A method for predicting the dynamic effective properties of piezoelectric composites

Abstract

The present application relates to a kind of prediction method of the dynamic effective property of piezoelectric composite based on the effective medium theory of mesoscopic mechanics.These dynamic effective properties refer to the effective material parameters and its combination parameters that piezoelectric composite is changed with frequency and is shown in macroscopically under the action of elastic wave.The prediction method described in the present application includes the following five steps: step one: obtaining the material parameters of each component phase;Step two: preparation or obtain piezoelectric composite sample;Step three: dynamic effective property prediction model-self-consistent equation set is established;Step four: calculate the dynamic effective parameter of composite material;Step five: draw the prediction curve of dynamic effective property and the verification of prediction model.The present application focuses on the influence of the alignment degree, radius, slenderness ratio, volume fraction, frequency and the like on the macroscopic effective performance of piezoelectric composite, and establishes a prediction model.The model is in good agreement with experimental results.In efficiency and economy, there are obvious advantages, which can solve the long cycle and high cost of piezoelectric composite dynamic characteristic test.

Description

Technical Field

[0001] The present invention relates to the technical field of dynamic homogenization methods for piezoelectric composite materials, and in particular to a method for predicting the dynamic effective properties of piezoelectric composite materials. Background Art

[0002] Piezoelectric materials can achieve the mutual conversion of mechanical energy and electrical energy on a macroscopic scale. This effect is called the piezoelectric effect, which is divided into the direct piezoelectric effect and the inverse piezoelectric effect. The direct piezoelectric effect converts mechanical quantities (deformation or force) into electrical signals; the inverse piezoelectric effect converts electrical signals into mechanical quantities (deformation or force). Based on the direct and inverse piezoelectric effects, piezoelectric materials are widely used in various fields. With the continuous breakthrough of technology, higher requirements are also placed on the performance of piezoelectric materials. Conventional piezoelectric monomer materials, such as piezoelectric single crystals and piezoelectric ceramics, can no longer meet the needs.

[0003] Piezoelectric composite materials are generally composed of piezoelectric inclusion phase and polymer matrix phase (see Appendix Figure 1 The piezoelectric composite material (which may or may not possess piezoelectricity) is composed of a specific connectivity pattern, volume or weight ratio, and spatial geometric distribution. This information constitutes the microstructure of the piezoelectric composite material and plays a crucial role in its macroscopic effective properties. The piezoelectric phase is used to produce the piezoelectric effect, while the polymer phase is mainly used to reduce the material's density and permittivity and increase its elastic compliance constant. This design can exponentially improve the material's macroscopic piezoelectric performance and also possess certain excellent properties that piezoelectric monomers do not have.

[0004] Based on classical elastic micromechanics theory, a micromechanical model of static piezoelectric composites has been established. This model predicts the static macroscopic effective properties of piezoelectric composites based on microstructural information, thereby optimizing the composite's macroscopic behavior through rational microstructural design. Under the action of elastic waves, the dynamic macroscopic effective properties of piezoelectric composites are no longer constant, but instead vary depending on the frequency. Currently, there are few theoretical methods for predicting the dynamic effective properties of piezoelectric composites, which to some extent restricts their widespread application as elastic wave control materials.

[0005] [1]Yue YP,Wan YP,Zhong Z.,The non-reciprocal wave propagation inrandom piezoelectric composites with aligned semi-spheroid inclusions.AppliedMathematical Modelling,2023,119,85-98

[0006] [2] Yue Y.P., Wan Y.P., Theoretical study on dynamic effective electroelastic properties of random piezoelectric composites with aligned inhomogeneities. Applied Mathematics and Mechanics (English Edition), 2023, 44(4), 525-546

[0007] The above two papers 1-2 are the papers published by the inventors in the early stage, which give the prediction method of piezoelectric composites containing aligned ellipsoidal inclusions and semi-ellipsoidal inclusions. However, the above research does not consider the influence of the distribution of the inclusion direction on the macroscopic dynamic effective properties of the piezoelectric composite.

[0008] In addition, the experimental test of the dynamic effective properties of the piezoelectric composite is also recorded in the related literature, such as:

[0009] [3] Gururaja T.R., Cross L.E., et al., Piezoelectric composite materials for ultrasonic transducer applications. Part II: Evaluation of ultrasonic medical applications. IEEE Transactions on Sonics and Ultrasonics, 1985, 32(4): 499-513

[0010] [4] Grewe M.G., Gururaja T.R., et al., Acoustic properties of particle / polymer composites for ultrasonic transducer backing applications. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1990, 37(6): 506-514

[0011] [5] Kinra V.K., Dispersive wave propagation in random particulate composites. In: Recent Advances in Composites in the United States and Japan, ASTM STP 864, American Society for Testing and Materials, Philadelphia, 1985, 309-325

[0012] The literatures 3-5 have the problems of great difficulty, long period and high cost in experimental testing of dynamic effective properties of piezoelectric composites. This is because the piezoelectric composites often work in a frequency range, which requires the designed piezoelectric composites to exhibit optimal dynamic effective properties in the working frequency range. In fact, there are many parameters affecting the dynamic effective properties of piezoelectric composites, including the material properties of the inclusion phase, the material properties of the matrix phase, the inclusion volume fraction, the inclusion cross-sectional coefficient, the inclusion direction, the inclusion alignment degree, the inclusion size, and these parameters jointly affect the macroscopic effective properties of the composites with frequency. If the optimal effective properties are found by experimental means, many parameters need to be controlled, which requires the preparation of a large number of samples for testing; in addition, the working frequency is a range, and multiple tests need to be done in the frequency range for each sample, so this traditional experimental method has the problems of great difficulty, long period and high cost. SUMMARY

[0013] The present application aims at the deficiencies of the prior art; a dynamic effective property prediction method of piezoelectric composites is established.

[0014] The comprehensive influence effect of the microstructure information of the inclusions in the composites and the frequency on the macroscopic dynamic effective properties will be mainly considered in the present application. These comprehensive influence factors include the directional angle distribution of the inclusions, the cross-sectional coefficient of the inclusions, the radius of the inclusions, the volume fraction of the inclusions, the relative relationship between the size of the inclusions and the wavelength, etc. They will jointly affect the macroscopic performance of the piezoelectric composites. The present application solves the problem of the comprehensive influence effect mechanism of the microstructure information and the frequency which is difficult to fully explore in the research method of the macroscopic dynamic effective performance of piezoelectric composites in the prior art.

[0015] The technical scheme of the present application: a dynamic effective property prediction method of piezoelectric composites, the piezoelectric composites containing inclusion materials and matrix materials, characterized in that: the prediction method comprises the following five steps;

[0016] Step one: determine the inclusion material and matrix material, and obtain the material parameters of each component phase

[0017] The piezoelectric tensor, dielectric tensor, elastic tensor and density of the inclusion material and the piezoelectric tensor, dielectric tensor, elastic tensor and density of the matrix material are obtained respectively; if a phase material does not have piezoelectricity, only its elastic tensor and density need to be obtained;

[0018] Step two: determine the inclusion material and matrix material, and prepare or obtain the piezoelectric composite material sample

[0019] According to the preparation or obtaining of the piezoelectric composite material sample, the experimental data parameters are obtained; the experimental data parameters include the inclusion phase volume fraction c n , the alignment degree parameter λ of the inclusion, the cross section coefficient δ of the inclusion, and the inclusion radius a; and then the effective phase velocity and effective acoustic impedance under the action of elastic wave are obtained;

[0020] Step three: dynamic effective property prediction model-self-consistent equation set

[0021] A self-consistent equation set for the dynamic effective parameters of piezoelectric composites is established based on the micromechanics effective medium theory;

[0022] Step four: calculation of the dynamic effective properties of the composite material

[0023] The material parameters of the inclusion phase and the matrix phase obtained in step one and the experimental data parameters obtained in step two are substituted into the prediction model, i.e. the self-consistent equation set established in step three, and the dynamic effective material parameters are obtained through numerical calculation;

[0024] Then the effective elastic wave propagation characteristics are calculated; and the complete prediction model is obtained;

[0025] Step five: drawing of the prediction curve of the dynamic effective properties and verification of the prediction model

[0026] Different frequencies are substituted into the established model to obtain the complete prediction curve of the dynamic effective properties of the piezoelectric composite material varying with frequency, the prediction curve is compared with the effective phase velocity or effective acoustic impedance under the action of elastic wave obtained in step two, and the prediction model is verified;

[0027] The steps one and two are not in any particular order.

