Seismic Isolation E ﬀ ect of Non-Water Reacted Two-Component Polymeric Material Coating on Tunnels

: An isolation layer is one of the countermeasures to promote the anti-seismic performance of tunnels. A newly invented polymeric material, non-water reacted two-component polymeric material (NRTCPM), is superior in impermeability and construction e ﬃ ciency. In this study, covering a tunnel with NRTCPM coating to mitigate the damage caused by an earthquake is discussed, and an Impact Resonance Test (IRT) is ﬁrstly used to obtain the damping ratios and dynamic elastic modulus of NRTCPM. By using inﬁnite element boundary, eight dynamic numerical modelsare made to study the isolation e ﬀ ects based on di ﬀ erent density, Poisson’s ratio, dynamic elastic modulus and thickness of isolation layer values. Three di ﬀ erent conditions are explored in this paper, namely (1) no NRTCPM layer coating around tunnel; (2) di ﬀ erent densities, Poisson’s ratios and dynamic elastic moduli of a polymeric layer; and (3) various thicknesses of polymeric isolation layers around the lining. Tensile and compressive stresses are compared under these di ﬀ erent conditions. The results show that retroﬁtting tunnel lining with this material has a good e ﬀ ect on seismic isolation. An optimum density and thickness of the NRTCPM layer is suggested considering cost and strength.


Introduction
The seismic response mitigation and seismic protection of structures have been discussed in recent decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. For tunnels, coating with soft seismic isolation materials has been studied by many researchers. The seismic isolation effect of a rectangular-shaped tunnel was investigated by Shimamura et al. (1999). In their work, the seismic isolation layers were installed in two patterns. An isolation layer wasused around a new tunnel, or the vertical isolation walls were equipped at both sides of an existing tunnel [1]. Kim and Konagai (2000), Kim and Konagai (2001), and Konagai and Kim (2001) showed that materials with a low Poisson's ratio and a low shear modulus were preferable for enhancing the seismic efficiency of an isolation layer. They studied the influence of key material properties, such as the reduction factor (TF coat /TF uncoat ), on the seismic isolation effect. TF coat and TF uncoat are the transferring factorsin the cases with and without coating [2][3][4]. Takemine (2004) combined the 3D finite element method and centrifuge model to demonstrate the effectiveness of the isolation layer and cement solidification surrounding the tunnel [5]. Kiryu et al. (2012) used a polymer isolation method to reduce the seismic effect on the tunnel, and the influences of shear stiffness and thicknesses of soil covering the tunnel were studied [6]. Zhao et al. (2013Zhao et al. ( , 2018 and Ma et al. (2018) retrofitted the tunnel with foamed concrete, andthe subsequent effectiveness of seismic isolation was evaluated by both Within this framework, this paper aims to apply this novel NRTCPM as the isolation layer material to cover the tunnel. The compressive and tensile strength of this polymeric material is investigated, and the damping ratios and dynamic elastic moduli are obtained byan Impact Resonance Test (IRT). Furthermore, a parametric study is conducted to evaluate the influence of material Within this framework, this paper aims to apply this novel NRTCPM as the isolation layer material to cover the tunnel. The compressive and tensile strength of this polymeric material is investigated, and the damping ratios and dynamic elastic moduli are obtained by an Impact Resonance Test (IRT). Furthermore, a parametric study is conducted to evaluate the influence of material properties (e.g., densities, Poisson's ratio, shear modulus, elastic modulus, etc.) on the seismic response of the lining.

Dynamic Elastic Modulus from Impact Resonance Test (IRT)
This section verifies the feasibility of using the Impact Resonance Test (IRT) to determine the dynamic elastic modulus of NRTCPM.

