A New Unified Solution for Circular Tunnels Based on Generalized SMP Criterion considering the Strain Softening and Dilatancy

According to the strain-softening characteristics of rock mass, an ideal elastic strain-softening model is developed, and the surrounding rock of tunnels is subdivided into the plastic broken zone, plastic strain-softening zone, and elastic zone. Based on the generalized spatially mobilized plane criterion, an elastic-plastic analytical solution of a circular tunnel is derived. .e effects of intermediate principal stress, strain softening, and dilatancy are considered in the unified solution. .e stress, displacement, and plastic zone radius of surrounding rock based on the SMP criterion are compared with those based on the Mohr–Coulomb criterion. Furthermore, the effects of parameters such as the softening modulus, dilatancy angle, and internal friction angle on the deformation and stress of tunnels are discussed. It has been found that the larger the dilatancy angle is, the larger the plastic zone displacement and the radius of the broken zone are. .e larger the internal friction angle, the smaller the sizes of the plastic zone, the strain-softening zone, and the broken zone are..e deformation of surrounding rock in the broken zone is more sensitive to the internal friction angle than that in the strain-softening zone. .e unified solution based on the SMP criterion provides a well understanding for the elastic-plastic state of tunnels, which can be the guidance for tunnel excavations and support designs.


Introduction
Rock tunnels are common underground constructions for mining engineering, traffic engineering, diversion engineering, and nuclear waste disposal engineering.e stress around tunnels redistributes due to the excavation disturbance.
e concentrated stress usually exceeds ultimate strengths of the surrounding rock, which make the rock properties transform from elastic state to plastic state.Based on the ideal elastic-plastic model, Fenner and Kastner analyzed the elastic and plastic zones of tunnels and derived Fenner formula and Kastner formula [1].Terzaghi and Peck first proposed that geotechnical materials have strainsoftening characteristics [2].en considerable researches have indicated that, after the stress of geomaterials exceeds the compressive strength, the stress of the geomaterials will gradually decrease to the residual strength with the strain increasing [3][4][5][6][7][8].
erefore, the strain-softening characteristics of the surrounding rock must be considered in the elastic-plastic analysis of tunnels.
Determination of the plastic zone of a tunnel is an important foundation for the stability analysis and support design of the tunnel.Analytical solutions of circular tunnels can provide a fast and direct understanding for tunnels elastic-plastic state, which is an important part in geotechnical analyses [9,10].Traditional elastic-plastic analytical solutions of tunnels are generally based on the Mohr-Coulomb criterion or the Hoek-Brown criterion, in which the effect of the intermediate principal stress is not considered.Although the Drucker-Prager criterion includes the intermediate principal stress, it does not consider the strength differential effect of geomaterials in the deviator stress plane [11][12][13][14][15][16][17].
Previous studies have shown that the geomaterials have a phenomenon of volume expansion during the postpeak deformation process, which is called "dilatancy." e dilatancy is closely related to the plastic shear strain and the strength parameters of rock mass [18][19][20].erefore, in the elastic-plastic analysis of tunnels, the effects of intermediate principal stress, strain softening, and dilatancy should be considered comprehensively.
In this paper, an ideal elastic strain-softening model is first developed and the surrounding rock of circular tunnels is subdivided into the plastic broken zone, plastic strainsoftening zone, and elastic zone.Based on the generalized SMP (i.e., spatially mobilized plane) criterion, a rational analytical solution of the circular tunnel is derived in consideration of the effects of the intermediate principal stress, strain softening, and dilatancy.e stress distribution, displacement distribution, and plastic zone radius of surrounding rock calculated by the solution based on the SMP criterion are compared with those based on the Mohr-Coulomb criterion.Finally, parametric studies of softening modulus, dilatancy angle, and internal friction angle are carried out.

