple expression form. However, the influence of intermediate principal stress on rock mass failure is ignored. In addition, the triaxial strength of rock mass obtained by MC criterion deviates greatly from the measured data under the high confining pressure conditions. The following is the governing equation for MC criterion based on the cohesion and internal friction angle [1, 14, 39]:where is a constant which is a function of the internal friction angle and is the uniaxial compressive strength (UCS). They can be expressed as follows:

##### 2.2. Tresca Criterion

The Tresca (TR) criterion is a simplified form of MC criterion and assumes that the failure will occur if the maximum shear stress inside any plane of rock mass reaches a critical value [1, 15]. As shown in Figure 1, the yield curve of TR criterion is a regular hexagon, which is inscribed in the circular yield curve of the von Mises (VM) criterion in *π*-plane. The expression form is

The TR criterion can be considered as a special case where the internal friction angle equals zero in the MC criterion. Equation (2) can be rewritten as

##### 2.3. Generalized Lade–Duncan Criterion

In early 1973, the Lade–Duncan criterion has been firstly proposed by considering the intermediate principal stresses based on the triaxial compression test for noncohesive soil. The initial expression form can be written as follows:where and are, respectively, the first and third principal stress invariants of friction material. is the material constant which is closely related to internal friction angle:

In 1999, the Lade–Duncan criterion was modified by Ewy so that it could reasonably describe the strength characteristics of cohesive soil by introducing bound stress () [35]. Its generalized expression is also given aswhere and are, respectively, the first and third principal stress invariants of cohesive-friction material:where is the bound stress: . For the noncohesive soils, under plane strain conditions, the internal principal stress can be expressed as

By substituting equations (7), (8), and (11) into equation (6), the Lade–Duncan criterion can be rewritten as

For the cohesive-friction material, according to the coordinate translation method [18, 19], the generalized Lade–Duncan (GLD) criterion will be obtained by integrating equation (12).

##### 2.4. Generalized SMP Criterion

Considering the effect of internal principal stress, H. Matsuoka and T. Nakai proposed the SMP criterion, which could be expressed by three principal stress invariants [31]:where is the second principal stress invariant and is the material constant in the SMP criterion:

For the frictional material, by introducing equations (7), (8), (11), and (16) into equation (14), another form of SMP criterion will be deduced.

As the GLD criterion, according to the coordinate translation method, the generalized SMP criterion will be given as [18]

As shown in Figure 2, compared with the yield curve of MC criterion in the *π*-plane, the GLD and GSMP criteria are nonlinear and the cross-sectional area of GLD criterion is the largest, followed by the GSMP criterion.

##### 2.5. Mogi–Coulomb Criterion

Based on Mog’s theory [24], Al-Ajmi and Zimmerman found that the polyaxial test data could be fitted by linear relationships in space.where is the mean normal stress and is octahedral shear stress. The parameters and can be related exactly to the Coulomb strength parameter:

As the von Mises criterion, we take . Then, the Mogi–Coulomb (GMC) criterion can be obtained by integrating equations (19), (20), and (21).

##### 2.6. Unified Strength Theory

Based on the twin shear yield criterion, the unified strength theory (UST) is established by considering the influence of all the stress components on the material yield failure [32–34]. In geotechnical engineering, the cohesion and internal friction angle are usually used to represent this yield theory. The yield function can be expressed as follows [5, 6]:

If ,

If ,where represents the yield parameter related to the intermediate principal stress, which can reflect the effect of the intermediate principal shear stress and the positive stress on the yield failure of the rock material, and . As shown in Figure 3, if , UST will translate into the MC criterion; if , UST is converted into general twin shear strength (GTSS) criterion; if , UST is a series of other ordered new strength criteria.

Just like Sections 2.3 and 2.6, if , can be judged by substituting into equation (24a) or equation (24b). Therefore, UST can be rewritten as

##### 2.7. Drucker–Prager Criterion

The Drucker–Prager (DP) criterion is an extension of the von Mises criterion, which takes into account the effect of intermediate principal stress and hydrostatic pressure on yield failure of the materials. It can be expressed aswhere the parameters and are the material constants, which can be determined from the slope and the intercept of the failure envelope. The parameter is related to the cohesion and internal friction angle of rock mass. The parameter is only related to the friction angle. Therefore, the Mohr–Coulomb parameter can be used to depict the DP criterion. By comparison with the yield curve of MC criterion in the *π*-plane (see Figure 4), the DP criteria can be divided into Circumscribe Drucker–Prager (CDP) criterion, Middle Circumscribe Drucker–Prager (MDP) criterion, and Inscribe Drucker–Prager (IDP) criterion.

As shown in Sections 2.6 and 2.7, we take . Then, the Drucker–Prager criterion can be deduced by solving equation (26).

##### 2.8. Establishment of Unified Criterion Equation

As shown in Table 1, the unified equation of different yield criterion can be summarized based on the above analysis under plane strain conditions.where the subscript “” represents different yield criteria and and are the material constants of different criteria, which can be divided into two different cases. When , equation (29) corresponds to the TR and VM criteria; when , equation (29) represents other criteria. Therefore, it can be regarded as a unified criterion equation to research the mechanical response of rock mass.

#### 3. Problem Description

##### 3.1. Establishment of Calculating Model

Figure 5 shows that a circular opening is excavated in a finite, isotropic, homogeneous elastic-brittle plastic rock mass subjected to an inner pressure at the inner radii and a hydrostatic pressure at the external radius . As gradually decreases, the displacement will occur and the plastic zone with the radii will firstly develop around the circular opening when the maximum principal stress and minimum principal stress satisfy the initial yield condition. The influence of the rock mass weight in the plastic zone on the radial displacement and inner pressure is ignored. In this paper, the brittle plastic rock mass is introduced to research the postpeak mechanical behavior of rock material. As shown in Figure 6, the strength of the rock mass suddenly drops after peak load and then the postpeak softening behavior of strength parameters will occur. In other words, the postpeak cohesion , internal friction angle , Young’s modulus , and Poisson’s ratio are used to solve the stress and displacement distributions in the plastic zone.