Preparation of Specimens
As shown in Figure 2a,b, the length of a seamless steel pipe is 300 mm, and the diameter is 100 mm. Two inner thread tube caps are separately set at the ends of the tube. A grouting head hole is drilled, and a grouting head is installed (Figure 2b). Before the mold is grouted, pipe caps from both ends are moved, and then the pipe inner wall is lubricated with silicone oil. Subsequently, two parts of the pipe are combined. After different amounts of two-component polymer are injected into the mold, various densities of specimens are produced. Finally, the 300 mm-long cylindrical NRTCPM is cut into specimens with 100 mm in diameter and 50 mm in thickness ( Figure 3). Within this framework, this paper aims to apply this novel NRTCPM as the isolation layer material to cover the tunnel. The compressive and tensile strength of this polymeric material is investigated, and the damping ratios and dynamic elastic moduli are obtained byan Impact Resonance Test (IRT). Furthermore, a parametric study is conducted to evaluate the influence of material properties (e.g., densities, Poisson's ratio, shear modulus, elastic modulus, etc.) on the seismic response of the lining.

Dynamic Elastic Modulus from Impact Resonance Test (IRT)
This section verifies the feasibility of using the Impact Resonance Test (IRT) to determine the dynamic elastic modulus of NRTCPM.

Preparation of Specimens
As shown in Figure 2a, 2b, the length of a seamless steel pipe is 300 mm, and the diameter is 100 mm. Two inner thread tube caps are separately set at the ends of the tube. A grouting head hole is drilled, and a grouting head is installed (Figure 2b). Before the mold is grouted, pipe caps from both ends are moved, and then the pipe inner wall is lubricated with silicone oil. Subsequently, two parts of the pipe are combined. After different amounts of two-component polymer are injected into the mold, various densities of specimens are produced. Finally, the 300 mm-long cylindrical NRTCPM is cut into specimens with 100 mm in diameter and 50 mm in thickness ( Figure 3).

Impact Resonance Test (IRT) and Theory
The damping ratio calculations are derived from the time domain and frequency domain response of the tested material. There are various methods to estimate the level of damping, such as logarithmic decrement and half-power bandwidth methods. The half-power bandwidth methods

Impact Resonance Test (IRT) and Theory
The damping ratio calculations are derived from the time domain and frequency domain response of the tested material. There are various methods to estimate the level of damping, such as logarithmic decrement and half-power bandwidth methods. The half-power bandwidth methods are based on the frequency domain and were used to generate the results of damping ratios from the Impact Resonance Test (IRT) system. Traditionally, the IRT system was used to obtain a damping ratio of the asphalt mixture, and it consists of a laptop, an acceleration sensor, a signal amplifier, and a software program [26][27][28].
The basic concept of the test is to determine the amplitude and attenuation of vibrations and relate these parameters to the material properties. A specimen was expanded to a disc shape by injecting different amounts of NRTCPM into a mold. Figure 3 shows that the specimen with free boundary conditions is struck with a rigid impulse tool at the impact location to excite a mechanical vibration. The vibration can be received by a signal amplifier connected to the edge of the specimen (f3). As is shown in Figure 3, when the edge of the specimen is impacted (f1), a sensor, connected to the signal amplifier, captures the waveform vibrations from the short-term mechanical impact, and the first vibration mode is generated. When impacting the center of the specimen (f2), the second vibration mode can be generated. Different vibration modes correspond to different frequencies [27].
By performing a Fast Fourier Transform (FFT) algorithm, the waveform in the time domain is converted to a response in the frequency domain, as illustrated in Figure 4 [29]. . Specimen on the Impact Resonance Test (IRT) system.

