Theoretical study on dynamic responses of an unlined circular tunnel subjected to blasting P-waves

In conventional studies, the blasting seismic wave is always treated as a time-harmonic wave, which is not suitable in some situations due to its short duration. In this paper, the blasting P-wave is simplified as a triangular impulse. The distribution functions of dynamic stress concentration factor (DSCF) and the radial and hoop vibration velocity scaling factors (RVSF and HVSF) around the circular tunnel are derived based on the Fourier–Bessel expansion method. Then, the effects of the rising duration, the total duration and Poisson’s ratio on DSCF, RVSF and HVSF are discussed. Results show that (1) the maximum RVSF and HVSF are located at the incident side, and the location of the maximum DSCF will move from the incident side to the shadow side when the total duration increases; (2) the maximum DSCF increases with the increasing total duration, decreases with the increasing Poisson’s ratio, but is immune to the ratio of the rising to total duration; (3) the maximum RVSF decreases with the increase of the total duration and the ratio of the rising to total duration, but increases with the increasing Poisson’s ratio; (4) the maximum HVSF decreases with the increase of the total duration and the ratio of the rising to total duration, but is immune to Poisson’s ratio; (5) the RVSFand HVSFunder triangular impulses gradually change from less than those under time-harmonic waves togreater than those under time-harmonic waves with the increasing total duration, but the DSCF shows the opposite trend.


INTRODUCTION
In the past few decades, the drilling and blasting method was widely used in the construction of underground structures, such as metro tunnels and hydropower stations. These structures are always not far from existing underground tunnels, whose safety is significantly influenced by the blasting-induced vibration. As a consequence, the study on blasting dynamic responses of underground structures becomes increasingly critical to evaluate their safety.
In conventional theoretical studies, the blasting seismic wave is always treated as a time-harmonic plane P-, SV-or SH-wave. Under this assumption, the dynamic response of underground structures subjected to blasting seismic waves has been sufficiently studied. Pao and Mow [1] and Yi et al. [2] studied the dynamic stress concentration of lined and unlined circular tunnels subjected to plane P-and S-waves. Wang and Sudak [3] studied the scattering of plane waves by multiple cylindrical cavities with imperfect interface. Liao et al. [4] analyzed the responses of an elastic half-space with a buried tunnel subjected to obliquely incident waves. Yi et al. [5] studied the peak particle velocity (PPV) threshold of the lining of a horseshoe-shaped tunnel subjected to plane P-waves. Manoogian and Lee [6] studied the interaction of plane SH-waves and arbitrarily shaped inclusions by conformal transforms, and Wang et al. [7] studied the scattering of plane waves by an arbitrarily shaped hole in porous medium by the similar means. Lee and Karl [8,9] studied the ground displacement induced by the scattering of plane P-and SV-waves by underground cylindrical tunnels. Smerzini et al. [10] studied the ground motions resulting from the diffraction of plane SH-waves by underground tunnels. Zhao and Qi [11] studied the scattering of plane SH-wave from a debonded cylindrical elastic inclusion in half-space. Lin et al. [12] studied the dynamic response of a circular underground tunnel in an elastic half-space under the incidence of plane P-waves. Qi et al. [13,14] carried out the dynamic analysis for circular inclusion near interface impacted by SH-wave. Yi et al. [15] studied the dynamic stress concentration factors (DSCFs) of circular lined tunnels with an imperfect interface subjected to cylindrical P-waves. Xu [16] studied the dynamic stress and vibration velocities around a circular tunnel induced by plane P-waves in the background of Luxin coal mine. Lu et al. [17,18] studied the dynamic stress concentration and the PPV response of a circular tunnel under the incidence of cylindrical P-waves in theory. Liu et al. [19,20] studied the dynamic response of multiple holes and liquid-filled pipes subjected to blasting waves in theory. Fu et al. [21] studied the soil-tunnel interactions under the incidence of plane SH-waves. In recent years, the experimental methods based on piezoceramic transducers were developed quickly, and they are always used to study the response of underground structures [22][23][24][25]. Du et al. [26][27][28] studied the damage, crack detection and corrosion of pipelines by using piezoceramic transducers.
However, the duration of blasting seismic waves is quite short and the approximation of blasting seismic waves is not suitable in some situations. It is necessary to consider the characteristic of short duration of blasting waves when studying the dynamic response of structures subjected to blasting waves. However, the related research is not sufficient. Li et al. [29] studied the dynamic stress concentration and energy evolution of deep-buried tunnels under triangular blasting loads.
The blasting waves are usually simplified as time-harmonic plane waves in theory to study the dynamic responses of structures under blasting waves. In fact, the pressure-time curve of blasting waves is more suitable to be considered as a doubleexponential curve or a triangle-shaped curve. In this paper, the blasting P-wave is simplified as a plane triangular impulse. Based on the Fourier-Bessel expansion method, the formulas for dynamic responses of a circular tunnel under a time-harmonic Pwave are derived. Then, the dynamic responses of a circular tunnel under a unit Heaviside step load are studied by the inverse Fourier transform and the formulas under the incidence of a triangular blasting wave are derived by the Duhamel integral. Finally, the effects of duration of the triangular impulse and Poisson's ratio of surrounding rock on the DSCF, and the hoop and radial velocity scaling factors (HVSF and RVSF) are investigated. The main goal is to reveal the dynamic responses of a tunnel subjected to blasting waves under different factors, including the rising duration, the total duration of the triangular impulse and the Poisson's ratio of surrounding rock.