[0028] Preferably, the material parameters of each component phase obtained in step one are carried out according to the following sub-steps:

[0029] 1.1 The fourth-order elastic tensor of the inclusion phase and the matrix phase is C x , wherein the superscript x=n represents the parameter of the inclusion phase, and the superscript x=m represents the parameter of the matrix phase; C xThere are 5 independent non-zero components, according to Voigt's rule, which are C 11 , C 12 , C 13 , C 33 , C 44 , which represent the elastic modulus in different directions, obtained by measurement or table lookup;

[0030] 1.2 The third-order piezoelectric tensor of the inclusion phase and the matrix phase is e x , where the superscript x = n represents the parameter of the inclusion phase, and the superscript x = m represents the parameter of the matrix phase; e x There are 3 independent non-zero components, according to Voigt's rule, which are e 15 , e 31 , e 33 , which represent the piezoelectric coefficient in different directions, obtained by measurement or table lookup; if a phase material does not have piezoelectricity, its piezoelectric coefficient is zero;

[0031] 1.3 The second-order dielectric tensor of the inclusion phase and the matrix phase is κ x , where the superscript x = n represents the parameter of the inclusion phase, and the superscript x = m represents the parameter of the matrix phase; κ x There are 2 independent non-zero components, according to Voigt's rule, which are κ 11 , κ 33 , which represent the dielectric constant in different directions, obtained by measurement or table lookup;

[0032] 1.4 The second-order dielectric density tensor of the inclusion phase and the matrix phase is ρ x , where the superscript x = n represents the parameter of the inclusion phase, and the superscript x = m represents the parameter of the matrix phase; ρ x is isotropic, and there is only one independent non-zero component ρ, which represents the static density of the inclusion phase or the matrix phase, obtained by measurement or table lookup.

[0033] Preferably, the experimental data parameters in step two are obtained by measurement or by referring to prior art materials, and the effective phase velocity and effective acoustic impedance under the action of elastic waves in step two are obtained by measurement or by referring to prior art materials.

[0034] Preferably, step two includes the following sub-steps:

[0035] 2.1 Prepare a piezoelectric composite material sample, and the volume fraction of the inclusion phase is c n , and the volume fraction of the matrix phase c m = 1-c n ;

[0036] If the piezoelectric composite material sample is obtained, the volume fraction of the inclusion phase c n of the existing piezoelectric composite material is obtained by referring to prior art materials;

[0037] 2.2 Measure the radius a and cross-section coefficient δ of the inclusions from the SEM image of the sample;

[0038] If the sample of the piezoelectric composite is obtained, the radius a and cross-section coefficient δ of the inclusions are obtained by referring to the prior art;

[0039] 2.3 Measure the probability of the inclusion orientation angle α taking different values from the SEM image of the sample, and obtain the parameter λ of the alignment of the inclusions from the following formula: λ (α) by back-calculation;

[0040]

[0041] If the sample of the piezoelectric composite is obtained, the parameter λ of the alignment of the inclusions is obtained by referring to the prior art;

[0042] 2.4 Measure the effective wave propagation characteristics of the sample of the piezoelectric composite under the action of elastic waves, including at least the effective phase velocity and effective acoustic impedance;

[0043] If the sample of the piezoelectric composite is obtained, the macroscopic effective wave propagation characteristics of the sample are obtained by referring to the prior art.

[0044] Preferably, the dynamic effective property prediction model-self-consistent equation set in step three is established according to the following sub-steps:

[0045] 3.1 In step three, the prediction model of the dynamic effective properties of the piezoelectric composite containing inclusions in any direction, i.e., the self-consistent equation set of the macroscopic dynamic effective parameters, is established; it is divided into 7 parts;

[0046] In 3.2, the microstructure information parameters of the piezoelectric composite are set;

[0047] In 3.3, the coordinate system conversion rules of the tensors are set;

[0048] In 3.4, the expression of the effective phase velocity of the piezoelectric composite is determined;

[0049] In 3.5, the expression of the structure coefficient h is determined;

[0050] In 3.6, the expression of the volume average tensor of the five effective Green's operators of the piezoelectric composite is determined;

[0051] In 3.7, the expression of the four combined parameter tensors in the self-consistent equation set is determined;

[0052] In 3.8, the self-consistent equations for predicting the dynamic effective properties of piezoelectric composites are established;

[0053] 3.2 Setting the microstructure information parameters of piezoelectric composites

[0054] In the global coordinate system o-x1x2x3, the piezoelectric composite macroscopically exhibits transverse isotropy, and the symmetry axis is the x3 axis; the piezoelectric composite is composed of a matrix phase and an inclusion phase, wherein the inclusion phase is the dispersed phase, and the volume fraction is c n , the matrix phase is the continuous phase, and the volume fraction is c m ; c n and c m are given in step two during the preparation of the piezoelectric composite; which can be expressed as

[0055] c m +c n =1 (1)

[0056] The position of the inclusion in the matrix is randomly distributed, but not overlapped; the shape of the inclusion is a rotating ellipsoid, and the rotation axis is the X3 axis, in the local coordinate system, the rotating ellipsoid equation is

[0057]

[0058] Where a is the inclusion radius, δ is the inclusion cross-section coefficient, which is obtained in step two; δ>1 can simulate elongated inclusions, δ=1 can simulate spherical inclusions, and δ<1 can simulate oblate spherical inclusions;

[0059] The direction of the inclusion in the matrix is randomly distributed, and the direction angle of the rotation axis X3 in the global coordinate system o-x1x2x3 is (α, β), wherein α is the angle between X3 axis and x3 axis, and β is the angle between the projection of X3 axis on o-x1x2 plane and x1 axis; since the composite is isotropic in the o-x1x2 plane, the direction angle β has no effect on the macroscopic properties; the probability distribution function of the inclusion direction angle α is expressed as

[0060]

[0061] Where λ and α are obtained in step two; λ=0 represents that the inclusion is completely randomly distributed; λ>0 represents that the inclusion begins to exhibit a certain directional preference; the larger the value of λ, the more obvious the directional preference of the inclusion, that is, the higher the alignment degree; λ→∞ represents that the rotation axis of the inclusion is completely aligned with the x3 axis;

[0062] 3.3 Setting the coordinate system conversion rule of the tensor

[0063] The tensor expressed in the local coordinate system needs to use the conversion tensor Q when converted to the global coordinate system

[0064]

[0065] where α and β are the directional angles of the inclusion in the global coordinate system, and without calculating the specific values of α and β, they will be integrated out in the following operations; if the fourth order tensor F, the third order tensor T, and the second order tensor S in the local coordinate system are assumed to be F', T', S' in the global coordinate system, then there are the following transformation relations between their components

[0066] F' ijkl = Q pi Q qj Q rk Q sl F pqrs (5)

[0067] T' ijk = Q pi Q qj Q rk T pqr (6)

[0068] S' ij = Q pi Q qj S pq (7)

[0069] All the subscript letters in the formulas 5-7 can take 1, 2, 3, and the Einstein summation convention is performed;

[0070] 3.4 Determining the expression of the effective phase velocity of the piezoelectric composite material

[0071] The effective phase velocity of the piezoelectric composite material to be solved is denoted by v N , and when the subscript N = 1, it is a quasi-P wave, when the subscript N = 2, it is a quasi-SR wave, and when the subscript N = 3, it is a quasi-SP wave; N is selected according to the type of the elastic wave used in step two;

[0072]

[0073] where the following combined parameters are used:

[0074]

[0075]

[0076] The effective phase velocity to be solved given by the formulas (8), (9) and (10) is a function of θ, C 0 , e 0 , κ 0 and ρ 0 ; θ is the effective phase velocity obtained in step two, the angle between the direction of the elastic wave propagation and the x3 axis; wherein the effective elastic tensor C 0The non-zero components of the effective piezoelectric tensor e and The non-zero components of the effective dielectric tensor κ 0 are and The non-zero components of the effective density tensor p 0 are and The non-zero components of the effective density tensor p 0 are and These unknowns are solved by the self-consistent equations of sub-step 3.8.

[0077] 3.5 Determination of the expression of the structural coefficient h

[0078] The structural coefficient h will be used in the following calculations, its expression is

[0079]

[0080] The expression of the parameter b is as follows:

[0081]

[0082] where a is the inclusion radius, δ is the inclusion cross-section coefficient, which is obtained in step 2; θ is the angle between the wave propagation direction and the x3 axis, and φ is the angle between the projection of the wave propagation direction on the x1-x2 plane and the x1 axis, which are obtained in step 2; α and β are the directional angles of the inclusion in the global coordinate system, and the specific values of α and β do not need to be calculated; k is the wave number, whose expression is:

[0083]

[0084] ω = 2πf (30)

[0085] where ω is the circular frequency, f is the frequency, whose value is the frequency of the elastic wave in step 2; v N is the macroscopic effective phase velocity of the piezoelectric composite material to be solved, subscript N = 1 for quasi-P wave, subscript N = 2 for quasi-SR wave, and subscript N = 3 for quasi-SP wave; N is selected according to the type of the elastic wave used in step 2; v N The expression of v

[0086] 3.6 Determination of the expression of the volume average tensor of the five effective Green's operators of the piezoelectric composite material

[0087] The volume average tensors of the five effective Green's operators of the piezoelectric composite material will be used in the following calculations, the expressions of their components are

[0088]

[0089] where all the subscripts of the integrand take the values 1, 2, 3, and follow Einstein's summation convention; n 下标 is the component of the unit spherical vector n;

[0090] n = (sin θ cos φ, sin θ sin φ, cos θ) (36)

[0091] Here θ and φ do not need to take specific values; is the component of the polarization vector U N ; N takes the values 1, 2, 3;

[0092]

[0093]

[0094] where M1is given in equation (11); α 下标 is the component of the vector α

[0095]

[0096] The remaining parameters in equations (31)-(35) are as follows:

