Dynamic homogenization method for elastic wave band gap and initial boundary value problem of piezoelectric composite material

Abstract

The invention discloses a dynamic homogenization method for an elastic wave band gap and an initial boundary value problem of a piezoelectric composite material, provides a non-local dynamic homogenization model of a piezoelectric periodic composite material with an elastic double-layer structure, and adopts a spatial multi-scale homogenization method based on eight-order displacement and an electric potential asymptotic expansion mode. A space-time non-local momentum balance equation which is the same as a gradient elastic model in structure is established by processing momentum balance equations of all orders of a piezoelectric composite material, and different from a traditional gradient elastic model needing to recognize model parameters related to the length scale. Model parameters are completely determined by the performance of the material and the geometrical shape of the microstructure and are accurately determined in the homogenization process, so that the model is more simplified and universal, the dependence on experimental measurement parameters is reduced, and the applicability of the model is remarkably improved.

Description

Technical Field

[0001] The invention belongs to the technical field of intelligent composite materials, and in particular relates to a dynamic homogenization method for analyzing elastic wave band gap characteristics and initial-boundary value problems in piezoelectric composite materials. Background Art

[0002] Piezoelectric composites are widely used in sensors, energy harvesting, seismic isolation, and acoustic devices due to their unique electromechanical coupling properties. In such materials, the propagation of elastic waves is affected by the periodic structure of the material and the electromechanical coupling effect, which may form a band gap inside the material, that is, a phenomenon in which elastic waves cannot propagate within a certain frequency range. The existence of a band gap is of great significance to the application of piezoelectric composites in seismic isolation, filtering, and other fields.

[0003] At present, the numerical methods for the dynamic behavior of periodic piezoelectric composites are mainly divided into two categories: (1) steady-state unit cell band gap characterization; (2) transient wave propagation analysis. The first category of methods is usually based on the Floquet-Bloch theorem, which assumes that the unit cells are arranged periodically in an infinite domain. Therefore, this type of method can provide important information about the position, size and wave velocity of the unit cell band gap, but cannot consider macroscopic phenomena such as the interaction between waves and boundaries or between waves. In the second category of methods, transient wave propagation analysis often uses finite element method or finite difference time domain method. Although these methods can study macroscopic phenomena, they often have high computational costs due to the need for high-resolution microstructures and small time steps.

[0004] To address this challenge, researchers have devoted themselves to the development of multiscale analysis methods over the past decade. Such methods can effectively simulate the steady-state and transient propagation behaviors of elastic waves in periodic composite materials. Multiscale models of inhomogeneous materials are generally divided into two categories: (1) independent of the scale separation assumption and (2) based on the scale separation assumption. The scale separation assumption assumes that the deformation wavelength is greater than or equal to the lattice constant. The first type of model is independent of this assumption and is applicable to a wider range of wave frequencies, but there are few studies on its application in transient wave propagation in composite materials, and there are few in-depth analyses of its computational efficiency. In the second type of model, researchers have proposed a variety of homogenization methods based on the scale separation assumption, such as computational homogenization methods and asymptotic homogenization methods. These methods can accurately and effectively predict the first band gap and transient wave propagation of macroscopic materials through microstructural properties. However, most of these methods are targeted at elastic composites, and relatively few studies on wave propagation in piezoelectric composites are available. Elastic composites are mainly used for passive modulation, and their band gap and wave characteristics are adjusted by changing the material composition or structural geometric parameters. However, this passive modulation has limitations and is difficult to meet the multi-band regulation requirements under complex engineering requirements, which limits its application in high-end equipment, such as submarine acoustic radiation control and aircraft acoustic transmission regulation. Therefore, it is crucial to develop smart composite materials that can actively adjust the band gap and other wave propagation characteristics. The use of multi-physical field coupling, such as thermal, mechanical, electrical, and magnetic coupling, is an effective way to achieve active regulation. Piezoelectric materials are widely used in the field of active regulation of smart composite materials due to their fast response characteristics and easy control. Therefore, a dynamic homogenization method specifically for the elastic wave band gap and initial boundary value problem of piezoelectric composites is proposed, which is of great significance for improving the computational efficiency of smart composite materials and achieving efficient analysis. Summary of the invention