INTER ACTION OF BL A STING P-WAVES AND SURROUNDING ROCK
According to the Duhamel integral equation, the entire loading history can be considered to consist of a succession of short impulses, each producing its own differential response, and the total response of a linear elastic system can then be obtained by summing all the differential responses developed during the loading history. Therefore, if we want to obtain the response of a circular tunnel subjected to blasting waves, we should obtain the transient response produced by an impulse load or a Heaviside step function load at first. The transient response can be obtained by the inverse Fourier transform of the response produced by a time-harmonic wave. So, the responses produced by a timeharmonic wave, by a unit Heaviside step load and by a blasting wave should be studied successively.

Dynamic responses of circular tunnel under time-harmonic incident P-waves
We suppose that a circular tunnel of radius a is in unbounded rock mass, whose axis is coincident with O, as shown in Fig. 1.
The potential function of the harmonic plane P-wave can be expressed in terms of the displacement potential as where ϕ 0 is the amplitude of incident wave, α is the wavenumber of P-waves and α = ω/C P , ω is the circular frequency of incident wave, C P is the wave speed of P-waves and i is the unit of complex number.
In terms of the Bessel function expansion method, the incident wave can be expanded as where A 0n = ∈ n i n ϕ 0 , ∈ 0 = 1 and ∈ n = 2 for n > 0. In general, when the incident P-wave arrives at the tunnel, two outward propagating reflected waves will be generated. They are the reflected P-wave (ϕ (r) ) and the reflected SV-wave (ψ (r) ). Both of them take the following forms: where A n and B n are undetermined constants, H (1) n is the first kind of Hankel function in nth order, β is the wavenumber of SV-waves and β = ω/C S , and C S is the wave speed of SV-waves.
Let ϕ = ϕ (i) + ϕ (r) and ψ = ψ (r) , then the relations between potentials, stresses and velocities can be expressed as where λ and μ are Lame constants.

Dynamic responses of circular tunnel under a unit Heaviside step load
The unit Heaviside step load can be expressed as wheret = t − (x + a)/C P and we choose time begins when the incident wave arrives at the tunnel boundary. The response of a linear system g h (x i ,t ) induced by the input H(t ) can be expressed as whereH(ω) is the Fourier transform of H(t ) and χ (x i ,ω) is the admittance function, which is defined as the steady-state response of the system for a unit input and given by Eq. (6). Consequently, for a unit Heaviside step load, the response can be obtained by the inverse Fourier transform and given by

Dynamic responses of circular tunnel under a blasting P-wave
For an arbitrary input P(τ ), the response g(x i ,τ ) of the system can be obtained by the Duhamel integral as The blasting P-wave is always simplified as a triangular impulse with linear loading and unloading processes as shown in Fig. 2, which can be expressed as where P m is the peak pressure of the blasting wave, andτ r andτ s are the rising and total durations, respectively.

