Viscoelastoplastic Displacement Solution for Deep Buried Circular Tunnel Based on a Fractional Derivative Creep Model

State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Chengdu University of Technology), Chengdu 610059, China College of Environment Geology and Civil Engineering, Chengdu University of Technology, Chengdu 610059, China Fifth Geological Brigade, Hebei Bureau of Geology and Mineral Resources, Tangshan 063000, China Tianjin Center, China Geological Survey, Tianjin 300000, China


Introduction
e rheological deformation of surrounding rock is a common issue encountered in the excavation of soft rock tunnels under the conditions of high in situ stress. When the rheological deformation exceeds a certain level, the surrounding rock will be squeezed, which causes the failure of supporting structures and adversely affects the safety of tunnel constructions and long-term operations [1]. erefore, the analysis regarding the rheological effect of surrounding rock and its influence on the viscoelastic-plastic process of tunnel deformation has important theoretical and practical significance.
Researchers have conducted a lot of analytical studies on the creep features of tunnels. Hong et al. [2] obtained the rheological deformation solution of an axisymmetric circular roadway tunnel based on the Nishihara model. Tang et al. [3] derived the time-dependent displacement solution of deep-buried tunnels considering the effects of strain strengthening and dilation of surrounding rock using a viscoelastic-plastic creep model which is comprised of Burgers body and Drucker-Prager yield criterion. Zhang et al. [4] adopted the improved Nishihara model to derive the viscoelastic-plastic creep displacement of seepage tunnels under the influence of the expansion strain of the surrounding rock. Gu and Yan [5] studied the creep behavior of surrounding rock under the action of seepage with the improved Burgers model. Deleruyelle et al. [6] used the Norton-Hoff criterion to describe the creep effect of surrounding rock and derived an analytical solution for tunnel displacement considering the postpeak effect of surrounding rock.
In former research, the creep behavior of surrounding rock is generally described by viscoelastic constitutive models. However, none of these models can well reflect the accelerated deformation stage of rocks [7,8]. Few researchers have done pioneer work in deriving the closedform solution of tunnel displacement using viscoelastoplastic models [9,10]. erefore, in this article, a creep model, which consists of an elastic element, a non-Newtonian viscous element, an Abel element based on the fractional calculus theory, and a plastic element, is adopted to simulate the whole creep deformation stage of rocks. en, the viscoelastic displacement solution of the deepburied tunnel is derived based on this creep model. Finally, taking the auxiliary tunnel of Jinping II Hydropower Station as an example, the influence of rheological parameters on the tunnel deformation is discussed.

Abel Element Based on Fractional eory.
Fractional calculus operation refers to that the order of integration or differentiation of a function is any real or complex number [11]. Among all relevant definitions, the Riemann-Liouville definition is most suitable for studying the viscosity characteristics of materials. e integral of function f(t) with an order of c is where c > 0 and n − 1 < c ≤ n (n is the smallest positive integer which is larger than c) and Γ(c) is the Gamma function with expression of ∞ 0 e − t t c− 1 dt. e Abel dashpot based on the theory of fractional calculus can well describe the viscous characteristics of the viscous body between the ideal elastic body and the ideal fluid, and its expression is When σ(t) is a constant, its creep equation is

Non-Newtonian Viscous
Element. e NVPB element [12] is put forward to describe the accelerated deformation process of rock after it enters the plastic stage, and its creep equation is where t 0 is reference time with a value of 1, n is the creep index, σ s is the long-term strength of rock, and H(σ) is unit step function:

e Creep Model.
A creep model ( Figure 1) combining the Abel dashpot and a non-Newton viscous element was adopted to simulate the whole creep deformation stage of rocks. e stress-strain relation of each element under the condition of viscoelastic state is [9] e stress-strain relation of each element under the condition of viscoplastic (the plastic behavior of rocks is described by the Hoke-Brown criterion) state is where σ and ε are the total stress and strain, respectively, σ 1 , σ 2 , and σ 3 are the stress of element 1, 2, and 3, respectively, ε 1 , ε 2 , and ε 3 are the strain of element 1, 2, and 3, respectively, and E 1 the elastic modulus of element 1.
(1) e stress-strain relationship of the elastic element: (2) e stress-strain relationship of the Abel dashpot [9]: (3) e stress-strain relationship of the NVPB element [12]: us, the constitutive model of the creep model is 2 Advances in Civil Engineering

