Theoretical Analysis of the Dynamic Properties of a 2-2 Cement-Based Piezoelectric Dual-Layer Stacked Sensor under Impact Load

Cement-based piezoelectric materials are widely used due to the fact that compared with common smart materials, they overcome the defects of structure-incompatibility and frequency inconsistency with a concrete structure. However, the present understanding of the mechanical behavior of cement-based piezoelectric smart materials under impact load is still limited. The dynamic characteristics under impact load are of importance, for example, for studying the anti-collision properties of engineering structures and aircraft takeoff-landing safety. Therefore, in this paper, an analytical model was proposed to investigate the dynamic properties of a 2-2 cement-based piezoelectric dual-layer stacked sensor under impact load based on the piezoelectric effect. Theoretical solutions are obtained by utilizing the variable separation and Duhamel integral method. To simulate the impact load and verify the theory, three types of loads, including atransient step load, isosceles triangle load and haversine wave load, are considered and the comparisons between the theoretical results, Li’s results and numerical results are presented by using the control variate method and good agreement is found. Furthermore, the influences of several parameters were discussed and other conclusions about this sensor are also given. This should prove very helpful for the design and optimization of the 2-2 cement-based piezoelectric dual-layer stacked sensor in engineering.


Introduction
Cement-based piezoelectric sensors, a new kind of functional structure developed in recent decades, are fabricated from a cement matrix and piezoelectric ceramic phase in different volume fractions and using various mixing rules [1,2]. Cement-based piezoelectric composites have very sensitive transduction properties as well as good compatibility with the most popular construction materials (such as cement and concrete) used in civil engineering. They have received much research attention in recent years and have great potential application as a novel kind of electromechanical sensor material in structural health monitoring, which makes it crucial to study the overall properties of cement-based piezoelectric composites for sensor design, practical engineering application and optimization [3][4][5].
Most of the studies have focused on the preparation of cement-based piezoelectric sensors and determining their relevant parameters by experimental methods. By using a cut-filling process, Huang et al. prepared 2-2 cement-based piezoelectric composites [6,7]. In their paper, the effects of ceramic volume fraction and water-cement ratio on the properties of the composites were studied. The results indicated that the piezoelectric strain constant increases rapidly with increasing volume are presented. This study should be very helpful for the design and optimization of 2-2 cement-based piezoelectric dual-layer stacked sensors in engineering. Figure 1a is a schematic of a 2-2 cement-based piezoelectric dual-layer stacked sensor with one fixed end and the other free. The bottom and top layers of the sensor, denoted as C#1 (thickness l 1 ) and P#2 (thickness h 2 ), are the cement and piezoelectric layer, respectively. The free end of the sensor is subjected to an impact load δ(t). The symbols D, E, ε and σ denote the electric displacement, electric field, strain and stress, respectively, with reference to the Cartesian coordinate system. Linear elastic material is assumed for the material of the cement and piezoelectric layer. Figure 1b  solutions. Finally, a summary and conclusions are presented. This study should be very helpful for the design and optimization of 2-2 cement-based piezoelectric dual-layer stacked sensors in engineering. Figure 1a is a schematic of a 2-2 cement-based piezoelectric dual-layer stacked sensor with one fixed end and the other free. The bottom and top layers of the sensor, denoted as C#1 (thickness l 1 ) and P#2 (thickness h 2 ), are the cement and piezoelectric layer, respectively. The free end of the sensor is subjected to an impact load δ(t). The symbols D, E, ε and σ denote the electric displacement, electric field, strain and stress, respectively, with reference to the Cartesian coordinate system. Linear elastic material is assumed for the material of the cement and piezoelectric layer. Figure 1b  According to Figure 1b and Newton's second law, we can get the equilibrium condition of the sensor longitudinal arbitrary unit body:

Basic Equations