[0097]

[0098] where v N in equation (51) is given in equations (8)-(10); a in equations (50) and (53) is the inclusion radius, δ is the inclusion cross-sectional coefficient, and is obtained in Step 2; k N in equation (53) is given in equation (29); i in equation (54) is the imaginary unit;

[0099] From this substep, it is known that the volume-averaged tensors of the effective Green's operator given by equations (31)-(35) are functions of ω, C 0 , e 0 , κ 0 , and ρ 0 ; ω is the circular frequency of the elastic wave in Step 2; the non-zero components of the sought effective elastic tensor C 0 are and the non-zero components of the sought effective piezoelectric tensor e 0 are and the non-zero components of the sought effective dielectric tensor κ 0 are and the non-zero components of the sought effective density tensor ρ 0 are and These sought quantities will be obtained by solving the self-consistent set of equations in Substep 3.8;

[0100] 3.7 Determination of the expressions of the four combined parameter tensors in the self-consistent equations

[0101] In the self-consistent equations, four combined parameter tensor functions V (1) , V (2) , V (3) and V (4) will be used, whose expressions are:

[0102]

[0103] where the expression of the structure coefficient h is given in sub-step 3.5; the fourth order tensor V (1) , the third order tensor V (2) , the second order tensor V (3) and V (4) are transversely isotropic with the symmetry axis x3; the tensors C n′ , e n′ , κ n ' and p n′ represent the elastic tensor C n , the piezoelectric tensor e n , the dielectric tensor κ n and the density tensor p n of the inclusion phase in the local coordinate system, which are transformed to the global coordinate system according to the coordinate transformation rule in sub-step 3.3; while the elastic tensor C n , the piezoelectric tensor e n , the dielectric tensor κ n and the density tensor p n of the inclusion phase in the local coordinate system will be obtained through step one; the expressions of the tensors J1-J5 are:

[0104] J1 = B1:R + F1 · W T (59)

[0105] J2 = A1 · W T - F1 T :R (60)

[0106] J3 = B1:W - F1 · Z (61)

[0107] J4 = F1 T :W + A1 · Z (62)

[0108]

[0109]

[0110] where:

[0111]

[0112] A1= [(e n′ -e 0 ):(C n′ -C 0 ) -1 :(e n′ -e 0 ) T +(κ n′ -κ 0 )] -1 (72)

[0113] B1= (C n′ -C 0 ) -1 -(C n′ -C 0 ) -1 :(e n′ -e 0 ) T ·A1· (e n′ -e 0 ):(C n′ -C 0 ) -1 (73)

[0114] F1= (C n′ -C 0 ) -1 :(e n′ -e 0 ) T ·A1 (74)

[0115] F1 T =A1· (e n′ -e 0 ):(C n′ -C 0 ) -1 (75)

[0116] The operator symbols "·" and "∶" in equations (55)-(75) represent the single and double dot product of tensors, respectively; the superscripts "T" and "-1" represent the tensor transposition and tensor inversion, respectively; the tensor I 2nd represents the second order unit tensor; the five tensors and in equations (63) and (68)-(71) can be obtained from the tensors and given in sub-step 3.6 equations (31)-(35) by coordinate transformation from the local coordinate system to the global coordinate system according to the coordinate transformation rule given in sub-step 3.3; the tensors C 0 , e 0 , κ 0 and p0 Represent the dynamic effective material parameters of the piezoelectric composite material to be determined; the tensor C n′ , e n′ , κ n ′ and ρ n′ Represent the material properties C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , according to the coordinate transformation rules of sub-step 3.3, it is converted to the tensor in the global coordinate system; the material properties C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , obtained in step 1;

[0117] 3.8 Establishing a self-consistent set of equations for the dynamic effective properties of piezoelectric composites

[0118] Based on the effective medium theory of micromechanics and using the above expressions, a self-consistent set of equations for the dynamic effective material parameters of piezoelectric composites can be established in the global coordinate system:

[0119] C 0 =C m +c n <V (1) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (76)

[0120] e 0 =e m +c n <V (2) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (77)

[0121] κ 0 =κ m +c n <V (3) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (78)

[0122] ρ 0 =ρ m +c n <V (4) (C 0e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (79)

[0123] where C 0 , e 0 , κ 0 and p 0 represent the dynamic effective properties of the piezoelectric composite, i.e. the effective elastic tensor, the effective piezoelectric tensor, the effective dielectric tensor, the effective density tensor, respectively; C 0 has 5 independent non-zero unknown components, denoted by and e 0 has 3 independent non-zero unknown components, denoted by and κ 0 has 2 independent non-zero unknown components, denoted by and p 0 has 2 independent non-zero unknown components, denoted by and C m , e m , κ m and p m represent the elastic tensor, the piezoelectric tensor, the dielectric tensor and the density tensor of the matrix phase, which have been obtained in Step 1; c n represents the volume fraction of the inclusion phase, which has been obtained in Step 2;

[0124] 4 combined parameter tensor functions V (1) , V (2) , V (3) and V (4) are given in Sub-step 3.7; the angle brackets < > in equations (76)-(79) represent the average of the function V with respect to the direction angles, which are expressed as follows:

[0125]

[0126] where the function P λ (α) is given in Sub-step 3.2; the parameter λ has been obtained in Step 2;

[0127] The equations set composed of equations (76)-(79) is the final equations set to be solved; since both sides of the equations set contain the unknown dynamic effective parameters, the equations set is a self-consistent equations set, which is solved by the iteration method.

[0128] Preferably, the calculation and extraction of the dynamic effective parameters of the piezoelectric composite in step four is carried out according to the following sub-steps:

[0129] 4.1 Using the material parameters C n , e n , κ n and ρ n of the inclusion phase obtained in step one, the material parameters C m , e m , κ m and ρ m of the matrix phase, and the volume fraction c n of the inclusions obtained in step two, the radius a of the inclusions, the cross-section coefficient δ and the parameter λ of the alignment degree of the inclusions, are substituted into the self-consistent equations (76)-(79) in step three, combined with the propagation direction angles θ and φ of the applied elastic wave and the frequency f, and the dynamic effective material parameters C 0 , e 0 , κ 0 and ρ 0 of the piezoelectric composite at the specified frequency f are calculated by iterative method; in the first iteration, the to-be-solved dynamic effective properties C 0 , e 0 , κ 0 and ρ 0 of the piezoelectric composite involved in all functions on the right side of the equation can be taken as the corresponding parameters C m , e m , κ m and ρ m of the matrix phase, and then the temporary C 0 , e 0 , κ 0 and ρ 0 are obtained, which are continuously substituted into the right side of the equation for calculation, and the cycle is repeated in turn until convergence, and finally the predicted dynamic effective parameters C 0 , e 0 , κ 0 and ρ 0 of the piezoelectric composite are obtained.

[0130] 4.2 Calculate the effective wave propagation properties of the piezoelectric composite; substitute the effective parameters C 0 , e 0 , κ 0 and ρ 0 obtained in sub-step 4.1 into equations (8)-(10) to obtain the effective phase velocity; then substitute into equation (29), and according to the attenuation formula Im(k N *a), the attenuation is obtained; then the effective acoustic impedance is obtained according to the acoustic impedance formula ; all the effective properties that can be predicted by this prediction model have been obtained.

[0131] Preferably, the drawing of the prediction curve of the dynamic effective property in step five and the validation of the prediction model are carried out according to the following scheme:

[0132] 5.1 The determined inclusion volume fraction is substituted into the established model to obtain a prediction curve of the dynamic effective property of the piezoelectric composite material varying with frequency, and the prediction curve is compared with the effective phase velocity or effective acoustic impedance under the action of the elastic wave obtained in step two to validate the prediction model;

[0133] 5.2 Another prediction curve of the dynamic effective property of the piezoelectric composite material varying with frequency is drawn by considering different inclusion volume fractions, and the prediction curve is compared with the effective phase velocity or effective acoustic impedance under the action of the elastic wave obtained in step two to validate the prediction model.

[0134] Advantages of the present application:

[0135] 1. The present application comprehensively considers the influence of the microstructure information of the piezoelectric composite material and the frequency of the elastic wave on the macroscopic dynamic effective property of the piezoelectric composite material, and has a clear physical meaning. By using experimental data parameters obtained by experiment or by referring to prior art materials, the macroscopic dynamic effective parameters of the piezoelectric composite material can be predicted, and the effective wave propagation characteristics such as attenuation, effective phase velocity and effective acoustic impedance when the elastic wave propagates in the piezoelectric composite material can also be predicted.

[0136] 2. The prediction method of the dynamic effective property of the piezoelectric composite material based on the effective medium theory of micromechanics has obvious advantages in efficiency and economy, and can solve the shortcomings of long test period and high cost of dynamic characteristic test experiments of the piezoelectric composite material.

[0137] 3. The present application is not only suitable for the prediction of the dynamic effective property of the piezoelectric composite material, but also suitable for the prediction of the dynamic effective property of the pure elastic composite material, which only needs to degenerate the piezoelectric coefficient in the model to zero.

[0138] 4. The present application is suitable for the case of completely aligned inclusions, completely random inclusions and inclusions with arbitrary direction and satisfying certain directional preference, and therefore has the characteristic of wide application range.