Impact Resonance Test (IRT) and Theory
The damping ratio calculations are derived from the time domain and frequency domain response of the tested material. There are various methods to estimate the level of damping, such as logarithmic decrement and half-power bandwidth methods. The half-power bandwidth methods are based on the frequency domain and were used to generate the results of damping ratios from the Impact Resonance Test (IRT) system. Traditionally, the IRT system was used to obtain a damping ratio of the asphalt mixture, and it consists of a laptop, an acceleration sensor,a signal amplifier, and a software program [26][27][28].
The basic concept of the test is to determine the amplitude and attenuation of vibrations and relate these parameters to the material properties. A specimen was expanded to a disc shape by injecting different amounts of NRTCPM into a mold. Figure 3 shows that the specimen with free boundary conditions is struck with a rigid impulse tool at the impact location to excite a mechanical vibration. The vibration can be received by a signal amplifier connected to the edge of the specimen (f3). As is shown in Figure 3, when the edge of the specimen is impacted (f1), a sensor, connected to the signal amplifier, captures the waveform vibrations from the short-term mechanical impact, and the first vibration mode is generated. When impacting the center of the specimen (f2), the second vibration mode can be generated. Different vibration modes correspond to different frequencies [27].
By performing a Fast Fourier Transform (FFT) algorithm, the waveform in the time domain is converted to a response in the frequency domain, as illustrated in Figure 4 [29]. The frequency of the maximum amplitude is the resonant frequency(fr), and the f1 and f2 are the frequencies of the 1/√2 maximum amplitude in Figure 4.
The damping ratio can be obtained from fr,f1, and f2, and it is written as: The frequency of the maximum amplitude is the resonant frequency(f r ), and the f 1 and f 2 are the frequencies of the 1/ √ 2 maximum amplitude in Figure 4. The damping ratio can be obtained from f r , f 1 , and f 2 , and it is written as: ξ-Damping ratio f r -Resonant frequency f 1 , f 2 -The frequencies of the1 √ 2 maximum amplitude. Table 1 presents the average damping ratio of the first and second mode. When the NRTCPM's density is 0.22 g/cm 3 , the average damping ratio is 0.04. The dynamic elastic modulus is an important parameter, and it can also be obtained from IRT using the following equations. The theoretical relation between the natural frequency and attenuation frequency can be written as: where, f d -Attenuation frequency, Hz; f n -Natural frequency, Hz; The storage modulus E can be calculated from the natural frequency f n , and it can be written as [30]: where, E -Storage modulus; ρ-Density of specimen, kg/m 3 ; R-Radius of specimen, m; -Material parameter.
is related to Poisson's ratio, the thickness, and the radius ratio. The values of the first and second mode are presented in Tables 2 and 3. R, h, and υ are the radius, thickness, and Poisson's ratio in Tables 2 and 3.  15 1.404 υ = 0. 30 1.482 The relation between the damping ratio and modulus can be expressed as follows: E"-Loss modulus; Therefore, the dynamic elastic modulus can be written as follows:

Results of the IRT
The dynamic elastic modulus can be calculated from Equation (5), and two kinds of specimens' dynamic elastic moduli are shown in Table 4. As shown in Table 4, the average dynamic elastic modulus of the first mode is 72.26 MPa, obtained from impacting the edge of specimens. The average dynamic elastic modulus of the second mode is 81.87 MPa, gained from impacting the center of specimens. According to the results of the dynamic elastic modulus obtained from the piezoceramic bender element done by Li [31], when the density of NRTCPM is 0.22 to 0.23 g/cm 3 , its dynamic elastic modulus is approximately 80 MPa. The results of the dynamic elastic modulus gained by the Impact Resonance Test (IRT) are very close to the results from Li's experiment [31]. Therefore, the results of NRTCPM's dynamic elastic modulus are used in the numerical models in Section 3 below.