Here A, ρ and w are, respectively, the cross section, density and displacement of the specific cement and piezoelectric layer. δ(t)δ(z  l 2 ) suggests that the sensor is subjected to the impact load at the free end. The expressions for δ(t) and δ(z  l 2 ) can be written as: Equation (1) can also be written as: It should be noted that the first expression on the right of Equation (3) represents the acceleration of the sensor generated by internal force, and the second expression represents the acceleration generated by external force. Therefore, we introduce the theoretical density ρ T , ρ T is expressed as follows: According to Figure 1b and Newton's second law, we can get the equilibrium condition of the sensor longitudinal arbitrary unit body: Here A, ρ and w are, respectively, the cross section, density and displacement of the specific cement and piezoelectric layer. δ(t)δ(z − l 2 ) suggests that the sensor is subjected to the impact load at the free end. The expressions for δ(t) and δ(z − l 2 ) can be written as: Equation (1) can also be written as: It should be noted that the first expression on the right of Equation (3) represents the acceleration of the sensor generated by internal force, and the second expression represents the acceleration generated by external force. Therefore, we introduce the theoretical density ρ T , ρ T is expressed as follows: Here ρ c , V c and ρ p , V p are, respectively, the density and volume fraction of the cement and piezoelectric material constituting the sensor. Then Equation (3) can be written as: For the cement layer C#1 (0 ≤ z ≤ l 1 ), according to Equation (5) and without considering the body force and body charge, the basic equations can be written as: Here C 33c , ρ c and w c are, respectively, the elastic stiffness coefficient, density and displacement of the cement material. Equations (2) and (6) are combined to give the following equation: Here C a = C 33c /ρ c represents the propagation velocity of the vibration wave in the cement layer. For the piezoelectric layer P#2 (l 1 ≤ z ≤ l 2 ), according to Equation (5) and without considering the body force and body charge, the basic equations can also be written as follows: Here C 33p , e 33 , ε S 33 and w p are, respectively, the elastic stiffness coefficient, piezoelectric coefficient, permittivity coefficient and displacement of the piezoelectric material.
Equations (2), (8) and (9) are combined to give the following equations: which could be rewritten as: Here C b = E 0 /ρ p represents the propagation velocity of the vibration wave in the piezoelectric layer, here, E 0 = C 33p +e 33 2 /ε S 33 . Considering the initial conditions and boundary conditions of the sensor, the equation of motion and the definite conditions are summarized as follows: In this section, the exact solution of a 2-2 cement-based piezoelectric dual-layer stacked sensor can be obtained by utilizing the variable separation method (also known as standing wave method) and the Duhamel integral. Firstly, the displacement of the cement and piezoelectric material can be decomposed as follows: Substituting the above equations into Equation (12), the eigenvalue problem of original definite problem and the frequency equations can be obtained as follows: Here C b = E 0 ∕ represents the propagation velocity of the vibration wave in the piezoelectric layer, here, E 0 = C 33p + e 33 2 ∕ ε 33 S .
Considering the initial conditions and boundary conditions of the sensor, the equation of motion and the definite conditions are summarized as follows: where E 0 is the modulus of elasticity of the piezoelectric material.

Theoretical Solutions of 2-2 Cement-Based Piezoelectric Dual-Layer Stacked Sensor under Impact Load
In this section, the exact solution of a 2-2 cement-based piezoelectric dual-layer stacked sensor can be obtained by utilizing the variable separation method (also known as standing wave method) and the Duhamel integral. Firstly, the displacement of the cement and piezoelectric material can be decomposed as follows: Substituting the above equations into Equation (12), the eigenvalue problem of original definite problem and the frequency equations can be obtained as follows: Combining Equation (12f) with Equation (15) gives the following relation: Combining Equation (12f) with Equation (15) gives the following relation: Solving Equations (14a) and (14b) obtains the following solutions: in which a 1n , b 1n , a 2n , b 2n are undetermined coefficients. Substitution of Equation (17) into Equations (14c) and (14d) leads to the following equations: Solving Equation (18) obtains the following solutions: In order to make a 1n , b 1n , a 2n , b 2n have untrivial solutions, let the coefficient determinant of Equation (18) equals to zero: That means that the following formula must be satisfied: where h 2 = l 2 −l 1 . On the basis of Equation (16), we can obtained the relation √ λ 1n C a = √ λ 2n C b = λ n . Then we define: Utilizing Equation (22) and relation (21) can be simplified as the dimensionless characteristic equation: After obtaining the value of λ n by the equation above, √ λ 1n and √ λ 2n can be obtained by , respectively, so the corresponding eigenfunctions can be obtained. According to the variable separation method: where n = 1, 2, 3, · · · . Substituting Equation (24) into Equations (12a) and (12b) leads to the following equations:  (24) where n = 1,2,3,⋯. Substituting Equation (24) into Equations (12a) and (12b) leads to the following equations: Using sin λ 1m z to multiplied both sides of Equation (25a), and taken definite integral dz using a 2m cos λ 2m z + b 2m sin λ 2m z to multiplied both sides of Equation (25b), and taken definite integral dz l 2 l 1 , then added these two equations. Combining the weighted orthogonality of the eigenfunctions and we can