[0139] 5. There is no dynamic homogenization theory based on the microstructure information of the material that links the distribution of the inclusion direction angle to the macroscopic dynamic effective property of the piezoelectric composite material. The present application introduces a function of the distribution of the inclusion direction angle to represent the alignment degree of the inclusions, which can be completely aligned, not aligned but with certain directional preference, or completely random direction. The present application establishes a prediction model of the macroscopic dynamic effective property of the piezoelectric composite material containing inclusions with arbitrary direction by using the self-consistent method, so as to obtain the continuous variation relationship of the dynamic effective property with frequency. BRIEF DESCRIPTION OF DRAWINGS

[0140] Figure 1 Schematic diagram of the geometry for a piezoelectric composite containing arbitrary orientation inclusions;

[0141] Figure 2 Schematic diagram of the geometry for transforming Figure 1 Schematic diagram of the geometry for transforming the scattering problem of a single inclusion in an equivalent medium.

[0142] Figure 3 Comparison of the predicted value of the dynamic effective acoustic impedance of PZT 501A / Spurrs epoxy 1-3 piezoelectric composite with the experimental value based on [3].

[0143] Figure 4 Comparison of the predicted value of the dynamic effective phase velocity of lead / EPON 828Z epoxy 0-3 purely elastic composite with volume fraction 0.05 with the experimental value based on [5].

[0144] Figure 5 Comparison of the predicted value of the dynamic effective phase velocity of lead / EPON 828Z epoxy 0-3 purely elastic composite with volume fraction 0.15 with the experimental value based on [5]. DETAILED DESCRIPTION

[0145] Embodiment one: the embodiment provides a method for predicting the dynamic effective properties of a piezoelectric composite containing an inclusion material and a matrix material, characterized in that the method comprises the following five steps:

[0146] Step one: determine the inclusion material and the matrix material, and obtain the material parameters of each component phase

[0147] Obtain the piezoelectric tensor, dielectric tensor, elastic tensor and density of the inclusion material and the piezoelectric tensor, dielectric tensor, elastic tensor and density of the matrix material respectively; if a phase material does not have piezoelectricity, only its elastic tensor and density are required;

[0148] The material parameters of each component phase obtained in step one are as follows:

[0149] 1.1 The fourth-order elastic tensor of the inclusion phase and the matrix phase is C x , where the superscript x=n represents the inclusion phase parameter and the superscript x=m represents the matrix phase parameter; C x has 5 independent non-zero components, which are C 11 , C 12 , C 13 , C 33 , C 44, which represent the elastic modulus in different directions, obtained by measurement;

[0150] 1.2 The third-order piezoelectric tensor of the inclusion phase and the matrix phase is e x , where the superscript x=n represents the parameter of the inclusion phase, and the superscript x=m represents the parameter of the matrix phase; e x has three independent nonzero components, which are e 15 , e 31 , and e 33 , respectively, according to the Voigt rule; they represent the piezoelectric coefficients in different directions, obtained by measurement; if a phase material does not have piezoelectricity, the piezoelectric coefficient thereof is zero;

[0151] 1.3 The second-order dielectric tensor of the inclusion phase and the matrix phase is κ x , where the superscript x=n represents the parameter of the inclusion phase, and the superscript x=m represents the parameter of the matrix phase; κ x has two independent nonzero components, which are κ 11 , and κ 33 , respectively, according to the Voigt rule; they represent the dielectric constants in different directions, obtained by measurement;

[0152] 1.4 The second-order dielectric density tensor of the inclusion phase and the matrix phase is ρ x , where the superscript x=n represents the parameter of the inclusion phase, and the superscript x=m represents the parameter of the matrix phase; ρ x is isotropic, and has only one independent nonzero component ρ, which represents the static density of the inclusion phase or the matrix phase, obtained by measurement;

[0153] Step two: according to step one, the inclusion material and the matrix material are determined, and a piezoelectric composite material sample is prepared or obtained

[0154] According to the preparation or obtaining of the piezoelectric composite material sample, experimental data parameters are obtained; the experimental data parameters include the volume fraction of the inclusion phase c n , the alignment degree parameter λ of the inclusion, the cross-sectional coefficient δ of the inclusion, and the inclusion radius a; then the effective phase velocity and the effective acoustic impedance under the action of the elastic wave are obtained;

[0155] The experimental data parameters in step two are obtained by measurement, and the effective phase velocity and the effective acoustic impedance under the action of the elastic wave in step two are obtained by measurement;

[0156] The step two includes the following sub-steps;

[0157] 2.1 A piezoelectric composite material sample is prepared, and the volume fraction of the inclusion phase is c n is known, then the volume fraction of the matrix phase c m =1-c n ;

[0158] 2.2 Prepared piezoelectric composite materials, based on the scanning electron microscope image of the sample, the radius a of the inclusion and the cross-sectional modulus δ were measured;

[0159] 2.3 Prepared piezoelectric composite materials, according to the scanning electron microscope image of the sample, statistical probability of inclusion direction angle α taking different values, according to the following formula: the probability distribution function of the inclusion direction P λ (α) Inversely calculate the parameter λ of the degree of inclusion alignment;

[0160]

[0161] 2.4 Measure the effective wave propagation characteristics of the prepared piezoelectric composite material sample under the action of elastic waves, including at least the effective phase velocity and effective acoustic impedance;

[0162] Step 3: Dynamic effective property prediction model - establishment of self-consistent equations

[0163] Based on the effective medium theory of micromechanics, a self-consistent set of equations for the dynamic effective parameters of the piezoelectric composite material is established. The dynamic effective property prediction model in step 3 - the self-consistent set of equations - is established according to the following sub-steps:

[0164] 3.1 In step 3, a prediction model for the dynamic effective properties of piezoelectric composite materials containing arbitrary orientation inclusions will be established, that is, a set of self-consistent equations about the macroscopic dynamic effective parameters; it is divided into 7 parts;

[0165] In 3.2, set the microstructure information parameters of the piezoelectric composite material;

[0166] In 3.3, set the coordinate system transformation rules of the tensor;

[0167] In 3.4, determine the expression for the effective phase velocity of the piezoelectric composite material;

[0168] In 3.5, determine the expression for the structural coefficient h;

[0169] In 3.6, determine the expression for the volume-averaged tensor of the five effective Green's operators for the piezoelectric composite material;

[0170] In 3.7, determine the expressions for the four combined parameter tensors in the self-consistent system of equations;

[0171] In 3.8, a self-consistent set of equations for predicting the dynamic effective properties of piezoelectric composites is established;

[0172] 3.2 Setting the microstructure information parameters of piezoelectric composite materials

[0173] In the global coordinate system o-x1x2x3, the piezoelectric composite material macroscopically exhibits transverse isotropy, and the symmetry axis is the x3 axis; the piezoelectric composite material is composed of a matrix phase and an inclusion phase, wherein the inclusion phase is a dispersed phase, and the volume fraction is c n , the matrix phase is a continuous phase, and the volume fraction is c m ; c n and c m are given in step two of the piezoelectric composite material preparation process; which can be expressed as

[0174] c m +c n =1 (1)

[0175] The position of the inclusion in the matrix is randomly distributed, but not overlapped; the shape of the inclusion is a rotating ellipsoid, and the rotation axis is the X3 axis. In the local coordinate system, the rotating ellipsoid equation is

[0176]

[0177] Where a is the inclusion radius, and δ is the inclusion cross-section coefficient, which is obtained in step two; δ>1 can simulate elongated inclusions, δ=1 can simulate spherical inclusions, and δ<1 can simulate oblate spherical inclusions;

[0178] The direction of the inclusion in the matrix is randomly distributed, and the direction angle of the rotating axis X3 axis in the global coordinate system o-x1x2x3 is (α, β), wherein α is the angle between the X3 axis and the x3 axis, and β is the angle between the projection of the X3 axis on the o-x1x2 plane and the x1 axis; since the composite material is isotropic in the o-x1x2 plane, the direction angle β has no effect on the macroscopic properties; the probability distribution function of the inclusion direction angle α is expressed as

[0179]

[0180] Where λ and α are obtained in step two; λ=0 represents that the inclusion is completely randomly distributed; λ>0 represents that the inclusion begins to exhibit a certain directional preference; the larger the value of λ, the more obvious the directional preference of the inclusion, that is, the higher the alignment degree; λ→∞ represents that the rotating axis of the inclusion is completely aligned with the x3 axis;

[0181] 3.3 Set the coordinate system conversion rule of the tensor

[0182] The tensor expressed in the local coordinate system needs to use the conversion tensor Q when converted to the global coordinate system

[0183]

[0184] where α and β are the directional angles of the inclusion in the global coordinate system, and there is no need to calculate the specific values of α and β, which will be integrated out in the following operations; if the fourth-order tensor F, the third-order tensor T, and the second-order tensor S in the local coordinate system are assumed to be F', T', and S' in the global coordinate system, then there are the following conversion relationships between their components

[0185] F' ijkl = Q pi Q qj Q rk Q sl F pqrs (5)

[0186] T' ijk = Q pi Q qj Q rk T pqr (6)