Infinite Element Model
The concept of infinite elements was first presented by Ungless (1973) and Bettess (1992) [32,33]. Further constitutive work has been done by some other researchers to apply the infinite elements to the solution of static and dynamic problems in engineering practice.
The main idea in forminga static infinite element is to derive it from an element displacement shape function, and the element displacement shape function is the product of the Lagrange interpolation function (decay function). The infinite element is mapped onto a finite one by using some mapping techniques, and the mapping techniques are also employed ondynamic infinite elements. But considering the difficult mechanism of wave propagation in infinite elements, the wave propagation function in its dynamic infinite elements is replaced by the decay function in the static infinite elements [32].
Although Zhao [34][35][36] have successfully used the finite and infinite elements coupled method and the related wave input procedure in infinite media to solve the dynamic response, the application of this method to study the dynamic response of polymeric seismic isolation around tunnels has not been done before.
2D finite element models with infinite element boundaries are made for the nonlinear numerical analysis by ABAQUS. According to the Code for the seismic design of urban rail transit structures [37], the distance between the numerical model boundary and underground structure should be three times larger than the structure height.
The finite element (FE) model in Figure 5 is 50 m wide and 50 m deep. The diameter of the tunnel is 5.5 m, and the thickness of the liner is 0.35 m, as shown in Figure 5.  Figure 5. These points' stress will be analyzed in the results.
To deal with contact, a tie constraint is adopted between the surrounding soil and NRTCPM coating, and is adopted between the NRTCPM coating and the lining in order to improve the calculation of the stability and efficiency.
In the mesh center, the characteristic dimension of the elements should satisfy the condition that the spacing of the finite element nodes must be smaller than around one-eighth to one-sixth of the wavelength related to the maximum frequency component of the input motion [38]. A detail of the mesh before excavation and after equipping the NRTCPM layer and lining is shown in Figures 6 and 7.  Figure 5. These points' stress will be analyzed in the results.
To deal with contact, a tie constraint is adopted between the surrounding soil and NRTCPM coating, and is adopted between the NRTCPM coating and the lining in order to improve the calculation of the stability and efficiency.
In the mesh center, the characteristic dimension of the elements should satisfy the condition that the spacing of the finite element nodes must be smaller than around one-eighth to one-sixth of the wavelength related to the maximum frequency component of the input motion [38].
A detail of the mesh before excavation and after equipping the NRTCPM layer and lining is shown in Figures 6 and 7.

Material Property
The surrounding rock is regarded as being homogeneous and isotropic, with an elastic, perfectly plastic behavior, and the Mohr-Coulomb yielding criterion is adopted in Table 5, while Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 16

Seismic Load Input
In the dynamic simulation, the horizontal and vertical time histories (a) and their Fourier amplitude spectra (b) are shown in Figures 7 and 8, respectively. The seismic ground motion was measured at the north-south direction at the Tianjin hospital in Tangshan China in 1976, and it is a horizontal input at the boundary between finite and infinite elements on the left side of the numerical model ( Figure 5). The vertical input is about four times the amplitude and acceleration of the horizontal input (Figure 7 and 8), and it is vertically input at the boundary between finite and infinite elements on the lower side of the numerical model ( Figure 5). The duration of the analysis is 20 s. The peak acceleration is 1.458 m/s 2 horizontally and 6.328 m/s 2 vertically. However, the acceleration of the ground motions is less than 0.5 g/m 2 horizontally and 2 g/m 2 vertically after 15 s and before 6 s. The damping constants of the structure and soil are considered to be those of Rayleigh damping, namely 5%.

Material Property
The surrounding rock is regarded as being homogeneous and isotropic, with an elastic, perfectly plastic behavior, and the Mohr-Coulomb yielding criterion is adopted in Table 5, while concrete damaged plasticity is used on the lining in Table 6. NRTCPM is evaluated using the crushable foam model. The parameters are in Table 7.

Analysis Step
The numerical simulation has been conducted by a step-by-step method; the digging operation step and the lining installation step are carried out, and the five steps are as follows: Step one is to reproduce the initial static stress. This load creates the vertical action of the soil weight under the consideration of a coefficient at rest equal to 0.5 in both horizontal directions.
Step two is named the reduce phase. The elastic modulus of the excavation area rockmass is reduced by 40%.
Step three is called the add phase. The lining is activated in this step. In step four, the excavation area is dug.
Step five is called the dynamic phase. The seismic load is applied, and the detailed descriptions are given in Section 3.5.

Numerical Scenarios
Eight kinds of mechanical properties of NRTCPM are presented from Case 1 to Case 8 in Table 7, aiming to analyze the influence of the NRTCPM layer density on the seismic isolation effect. In these cases, the thickness of the NRTCPM layer is 20 cm. The dynamic elastic modulus and damping ratio of NRTCPM under the density of 0.22 g/cm 3 are from the Impact Resonance Test (IRT) done before. Li used the same polymeric material. Therefore, the dynamic elastic modulus of other densities can be obtained from the piezoceramic bender test done by Li et al. (2017) [31]. The tensile and compressive strength is obtained from Figure 1.
The density of the NRTCPM layer in Cases 9 to 16 is 0.22 g/cm 3 , and these eight cases are set to research the impact of different layer thicknesses in Table 8. Case 2 and Case 12 are the same model with the same parameters and the same NRTCPM layer thickness. The numerical model without NRTCPM coating was taken as the benchmark.