RESULTS AND DISCUSSION
The PPV is easy to be measured and has been an index to evaluate the vibration intensity and structural damage [30], which has been one of the major concerns in the blasting field [31]. In fact, the dynamic stress is the most essential index. When the stress induced by external loadings exceeds the maximum that the structure can bear, the structure is considered to be damaged. Therefore, both the stress and the PPV responses of the tunnel should be considered. In order to obtain general results, some dimensionless parameters should be defined.

Definitions of dynamic stress concentration and velocity scaling factors
The DSCF is defined as follows: The RVSF is defined as follows [14]: where v 0 is the maximum vibration velocity of the incident wave in its propagating direction, and v 0 = P m /ρC P . The HVSF is defined as follows:

Parametric studies and discussion
The DSCF, RVSF and HVSF are relevant to the total duration τ s , the rising durationτ r and Poisson's ratio ν. As a consequence, the influence ofτ s , ν andτ r /τ s on the DSCF, RVSF and HVSF is investigated. The distributions of DSCF, RVSF and HVSF around the tunnel under differentτ s are shown in Fig. 3, where ν = 0.25. It can be seen from Fig. 3 that the distributions of DSCF, RVSF and HVSF at the tunnel boundary are similar under differentτ s . There are two same peak values of DSCF occurring at about θ = π/2 and θ = 3π/2 due to the symmetry. However, the RVSF at θ = π is obviously larger than that at other locations. The locations of the maximum HVSF are at the incidence side. It can also be observed that the DSCF, RVSF and HVSF all increase with the increasingτ s , but the DSCF changes most apparently.
The variations of DSCF, RVSF and HVSF at different locations with the change ofτ s are shown in Fig. 4. It can be found from Fig. 4a that the DSCF increases with the increasingτ s except that at θ = 0. The DSCFs at θ = π/3 and π/2 increase rapidly when τ s < 0.1π, which suggests that the DSCFs at the two locations are greatly affected byτ s . Whenτ s ≥ 0.1π, the increasing rates of DSCF at θ = π/3 and π/2 gradually decrease. The DSCFs at θ = π/3, π/2 and 2π/3 are obviously larger than those at other locations. It can be seen from Fig. 4b that the RVSFs at θ = 0, π/6 and π/3 increase withτ s , increasing first and then tend to constant values, which is contrary to that at θ = π/2. The RVSFs at θ = 2π/3, 5π/6 and π decrease linearly with the increasinĝ τ s . It can be seen from Fig. 4c that the variation of HVSF with the change ofτ s is not apparent, which indicates that HVSF is immune toτ s .
The distributions of DSCF, RVSF and HVSF around the circular tunnel under different ν are shown in Fig. 5. The rising and total durations of the triangular impulse are chosen asτ r = 2π andτ s = 10π, respectively. It can be observed from Fig. 5a that there are two peak values of DSCF occurring at about θ = π/2 and θ = 3π/2 when ν = 0.3 and ν = 0.4, but there are four peak values when ν = 0.1 and ν = 0.2, which suggests that ν exerts a significant influence on DSCF. It can be observed from Fig. 5b that two peak values of RVSF occur at θ = 0 and θ = π, and the maximum RVSF appears at θ = π. The distribution of RVSF is similar under different ν. However, the value is basically identical at the shadow side and changes comparatively larger at the incident side. It can be observed from Fig. 5c that the two peak values of HVSF both occur at the incident side. The basically identical distribution of HVSF indicates that ν has little influence on HVSF.
The variations of DSCF, RVSF and HVSF at different locations with ν are shown in Fig. 6. It can be observed from Fig. 6a that the DSCF increases with the increasing ν when θ = π/6, π/3, 2π/3 and 5π/6. Although the DSCF at θ = π/2 decreases with the increasing ν, it is larger than that at other locations. The DSCFs at θ = 0 and θ = π both decrease first when ν < 0.25, and then increase. It can be observed from Fig. 6b that the RVSFs at θ = 2π/3, 5π/6 and π increase with the increasing ν. The RVSF at θ = π/2 slightly decreases when ν is <0.25 and then increases rapidly. When θ = 0, π/6 and π/3, the RVSF is little influenced by ν. It can be observed from Fig. 6c that the HVSF varies slightly with the change of ν except at θ = π/3, which increases apparently when ν ≥ 0.25.
The distributions of DSCF, RVSF and HVSF around the circular tunnel under differentτ r /τ s are shown in Fig. 7, wherê τ s =10π and ν = 0.25. Figure 7a shows that the distributions of DSCF are nearly identical, which suggests that DSCF is little affected byτ r /τ s . Figure 7b and c shows that the distributions of RVSF and HVSF at the shadow side are also basically identical, but they change obviously at the incident side.
The variations of DSCF, RVSF and HVSF withτ r /τ s at different locations are shown in Fig. 8. Figure 8a shows that the DSCF at different locations changes little. Figure 8b and c shows that only the RVSF and HVSF at θ = π/2, 2π/3 and 5π/6 decrease quickly whenτ r /τ s is <0.3 and then change little withτ r /τ s . The variations of DSCF, RVSF and HVSF indicate that only the RVSF and HVSF at the incident side are influenced byτ r /τ s , and the DSCF is immune toτ r /τ s .