[0187] S' ij = Q pi Q qj S pq (7)

[0188] All the subscript letters in the formulas 5-7 can take 1, 2, 3, and the Einstein summation convention is performed;

[0189] 3.4 Determining the expression of the effective phase velocity of the piezoelectric composite material

[0190] The effective phase velocity of the piezoelectric composite material to be solved is denoted by v N , where subscript N = 1 is a quasi-P wave, subscript N = 2 is a quasi-SR wave, and subscript N = 3 is a quasi-SP wave; N is selected according to the type of the elastic wave used in step two;

[0191]

[0192] where the following combined parameters are used:

[0193]

[0194]

[0195] The effective phase velocity to be solved given by the formulas (8), (9), and (10) is a function of θ, C 0 , e 0 , κ 0 , and p 0 ; θ is the effective phase velocity obtained in step two, and the angle between the direction of the elastic wave propagation and the x3 axis; the non-zero components of the effective elastic tensor C 0 to be solved are and The effective piezoelectric tensor e is to be determined 0 The non-zero components of and The effective dielectric tensor κ is to be determined 0 The non-zero components of and The effective density tensor ρ is to be determined 0 The non-zero components of and These quantities are obtained by solving the self-consistent equations in sub-step 3.8;

[0196] 3.5 Determine the expression for the structural coefficient h

[0197] The structural coefficient h will be used in subsequent calculations and its expression is

[0198]

[0199] The expression of parameter b is as follows:

[0200]

[0201] Where a is the inclusion radius, δ is the inclusion cross-sectional coefficient, which is obtained in step 2; θ is the angle between the wave propagation direction and the x3 axis, and φ is the angle between its projection on the x1-x2 plane and the x1 axis, which is obtained in step 2; α and β are the direction angles of the inclusion in the global coordinate system, and the specific values ​​of α and β do not need to be calculated; k is the wave number, and its expression is:

[0202]

[0203] ω=2πf (30)

[0204] Where ω is the circular frequency, f is the frequency, and its magnitude is the frequency of the elastic wave in step 2; v N is the macroscopic effective phase velocity of the piezoelectric composite material to be determined, where subscript N = 1 indicates a quasi-P wave, subscript N = 2 indicates a quasi-SR wave, and subscript N = 3 indicates a quasi-SP wave; N is selected according to the type of elastic wave used in step 2; v N The expression of is shown in sub-step 3.4;

[0205] 3.6 Determine the expression of the volume average tensor of the five effective Green operators of the piezoelectric composite material

[0206] The volume average tensors of the five effective Green operators of piezoelectric composite materials will be used in subsequent calculations, and their component expressions are:

[0207]

[0208] In which, all subscripts of the integrand are 1, 2, 3 and follow the Einstein summation convention; n下标 is the component of the unit spherical vector n;

[0209] n=(sinθcosφ,sinθsinφ,cosθ) (36)

[0210] Here, θ and φ do not need specific values; is the polarization vector U N The component of , N is 1, 2, 3;

[0211]

[0212] Q1=B4 sin 4 θ+B5 sin 2 θcos 2 θ (46)

[0213] Q2=-B1 sin 4 θ+B2 sin 2 θcos 2 θ+B3 cos 4 θ (47)

[0214] Where M1 is shown in formula (11); α 下标 is the component of vector α

[0215] The remaining parameters in formulas (31)-(35) are as follows:

[0216]

[0217] Among them, v in formula (51) N See formulas (8)-(10); a in formulas (50) and (53) is the inclusion radius, δ is the inclusion cross-sectional coefficient, obtained in step 2; k in formula (53) N See formula (29); i in formula (54) is an imaginary unit;

[0218] From this sub-step, we know that the volume average tensor of the effective Green operator of the piezoelectric composite material given by formulas (31)-(35) is ω, C 0 , e 0 , κ 0 and ρ 0 function; ω is the circular frequency of the elastic wave in step 2; where the effective elastic tensor C is to be determined 0 The non-zero components of and The effective piezoelectric tensor e is to be determined 0 The non-zero components of and The effective dielectric tensor κ is to be determined 0 The non-zero components of and the unknown effective density tensor p 0 The nonzero components of p and These unknowns will be solved from the self-consistent equations in substep 3.8.

[0219] 3.7 Expressions of the four combined parameter tensors in the self-consistent equations

[0220] Four combined parameter tensor functions V (1) , V (2) , V (3) and V (4) will be used in the self-consistent equations, whose expressions are:

[0221]

[0222] where the expression of the structure coefficient h is given in substep 3.5; the fourth order tensor V (1) , the third order tensor V (2) , the second order tensor V (3) and V (4) are transversely isotropic with the symmetry axis along the x3 axis; the tensors C n′ , e n′ , κ n ' and p n ' represent the elastic tensor C n , the piezoelectric tensor e n , the dielectric tensor κ n and the density tensor p n of the inclusion phase in the local coordinate system, which are transformed to the global coordinate system according to the coordinate transformation rule in substep 3.3; while the elastic tensor C n , the piezoelectric tensor e n , the dielectric tensor κ n and the density tensor p n of the inclusion phase in the local coordinate system are obtained from step one; the expressions of the tensors J1-J5 are:

[0223] J1 = B1 : R + F1 · W T (59)

[0224] J2 = A1 · W T - F1 T : R (60)

[0225] J3 = B1 : W - F1 · Z (61)

[0226] J4 = F1 T : W + A1 · Z (62)

[0227]

[0228]

[0229] where:

[0230]

[0231] A1= [(e n′ -e 0 ):(C n′ -C 0 ) -1 :(e n′ -e 0 ) T +(κ n′ -κ 0 )] -1 (72)

[0232] B1= (C n ′-C 0 ) -1 -(C n ′-C 0 ) -1 :(e n ′-e 0 ) T ·A1·(e n ′-e 0 ):(C n ′-C 0 ) -1 (73)

[0233] F1= (C n ′-C 0 ) -1 :(e n ′-e 0 ) T ·A1 (74)

[0234] F1 T =A1·(e n ′-e 0 ):(C n ′-C 0 ) -1 (75)

[0235] The operator symbols "•" and ":" in equations (55)-(75) represent the single and double dot products of tensors, respectively; the superscripts "T" and "-1" represent the tensor transposition and tensor inversion operations, respectively; the tensor I 2nd represents the second-order unit tensor; the five tensors and The tensor given by formulas (31)-(35) in substep 3.6 is and According to the coordinate transformation rules given in sub-step 3.3, the local coordinate system is transformed into the global coordinate system; the tensor C 0 , e 0 , κ 0 and ρ 0 Represent the dynamic effective material parameters of the piezoelectric composite material to be determined; the tensor C n′ , e n′ , κ n ′ and ρ n ′ respectively represent the material properties C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , according to the coordinate transformation rules of sub-step 3.3, it is converted to the tensor in the global coordinate system; the material properties C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , obtained in step 1;

[0236] 3.8 Establishing a self-consistent set of equations for the dynamic effective properties of piezoelectric composites

[0237] Based on the effective medium theory of micromechanics and using the above expressions, a self-consistent set of equations for the dynamic effective material parameters of piezoelectric composites can be established in the global coordinate system:

[0238] C 0 =C m +c n <V (1) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (76)

[0239] e 0 =e m +c n <V (2) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (77)

[0240] κ 0 =κ m +c n <V (3) (C 0 ,e 0 ,κ0 ,ρ 0 ,θ,φ,ω,α,β)> (78)

[0241] ρ 0 =ρ m +c n <V (4) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (79)

[0242] where C 0 , e 0 , κ 0 and p 0 represent the dynamic effective properties of the piezoelectric composite, i.e. the effective elastic tensor, the effective piezoelectric tensor, the effective dielectric tensor, the effective density tensor, respectively; C 0 has 5 independent non-zero unknown components, denoted by and e 0 has 3 independent non-zero unknown components, denoted by and κ 0 has 2 independent non-zero unknown components, denoted by and p 0 has 2 independent non-zero unknown components, denoted by and C m , e m , κ m and p m represent the elastic tensor, the piezoelectric tensor, the dielectric tensor and the density tensor of the matrix phase, which have been obtained in Step 1; c n represents the volume fraction of the inclusion phase, which has been obtained in Step 2;

[0243] 4 combined parameter tensor functions V (1) , V (2) , V (3) and V (4) are given in Sub-step 3.7; the angle brackets < > in equations (76)-(79) represent the average of the function V over the direction angles, which is expressed as follows:

[0244]

[0245] where the function P λ (α) is given in Sub-step 3.2; the parameter λ has been obtained in Step 2;

[0246] The system of equations formed by formulas (76)-(79) is the final system of equations to be solved. Since both sides of the equal sign of the system of equations contain dynamic effective parameters to be solved, the system of equations is self-consistent and can be numerically solved by an iterative method.

[0247] Step 4: Calculate the dynamic effective properties of the composite material

[0248] The material parameters of the inclusion phase and matrix phase obtained in step 1 and the experimental data parameters obtained in step 2 are substituted into the prediction model established in step 3, i.e., the self-consistent equations, and the dynamic effective material parameters are obtained through numerical calculation.