Seismic Load Input
In the dynamic simulation, the horizontal and vertical time histories (a) and their Fourier amplitude spectra (b) are shown in Figures 7 and 8, respectively. The seismic ground motion was measured at the north-south direction at the Tianjin hospital in Tangshan China in 1976, and it is a horizontal input at the boundary between finite and infinite elements on the left side of the numerical model ( Figure 5). The vertical input is about four times the amplitude and acceleration of the horizontal input (Figures 7  and 8), and it is vertically input at the boundary between finite and infinite elements on the lower side of the numerical model ( Figure 5). The duration of the analysis is 20 s. The peak acceleration is 1.458 m/s 2 horizontally and 6.328 m/s 2 vertically. However, the acceleration of the ground motions is less than 0.5 g/m 2 horizontally and 2 g/m 2 vertically after 15 s and before 6 s. The damping constants of the structure and soil are considered to be those of Rayleigh damping, namely 5%. the horizontal input (Figure 7 and 8), and it is vertically input at the boundary between finite and infinite elements on the lower side of the numerical model ( Figure 5). The duration of the analysis is 20 s. The peak acceleration is 1.458 m/s 2 horizontally and 6.328 m/s 2 vertically. However, the acceleration of the ground motions is less than 0.5 g/m 2 horizontally and 2 g/m 2 vertically after 15 s and before 6 s. The damping constants of the structure and soil are considered to be those of Rayleigh damping, namely 5%.

Results and Discussionof Numerical Analysis
According to the results of the finite element method, the accelerations and displacements at the same point with coating and without coating NRTCPM are very close. But the maximum principle stress varies largely. Concrete is prone to yield after tensile stress. Therefore, the isolation effect is compared here based on the maximum principle stress.
All the peak values ofdifferent points on thelining and NRTCPM coating are selected for a comparison during earthquakes.The different layer densities anddynamic elastic moduli used in the numerical model from Case 1 to Case 8 are discussed in Section 4.1, while the various thicknessesfrom Case 9 to Case 16 are explored in Section 4.2.

Effect of Layer Density and Elastic Modulus(Case 1 to 8)
The tensile stresses and absolute values of compressive stress on the lining and NRTCPM layer decrease obviously when the layer density decreases, and their variations tend to be similar, as shown in Figures 9-12.

Results and Discussion of Numerical Analysis
According to the results of the finite element method, the accelerations and displacements at the same point with coating and without coating NRTCPM are very close. But the maximum principle stress varies largely. Concrete is prone to yield after tensile stress. Therefore, the isolation effect is compared here based on the maximum principle stress.
All the peak values of different points on the lining and NRTCPM coating are selected for a comparison during earthquakes. The different layer densities and dynamic elastic moduli used in the numerical model from Case 1 to Case 8 are discussed in Section 4.1, while the various thicknesses from Case 9 to Case 16 are explored in Section 4.2.