**(a)**

**(b)**

Under the axisymmetric plane strain conditions, when , the hoop stress and radial stress are, respectively, maximum principal stress and minimum principal stress; the tangential strain and radial strain are, respectively, maximum principal strain and minimum principal strain. Accordingly, the unified criterion equation can be rewritten as follows:where the subscript “” represents different zones ( = 1, 2), and are the initial strength parameters, respectively, and and are the residual strength parameters, respectively.

##### 3.2. Based Equation

For the axisymmetric plane strain problem, the equilibrium differential equation can be expressed as (ignoring the body force of rock masses) [29, 30, 39]

The geometric equation, based on the small deformation assumption, can be denoted aswhere is the radial displacement. Both the radial displacement and radial contact stress should be continuous at the elastic-plastic interface, respectively. Therefore, the boundary conditions around the circular opening can be summarized as

#### 4. The New Unified Solution of Circular Opening

##### 4.1. A Semianalytical Solution with the Finite External Boundary

According to the elastic mechanics theory, the stresses and displacement in the elastic zone can be deduced by regarding it as a hollow thick-wall cylinder subjected to an inner pressure at the internal boundary and a hydrostatic pressure at the external boundary as follows:where and are the initial Young’s modulus and initial Poisson’s ratio, respectively. The initial yield failure condition should be satisfied at the elastic-plastic interface. Therefore, the radial contact stress can be derived by substituting equations (34a) and (34b) into equation (30a) under different yield criteria:

Obviously, the stresses in the plastic zone should satisfy the equilibrium differential equation and are easily deduced by submitting equation (30b) into equation (31) as well as combining with the boundary condition at . However, for and , the stress solutions in the plastic zone are inconsistent.(i)f , the unified stresses solution in the plastic zone will be obtained as Here, the subscript “” represents MC, GMC, GLD, GSMP, CDP, MDP, IDP, UST, and GTSS criteria. Then, the radii of plastic zone can be determined by considering the boundary condition at the elastic-plastic interface. By integrating equation (35) and equation (36a), the nonlinear function between and can be established as follows:(ii)If , the unified stress solutions of plastic zone based on the TR and VM criteria are

As the solution for equation (41), the nonlinear function between and based on the VM and TR criterion can be also deduced as follows:

In practice, the external radius is a constant. The approximate solution for in equations (37) and (39) can be obtained by using the numerical calculation methods such as the least square method and Newton iteration method. Then, the stresses and displacement in elastic zone can be determined by equations (34a), (34b), and (34c).

In the plastic zone, the total hoop strain and radial strain are, respectively, composed of elastic strain and plastic strain. Therefore, the total strain can be expressed aswhere and are, respectively, the hoop elastic strain and radial elastic strain in the plastic zone; and are the hoop plastic strain and radial plastic strain of plastic zone, respectively.

For axisymmetric plane strain problems, the plastic strain relationships can be established by considering the small strain theory and nonassociated flow rule [5, 9].where is the dilation coefficient: . is the dilation angle.

By substituting equations (32) and (41) into equation (44), the following differential equation for the radial displacement in the plastic zone can be derived as

From equation (42), it can be seen that the radial displacement is closely related to the elastic strain form in the plastic zone.

Then, the following function can be obtained by solving equation (42) as follows:where is the radial displacement at the elastic-plastic interface. It can be easily determined by taking in equation (34c) as follows:

In order to obtain the radial displacement of plastic zone, the expression for elastic strain should be firstly determined. Generally, four different definitions for elastic strain can be used to research the deformation behavior of rock mass in the plastic zone.(1)Case 1: it is assumed that the elastic strain in the plastic region is equal to that on the elastic-plastic interface. Then, the elastic strains can be expressed as [36] Then, the function is By substituting equation (46) into equation (43), the radial displacement in the plastic zone can be derived as follows:(2)Case 2: by regarding the plastic zone as the thick-wall cylinder subjected to the inner pressure at and radial contact stress at , the elastic strain in the plastic zone can be written as [19, 39] where and . Therefore, the function is where and . By submitting equation (49) into equation (43), we can obtain the radial displacement of plastic zone as follows:(3)Case 3: adopting Hooke’s law, the elastic strains in the plastic zone can be written as follows [27]: Then, the function can be expressed as follows:(i)If , the function will be rewritten by substituting equations (36a) and (36b) into equation (52) as follows: where and . By substituting equation (53) into equation (43), the radial displacement in the plastic zone can be derived as follows:(ii)If , the function can be obtained by substituting equations (38a) and (38b) into equation (52) as follows: where and . By submitting equation (55) to equation (43), the radial displacements for TR and VM criterion in the plastic zone are(4)Case 4: the mechanical behavior of rock mass is closely related to its damage degree in the plastic zone. The higher damage usually leads to larger deformation behavior. Then, the attenuation of Young’s modulus should be considered in the plastic zone. In this case, the power function attenuation model of Young’s modulus along the radius direction is introduced to research the deformation behavior in the plastic zone [6]:(i)If , the function will be rewritten by substituting equation (57) into equation (53) as follows: By substituting equation (58) into equation (43), the radial displacement in the plastic zone can be derived. In fact, the expression form is the same as that in equation (54):(ii)If