[0249] Then the effective elastic wave propagation characteristics are calculated and a complete prediction model is obtained;

[0250] The calculation and extraction of the dynamic effective parameters of the piezoelectric composite material in step 4 are carried out according to the following sub-steps:

[0251] 4.1 Using the material parameters C of the inclusion phase obtained in step 1 n , e n , κ n and ρ n , the material parameter C of the matrix phase m , e m , κ m and ρ m , and the volume fraction of the inclusion phase obtained in step 2 is c n , the inclusion radius a, the cross-sectional coefficient δ and the parameter λ of the inclusion alignment are substituted into the self-consistent equations (76)-(79) in step 3, and then combined with the propagation direction angles θ and φ of the applied elastic wave, as well as the frequency f, the dynamic effective material parameter C of the piezoelectric composite material at the specified frequency f is numerically calculated by the iterative method. 0 , e 0 , κ 0 and ρ 0 ; In the first iteration, the dynamic effective properties C of the piezoelectric composite material involved in all functions on the right side of the equal sign are 0 , e 0 , κ 0 and ρ 0 The parameter C corresponding to the matrix can be taken m , e m , κ m and ρ m , and then a temporary C 0 , e 0 , κ 0 and ρ 0 , continue to substitute into the right side of the equal sign to calculate, repeat in sequence until convergence, and the final predicted dynamic effective parameter C of the piezoelectric composite material is obtained.0 , e 0 , K 0 and p 0 ;

[0252] 4.2 Calculate the effective wave propagation properties of the piezoelectric composite; substitute the effective parameters C 0 , e 0 , K 0 and p 0 obtained from sub-step 4.1 into equations (8)-(10) to obtain the effective phase velocity; then substitute into equation (29) and obtain the attenuation according to the attenuation equation Im(k N *a); then obtain the effective acoustic impedance according to the acoustic impedance equation ; all the effective properties that can be predicted by the prediction model have been obtained.

[0253] The steps one and two are not in order.

[0254] Step five: draw the prediction curve of the dynamic effective properties and verify the prediction model

[0255] Substitute different frequencies into the established model to obtain the complete prediction curve of the dynamic effective properties of the piezoelectric composite with frequency, and compare the prediction curve with the effective phase velocity or effective acoustic impedance obtained in step two under the action of elastic wave to verify the prediction model.

[0256] The drawing of the prediction curve of the dynamic effective properties and the verification of the prediction model in step five are carried out according to the following scheme:

[0257] 5.1 Substitute the determined inclusion phase volume fraction into the established model considering different frequencies to obtain the complete prediction curve of the dynamic effective properties of the piezoelectric composite with frequency, and compare the prediction curve with the effective phase velocity or effective acoustic impedance obtained in step two under the action of elastic wave to verify the prediction model;

[0258] 5.2 Consider different inclusion phase volume fractions to draw another prediction curve of the dynamic effective properties of the piezoelectric composite with frequency, and compare the prediction curve with the effective phase velocity or effective acoustic impedance obtained in step two under the action of elastic wave to verify the prediction model.

[0259] Example two: example two is basically the same as example one, the same part is not repeated, and the different part is:

[0260] The material parameters of each component phase obtained in step one in example two are carried out according to the following sub-steps:

[0261] 1.1 The fourth-order elastic tensor of the inclusion phase and the matrix phase is C x, where the superscript x=n denotes the inclusion phase parameters and the superscript x=m denotes the matrix phase parameters; C x There are five independent nonzero components, which are C 11 , C 12 , C 13 , C 33 , C 44 , which represent the elastic modulus in different directions, obtained by looking up the table;

[0262] 1.2 The third-order piezoelectric tensor of the inclusion phase and the matrix phase is e x , where the superscript x=n denotes the inclusion phase parameters and the superscript x=m denotes the matrix phase parameters; e x There are three independent nonzero components, which are e 15 , e 31 , e 33 , which represent the piezoelectric coefficient in different directions, obtained by looking up the table; if a phase material does not have piezoelectricity, its piezoelectric coefficient is zero;

[0263] 1.3 The second-order dielectric tensor of the inclusion phase and the matrix phase is κ x , where the superscript x=n denotes the inclusion phase parameters and the superscript x=m denotes the matrix phase parameters; κ x There are two independent nonzero components, which are κ 11 , κ 33 , which represent the dielectric constant in different directions, obtained by looking up the table;

[0264] 1.4 The second-order dielectric density tensor of the inclusion phase and the matrix phase is ρ x , where the superscript x=n denotes the inclusion phase parameters and the superscript x=m denotes the matrix phase parameters; ρ x is isotropic, and there is only one independent nonzero component ρ, which represents the static density of the inclusion phase or the matrix phase, obtained by looking up the table;

[0265] Step 2 in Example 2: The experimental data parameters in step 2 are obtained by referring to existing technical materials, and the effective phase velocity and effective acoustic impedance under the action of elastic waves in step 2 are obtained by referring to existing technical materials;

[0266] The step 2 includes the following sub-steps:

[0267] 2.1 Obtain the piezoelectric composite material sample, and obtain the volume fraction c n of the inclusion phase of the existing piezoelectric composite material by referring to existing technical materials;

[0268] 2.2 Obtain the piezoelectric composite material sample, and obtain the radius a and the cross-sectional coefficient δ of the inclusion by referring to existing technical materials;

[0269] 2.3 Obtain the piezoelectric composite sample, and obtain the alignment degree parameter λ of the inclusion by referring to the prior art.

[0270] 2.4 Obtain the piezoelectric composite sample, and obtain the macroscopic effective wave propagation characteristics of the sample by referring to the prior art. At least include effective phase velocity, effective acoustic impedance.

[0271] In Example 2, the material parameters of each component phase are obtained by referring to the table, and the experimental data parameters, effective phase velocity and effective acoustic impedance of the composite material are obtained by referring to the prior art. Although they are different from those measured by experiments in Example 1, they are also data measured by experiments of those skilled in the art before the technical scheme of the present application is proposed, and they are also a kind of experimental data which can be directly used.

[0272] Verification example 1 based on example 2:

[0273] In order to facilitate understanding of the present application, the present application will be described in detail below with the example of PZT501A / Spurrs epoxy 1-3 type piezoelectric composite material.

[0274] A prediction method of piezoelectric composite material dynamic effective properties based on the self-consistent method of micromechanics according to an embodiment of the present application, the prediction method comprises the following steps:

[0275] 1. According to the record in literature [3], the inclusion phase is PZT501A piezoelectric material, and the non-zero components of its elastic tensor C n and The non-zero components of its piezoelectric tensor e n and The non-zero components of its dielectric tensor κ n and κ0=8.85*10 -12 C 2 / Nm 2 ; The non-zero components of its density tensor ρ n

[0276] 2. According to the record in literature [3], the matrix phase is Spurrs epoxy, and the non-zero components of its elastic tensor C m and The non-zero components of its dielectric tensor κ m The non-zero components of its density tensor ρ m ​​​​​​​

[0277] 3. According to the record in reference [3], the radius of inclusion a = 0.225 mm, the cross-sectional coefficient δ = 4.4; the parameter λ of the inclusion alignment degree is 10; in reference [3], the samples with inclusion volume fractions of 10% and 20% were experimentally measured, so the inclusion phase volume fraction c is taken in the theoretical model respectively. n =0.1 and c n =0.2;

[0278] 4. Reference [3] experimentally measured the effective acoustic impedance of PZT501A / Spurrsepoxy 1-3 piezoelectric composite materials with 10% and 20% inclusion volume fractions in the frequency range of 0.3MHz-0.6MHz when elastic longitudinal waves propagated along the direction n=(0,0,1) or θ=0, respectively. Figure 3 As shown by the hollow triangle and hollow circle in .

[0279] 5. Substitute the above microstructure information (experimental data parameters) and frequency parameters into the prediction model established in the second embodiment of the present invention, that is, the self-consistent equations. First, numerically calculate the effective material parameter C by the iterative method. 0 , e 0 , κ 0 and ρ 0 Continuous change in the frequency range of 0.3MHz-0.6MHz; then calculate the effective longitudinal wave velocity according to formula (8); finally, according to the effective acoustic impedance formula in sub-step 4.2 Predict the discrete changes of effective acoustic impedance in the frequency range of 0.3MHz-0.6MHz. The predicted data are as follows: Figure 3 The solid triangle and solid circle in the coordinate system are shown in FIG; the solid triangle is fitted to obtain c n = 0.1 to get the curve of the predicted data, and fit the solid circle to get c n =0.2The curve of the predicted data, such as Figure 3 shown.

[0280] 6. If Figure 3 As shown in the figure, the curve of the predicted data of this prediction model is consistent with the effective acoustic impedance measured experimentally.

[0281] Verification Example 2 based on Example 2:

[0282] To facilitate understanding of the present invention, the present invention will be fully described below with reference to an example of a PZT501A / Spurrs epoxy 0-3 piezoelectric composite material.