Basic Assumption.
e derivation of the visco-elastoplastic displacement of tunnels is based on the following assumptions ( Figure 3): (1) e radius of deep-buried tunnel is R 0 . e hydrostatic pressure is p 0 and the tunnel is in the plane strain state. (2) e creep deformation of surrounding rock occurs after the tunnel excavation. At this time, the stress redistribution of surrounding rock has been completed, and the secondary stress field of the surrounding rock is constant.

e Initial Stress Field before Creep Deformation of Tunnels.
e surrounding rock of deep tunnels has nonlinear characteristics. erefore, the Hoek-Brown criterion is suitable to describe the nonlinearity of rock mass. Its constitutive formula is where σ 1 and σ 3 are the maximum and minimum principal stress after the rock mass enters into the yield failure (MPa), σ c is the uniaxial compression strength of intact rock (MPa), and m and s are parameters related to the quality of rock mass. e stress in the elastic zone after tunnel excavation is [10] σ e r � p 0 − e stress in the plastic zone is [10] σ p r � mσ c 4 In where σ e r , σ e θ , and σ e z are the radial, tangential, and normal elastic stress, respectively, σ p r , σ p θ , and σ p z are the radial, tangential, and normal plastic stress, respectively, R p is the radius of plastic zone, r is the distance between any point within the surrounding rock and the circle center, and ψ is the dilation angle. Figure 1 shows that the NVPB plastic element does not contribute to the creep deformation of tunnels when the surrounding rock is at the viscoelastic state. us, the creep model degrades into a creep model in which the elastic element connects with the Abel dashpot. e creep feature of surrounding rock at the elastic zone is described using this degraded model.  e relationship between deformation and strain of surrounding rock at the viscoelastic zone is

Viscoelastic Zone.
where ε θ and ε r are the tangential and radial strain of surrounding rock and u is the radial deformation of surrounding rock. us, the creep deformation is

Viscoplastic Zone.
e nonassociated rule is adopted to calculate the deformation at the viscoplastic zone: where u ηe r is the radial displacement at the viscoplastic zone and k ψ is the dilated coefficient: where φ is the frictional angle of rock mass. Substituting equation (15) into equation (18) yields e displacement at the boundary between the viscoelastic and viscoplastic zone meets the following relation: us, the displacement at the viscoplastic zone, u ηp r , is where g 1 (r) is

Validation of the Closed-Form Solution.
e Jinping number II hydropower station was built on the Jinping Dahe Bay in Sichuan Province (Figure 4(a)). is project consists of 4 diversion tunnels, 2 auxiliary tunnels, and 1 construction drainage tunnel. e average length of the tunnel is about 16.8 km, and the average buried depth is between 1500 and 2000 m. e stratum that the auxiliary tunnel passes through is mainly composed of marble, and its uniaxial compressive strength is 141.17 MPa, GSI � 50, and mi � 9. According to the test results, the rock rheological parameters obtained by inversion are shown in Table 1. e initial in situ stress field of the tunnel site is simplified to the hydrostatic pressure field, that is, p 0 � 40 MPa. e arrangement of the measurement points of the convergence monitoring section is shown in Figure 4(b). e displacement curves of field monitoring and theoretical calculation is shown in Figure 5. It can be seen that the average value of the monitoring curve is in good agreement with the theoretical calculation result, which verifies the validity of the theoretical solution in this paper.

e Influence of Fractional Order.
e relationship between the order c of the Abel dashpot and the displacement of surrounding rock is shown in Figure 6. It shows that there is a positive correlation between the displacement of the surrounding rock and the magnitude of c.
is is because that, as the value of c increases, the viscous characteristic of the Abel dashpot becomes more and more    Advances in Civil Engineering obvious, which leads to an increase in the creep displacement of the surrounding rock.

Influence of the Rheological Index of NVPB Element.
e influence of the rheological index of NVPB element on surrounding rock displacement is shown in Figure 7. It reveals that the displacement of surrounding rock increases with the increase of the rheological index. is is because, with the increase of the rheological index, the accelerating rheological process of the surrounding rock becomes more obvious, leading to an increase in the creep displacement of surrounding rock.

Conclusion
A creep model combining the Abel dashpot and a non-Newton viscous element was adopted, and the analytical solution about the visco-elastoplastic deformation for circular tunnel was derived based on this creep model. Major conclusions are as follows: (1) e creep model can well describe the whole creep stage of rocks, that is, the decay, constant, and accelerated creep stages. Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.  Advances in Civil Engineering