A viscousslip interface model is proposed to simulate the contact state between a tunnel lining structure and the surrounding rock. The boundary integral equation method is adopted to solve the scattering of the plane SV wave by a tunnel lining in an elastic halfspace. We place special emphasis on the dynamic stress concentration of the lining and the amplification effect on the surface displacement near the tunnel. Scattered waves in the lining and halfspace are constructed using the fictitious wave sources close to the lining surfaces based on Green’s functions of cylindrical expansion and the shear wave source. The magnitudes of the fictitious wave sources are determined by viscousslip boundary conditions, and then the total response is obtained by superposition of the free and scattered fields. The slip stiffness and viscosity coefficients at the liningsurrounding rock interface have a significant influence on the dynamic stress distribution and the nearby surface displacement response in the tunnel lining. Their influence is controlled by the incident wave frequency and angle. The hoop stress increases gradually in the inner wall of the lining as sliding stiffness increases under a lowfrequency incident wave. In the highfrequency resonance frequency band, where incident wave frequency is consistent with the natural frequency of the soil column above the tunnel, the dynamic stress concentration effect is more significant when it is smaller. The dynamic stress concentration factor inside the lining decreases gradually as the viscosity coefficient increases. The spatial distribution and the displacement amplitudes of surface displacement near the tunnel change as incident wave frequency and angle increase. The effective dynamic analysis of the underground structure under an actual strong dynamic load should consider the slip effect at the liningsurrounding rock interface.
Analyses of seismic damage incurred by disasters such as the Kobe earthquake, ChiChi earthquake, and Wenchuan earthquake have shown that underground structures might be severely damaged during strong earthquakes, resulting in massive economic and societal losses [
In general, the calculation methods include the analytical method [
It should be noted that the abovementioned research works based on interface contact models were mainly limited to deepburied tunnels, while the response of shallowburied tunnels differs significantly from that of deepburied ones [
However, until now, few studies have explored the seismic response of shallow tunnels under incident SV waves with the interfaceslipping model due to the complexity of multimode coupling and hybrid boundary conditions. We used an indirect boundary integral equation method (IBIEM) to solve the scattering of the plane SV wave by a tunnel lining in a halfspace based on the viscousslip interface model. This method had been used effectively to solve the dynamic response of the tunnel structure [
This study aims to investigate the dynamic stress concentration effect of the tunnel lining and the surface displacement amplification near the tunnel with a viscousslip interface. We assessed the influence of parameters such as incident wave frequency and angle, viscousslip interface stiffness, and viscosity coefficient on the overall dynamic response of the lining and surrounding rock. This study can provide a theoretical basis for the seismic design of actual underground engineering structures under intense dynamic loads.
As shown in Figure
Seismic response model of tunnel lining with the viscousslip interface.
Halfspace and tunnel parameters.
Domain  Shear modulus  Poisson’s ratio  Density  Longitudinal wave speed  Transverse wave speed  Virtual source surface  