Generalized SMP Criterion
e theory of spatially mobilized plane for noncohesive soils is proposed by Matsuoka and Nakai, referred to as the SMP failure criterion [21].e SMP criterion has a good agreement with the Mohr-Coulomb criterion.Moreover, it overcomes the singularity of the Mohr-Coulomb criterion in the deviator stress plane and the equivalence of strengths of the Drucker-Prager criterion.In addition, the SMP criterion also reflects the effect of intermediate principal stress.
e SMP criterion is expressed as [22] τ where τ SMP and σ SMP are the shear and normal stresses on the SMP, respectively, and σ 1 , σ 2 , and σ 3 are the large, medium, and small principal stresses of the noncohesive soil, respectively.
Figure 1 shows the mutual relationship between the Mohr-Coulomb criterion and SMP criterion in the π plane.It can be seen that, the limiting trajectory of the SMP criterion in the π plane is a smoothly convex curve circumscribing the irregularly hexagon of the Mohr-Coulomb criterion.
Matsuoka et al. revised the SMP criterion by introducing a bond stress σ 0 and further extended the SMP criterion to cohesive soil [22,23].According to the SMP criterion, when the noncohesive soil is destroyed in the plane strain state, the common tangent of the Mohr circle of limit stress on the τ-σ space just passes through the coordinate origin.When the general clay soil is destroyed in the plane strain state, the common tangent of the Mohr circle of limit stress on the τ-σ space just passes the bond stress point σ 0 [24,25].e limit stress state of generalized SMP criterion in the plane strain condition is shown in Figure 2.
For the materials containing cohesion c and internal friction angle φ, the failure criterion in the plane strain condition can be written as ( e bond stress σ 0 can be obtained as e generalized SMP criterion expressed by stress invariants is as follows: where φ is the internal friction angle obtained by the triaxial test; K SMP is the material constant of the SMP criterion; and  I 1 ,  I 2 , and  I 3 are the generalized first, second, and third effective stress invariants, respectively.e effective stress invariants are given as follows by using the principal stress σ i and the bond stress σ 0 : According to the associated flow rule and the generalized SMP criterion, the relationship among σ 1 , σ 2 , and σ 3 in the plane strain condition can be expressed as follows: Substituting ( 6) in (4), the expression of the generalized SMP criterion in the plane strain condition is as follows:  Advances in Civil Engineering e tangential stress σ θ , radial stress σ r , and axial stress σ z of the tunnel are orthogonal to each other.When the lateral pressure coefficient λ � 1, the stresses σ θ , σ r , and σ z can be regarded as three principal stresses.It can be regarded that the three principal stresses in surrounding rock of circular tunnels are σ 1 � σ θ , σ 3 � σ r , and σ 2 � σ z .Formula (7) can be written as where

Description of the Elastic-Plastic Model
3.1.Strain-Softening Model.Considerable laboratory tests and field observations have indicated that, after the stress of the rock exceeds its compressive strength, the stress will gradually decrease to the residual strength [3][4][5][6][7][8].According to the strain-softening characteristics of the surrounding rock, the strain-softening model is established as shown in Figure 3. e rock mass in the elastic zone is far away from the tunnel.e rock mass in the broken zone is near the surface of the tunnel with residual strength.And the plastic strain-softening zone is the area between the elastic zone and the broken zone.

Calculation Model of Circular Tunnels.
In order to carry out the elastic-plastic analysis of surrounding rock, the following assumptions are made: (1) e cross section of the tunnel is in circular shape and the length is infinite.So it can be simplified as a plane strain problem.(2) Rock mass is a continuous, homogeneous, and isotropic elastic-plastic material.
(3) Ignoring the effects of the surrounding rock weight on the yield, the original rock stress can be simplified as a uniform stress distribution.e tunnel is under the conditions of uniform stress p 0 and support resistance p i .(4) e plastic zone induced by the excavation is subdivided into two parts: the plastic strain-softening zone and the plastic broken zone.e radius of the broken zone and the strain-softening zone are R b and R s , respectively.
e radius of the circular tunnel is r 0 .Based on the above assumptions, the mechanical model of the tunnel is shown in Figure 4.

Softening Model of the Strength Parameter.
e strainsoftening characteristics of the rock mass can be reflected by strength parameters such as cohesion c or internal friction angle φ. e internal friction angle φ is not sensitive to the strain-softening characteristics of the rock mass, while the cohesion c is obviously affected by the strain-softening characteristics of rock mass [5,26,27].erefore, the cohesion c is used to represent the strain-softening characteristics of rock mass.e softening process of cohesion c is mainly related to the shear strain ε s θ of the strain-softening zone.e softening process is assumed as linear.e degree of softening is represented by the angle α and the corresponding softening modulus H c � tan α. erefore, the softening model of the strength parameter is shown as Figure 5.
c 0 and c * in Figure 5 is the initial cohesion and residual cohesion, respectively.
e shear strain at the interface between the elastic zone and the strain-softening zone is ε e−s θ , and the shear strain at the interface of the strain-softening zone and the broken zone is ε s−b θ .e cohesion c of a point in the surrounding rock of the circular tunnel can be expressed as follows: where