obtained the following equations: So when n = m, the following equation can be obtained (n ≠ m meaningless): where: Since the initial conditions of the sensor is zero, the Equation (28) can be solved by the Duhamer integral formula: Using sin √ λ 1m z to multiplied both sides of Equation (25a), and taken definite integral z to multiplied both sides of Equation (25b), and taken definite integral l 2 l 1 dz, then added these two equations. Combining the weighted orthogonality of the eigenfunctions and we can obtained the following equations: So when n = m, the following equation can be obtained (n = m meaningless): where: Since the initial conditions of the sensor is zero, the Equation (28) can be solved by the Duhamer integral formula: Therefore, the exact solutions of the displacement of 2-2 cement-based piezoelectric dual-layer stacked sensor under impact load can be obtained as: ; n = 1, 2, 3, · · · . Combining Equations (6), (8), (9) and (12h), the exact precise solutions of the mechanical and electrical quantities of 2-2 cement-based piezoelectric dual-layer stacked sensor under impact load can be obtained as follows: Stress functions: Strain functions: Velocity functions: Acceleration functions: Electric potential of piezoelectric layer: Electric field intensity of piezoelectric layer: Thus, the precise mechanical and electrical fields of 2-2 cement-based piezoelectric dual-layer stacked sensor under impact load have been fully determined by the variable separation method and Duhamel integral.

Comparison and Discussion
In this section, a numerical simulation of the 2-2 cement-based piezoelectric dual-layer stacked sensor under impact load is presented and compared with the theoretical solutions obtained in the previous sections and Li's results [25]. The total thickness of the sensor l 2 is taken as 0.015 m. It is defined that the cement layer and piezoelectric layer are made of ordinary Portland cement and piezoelectric ceramics, respectively. The main material parameters of piezoelectric ceramics are based on Li's experiments [26]. The related structural and material parameters take the values summarized in Table 1. The numerical simulation analysis is modeled by the finite element analysis software, and the size of the model is 0.001 m × 0.001 m × 0.015 m. The direction of polarization is z-axis. By using the free meshing method, the unit partition of the analysis model is divided into 10, 10 and 300 segments along x, y and z axis, respectively. The upper and lower surfaces of the piezoelectric layer in the z-axis direction are subjected to the piezoelectric coupling. The electric potential of the lower surface of the piezoelectric layer is set to zero. The model is loaded and solved after the symmetrical boundary conditions are set on the four sides of the model. The impact load Q(t) used in this numerical simulation analysis includes three types, namely, the transient step load (denoted as load A), transient isosceles triangle load (denoted as load B) and transient haversine wave load (denoted as load C). The three types of loads are shown in Figure 2 and they all satisfy previous sections and Li's results [25]. The total thickness of the sensor l 2 is taken as 0.015 m. It is defined that the cement layer and piezoelectric layer are made of ordinary Portland cement and piezoelectric ceramics, respectively. The main material parameters of piezoelectric ceramics are based on Li's experiments [26]. The related structural and material parameters take the values summarized in Table 1. The numerical simulation analysis is modeled by the finite element analysis software, and the size of the model is 0.001 m × 0.001 m × 0.015 m. The direction of polarization is z-axis. By using the free meshing method, the unit partition of the analysis model is divided into 10, 10 and 300 segments along x, y and z axis, respectively. The upper and lower surfaces of the piezoelectric layer in the z-axis direction are subjected to the piezoelectric coupling. The electric potential of the lower surface of the piezoelectric layer is set to zero. The model is loaded and solved after the symmetrical boundary conditions are set on the four sides of the model. The theoretical influences of the impact load δ(t) on the displacements w p (l 2 ,t) and w c (l 1 ,t) at the free end of the sensor and the interface between the piezoelectric and cement layer when n = 1, n = 2, n = 10 and n = 1000 are shown in Figure 3. It is noted that the displacement functions of the sensor agrees well for n = 1, 2, 10 and 1000. For convenience and without loss of generality, the theoretical solution with n = 2 is selected for the following analysis except for special instructions. The theoretical influences of the impact load δ(t) on the displacements w p (l 2 , t and w c (l 1 , t) at the free end of the sensor and the interface between the piezoelectric and cement layer when n = 1, n = 2, n = 10 and n = 1000 are shown in Figure 3. It is noted that the displacement functions of the sensor agrees well for n = 1, 2, 10 and 1000. For convenience and without loss of generality, the theoretical solution with n = 2 is selected for the following analysis except for special instructions.