Halfspace 







Tunnel 







Let the buried depth of the tunnel be
In this study, we considered the cylindrical shear wave source in the halfspace as the fundamental solution. The indirect boundary integral equation method and viscousslip boundary condition were used to solve the scattering of plane waves by tunnel linings and the dynamic response around the tunnel lining [
The total halfspace wave field can be viewed as a superposition of a halfspace free field (without tunnel linings) and a scattering field. We first carry out a free field analysis. The shear wave potential function in the halfspace is denoted as
For the sake of simplicity, the time factor
Scattered fields are generated in the halfspace of a lined tunnel and in the interior of the lining. The fields can be determined by superimposing all the expansion wave and shear wave sources on the virtual wave source surface inside and outside the lining, respectively. Assuming that the scattered field in the halfspace is generated by the virtual source face
The internal scattering field of the lining is obtained by superimposing the action of all the expansion wave sources and shear wave sources on the virtual wave source surfaces
The total displacement and stress fields in the halfspace are obtained by superposition of the scattered wave field and the free field in the halfspace. The internal reactions of the lining are all generated by the scattered fields within it.
We built a viscousslip interface model to determine the influence of the interface effect on the dynamic response. The lining and the halfspace were connected by linear springs and dampers (Figure
A linear system of equations can be obtained by synthesizing the above formulas:
In this section, the ratio of the buried depth of the tunnel to the inner radius of the lining is
We define the dimensionless dynamic stress concentration factor (DSCF) as
In order to verify the correctness of the IBIEM method, we first let
DSCF spectrum distribution on the inner lining wall surface.
Figures
DSCF distribution on lining the inner wall under the incident SV wave (
DSCF distribution on lining the outer wall under the incident SV wave (
DSCF distribution on lining the inner wall under the incident SV wave (
DSCF distribution on lining the outer wall under the incident SV wave (
As shown in Figures
When the SV wave is obliquely incident, the increase and decrease amplitudes of the DSCF of the inner and outer wall surfaces is smaller than the normal incidence. When
When SV wave is incident with frequency
When the SV wave is incident at high frequency (
Figure
DSCF distribution on outer lining wall surfaces under incident SV waves: (a)
As shown in Figure
Compared to the normal incidence, the influence of the viscosity factor on the amplitude distribution of the circumferential stress of the lining weakens at oblique incidence. When the SV wave is incident at an angle of 30°, the spatial distribution of the circumferential stress of the lining is relatively gentle along the circumference of the hole. In the lowfrequency region (
Figures
DSCF spectrum distribution on the inner lining wall surface under the vertically incident SV wave: (a)
DSCF spectrum distribution on the outer lining wall surface under the vertically incident SV wave: (a)
As shown in Figure
The resonance frequency point also is offset to a certain extent as the sliding stiffness factor increases. When
As shown in Figure
Figure
DSCF spectrum distribution on the outer lining wall surface under the vertically incident SV wave (
Figures
Surface displacement amplitudes near tunnel lining under the incident SV wave (
Surface displacement amplitude near tunnel lining under the incident SV wave (
Figures
When the SV wave is incident at an angle of 30° and at a low frequency (
The boundary integral equation method was applied to solve the seismic response of a tunnel lining in an elastic halfspace under incident plane SV waves based on a viscousslip interface model. The effects of key factors such as incident wave frequency and angle, interface slip stiffness, and interfacial viscosity coefficient on the dynamic stress response of the tunnel lining in elastic halfspace and the surface displacement near the tunnel lining were analyzed. Main conclusions can be summarized as follows:
The interface slip stiffness factor significantly affects the dynamic stress distribution of the tunnel lining; the response characteristics are controlled by the incident wave frequency. When the slip stiffness is small, the internal stress of the lining changes very radically along the space around the hole, and the spatial distribution of the dynamic stress is highly complex. When the slip stiffness is large (
Under an incident lowfrequency wave, increase in interface slip stiffness causes a gradual increase in the circumferential stress of the inner wall. The dynamic stress concentration is more significant in the noslip state than the slip state. When the SV wave is incident with
The viscosity factor of the viscousslip interface also has a significant influence on the dynamic stress distribution of the tunnel lining. As the viscosity factor increases, the DSCF of the lining outer wall decreases gradually; however, this effect gradually weakens as the incident wave frequency increases. The influence of the viscosity coefficient under a normally incident plane SV wave is greater than that under a wave with oblique incidence.
The interface slip stiffness factor has a significant effect on the DSCF spectrum characteristics of the tunnel lining surface. When the slip stiffness is small (
When the SV wave is incident at a low frequency, the spatial distribution of surface displacements above the lining under different slip stiffness is essentially constant, but the displacement amplitudes are quite different. Any increase in SV wave incident frequency and incidence angle has a significant effect on both the spatial distribution of surface displacement near the lining and the amplitude of said surface displacement.
In this study, we analyzed only the 2D seismic response of a shallow lined tunnel based on the viscousslip interface model of a uniform space in halfspace. Similar seismic response analyses of uneven sites and 3D tunnels merit further research.
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
The research described in this paper was financially supported by the National Natural Science Foundation of China under grant nos. 51678389 and 51678390 and the National Basic Research Program of China under grant no. 2015CB058002.