Dilatancy Model.
With the surrounding rock deformation and crushing, the volume of rock will expand.
According to the plastic potential theory, the plastic strain increment can be expressed as follows [28]: where dε p ij is the plastic strain increment; dλ is the plastic factor increment; g is the plastic potential function; and σ ij is the stress tensor.
In order to reflect the dilatancy characteristics of the surrounding rock, it is assumed that the plastic potential function g has the same form as the yield function f.And the internal friction angle φ in the yield function is replaced by the dilatancy angle ψ. us, the plastic potential function g can be expressed as follows: where Substituting ( 12) into (11), the plastic strain increment can be obtained as Based on the linear nonassociative flow rule, the relationship of plastic strain increment can be expressed as [29] Δε p r + χΔε e expression of dilatancy coefficient in the strainsoftening zone and broken zone is obtained by simultaneous ( 14) and ( 15): where M s * and M b * are the parameters for replacing the internal friction angle φ with the dilatancy angle ψ s in the strain-softening zone and the dilatancy angle ψ b in the broken zone, respectively.When ψ � φ, the uncorrelated flow law is transformed into the associated flow rule.When ψ � 0 °, the dilatancy characteristics of the rock are not considered.

Basic Equation.
Based on the elastic-plastic theory, the equilibrium equation of the infinitesimal body around the tunnel can be expressed as In addition, strain-displacement relations of the surrounding rock are given as After the strain components and displacement are obtained, the stress-strain relation involved in the plane strain problem can be expressed as

Elastic zone
Strain so ening zone 4 Advances in Civil Engineering

Stress and Displacement in the Elastic Zone.
Boundary conditions of the elastic zone can be given by where σ e−s r is the radial stress at the interface between the elastic zone and the plastic strain-softening zone.
At the elastic-plastic interface, both failure criterion (8) and the following relationship are satisfied: e radial stress at the elastic-plastic interface can be obtained as where M e and N e are the parameters M and N in the elastic zone, respectively.According to the elastic-plastic theory, the elastic solutions of stress components can be expressed as According to the small deformation theory, the elastic solutions of strain components can be expressed as Substituting equation (24) into equation ( 18), the radial displacement component u e of the tunnel surrounding rock in the elastic zone can be expressed as follows: where G � E/2(1 + ]).

Stress and Displacement in the Plastic Strain-Softening
Zone.According to the elastic-plastic theory, the strain in the strain-softening zone is composed of elastic strain and plastic strain which can be written as where se and sp represents the elastic part and plastic part in the strain-softening zone, respectively.By letting and combining equations ( 15) and ( 26), the equation for f(r) can be obtained as follows: Ignoring the effect of stress redistribution on the elastic strain in the strain-softening zone and regarding the elastic strain of the strain-softening zone as the same with the strain of the elastic zone at the elastic-plastic interface, the elastic strain in the strain-softening zone can be expressed as Substituting equations ( 28) and ( 29) into equation ( 18), f(r) can be obtained as In combination with the boundary condition at the elastic-plastic interface: e displacement expression of the strain-softening zone is calculated as Combining equations ( 18) and (32), the strain in the strain-softening zone can be obtained: Furthermore, the following equation can be obtained: According to equation (10), the expression of the cohesive force in the strain-softening zone can be obtained: Substitute equation (35) into the yield equation ( 8), replace the initial cohesion c 0 , combined with (17) and the boundary conditions at the elastic-plastic interface, Advances in Civil Engineering e radial and stress in the strain-softening zone can be obtained: where M s and N s are the parameters M and N in the strainsoftening zone, respectively.

Analysis of the Plastic Broken
Zone.Substituting equation (8) into equation ( 17), e radial and stress in the broken zone can be solved by considering the boundary condition σ b r � p i at r � r 0 : where M b and N b are the parameters M and N in the broken zone, respectively.e strain in the broken zone is also composed of elastic strain and plastic strain, and it satisfies ε be r � ε s r and ε be θ � ε s θ at r � R b : By equation (33), the strain in the broken zone can be obtained as According to the same radial displacement at the interface between the strain-softening zone and the broken zone, the following equation can be obtained: We can get the expression of the displacement in the broken zone: e displacement at the wall of the circular tunnel can be obtained by substituting r � r 0 into equation (42).