The theoretical influences of the impact load δ(t) on the displacements w p (l 2 ,t) and w c (l 1 ,t) at the free end of the sensor and the interface between the piezoelectric and cement layer when n = 1, n = 2, n = 10 and n = 1000 are shown in Figure 3. It is noted that the displacement functions of the sensor agrees well for n = 1, 2, 10 and 1000. For convenience and without loss of generality, the theoretical solution with n = 2 is selected for the following analysis except for special instructions.  The comparison between the theoretical and numerical solutions of the time-dependent displacement function w p (l 2 ,t) at the free end of the sensor is shown in Figure 4. It can be found from Figure 4a-c that the numerical simulation and the theoretical solutions are closer when the peak value of the impact loads A, B and C is 800 kPa. Furthermore, when the peak value of impact load is larger, the simulation results become closer to the theoretical solutions. The comparison between the theoretical and numerical solutions of the time-dependent displacement function w p (l 2 , t at the free end of the sensor is shown in Figure 4. It can be found from Figure 4a-c that the numerical simulation and the theoretical solutions are closer when the peak value of the impact loads A, B and C is 800 kPa. Furthermore, when the peak value of impact load is larger, the simulation results become closer to the theoretical solutions. The theoretical influences of the impact load δ(t) on the displacements w p (l 2 ,t) and w c (l 1 ,t) at the free end of the sensor and the interface between the piezoelectric and cement layer when n = 1, n = 2, n = 10 and n = 1000 are shown in Figure 3. It is noted that the displacement functions of the sensor agrees well for n = 1, 2, 10 and 1000. For convenience and without loss of generality, the theoretical solution with n = 2 is selected for the following analysis except for special instructions.  The comparison between the theoretical and numerical solutions of the time-dependent displacement function w p (l 2 ,t) at the free end of the sensor is shown in Figure 4. It can be found from Figure 4a-c that the numerical simulation and the theoretical solutions are closer when the peak value of the impact loads A, B and C is 800 kPa. Furthermore, when the peak value of impact load is larger, the simulation results become closer to the theoretical solutions.   Figure 4d indicates that the numerical simulation under the load A is slightly better than those obtained under the loads B and C. Therefore, the numerical simulations indicate that the transient step load A (shown in Figure 2a) behaves more closely to the function δ(t) in this theoretical solution.
For the sensor under the impact load δ(t) and the load A, B, C with peak value 800 kPa, the displacement w c (l 1 ,t), electric potential ϕ(l 2 ,t) and stress (l 1 ,t) are plotted in Figure 5. It can also be seen that the results with the load A are closer to the theoretical solutions as compared to the  Figure 4d indicates that the numerical simulation under the load A is slightly better than those obtained under the loads B and C. Therefore, the numerical simulations indicate that the transient step load A (shown in Figure 2a) behaves more closely to the function δ(t) in this theoretical solution.