[0005] The purpose of the present invention is to address the limitations of the prior art. The present invention provides a dynamic homogenization method with high computational efficiency and accurate results for the elastic wave band gap and initial boundary value problems of piezoelectric composite materials, thereby facilitating the design of vibration and noise reduction of intelligent piezoelectric composite materials to solve the problems raised in the above-mentioned background technology.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] A dynamic homogenization method for elastic wave band gap and initial boundary value problem of piezoelectric composite materials comprises the following steps:

[0008] S1: In the framework of multiscale analysis, the displacement and potential are asymptotically expanded in the Laplace domain;

[0009] The expressions of periodic boundary conditions and continuity conditions of each order are obtained through the periodic boundary conditions of displacement, stress, potential and electric displacement field in asymptotic expansion form and the continuity conditions of displacement, stress, potential and electric displacement field at the interface of two-phase materials;

[0010] Substitute the asymptotic expansion of displacement and potential into the dynamic equilibrium equation of the piezoelectric material, collect the terms of equal order, and obtain the dynamic equilibrium equation of each order;

[0011] S2: recursively process the dynamic equilibrium equations of each order obtained in step S1 to obtain the expressions of displacement and potential of each order containing unknown microscopic functions;

[0012] S3: Obtaining each order homogeneous dynamic equilibrium equation by applying an average operator to each order dynamic equilibrium equation obtained in step S1;

[0013] The homogeneous dynamic equilibrium equations of each order are combined into a single homogeneous dynamic equilibrium equation through macroscopic displacement / potential expression;

[0014] S4: Apply open or short circuit electrical boundary conditions, and use the formula structure of gradient elasticity theory to process a single homogeneous dynamic equilibrium equation, accurately predict the band gap width and position, and derive the spatiotemporal nonlocal dynamic homogeneous equilibrium equation;

[0015] S5: Considering the propagation form of steady-state harmonics, the dispersion function of the piezoelectric composite material can be obtained by substituting the above formula into the spatiotemporal nonlocal dynamic homogeneous balance equation in the time domain;

[0016] By analyzing the periodic dispersion curve, the starting frequency, cutoff frequency and band gap width of the first band gap of the piezoelectric composite material are obtained;

[0017] S6: Solving the initial boundary value problem by applying initial boundary conditions and then using the space-time nonlocal dynamic equilibrium equation in the Laplace domain obtained in step S4;

[0018] S7: All relevant contents in the Laplace domain are obtained, and the Laplace domain is converted into the time domain through fast Fourier transform to obtain the transient wave propagation in the one-dimensional piezoelectric composite material.

[0019] Preferably, in step S1, the asymptotic expansion of displacement and potential is substituted into the control equation of the piezoelectric material, so as to obtain the dynamic equilibrium equation of each order.

[0020] Preferably, in step S2, when recursively processing each order dynamic equilibrium equation, a control differential equation about the microscopic function is obtained at the same time, and the microscopic function is solved by integrating the above control differential equation and applying periodic boundary and interface conditions.

[0021] Preferably, in step S5, by selecting different electrical boundary conditions, active regulation of the band gap width of the smart piezoelectric composite material is achieved, and the band gap prediction results of the piezoelectric composite material under different electrical boundary conditions are obtained.

[0022] Preferably, based on the initial boundary conditions and the single homogeneous dynamic equilibrium equation in step S3, transient displacement and potential expressions containing unknown coefficients are obtained;

[0023] By processing the derived expressions of the low-order model, the high-order boundary conditions are obtained;

[0024] By using high-order boundary conditions, the unknown coefficients of transient displacement and potential expressions are solved, and finally the transient displacement and potential expressions are obtained.