Radius of the Strain-Softening Zone.
e radial stress at the interface between the strain-softening zone and the broken zone is continuous.And the cohesion c 0 is reduced to the residual value at the interface.By equations ( 36) and (37), the expression of the ratio of the strain-softening zone radius R s to the circular tunnel radius r 0 can be obtained as 4.6.Radius of the Plastic Broken Zone.According to equation (10), the residual cohesion c * at the interface between the strain-softening zone and the broken zone can be expressed as e ratio of the strain-softening zone radius R s to the broken zone radius R b can be obtained as

Worked Examples and Parametric Studies
Taking a circular tunnel with the radius r 0 � 3 m as an example, the geometrical and mechanical parameters are shown in Table 1.e developed elastic-plastic model is used to analyze the stress and displacement of the tunnel.e Mohr-Coulomb criterion is one of the most commonly used strength criteria in geotechnical engineering.
e stress distribution of surrounding rock for the SMP criterion and the Mohr-Coulomb criterion are shown as Figures 6 and 7.It can be seen that the radial tangential stress distributions based on the SMP criterion and the Mohr-Coulomb criterion are different.e radial stress increases with radius increasing, but the growth rate gradually slows down.With the radius increasing, the tangential stress increases in the plastic zone, reaches a maximum value at the elastic-plastic interface, and then gradually decreases.Both radial and tangential stress will eventually be close to the initial rock stress.In addition, it can be seen that the radius of the plastic zone calculated by the SMP criterion is smaller than that of the Mohr-Coulomb criterion.
Figure 8 shows the displacement distribution and the radius of the plastic zone based on the SMP criterion and the Mohr-Coulomb criterion.e dotted vertical line indicates the radius of the broken zone, and the solid vertical line indicates the radius of the strain-softening zone.It can be seen that the displacement of the surrounding rock decreases with the radius increasing.
e displacement of the surrounding rock based on the SMP criterion is generally smaller than that based on the Mohr-Coulomb criterion, 6 Advances in Civil Engineering and the difference between them decreases with the increase of the radius.e displacement will eventually tend to be 0 m as the radius increases.At the tunnel surface, the displacement based on the SMP criterion is 16.67 mm. e maximum displacement based on the Mohr-Coulomb criterion is 28.85 mm, which is 1.73 times larger than that based on the SMP criterion.e radius of the broken zone and the strain-softening zone based on the SMP criterion are both smaller than those based on the Mohr-Coulomb criterion.e radius of the broken zone for the SMP criterion is 4.39 m, which is about 75.82% of that for the Mohr-Coulomb criterion.e radius of the strain-softening zone of the Mohr-Coulomb criterion is 5.70 m, which is about 73.74% of that for the Mohr-Coulomb criterion.

Effect of the Softening Modulus.
In order to analyze the influence of cohesive softening modulus on the deformation of tunnel surrounding rock, the cohesive softening modulus H c are selected as 200 MPa, 300 MPa, 400 MPa, 500 MPa, and 600 MPa for comparative analysis.
e variation of surrounding rock displacement is shown in Figure 9. e dotted vertical line indicates the radius of the broken zone, and the solid vertical line indicates the radius of the strainsoftening zone.It can be seen that the cohesive softening modulus H c has a little effect on the displacement of tunnel surrounding rock.e larger the H c , the smaller the displacement of the surrounding rock near the tunnel is.H c has no effect on the radius of the strain-softening zone but has a significant effect on the radius of the broken zone.With H c increasing, the radius of the broken zone gradually expands, but the extent of expansion becomes smaller and smaller.With H c increasing, the range of the total plastic zone remains unchanged.

Effect of the Dilatancy.
e dilatancy properties of the tunnel surrounding rock are mainly manifested by the dilatancy angle and the plastic potential function.In this paper, for the associated flow rule (ψ � φ), nonassociative flow rule (ψ ≠ φ), and irrespective of dilatancy (ψ � 0), the corresponding value of the dilatancy coefficient χ is obtained.e softening zone radius, the broken zone radius, and the displacement distribution corresponding to different dilatancy angles are illustrated in Figure 10.It can be seen that the dilatancy angle has a significant influence on the displacement of the plastic zone.e displacement of the tunnel surface for the associated flow rule (ψ � φ) and for the nonassociative flow rule (ψ � φ/2) are 21.97 mm and 16.67 mm, respectively.e displacement of the tunnel surface for the dilatancy (ψ � 0) is 14.21 mm, which is 85.24% of that for the nonassociative flow rule (ψ � φ/2).In other words, the associated flow rule takes into account the influence of dilatancy too much, and the actual deformation of the surrounding rock is underestimated without considering the dilatancy.e dilatancy angle has no effect on the radius of the strain-softening zone.However, the larger the dilatancy angle is, the larger the radius of the broken zone is.