For the sensor under the impact load δ(t) and the load A, B, C with peak value 800 kPa, the displacement w c (l 1 , t), electric potential φ(l 2 , t) and stress σ c (l 1 , t) are plotted in Figure 5. It can also be seen that the results with the load A are closer to the theoretical solutions as compared to the loads B and C.
(c) (d) Figure 4. Influences of the impact loads δ(t) and loads A, B, C in theoretical and numerical solutions on the displacement at the free end of the sensor: (a) Influence of impact loads δ(t) and load A on w p (l 2 ,t); (b) Influence of impact loads δ(t) and load B on w p (l 2 ,t); (c) Influence of impact loads δ(t) and load C on w p (l 2 ,t); (d) Influence of impact loads δ(t) and loads A, B, C on w p (l 2 ,t). Figure 4d indicates that the numerical simulation under the load A is slightly better than those obtained under the loads B and C. Therefore, the numerical simulations indicate that the transient step load A (shown in Figure 2a) behaves more closely to the function δ(t) in this theoretical solution.
For the sensor under the impact load δ(t) and the load A, B, C with peak value 800 kPa, the displacement w c (l 1 ,t), electric potential ϕ(l 2 ,t) and stress (l 1 ,t) are plotted in Figure 5. It can also be seen that the results with the load A are closer to the theoretical solutions as compared to the loads B and C.   [24]. Comparing the theory at present paper with their theories, as shown in Figure 6b,c, and it can be found that the displacement of the free end of pure cement structure and pure piezoelectric structure are in good agreement with Li's and Zhang's theories, respectively. It also shows the rationality of using the theoretical density in Section 2, and the correctness of the theory presented in this paper. rod under the impact load ( [25], pp. 70-74). Zhang et al. have also researched the dynamic characteristics of the pure piezoelectric structure under the impact loading [24]. Comparing the theory at present paper with their theories, as shown in Figure 6b,c, and it can be found that the displacement of the free end of pure cement structure and pure piezoelectric structure are in good agreement with Li's and Zhang's theories, respectively. It also shows the rationality of using the theoretical density in Section 2, and the correctness of the theory presented in this paper. The distributions of the displacement w(z,t 0 ), electric potential ϕ(z,t 0 ) and stress σ(z,t 0 ) along the z-axis are shown in Figure 7a-c, respectively. The displacement at the free end of the sensor firstly peaks at t 0 and t 0 = 0.74 × 10 5 s. It can be found that with the increases of h 2 , the displacement w(z,t 0 ) and stress σ(z,t 0 ) of the composite structure and pure piezoelectric structure decreases, and the electric potential of the piezoelectric layer increases as h 2 increases. When the cement layer is thicker, the overall displacement change is larger; for a pure piezoelectric structure The distributions of the displacement w(z, t 0 ), electric potential φ(z, t 0 ) and stress σ(z, t 0 ) along the z-axis are shown in Figure 7a-c, respectively. The displacement at the free end of the sensor firstly peaks at t 0 and t 0 = 0.74 × 10 −5 s. It can be found that with the increases of h 2 , the displacement w(z, t 0 ) and stress σ(z, t 0 ) of the composite structure and pure piezoelectric structure decreases, and the electric potential of the piezoelectric layer increases as h 2 increases. When the cement layer is thicker, the overall displacement change is larger; for a pure piezoelectric structure (l 1 = 0 m), the overall displacement of the structure is minimized. This shows that the actuating capability of the cement-based piezoelectric dual-layer stacked sensor is better than that of a pure piezoelectric structure.
The influences of the elastic stiffness C 33p on the displacement amplitude w p (l 2 , t 0 , electric potential amplitude φ(l 2 , t 0 ) and stress amplitude σ c (l 1 , t 0 ) of the sensor are shown in Figure 8a−c, respectively. It can be found that w p (l 2 , t 0 , φ(l 2 , t 0 ) and σ c (l 1 , t 0 ) decrease as the elastic stiffness increases. In addition, with the increasing elastic stiffness, the changing rates of the displacement and electric potential amplitudes of the free end tend to decrease. As for the thicker piezoelectric layer, the rates of change of the displacement amplitude and electric potential amplitude are larger.