Effect of Layer Density and Elastic Modulus(Case 1 to 8)
The tensile stresses and absolute values of compressive stress on the lining and NRTCPM layer decrease obviously when the layer density decreases, and their variations tend to be similar, as shown in Figures 9-12.
Taking the eight points on the NRTCPM layer as an example, as the layer density decreases from 0.54 g/cm 3 to 0.17 g/cm 3 , the elastic modulus decreases from 228 MPa to 17.2 MPa and the Poisson's ratio increases from 0.1 to 0.25, while the compressive and tensile principal stresses go down at the same time (Figures 9 and 10). When the coating layer density decreases from 0.33 g/cm 3 to 0.28 g/cm 3 and from 0.28 g/cm 3 to 0.2 g/cm 3 , the tensile stresses and absolute values of the compressive stress decrease sharply, while when the density decreases from 0.2 g/cm 3 to 0.17 g/cm 3 , the tensile stresses and absolute values of the compressive stress decrease gradually. The reduction rate of the tensile stresses and absolute values of the compressive stress differs at different stages. We can see from Figure 9 that the tensile stress declines slowly at every point as the density of NRTCPM decreases from 0.54 g/cm 3 to 0.4 g/cm 3 , and then rapidly within the medium density. Finally, the tensile stress at every point declines gently again. To be specific, the largest tensile stress drop value from Point No. 9 to Point No. 16 is 261 KPa at Point No. 14 when the density of the NRTCPM layer decreases from 0.33 g/cm 3 to 0.28 g/cm 3 , and 125 KPa at Point No. 12 when its density decreases from 0.28 g/cm 3 to 0.2 g/cm 3 , respectively. As shown in Figure 10, the compressive stress increases by 326 KPa at Point No. 12 when the density of the NRTCPM layer decreases to 0.28 g/cm 3 from 0.33 g/cm 3 , and the compressive stress goes up 127 KPa at Point No. 15 when the density of the NRTCPM layer decreases to 0.2 g/cm 3 from 0.28 g/cm 3 .
Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 16         Taking the eight points on the NRTCPM layer as an example, as the layer density decreases from 0.54 g/cm 3 to 0.17 g/cm 3 , the elastic modulus decreases from 228 MPa to 17.2 MPa and the Poisson's ratio increases from 0.1 to 0.25, while the compressive and tensile principal stresses go down at the same time (Figures 9 and 10). When the coating layer density decreases from 0.33 g/cm 3 to 0.28 g/cm 3 and from 0.28 g/cm 3 to 0.2 g/cm 3 , the tensile stresses and absolute values of the compressive stress decreasesharply, while when the density decreases from 0.2 g/cm 3 to 0.17 g/cm 3 , the tensile stresses and absolute values of the compressive stress decrease gradually. The reduction rate of the tensile stresses and absolute values of the compressive stress differs at different stages. We can see from Figure 9 that the tensile stress declines slowly at every point as the density of NRTCPM decreases from 0.54 g/cm 3 to 0.4 g/cm 3 , and then rapidly within the medium density. Finally, the tensile stress at every point declines gently again. To be specific, the largest tensile stress drop value from Point No. 9 to Point No. 16 is 261 KPa at Point No. 14 when the density of the NRTCPM layer decreases from 0.33 g/cm 3 to 0.28 g/cm 3 , and 125 KPa at Point No. 12 when its density decreases from 0.28 g/cm 3 to 0.2 g/cm 3 , respectively. As shown in Figure 10, the compressive stress increasesby 326 KPa at Point No. 12 when the density of the NRTCPM layer decreases to 0.28 g/cm 3 from 0.33 g/cm 3 , and the compressive stress goes up 127 KPa at Point No. 15 when the density of the NRTCPM layer decreases to 0.2 g/cm 3 from 0.28 g/cm 3 .
The changing trends of the tensile stresses and absolute values of the compressive stress on the lining shown in Figures 11 and 12 are the same as the changing trends of these stresses on the NRTCPM layer from Figures 9 and 10. When the density of the isolation layer is 0.2 g/cm 3 , the tensile stress and absolute value of the compressive stress on the lining can be reduced by 80% on average compared witha tunnel without the condition of an isolation layer around it. It can be seen from Figure 11 that the compressive stresses increaseby an average of 2902 KPa and 1304 KPa on all points when the density of the NRTCPM layer changes from 0.33 g/cm 3 to 0.28 g/cm 3 and from 0.28 g/cm 3 to 0.2 g/cm 3 . Figure 12 shows that the tensile stress values decrease by 2902 KPa and 1088 KPa when the density of the NRTCPM layer changes from 0.33 g/cm 3 to 0.28 g/cm 3 and from 0.28 g/cm 3 to 0.2 g/cm 3 . In conclusion, the lower the density of the NRTCPM layer is, the more the seismic energy is taken in, and the better the seismic isolation effect performs. As the density decreases, the elastic modulus decreases too (Table 4). 0.2 g/cm 3 is suggested as an optimum density in view of the compressive and tensile strength reduction rate. Furthermore, the lower the density is, the lower the cost is. The changing trends of the tensile stresses and absolute values of the compressive stress on the lining shown in Figures 11 and 12 are the same as the changing trends of these stresses on the NRTCPM layer from Figures 9 and 10. When the density of the isolation layer is 0.2 g/cm 3 , the tensile stress and absolute value of the compressive stress on the lining can be reduced by 80% on average compared with a tunnel without the condition of an isolation layer around it. It can be seen from Figure 11 that the compressive stresses increase by an average of 2902 KPa and 1304 KPa on all points when the density of the NRTCPM layer changes from 0.33 g/cm 3 to 0.28 g/cm 3 and from 0.28 g/cm 3 to 0.2 g/cm 3 . Figure 12 shows that the tensile stress values decrease by 2902 KPa and 1088 KPa when the density of the NRTCPM layer changes from 0.33 g/cm 3 to 0.28 g/cm 3 and from 0.28 g/cm 3 to 0.2 g/cm 3 . In conclusion, the lower the density of the NRTCPM layer is, the more the seismic energy is taken in, and the better the seismic isolation effect performs. As the density decreases, the elastic modulus decreases too (Table 4). 0.2 g/cm 3 is suggested as an optimum density in view of the compressive and tensile strength reduction rate. Furthermore, the lower the density is, the lower the cost is.