Effect of Internal Friction Angle.
To analyze the influence of the internal friction angle on the displacement of surrounding rock, different internal friction angles in the strainsoftening zone and the broken zone were selected for analysis.
e internal friction angles in the strain-softening zone are selected as 22 °, 24 °, 26 °, 28 °, and 30 °, respectively.e internal friction angles in the broken zone are selected as 10 °, 12 °, 14 °, 16 °, and 18 °, respectively.e rock displacement and plastic zone radius are shown in Figure 11.e dotted vertical line indicates the radius of the broken zone, and the solid vertical line indicates the radius of the strain-softening zone.
It can be seen from Figure 11(a) that, as the internal friction angle in the strain-softening zone increases, the displacement of the tunnel surrounding rock decreases.When the internal friction angle in the strain-softening zone is increased from 22 °to 30 °, the displacement at the tunnel wall is reduced from 19.34 mm to 14.06 mm, which is reduced by 27.30%.As the internal friction angle in the strainsoftening zone increases, the radius of the strain-softening zone and the radius of the broken zone decrease.When the internal friction angle in the strain-softening zone is increased from 22 °to 30 °, the radius of the strain-softening zone is reduced from 6.39 m to 5.08 m, and the radius of the broken zone was reduced from 4.81 m to 3.98 m.It can be seen from Figure 11(b) that, as the internal friction angle increases in the broken zone, the displacement of the tunnel surrounding rock decreases.When the internal friction angle in the broken zone is increased from 10 °to 18 °, the displacement at the tunnel wall is reduced from 21.85 mm to 13.25 mm, which is reduced by 39.36%.As the internal friction angle in the strain-softening zone increases, the radius of the strain-softening zone and the radius of the broken zone are reduced.When the internal friction angle in the broken zone increases from 10 °to 18 °, the radius of the strain-softening zone decreases from 6.47 m to 5.11 m and the radius of the broken zone decreases from 4.97 m to 3.96 m. erefore, the change of the internal friction angle in the strain-softening zone and the broken zone will affect the displacement distribution of the tunnel surrounding rock.
e larger the internal friction angle is, the smaller the displacement of the surrounding rock and the radius of the plastic zone are.e surrounding rock of the broken zone is more sensitive to the internal friction angle than that of the strain-softening zone.In the engineering practice, the deformation of the surrounding rock can be effectively  8 Advances in Civil Engineering controlled by increasing the internal friction angle in the broken zone of surrounding rock by grouting or other means.

Conclusions
According to the strain-softening characteristics of rock mass, an ideal elastic strain-softening model was developed.e circular tunnel surrounding rock is subdivided into plastic broken zone, plastic strain-softening zone, and elastic zone.Based on the generalized SMP criterion, the elasticplastic analytical solution of the surrounding rock of the circular tunnel was obtained.
e effects of intermediate principal stress, strain softening, and dilatancy were considered in the unified solution.
e stress distribution, displacement distribution, and plastic zone radius of surrounding rock based on the SMP criterion are compared with those based on the Mohr-Coulomb criterion.e sensitivity of parameters such as softening modulus, dilatancy angle, and internal friction angle were discussed.e results summarized as follows: (1) e displacement of surrounding rock based on the SMP criterion is generally smaller than that of Mohr-Coulomb criterion.e radius of the broken zone and the strain-softening zone based on the SMP criterion are also smaller than that based on the Mohr-Coulomb criterion.(2) e softening modulus has a little effect on the displacement of the tunnel surrounding rock but has a significant influence on the radius of the plastic broken zone.With the softening modulus increasing, the range of the broken zone becomes larger, the range of the strain-softening zone decreases, and the range of the total plastic zone remains unchanged.
(3) e dilatancy angle and the internal friction angle have a great significance on the displacement of the tunnel.e deformation of surrounding rock will be underestimated without considering the dilatancy.e larger the internal friction angle is, the smaller the displacement and the radius of the plastic zone are.e surrounding rock in the broken zone is more sensitive to the internal friction angle than the surrounding rock in the strain-softening zone.Advances in Civil Engineering

2 Figure 1 :
Figure 1: Mutual relationship between the Mohr-Coulomb criterion and SMP criterion in the π plane.

Figure 5 :
Figure 5: Softening model of the strength parameter.

Figure 4 :
Figure 4: Mechanical model of the circular tunnel.

Figure 6 :
Figure 6: Surrounding rock radial stress distribution of the Mohr-Coulomb and SMP criteria.

Figure 7 :
Figure 7: Surrounding rock tangential stress distribution of the Mohr-Coulomb and SMP criteria.

Figure 8 :
Figure 8: Plastic zone radius and displacement of the Mohr-Coulomb and SMP criteria.

H
Figure 9: e influence of strain softening on plastic zone radius and displacements.

Figure 10 :
Figure 10: e influence of dilatancy on the plastic zone radius and displacements.

Figure 11 :
Figure 11: e influence of internal friction angle on plastic zone radius and displacements.(a) Strain-softening zone.(b) Broken zone.

Table 1 :
Geometrical and mechanical parameters.