(l 1 = 0 m), the overall displacement of the structure is minimized. This shows that the actuating capability of the cement-based piezoelectric dual-layer stacked sensor is better than that of a pure piezoelectric structure. The influences of the elastic stiffness C 33p on the displacement amplitude w p (l 2 ,t 0 ), electric potential amplitude ϕ(l 2 ,t 0 ) and stress amplitude (l 1 ,t 0 ) of the sensor are shown in Figure 8a−c, respectively. It can be found that w p (l 2 ,t 0 ), ϕ(l 2 ,t 0 ) and (l 1 ,t 0 ) decrease as the elastic stiffness increases. In addition, with the increasing elastic stiffness, the changing rates of the displacement and electric potential amplitudes of the free end tend to decrease. As for the thicker piezoelectric layer, the rates of change of the displacement amplitude and electric potential amplitude are larger.  The influences of the elastic stiffness C 33p on the displacement amplitude w p (l 2 ,t 0 ), electric potential amplitude ϕ(l 2 ,t 0 ) and stress amplitude (l 1 ,t 0 ) of the sensor are shown in Figure 8a−c, respectively. It can be found that w p (l 2 ,t 0 ), ϕ(l 2 ,t 0 ) and (l 1 ,t 0 ) decrease as the elastic stiffness increases. In addition, with the increasing elastic stiffness, the changing rates of the displacement and electric potential amplitudes of the free end tend to decrease. As for the thicker piezoelectric layer, the rates of change of the displacement amplitude and electric potential amplitude are larger. The influences of the piezoelectric stress constant e 33 on the displacement amplitude w p (l 2 ,t 0 ), electric potential amplitude ϕ(l 2 ,t 0 ) and stress amplitude (l 1 ,t 0 ) of the sensor are shown in Figure 9a-c, respectively. It can be found that with the increases of e 33 , w p (l 2 ,t 0 ) decreases, with the change smaller for thinner piezoelectric layer. This could be explained as below. From the expression of C b the numerical change of e 33 has no effect on C b and the influence on the displacement solution in Equation (31) is also small. Besides, with the increasing e 33 , the electric potential amplitude ϕ(l 2 ,t 0 ) of the free end tends to flatten; meanwhile, the variation of stress amplitude σ (l 1 ,t 0 ) is approximately linear with the piezoelectric stress constant e 33 . Moreover, Figure 9b shows that e 33 has a great influence on the amplitude of ϕ(l 2 ,t 0 ) of the sensor, which ensures that the sensor can produce large electric potential. Like the cases in Figure 8, for a thicker piezoelectric layer, the influence of e 33 on ϕ(l 2 ,t 0 ), w p (l 2 ,t 0 ) and (l 1 ,t 0 ) is larger. The influences of the piezoelectric stress constant e 33 on the displacement amplitude w p (l 2 , t 0 , electric potential amplitude φ(l 2 , t 0 ) and stress amplitude σ c (l 1 , t 0 ) of the sensor are shown in Figure 9a-c, respectively. It can be found that with the increases of e 33 , w p (l 2 , t 0 decreases, with the change smaller for thinner piezoelectric layer. This could be explained as below. From the expression of C b the numerical change of e 33 has no effect on C b and the influence on the displacement solution in Equation (31) is also small. Besides, with the increasing e 33 , the electric potential amplitude φ(l 2 , t 0 ) of the free end tends to flatten; meanwhile, the variation of stress amplitude σ c (l 1 , t 0 ) is approximately linear with the piezoelectric stress constant e 33 . Moreover, Figure 9b shows that e 33 has a great influence on the amplitude of φ(l 2 , t 0 ) of the sensor, which ensures that the sensor can produce large electric potential. Like the cases in Figure 8, for a thicker piezoelectric layer, the influence of e 33 on φ(l 2 , t 0 ), w p (l 2 , t 0 and σ c (l 1 , t 0 ) is larger. The influences of the piezoelectric stress constant e 33 on the displacement amplitude w p (l 2 ,t 0 ), electric potential amplitude ϕ(l 2 ,t 0 ) and stress amplitude (l 1 ,t 0 ) of the sensor are shown in Figure 9a-c, respectively. It can be found that with the increases of e 33 , w p (l 2 ,t 0 ) decreases, with the