Effect of Layer Thickness(Case 9 to 16)
As shown in Figure 13, the thicker the NRTCPM layer is, the lower the tensile stresses and absolute values of the compressive stress on the NRTCPM layer and lining are. Taking the eight points on the lining as an example, the tensile stresses and absolute values of the compressive stress on the lining decrease sharply when the isolation layer increases from a thickness of 5 cm to 20 cm; however, they change slightly when the thickness of the isolation layer increases from 20 cm to 40 cm (Figures 13  and 14). We can see from Figure 13 that the drop values of the tensile stress are 2304 KPa, 1024 KPa, and 588 KPa on average for all points when the thickness of the NRTCPM layer increases from 5 mm to 10 mm, from 10 mm to 15 mm, and from 15 mm to 20 mm, respectively. In Figure 14, the compressive stresses have a similar trend. The increased values of the compressive stress are 2313 KPa and 1204 KPa on average for all points when the thickness of the NRTCPM layer increases from 5 mm to 10 mm and from 10 mm to 15 mm.
The absolute values of the compressive stressesand tensile stresses from number 9 to number 16 on the NRTCPM layer tend to decline in a similar trend as the points on the lining (Figures 15 and 16). According to Figure 15, the average increases in the values of the compressive stresses on eight points of the NRTCPM layer are 224 KPa, 134 KPa, 77 KPa, and 47 KPa, when the thickness of the NRTCPM layer increases from 5 mm to 10 mm, from 10 mm to 15 mm, from 15 mm to 20 mm, and from 25 mm to 30 mm, respectively. On the contrary, the average decrease in the values of the tensile stress on eight points on the NRTCPMlayer are 204 KPa, 124 KPa, 77 KPa, and 42 KPa in Figure 16. We can conclude that a further increase of theNRTCPM layer's thickness would not have a significant effect on the seismic response of the lining and the seismic is olation when the NRTCPM layer's thickness reaches a certain value. on the lining decrease sharply when the isolation layer increasesfrom a thickness of 5 cm to 20 cm; however, they change slightly when the thickness ofthe isolation layer increasesfrom 20 cm to 40 cm (Figures 13 and 14). We can see from Figure 13 that the drop values of the tensile stress are 2304 KPa, 1024KPa, and 588 KPa on average for all points when the thickness of the NRTCPM layer increases from 5 mm to 10 mm, from 10 mm to 15 mm, and from 15 mm to 20 mm, respectively. In Figure 14, the compressive stresses havea similar trend. The increasedvalues of thecompressive stress are 2313 KPa and 1204KPa on average for all points when the thickness of the NRTCPM layer increases from 5 mm to 10 mm and from 10 mm to 15 mm.     1024KPa, and 588 KPa on average for all points when the thickness of the NRTCPM layer increases from 5 mm to 10 mm, from 10 mm to 15 mm, and from 15 mm to 20 mm, respectively. In Figure 14, the compressive stresses havea similar trend. The increasedvalues of thecompressive stress are 2313 KPa and 1204KPa on average for all points when the thickness of the NRTCPM layer increases from 5 mm to 10 mm and from 10 mm to 15 mm.    The absolute values of the compressive stressesand tensile stresses from number 9 to number 16 on the NRTCPM layer tend to decline in a similar trend as the points on the lining (Figures 15 and  16). According to Figure 15, the average increases in the valuesof the compressive stresses on eight points of the NRTCPM layer are 224 KPa, 134 KPa, 77 KPa, and 47 KPa, when the thickness of the NRTCPM layer increases from 5 mm to 10 mm, from 10 mm to 15 mm, from 15 mm to 20 mm, and from 25 mm to 30 mm, respectively. On the contrary, the average decrease in the values of the tensile stress on eight points on the NRTCPMlayer are 204 KPa, 124 KPa, 77 KPa, and 42 KPa in Figure 16. We can conclude that a further increase of theNRTCPM layer's thickness would not have a significant effect on the seismic response of the lining and the seismic isolationwhen the NRTCPM layer's thickness reaches a certain value.  Points numbers Figure 15. Compressive stress of eight points on the NRTCPM layer.  When the thickness of the isolation layer is 20 cm, the tensile stresses and absolute values of the compressive stress can reduce70% of stress on average compared with the condition without the NRTCPM isolation layer. Therefore, we recommend an optimum NRTCPM layer thickness of 20 cm considering the stress reduction rate and the construction costs of the polymeric material.