[0025] Compared with the prior art, the present invention has the following beneficial effects:

[0026] 1. The present invention proposes a non-local dynamic homogenization model of a piezoelectric periodic composite material with an elastic double-layer structure, and adopts a spatial multi-scale homogenization method based on the eighth-order displacement and potential asymptotic expansion. By processing the momentum balance equations of each order of the piezoelectric composite material, a spatiotemporal non-local momentum balance equation with the same structure as the gradient elastic model is established. Unlike the traditional gradient elastic model that requires the identification of model parameters related to the length scale, the model parameters of the present invention are completely determined by the properties of the material and the geometry of the microstructure, and are accurately determined during the homogenization process. This not only makes the model simpler and more universal, but also reduces the dependence on experimental measurement parameters, significantly improving the applicability of the model.

[0027] 2. The present invention can effectively predict the wave propagation and attenuation behavior in piezoelectric composites by establishing a spatiotemporal nonlocal momentum balance equation. Compared with traditional numerical methods, the present invention exhibits higher computational efficiency when the complexity of the material microstructure increases. Therefore, especially when dealing with piezoelectric composites with complex microstructures, the present invention provides an efficient and accurate analysis tool that can be widely used in seismic isolation design and optimization design of acoustic devices.

[0028] 3. The present invention also studies the regulation effect of external circuits on the band gap of piezoelectric composite materials under open circuit and short circuit electrical boundary conditions. This study provides a theoretical basis and technical means for further optimizing the acoustic performance and seismic isolation effect of piezoelectric materials in practical applications, greatly expanding the application potential of such materials in the engineering field. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 is a calculation flow chart of the dynamic homogenization method of the present invention;

[0030] Figure 2Macro- and microstructures of inhomogeneous piezoelectric composites under open-circuit and short-circuit conditions;

[0031] Figure 3 Dispersion curve of piezoelectric composite composed of PZT-7A / Al;

[0032] Figure 4 Dispersion curves of piezoelectric composites under different electrical boundary conditions of open circuit and short circuit;

[0033] Figure 5 Historical displacement and historical potential at different external load frequencies;

[0034] Figure 6 Transmission amplitude at different repeating unit cell numbers. DETAILED DESCRIPTION

[0035] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0036] See also Figure 1 The present invention provides a method for dynamically homogenizing elastic band gap and initial-boundary value problems of piezoelectric composite materials, comprising the following steps:

[0037] Step 1: Consider Figure 2 One-dimensional inhomogeneous piezoelectric composite with periodic layered microstructure shown. Figure 2 The repeating unit in (b) is composed of two phase materials, Θ1 and Θ2, where the volume fraction of Θ1 is represented by α. The left end of the structure is fixed, and a dynamic displacement load is applied to the right end. In order to study the active control characteristics of piezoelectric composites, we set different boundary conditions for the piezoelectric material, namely open circuit and short circuit, such as Figure 2 (a) and Figure 2 (c) In this example, the two-phase materials are PZT-7A piezoelectric material as Θ1 and Al elastic material as Θ2, and the volume fraction of Θ1 is 0.5. The properties of the two-phase materials are shown in Table 1. In addition, in order to study the initial boundary value problem, a sinusoidal load is applied to the right end of the structure. Where U0 is the amplitude and f0 is the external load frequency. In order to avoid wave reflection at the fixed end during the observation time, we ensure that the structure length is large enough, set to L = 0.5m, and the number of repeating unit cells is Nb = 50.

[0038] Table 1 Material parameters of PZT-7A and Al

[0039]

[0040] This method does not directly solve the dynamic response problem of piezoelectric composites, but develops an effective dynamic homogenization method within the framework of multiscale analysis. To this end, two coordinates are introduced, namely x for the macroscale coordinate and y for the microscale coordinate. These two coordinates are linked by a small scale parameter δ, i.e. y = x / δ, where 0 < δ < 1 is the ratio of the characteristic size l of the microstructure to the traveling wavelength λ, i.e. δ = l / λ. Therefore, the following chain rule applies to the spatial derivative:

[0041]