change smaller for thinner piezoelectric layer. This could be explained as below. From the expression of C b the numerical change of e 33 has no effect on C b and the influence on the displacement solution in Equation (31) is also small. Besides, with the increasing e 33 , the electric potential amplitude ϕ(l 2 ,t 0 ) of the free end tends to flatten; meanwhile, the variation of stress amplitude σ (l 1 ,t 0 ) is approximately linear with the piezoelectric stress constant e 33 . Moreover, Figure 9b shows that e 33 has a great influence on the amplitude of ϕ(l 2 ,t 0 ) of the sensor, which ensures that the sensor can produce large electric potential. Like the cases in Figure 8, for a thicker piezoelectric layer, the influence of e 33 on ϕ(l 2 ,t 0 ), w p (l 2 ,t 0 ) and (l 1 ,t 0 ) is larger.  -c show the influences of the relative dielectric constant ε S 33 /ε 0 on the displacement amplitude w p (l 2 , t 1 , electric potential amplitude φ(l 2 , t 1 ) and stress amplitude σ c (l 1 , t 0 ) of the sensor, respectively. It can be found that with the increase of ε S 33 /ε 0 , w p (l 2 , t 1 increases, though the change is quite small; meanwhile, φ(l 2 , t 1 ) decreases rapidly at the beginning and then tends to flatten, and the influence on σ c (l 1 , t 0 ) is negligible. Furthermore, the influence of ε S 33 /ε 0 on φ(l 2 , t 1 ) is larger for a thicker piezoelectric layer.  Figure 10a-c show the influences of the relative dielectric constant ε 33 S /ε 0 on the displacement amplitude w p (l 2 ,t 1 ) , electric potential amplitude ϕ(l 2 ,t 1 ) and stress amplitude (l 1 ,t 0 ) of the sensor, respectively. It can be found that with the increase of ε 33 S /ε 0 , w p (l 2 ,t 1 ) increases, though the change is quite small; meanwhile, ϕ(l 2 ,t 1 ) decreases rapidly at the beginning and then tends to flatten, and the influence on (l 1 ,t 0 ) is negligible. Furthermore, the influence of ε 33 S /ε 0 on ϕ(l 2 ,t 1 ) is larger for a thicker piezoelectric layer.  Figure 1a. These results are quite helpful for the design and optimization ofthe 2-2 cement-based piezoelectric dual-layer stacked sensors.

Conclusions
An analytical study of a 2-2 cement-based piezoelectric dual-layer stacked sensor under impact load is presented based upon the theory of piezoelasticity. Theoretical solutions are obtained by combining all the equations and boundary conditions and utilizing the variable separation method and Duhamel integral. It is found that: 1. The compliance of numerical simulations under the transient step load is slightly better than that under transient isosceles triangle load and transient haversine wave load. The theoretical Figure 10. Influences of ε S 33 /ε 0 on the displacement, electric potential and stress amplitudes with the different thickness of piezoelectric layer h 2 : (a) Influence on the displacement amplitude w p (l 2 , t 0 ; (b) Influence on the electric potential amplitude φ(l 2 , t 0 ); (c) Influence on the stress amplitude σ c (l 1 , t 0 ).  Figure 1a. These results are quite helpful for the design and optimization ofthe 2-2 cement-based piezoelectric dual-layer stacked sensors.

Conclusions
An analytical study of a 2-2 cement-based piezoelectric dual-layer stacked sensor under impact load is presented based upon the theory of piezoelasticity. Theoretical solutions are obtained by combining all the equations and boundary conditions and utilizing the variable separation method and Duhamel integral. It is found that: 1.
The compliance of numerical simulations under the transient step load is slightly better than that under transient isosceles triangle load and transient haversine wave load. The theoretical results show overall good agreement with the numerical results. The numerical simulation are closest to the theoretical results for the larger peak value of the impact load; 2.
The displacement amplitude and period of the free end are both larger than that of the pure piezoelectric structures, and the thinner the piezoelectric layer, the larger the displacement amplitude and period;