[0283] A method for predicting the dynamic effective properties of a piezoelectric composite material based on a micromechanical self-consistent method according to an embodiment of the present invention comprises the following steps:

[0284] 1. According to the literature [4], the inclusion phase is PZT501A piezoelectric material, the non-zero components of its elastic tensor C n The non-zero components of its piezoelectric tensor e n The non-zero components of its dielectric tensor κ n The non-zero components of its density tensor ρ n

[0285] 2. According to the literature [4], the matrix phase is Spurrs epoxy, the non-zero components of its elastic tensor C m The non-zero components of its dielectric tensor κ m The non-zero components of its density tensor ρ m

[0286] 3. The literature [4] measured the effective acoustic impedance of the 0-3 type piezoelectric composite material when the elastic longitudinal wave with a frequency of 5 MHz propagates along the direction n = (0, 0, 1) that is θ = 0, the inclusion radius a = 0.75 μm, the inclusion volume fraction c n = 0.1 and 0.2, and the measured values are shown in Table 1.

[0287] 4. According to the literature [4], the radius of the inclusion a = 0.75 μm, the cross-sectional coefficient δ = 1, and the parameter λ of the inclusion alignment degree is 0.

[0288] The above microstructure information and frequency parameters are substituted into the prediction model, that is, the self-consistent equation group, established by the present application. The finally predicted effective acoustic impedance data are shown in Table 1. It can be seen that the prediction data of the present prediction model are in good agreement with the experimental data.

[0289] Table 1 Comparison of acoustic impedance of 0-3 type piezoelectric composite material

[0290]

[0291] Verification example 3 based on example 2:

[0292] In order to facilitate the understanding of the present application, the present application will be comprehensively described below in combination with the example of lead / EPON828Z epoxy 0-3 type pure elastic composite material.

[0293] ​​​​​​​​​​​A method for predicting the dynamic effective properties of piezoelectric composites based on the self-consistent method of micromechanics according to an embodiment of the present application, the method comprising the following steps:

[0294] 1. According to the literature [5], the inclusion phase is a lead particle material, and the non-zero components of the elastic tensor C n of the inclusion phase are and The non-zero components of the density tensor p n of the inclusion phase are

[0295] 2. According to the literature [5], the matrix phase is EPON828Z epoxy, and the non-zero components of the elastic tensor C m of the matrix phase are and The non-zero components of the density tensor p m of the matrix phase are

[0296] 3. The literature [5] experimentally measured the variation of the effective longitudinal wave velocity v1 in the frequency range of 0-1.5 MHz. The effective longitudinal wave velocity v1 under the conditions of an inclusion radius a = 660 μm and inclusion volume fractions c n = 0.05 and 0.15 is shown by the hollow circles in Figure 4 and Figure 5 .

[0297] 4. According to the literature [5], the inclusion radius a = 660 μm and the cross-sectional coefficient δ = 1 are taken, and the parameter λ of the inclusion alignment degree is taken as 0. The above microstructure information and the frequency parameter are substituted into the prediction model, i.e. the self-consistent equation group, established by the present application. The fitting curve of the finally predicted effective longitudinal wave velocity v1 is shown by the solid line in Figure 4 and Figure 5 .

[0298] The comparison of the hollow circles in Figure 4 Figure 5 and 5 with the fitting curve shows that the predicted effective longitudinal wave velocity v1 by the prediction model is in good agreement with the experimentally measured effective longitudinal wave velocity v1.

Claims

1. A method for predicting the dynamic effective properties of a piezoelectric composite material, the piezoelectric composite material comprising an inclusion material and a matrix material, characterized in that: The prediction method includes the following five steps: Step 1: Determine the inclusion material and matrix material, and obtain the material parameters of each component phase Obtain the piezoelectric tensor, dielectric tensor, elastic tensor and density of the inclusion material and the piezoelectric tensor, dielectric tensor, elastic tensor and density of the matrix material respectively; if a phase material does not have piezoelectricity, only its elastic tensor and density need to be obtained; Step 2: Determine the inclusion material and matrix material, and prepare or obtain the piezoelectric composite material sample According to the preparation or acquisition of piezoelectric composite material samples, experimental data parameters are obtained; the experimental data parameters include the volume fraction of the inclusion phase c n , the alignment parameter λ of the inclusion, the section coefficient δ of the inclusion, and the inclusion radius a; then the effective phase velocity and effective acoustic impedance under the action of elastic waves are obtained; Step 3: Dynamic effective property prediction model - establishment of self-consistent equations Based on the effective medium theory of micromechanics, a self-consistent set of equations for the dynamic effective parameters of piezoelectric composite materials is established; Step 4: Calculate the dynamic effective properties of the composite material The material parameters of the inclusion phase and matrix phase obtained in step 1 and the experimental data parameters obtained in step 2 are substituted into the prediction model established in step 3, i.e., the self-consistent equations, and the dynamic effective material parameters are obtained through numerical calculation. Then the effective elastic wave propagation characteristics are calculated and a complete prediction model is obtained; Step 5: Draw the prediction curve of dynamic effective properties and verify the prediction model Substituting different frequencies into the established model, a complete prediction curve of the dynamic effective properties of the piezoelectric composite material as a function of frequency is obtained. This prediction curve is compared with the effective phase velocity or effective acoustic impedance under the action of elastic waves obtained in step 2 to verify the prediction model. The steps one and two are described in no particular order.

2. The method for predicting dynamic effective properties of a piezoelectric composite material according to claim 1, wherein: In step 1, the material parameters of each component phase are obtained according to the following sub-steps: 1.1 The fourth-order elastic tensor of the inclusion phase and the matrix phase is C x , superscript x=n represents the inclusion phase parameter, superscript x=m represents the matrix phase parameter; C x There are 5 independent non-zero components, according to Voigt's rule, which are C 11 , C 12 , C 13 , C 33 , C 44 , which represent the elastic modulus in different directions and are obtained by measurement or table lookup; 1.2 The third-order piezoelectric tensor of the inclusion phase and the matrix phase is e x , superscript x=n represents the inclusion phase parameter, superscript x=m represents the matrix phase parameter; e x There are three independent non-zero components, according to Voigt's rule, namely e 15 , e 31 , e 33 , which represent the piezoelectric coefficients in different directions, obtained by measurement or table lookup; if a phase material does not have piezoelectricity, its piezoelectric coefficient is set to zero; 1.3 The second-order dielectric tensor of the inclusion phase and the matrix phase is κ x , superscript x=n represents the inclusion phase parameter, superscript x=m represents the matrix phase parameter; κ x There are two independent non-zero components, according to Voigt's rule, which are κ 11 , κ 33 , which represent the dielectric constant in different directions and are obtained by measurement or table lookup; 1.4 The second-order dielectric density tensor of the inclusion phase and the matrix phase is ρ x , superscript x=n represents the inclusion phase parameter, superscript x=m represents the matrix phase parameter; ρ x It is isotropic and has only one independent non-zero component ρ, which represents the static density of the inclusion phase or matrix phase and can be obtained by measurement or table lookup.

3. The method for predicting dynamic effective properties of a piezoelectric composite material according to claim 1, wherein: The experimental data parameters in step 2 are obtained by measurement or by referring to existing technical data. The effective phase velocity and effective acoustic impedance under the action of elastic waves in step 2 are obtained by measurement or by referring to existing technical data.

4. The method for predicting dynamic effective properties of a piezoelectric composite material according to claim 3, wherein: Step 2 includes the following sub-steps: 2.1 Preparation of piezoelectric composite material samples with an inclusion phase volume fraction of c n Known, the volume fraction of the matrix phase c m =1-c n ; If the piezoelectric composite material sample is obtained, the volume fraction of the inclusion phase c of the existing piezoelectric composite material can be obtained by consulting the existing technical data. n ; 2.2 Prepared piezoelectric composite materials, based on the scanning electron microscope image of the sample, the radius a of the inclusion and the cross-sectional modulus δ were measured; If the piezoelectric composite material sample is obtained, the radius a and the cross-sectional modulus δ of the inclusion are obtained by consulting the existing technical data; 2.3 Prepared piezoelectric composite materials, according to the scanning electron microscope image of the sample, statistical probability of inclusion direction angle α taking different values, according to the following formula: the probability distribution function of the inclusion direction P λ (α) Inversely calculate the parameter λ of the degree of inclusion alignment; If the obtained piezoelectric composite material sample is obtained, the alignment degree parameter λ of the inclusion is obtained by consulting the existing technical data; 2.4 Measure the effective wave propagation characteristics of the prepared piezoelectric composite material sample under the action of elastic waves, including at least the effective phase velocity and effective acoustic impedance; If a piezoelectric composite material sample is obtained, the macroscopic effective wave propagation characteristics of the sample are obtained by consulting existing technical data.