[0042] In the framework of multiscale analysis, we consider the displacement and potential The asymptotic expansion in the Laplace domain is as follows:

[0043]

[0044] Where u i and φ i are the i-th order displacement and electric potential respectively. The corresponding periodic boundary conditions of displacement, stress, electric potential and electric displacement field in the asymptotic expansion form are:

[0045]

[0046] Where l = L / δ, L is the length of the structure. The continuity conditions of displacement, stress, potential and electric displacement field at the interface of two-phase materials are as follows:

[0047]

[0048] From the above periodic boundary conditions and interface conditions, it can be observed that the satisfaction of the periodic or interface conditions requires that the corresponding expansion terms of the same order are equal:

[0049] Periodic continuity conditions:

[0050] Interface continuity conditions:

[0051] The corresponding initial boundary conditions of the structure are:

[0052]

[0053] Substitute the asymptotic expansion of displacement and potential into the dynamic equilibrium equation of the piezoelectric material, collect the terms of equal order, and obtain the dynamic equilibrium equation of each order:

[0054]

[0055]

[0056]

[0057] Where c represents the elastic modulus, e represents the piezoelectric constant, k represents the dielectric constant, and ρ represents the material density;

[0058] Step 2: By recursively processing the dynamic equilibrium equations of each order obtained in step 1, the expressions of displacement and potential of each order containing unknown microscopic functions can be obtained:

[0059]

[0060] In the formula represents the unknown microscopic function in each order of displacement and potential expression. Where k is an integer and should satisfy (2k+1) <i. Appears in 3rd order displacement, Appears in 5th order displacement, appears in the 7th-order displacement. In addition, when recursively processing each order of dynamic equilibrium equation, the governing differential equation of the microscopic function can be obtained at the same time. The governing differential equation is as follows:

[0061]

[0062] in and

[0063]

[0064] about The governing differential equation is as follows:

[0065]

[0066] where i≥(2k-1), k=1,2,3,

[0067] The solution of the microscopic function can be obtained by integrating the above governing differential equations and applying periodic boundary and interface conditions. In addition, in order to ensure the uniqueness of the microscopic function, the following normality conditions need to be applied:

[0068]

[0069] Where <.> is a MacCauley bracket, which represents the spatial average value of the microstructure:

[0070]

[0071] Step 3: By applying the average operator to each order dynamic equilibrium equation obtained in step 1, each order homogeneous dynamic equilibrium equation is obtained:

[0072]

[0073]

[0074]

[0075]

[0076]

[0077]

[0078] In the formula Represents the uniform effective modulus of each order. Approximate expression for macroscopic displacement / potential:

[0079]

[0080] By considering the above equation, each order of homogeneous dynamic balance equation can be combined into a single homogeneous dynamic balance equation:

[0081]

[0082] Step 4: As with standard piezoelectric media, further derivations depend on the form of the electric field. Here we set two forms: (i) For open circuit, (ii) For short circuit. In addition, it is worth mentioning that when the piezoelectric material is in a short circuit state, its performance is no different from that of ordinary elastic materials. To obtain the elastic wave propagation of the piezoelectric composite material under the short circuit state, just set the dielectric constant and piezoelectric constant under the open circuit to 0, and you can get the elastic wave propagation results under the short circuit electrical boundary conditions. Therefore, the subsequent derivation will mainly be analyzed based on the open circuit electrical boundary conditions.

[0083] In order to more accurately predict the band gap width and position, we processed the single homogeneous dynamic equilibrium equation by using the formula structure of the gradient elastic theory. The formula structure of the gradient elastic theory is expressed as:

[0084]

[0085] In the formula represents uniform wave velocity, (l1, l2, l3) represents length scale parameters; according to the above structure, the time-space non-local dynamic homogeneous equilibrium equation can be derived as follows:

[0086]

[0087] Where v is a high-order correction coefficient. The spatiotemporal nonlocal dynamic homogeneous balance equation in the corresponding time domain is expressed as:

[0088]

[0089] Unlike the gradient elastic model where the length scale parameters need to be calibrated, the parameters of the nonlocal dynamic homogeneous model are completely determined by the homogenization process.