Conclusions
The main conclusions are as follows: An impact resonance test is a reasonable and effective method to gain a damping ratio and dynamic elastic modulus of NRTCPM. When NRTCPM's density is approximately 0.2 g/cm 3 , the average damping ratio is 0.0375, and the average dynamic elastic modulus is 359.025 MPa from the results of IRT. In future studies, IRT can be used to gain the damping ratios and dynamic elastic moduli of other densities' NRTCPM.
The NRTCPM is found to have good capabilities to reduce the seismic demand and attenuate the seismic load. It offers superior capabilities to protect tunnels under seismic load.
The compressive and tensile strength of NRTCPM increases as its density increases. The seismic response of NRTCPM is highly influenced by its density, elastic modulus, Poisson's ratio, and thickness. To be specific, the tensile stresses and absolute values of the compressive stress on a NRTCPM layer can be reduced by about 90% as the density decreases from 0.54 g/cm 3 to 0.17g/cm 3 and the elastic modulus from 228 MPa to 17.2 MPa, and the tensile stresses and absolute values of the compressive stress on the lining can be reduced by 75%. When the density decreases, the elastic modulus decreases at the same time and the Poisson's ratio increases steadily. By retrofitting NRTCPM layers from 5 cm to 40 cm, the tensile stresses and the absolute values of the compressive stress on the NRTCPM layer and lining can be reduced by 80% on average.
The NRTCPM layer properties, including the layer density, thickness, dynamic elastic modulus, and Poisson's ratio, are determinative factors for the seismic isolation effects of tunnels. Ultimately, the seismic isolation effect performs better at a lower density, smaller elastic modulus, and higher Poisson's ratio and thicker layer. The influences of these parameters are all significant for seismic isolation. The optimum density is 0.2 g/cm 3 and thickness is 20 cm under the consideration of cost and strength.
Author Contributions: Conceptualization, C.G. and X.M.; data curation, X.M. and B.S.; writing-original draft preparation, X.M.; writing-review and editing, C.G. and X.M.; funding acquisition, C.G. and F.W. All authors have read and agreed to the published version of the manuscript.