Comparison study of dynamic responses
In order to compare the dynamic responses of a tunnel under time-harmonic wave and triangular impulse wave, the variations of DSCF, RVSF and HVSF of the circular tunnel subjected to the two blasting waves with different ν andτ s are investigated.

Comparison of dynamic responses under different durations
When the Poisson's ratio of the surrounding rock remains unchanged, the variations of DSCF, RVSF and HVSF around the circular tunnel subjected to the time-harmonic wave and triangular impulse wave withτ s are shown in Figs 9-11. The Poisson's ratio of the surrounding rock is set as 0.25. It can be seen that when the total duration of blasting wave is short (such asτ s = 0.5π andτ s = π), the RVSF and HVSF of the tunnel under the time-harmonic wave are larger than those under the triangular impulse wave. With the increasing total duration (e.g.τ s = 10π), the RVSF and HVSF of the tunnel under the triangular impulse wave are larger than those under the timeharmonic wave. The DSCF shows the opposite trend.

Comparison of dynamic responses under different Poisson' s ratios
When the total durationτ s of blasting waves remained unchanged, the variations of DSCF, RVSF and HVSF around the circular tunnel under the time-harmonic wave and triangular im- It can be found that the DSCF and HVSF of the tunnel under the time-harmonic wave are basically the same as those under triangular impulse wave when the Poisson's ratio increases, which suggests that the DSCF and HVSF of the tunnel obtained from the time-harmonic wave and triangular impulse wave are little influenced by the Poisson's ratio. The RVSF of the tunnel subjected to the triangular impulse wave is larger than that subjected to the time-harmonic wave under different Poisson's ratios. The difference between the RVSF obtained from the timeharmonic wave and triangular impulse wave increases with the increasing Poisson's ratio. This indicates that the time-harmonic wave has an amplification effect on the RVSF of the tunnel com-pared with the triangular impulse wave, and the amplification effect increases with the increasing Poisson's ratio of surrounding rock.

CONCLUSIONS
The following conclusions can be made from the above results: (1) The formulas of DSCF, RVSF and HVSF are derived based on the Bessel-Fourier expansion and the Duhamel integral. (2) The locations of the maximum DSCF are different from those of the maximum RVSF and HVSF. The maximum RVSF and HVSF are located at the incident side, and the location of the maximum DSCF will move from the incident side to the shadow side when the total duration increases. (3) The maximum DSCF increases with the increasing total duration of the incident triangular impulse, but the maximum RVSF and HVSF decrease. (4) The maximum DSCF decreases with the increasing Poisson's ratio, the maximum RVSF increases and the maximum HVSF is immune. (5) Only the RVSF and HVSF at the incident side are obviously influenced by the ratio of the rising to total duration, but the DSCF is immune. (6) The RVSF and HVSF under triangular impulses gradually change from less than those under time-harmonic waves to greater than those under time-harmonic waves with the increasing total duration, but the DSCF shows the opposite trend. (7) The time-harmonic wave has an increasing amplification effect on the RVSF with the increasing Poisson's ratio, but the DSCF and HVSF are immune.