[0090] Step 5: In order to obtain the dispersion relation of the piezoelectric composite material, it is necessary to consider the propagation form of the steady-state harmonics:

[0091] U(x,t)=U0e i(ωt-kx)

[0092] Where U0 is the displacement amplitude, k is the Bloch wave vector defined in the Brillouin zone, and ω is the angular frequency of the wave. By substituting the above equation into the spatiotemporal nonlocal dynamic homogeneous equilibrium equation in the time domain, the dispersion function of the piezoelectric composite material can be obtained:

[0093] A k k 4 +B k k 2 +C k =0

[0094] In the formula

[0095] By analyzing the periodic dispersion curve, the starting frequency, cutoff frequency and band gap width of the first band gap of the piezoelectric composite material are obtained. In addition, by selecting different electrical boundary conditions, the active regulation of the band gap width of the smart piezoelectric composite material can be achieved.

[0096] Figure 3 and Figure 4 The band gap prediction results of piezoelectric composites under different electrical boundary conditions obtained according to the above steps are given and will be explained below.

[0097] In order to evaluate the ability of the dynamic homogenization method (DH) in predicting elastic wave dispersion, Figure 3 The comparison of the theoretical predictions of the dispersion relation between DH and the plane wave expansion method (PWE) under open circuit conditions is shown. Figure 3 In , the x-axis and y-axis represent the normalized wave number and frequency, respectively. Figure 3 In the above figure, the two physical wavenumber solutions k1 and k2 of DH are consistent with the wavenumber solutions of PWE in the first and second passbands. Figure 3The nonphysical solution for negative group velocity is shown, but these branches of the dispersion curve do not actually exist. Therefore, DH is able to accurately capture the dispersion characteristics of the first and second passbands and the start and end of the first band gap. In addition, when the normalized wave number reaches 1, the second band gap begins to appear, but the dispersion curves predicted by the DH method gradually deviate from the reference solution. This deviation is mainly due to the fact that the scale separation assumption of dynamic homogenization no longer holds when the scale ratio reaches 1.

[0098] The band gap width of piezoelectric composites is not only affected by the elastic impedance ratio of the constituent materials, but also significantly restricted by the electrical boundary conditions. By properly selecting the electrical boundary conditions, the band gap can be actively adjusted. Figure 4 The prediction of the band gap of periodic piezoelectric composites by the DH model under open-circuit and short-circuit electrical boundary conditions is demonstrated. Figure 4 It can be seen that the band gap frequency ranges under open circuit and short circuit electrical boundary conditions are f0∈[167,311]kHz and f0∈[165,292]kHz, respectively, and the corresponding band gap widths are 144 and 127, respectively, indicating that the band gap width can be increased by 13.3% by changing the electrical boundary conditions of the piezoelectric composite material. This feature provides an effective tool for actively controlling the elastic wave band of phononic crystals.

[0099] Step 6: Solve the initial boundary value problem by using the space-time nonlocal dynamic equilibrium equation in the Laplace domain obtained in step 4, and then use the inverse Laplace factor transformation to obtain the time domain solution. According to the boundary conditions and the single homogeneous dynamic equilibrium equation in step 3, the transient displacement and potential expressions can be expressed as:

[0100]

[0101] Where K1 and K2 have the following form:

[0102]

[0103] By substituting the transient displacement and potential expressions into DH, expressions for ξ and η are obtained:

[0104]

[0105] in Since DH involves fourth-order derivatives, the existing initial boundary conditions are not sufficient, so we need to construct higher-order boundary conditions. We obtain higher-order boundary conditions by processing the expressions derived from the low-order model:

[0106]

[0107] In the formula Using the above high-order boundary conditions, we can get the expressions of C(s) and D(s) in the transient displacement and potential expressions:

[0108]

[0109] Step 7: The above process obtains all relevant contents in the Laplace domain, which can be converted to the time domain by the inverse Laplace transform. The response field is converted to the time domain using the fast Fourier transform. The inverse Laplace transform is defined as:

[0110]

[0111] The basic assumption is |f(t)|≤Ke γt , K represents a positive real number, and γ represents a real function. The numerical inverse Laplace transform is calculated using the N-term truncation method, F N The transformed function value is F(s), where N represents the dimension of the vector:

[0112]

[0113] in represents the Hadamard product of matrices; E{·} represents the operator of the ε-algorithm. is the M-unit constant vector of value c, which can be calculated as:

[0114]

[0115] Ω=π(1-1 / M) / (M-1)T s

[0116] Where M = N / 2, T s is the original sampling period, E r represents the expected relative error, and:

[0117]

[0118] F N (n)=F(c-inΩ),n=0,1,…,N-1

[0119] By using the inverse fast Fourier transform, the initial boundary value problem in the frequency domain can be converted to the time domain, thereby obtaining the transient wave propagation in the one-dimensional piezoelectric composite material. This method effectively solves the initial boundary value problem, making it more feasible and efficient to solve the wave problem in piezoelectric composite materials.

[0120] Figures 5 and 6 The initial boundary value problem analysis of the piezoelectric composite material obtained according to the above steps is given and will be explained below.

[0121] To evaluate the accuracy of DH for initial-boundary value problems, we compare it with the direct numerical method (DNS) for sinusoidal loads. The predicted transient response is compared below. Figure 5 The historical displacement and electric potential of the structure at x = 0.9L are shown under sinusoidal loads with external load frequencies of 140kHz and 250kHz respectively. Figure 5 It is shown that at f0 = 140kHz, the response falls within the first dispersion passband (see Figure 3 ), so there is no wave attenuation. In addition, the displacement histories and potentials predicted by DH and DNS are in good agreement. At f0 = 250 kHz, the transient response falls within the first dispersion band gap, resulting in significant wave attenuation in the displacement histories and potentials predicted by DH and DNS.

[0122] In addition to analyzing the computational efficiency of the DH method, we also analyzed the computational efficiency of DH and its compliance with the repeating unit cell (RUC) quantity. Figure 6 The maximum transmitted displacement amplitude at x = 0.9L for a sinusoidal load within a certain frequency range is shown. Figure 6 It can be seen that for Nb = 5, the starting and ending positions of the first band gap predicted by DH and DNS are the same, but the wave attenuation amplitudes in the band gap are slightly different for DH and DNS. However, when Nb = 20, the two methods can reach a consensus on the prediction of the band gap width and attenuation amplitude of the first band gap. This shows that if only the position and width of the first band gap need to be predicted, the DH method with Nb = 5 is sufficient; when Nb = 20, DH can accurately predict the position of the first band gap and the attenuation degree of the piezoelectric composite material.

[0123] Table 2 compares the computational time of the DH and DNS methods for the initial boundary value problem with different numbers of RUCs. These tests were performed on a PC running Windows 7 Ultimate 64-bit operating system with 16GB memory and Intel(R) Core(TM) i5-3320M CPU @ 2.6GHz. As can be seen from Table 2, the DNH method is able to quickly predict transient wave propagation in materials when the number of RUCs is small (such as Nb = 5 and 20). However, the efficiency of the DNH method decreases significantly with the increase in the number of RUCs. In contrast, the computational efficiency of DH is lower than that of DNS when Nb = 5 and 20, but gradually surpasses DNS as the number of RUCs increases. When Nb = 80, the computational efficiency of DH is almost an order of magnitude higher than that of DNS. It can be expected that the computational efficiency of DH will further outperform DNS as the number of RUCs further increases. The reason is that DNS requires fine meshing of each microstructure, while the dynamic homogenization method works in a multi-scale framework, providing an effective homogenous medium for heterogeneous materials and avoiding fine meshing of structures. Dynamic homogenization improves computational efficiency and facilitates rapid understanding of wave propagation in piezoelectric composites and accelerates the design of complex microstructures of phononic crystals, which is why we developed this model.

[0124] Table 2 compares the computation time of DNS and DH under different RUC numbers.

[0125]

[0126] Although embodiments of the present invention have been shown and described, for those skilled in the art, these descriptions are merely to illustrate the principles and implementation methods using specific examples, and are not intended to limit the application of the present invention. It is understood that various changes, modifications, substitutions and variations may be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A dynamic homogenization method for elastic wave band gap and initial boundary value problem of piezoelectric composite materials, characterized in that: The method comprises the following steps: S1: expanding the displacement and potential asymptotically in the Laplace domain within the framework of multi-scale analysis; The expressions of periodic boundary conditions and continuity conditions of each order are obtained through the periodic boundary conditions of displacement, stress, potential and electric displacement field in asymptotic expansion form and the continuity conditions of displacement, stress, potential and electric displacement field at the interface of two-phase materials; Substitute the asymptotic expansion of displacement and potential into the dynamic equilibrium equation of the piezoelectric material, collect the terms of equal order, and obtain the dynamic equilibrium equation of each order; S2: recursively process the dynamic equilibrium equations of each order obtained in step S1 to obtain the expressions of displacement and potential of each order containing unknown microscopic functions; S3: Obtaining each order homogeneous dynamic equilibrium equation by applying an average operator to each order dynamic equilibrium equation obtained in step S1; The homogeneous dynamic equilibrium equations of each order are combined into a single homogeneous dynamic equilibrium equation through macroscopic displacement / potential expression; S4: Apply open or short circuit electrical boundary conditions, and use the formula structure of gradient elasticity theory to process a single homogeneous dynamic equilibrium equation, accurately predict the band gap width and position, and derive the spatiotemporal nonlocal dynamic homogeneous equilibrium equation; S5: Considering the propagation form of steady-state harmonics, the dispersion function of the piezoelectric composite material can be obtained by substituting the above formula into the spatiotemporal nonlocal dynamic homogeneous balance equation in the time domain; By analyzing the periodic dispersion curve, the starting frequency, cutoff frequency and band gap width of the first band gap of the piezoelectric composite material are obtained; S6: Solving the initial boundary value problem by applying initial boundary conditions and then using the space-time nonlocal dynamic equilibrium equation in the Laplace domain obtained in step S4; S7: All relevant contents in the Laplace domain are obtained, and the Laplace domain is converted into the time domain through fast Fourier transform to obtain the transient wave propagation in the one-dimensional piezoelectric composite material.

2. The method for dynamic homogenization of elastic band gap and initial boundary value problem of piezoelectric composite material according to claim 1, characterized in that: In step S1, the asymptotic expansion of displacement and potential is substituted into the control equation of the piezoelectric material to obtain the dynamic equilibrium equation of each order.

3. The method for dynamic homogenization of elastic band gap and initial boundary value problem of piezoelectric composite material according to claim 1, characterized in that: In step S2, when recursively processing each order of dynamic equilibrium equation, the control differential equation about the microscopic function is obtained at the same time. The microscopic function is solved by integrating the above control differential equation and applying periodic boundary and interface conditions.

4. The method for dynamic homogenization of elastic band gap and initial boundary value problem of piezoelectric composite material according to claim 1, characterized in that: In step S5, by selecting different electrical boundary conditions, active regulation of the band gap width of the smart piezoelectric composite material is achieved, and the band gap prediction results of the piezoelectric composite material under different electrical boundary conditions are obtained.

5. The method for dynamic homogenization of elastic band gap and initial boundary value problem of piezoelectric composite material according to claim 1, characterized in that: According to the initial boundary conditions and the single homogeneous dynamic equilibrium equation in step S3, the transient displacement and potential expressions containing unknown coefficients are obtained; By processing the derived expressions of the low-order model, the high-order boundary conditions are obtained; By using high-order boundary conditions, the unknown coefficients of transient displacement and potential expressions are solved, and finally the transient displacement and potential expressions are obtained.

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