5. The method for predicting dynamic effective properties of a piezoelectric composite material according to claim 1, wherein: The dynamic effective property prediction model in step 3 - the self-consistent equation system - is established according to the following sub-steps: 3.1 In step 3, a prediction model for the dynamic effective properties of piezoelectric composite materials containing arbitrary orientation inclusions will be established, that is, a set of self-consistent equations about the macroscopic dynamic effective parameters; it is divided into 7 parts; In 3.2, set the microstructure information parameters of the piezoelectric composite material; In 3.3, set the coordinate system transformation rules of the tensor; In 3.4, determine the expression for the effective phase velocity of the piezoelectric composite material; In 3.5, determine the expression for the structural coefficient h; In 3.6, determine the expression for the volume-averaged tensor of the five effective Green's operators for the piezoelectric composite material; In 3.7, determine the expressions for the four combined parameter tensors in the self-consistent system of equations; In 3.8, a self-consistent set of equations for predicting the dynamic effective properties of piezoelectric composites is established; 3.2 Setting the microstructure information parameters of piezoelectric composite materials In the overall coordinate system o-x1x2x3, the piezoelectric composite material exhibits transverse isotropy in the macroscopic view, with the symmetry axis being the x3 axis; the piezoelectric composite material consists of a matrix phase and an inclusion phase, wherein the inclusion phase is a dispersed phase with a volume fraction of c n , the matrix phase is the continuous phase, and its volume fraction is c m ;c n and c m Given in step 2 during the preparation of the piezoelectric composite material; it can be expressed as c m +c n =1 (1) The positions of inclusions in the matrix are randomly distributed but do not overlap; the inclusion shape is a rotating ellipsoid with the rotation axis being the X3 axis. In the local coordinate system, the equation of the rotating ellipsoid is Where a is the inclusion radius, δ is the inclusion cross-sectional coefficient, obtained in step 2; δ>1 can simulate slender inclusions, δ=1 can simulate spherical inclusions, and δ<1 can simulate oblate spherical inclusions; The directions of inclusions in the matrix are randomly distributed. The orientation angle of the X3 axis of the inclusion in the global coordinate system o-x1x2x3 is (α, β), where α is the angle between the X3 axis and the x3 axis, and β is the angle between the projection of the X3 axis on the o-x1x2 plane and the x1 axis. Since the composite material is isotropic in the o-x1x2 plane, the orientation angle β has no effect on the macroscopic properties. The probability distribution function of the inclusion orientation angle α is expressed as Where λ and α are obtained in step 2; λ = 0 means that the inclusion orientation is completely randomly distributed; λ > 0 means that the inclusion orientation begins to show a certain directional preference; the larger the λ value, the more obvious the directional preference of the inclusion, that is, the higher the degree of alignment; λ → ∞ means that the rotation axis of the inclusion is completely aligned with the x3 axis; 3.3 Setting the coordinate system transformation rules for tensors When converting a tensor expressed in a local coordinate system to a global coordinate system, the conversion tensor Q required is Among them, α and β are the direction angles in the global coordinate system. There is no need to calculate the specific values ​​of α and β, as they will be integrated out in the subsequent calculations. If we assume that the fourth-order tensor F, third-order tensor T, and second-order tensor S in the local coordinate system are F', T', and S' in the global coordinate system, then there is the following conversion relationship between their components F′ ijkl =Q pi Q qj Q rk Q sl F pqrs (5) T′ ijk =Q pi Q qj Q rk T pqr (6) S′ ij =Q pi Q qj S pq (7) All subscripts in formulas 5-7 can be 1, 2, or 3, and the Einstein summation convention is implemented; 3.4 Expression for determining the effective phase velocity of piezoelectric composites The effective phase velocity of the piezoelectric composite material to be determined is expressed as v N Indicates that when subscript N = 1, it is a quasi-P wave, when subscript N = 2, it is a quasi-SR wave, and when subscript N = 3, it is a quasi-SP wave; N is selected according to the type of elastic wave used in step 2; Among them, the following combination parameters are used: Formulas (8), (9) and (10) give the effective phase velocities to be determined, which are θ, C 0 , e 0 , κ 0 and ρ 0 function; θ is the effective phase velocity obtained in step 2, the angle between the elastic wave propagation direction and the x3 axis; where the effective elastic tensor C is to be determined 0 The non-zero components of and The effective piezoelectric tensor e is to be determined 0 The non-zero components of and The effective dielectric tensor κ is to be determined 0 The non-zero components of and The effective density tensor ρ is to be determined 0 The non-zero components of and These quantities are obtained by solving the self-consistent equations in sub-step 3.8; 3.5 Determine the expression for the structural coefficient h The structural coefficient h will be used in subsequent calculations and its expression is The expression of parameter b is as follows: Where a is the inclusion radius, δ is the inclusion cross-sectional coefficient, which is obtained in step 2; θ is the angle between the wave propagation direction and the x3 axis, and φ is the angle between its projection on the x1-x2 plane and the x1 axis, which is obtained in step 2; α and β are the direction angles of the inclusion in the global coordinate system, and the specific values ​​of α and β do not need to be calculated; k is the wave number, and its expression is: ω=2πf (30) Where ω is the circular frequency, f is the frequency, and its magnitude is the frequency of the elastic wave in step 2; v N is the macroscopic effective phase velocity of the piezoelectric composite material to be determined, where subscript N = 1 indicates a quasi-P wave, subscript N = 2 indicates a quasi-SR wave, and subscript N = 3 indicates a quasi-SP wave; N is selected according to the type of elastic wave used in step 2; v N The expression of is shown in sub-step 3.4; 3.6 Determine the expression of the volume average tensor of the five effective Green operators of the piezoelectric composite material The volume average tensors of the five effective Green operators of piezoelectric composite materials will be used in subsequent calculations, and their component expressions are: In which, all subscripts of the integrand are 1, 2, 3 and follow the Einstein summation convention; n 下标 is the component of the unit spherical vector n; n=(sinθcosφ,sinθsinφ,cosθ) (36) Here, θ and φ do not need specific values; is the polarization vector U N The component of , N is 1, 2, 3; Q1=B4sin 4 θ+B5sin 2 θcos 2 i (46) Q2=-B1sin 4 θ+B2sin 2 θcos 2 θ+B3cos 4 i (47) Where M1 is shown in formula (11); α 下标 is the component of vector α The remaining parameters in formulas (31)-(35) are as follows: F N =T[1-χ(z N )] (52) Among them, v in formula (51) N See formulas (8)-(10); a in formulas (50) and (53) is the inclusion radius, δ is the inclusion cross-sectional coefficient, obtained in step 2; k in formula (53) N See formula (29); i in formula (54) is an imaginary unit; From this sub-step, we know that the volume average tensor of the effective Green operator of the piezoelectric composite material given by formulas (31)-(35) is ω, C 0 , e 0 , κ 0 and ρ 0 function; ω is the circular frequency of the elastic wave in step 2; where the effective elastic tensor C is to be determined 0 The non-zero components of and The effective piezoelectric tensor e is to be determined 0 The non-zero components of and The effective dielectric tensor κ is to be determined 0 The non-zero components of and The effective density tensor ρ is to be determined 0 The non-zero components of and These quantities will be obtained by solving the self-consistent equations in sub-step 3.8; 3.7 Determine the expressions of the four combined parameter tensors in the self-consistent system of equations In the self-consistent equation system, four combined parameter tensor functions V will be used (1) , V (2) , V (3) and V (4) , their expressions are: V (1) (C 0 ,e 0 ,k 0 ,r 0 ,θ,φ,ω,α,β)=h 2 [(C n′ -C m ):J1-(e n′ -e m ) T ·J2] (55) V (2) (C 0 ,e 0 ,k 0 ,r 0 ,θ,φ,ω,α,β)=h 2 [(e n′ -e m ):J1+(k n′ -k m )·J2] (56) V (3) (C 0 ,e 0 ,k 0 ,r 0 ,θ,φ,ω,α,β)=h 2 [(k n′ -k m )·J4-(e n′ -e m ):J3] (57) V (4) (C 0 ,e 0 ,k 0 ,r 0 ,θ,φ,ω,α,β)=h 2 (r n′ -r m )·J5 (58) The expression of the structural coefficient h is shown in sub-step 3.5; the fourth-order tensor V (1) , the third-order tensor V (2) , the second-order tensor V (3) and V (4) They are all transversely isotropic, with the axis of symmetry being the x3 axis; the tensor C n′ , e n′ , κ n′ and ρ n′ Represent the elastic tensor C of the inclusion phase in the local coordinate system n , the piezoelectric tensor e n , dielectric tensor κ n and the density tensor ρ n , according to the coordinate transformation rule of sub-step 3.3, it is converted to the tensor in the global coordinate system; and the elastic tensor C of the inclusion phase in the local coordinate system n , the piezoelectric tensor e n , dielectric tensor κ n and the density tensor ρ n It will be obtained through step 1; the expression of tensor J1-J5 is: J1=B1:R+F1·W T (59) J2=A1·W T -F1 T :R (60) J3=B1:W-F1·Z (61) J4=F1 T :W+A1·Z (62) in: A1=[(e n′ -e 0 ):(C n′ -C 0 ) -1 :(he n′ -e 0 ) T +(κ n′ -κ 0 )] -1 (72) B1=(C n′ -C 0 ) -1 -(C n′ -C 0 ) -1 :(and n′ -and 0 ) T ·A1·(e n′ -and 0 ):(C n′ -C 0 ) -1 (73) F1=(C n′ -C 0 ) -1 :(e n′ -e 0 ) T ·A1 (74) F1 T =A1·(e n′ -e 0 ):(C n′ -C 0 ) -1 (75) In formulas (55)-(75), the operator symbols "·" and "∶" represent the single dot product and double dot product of tensors respectively; the superscripts "T" and "-1" represent the tensor transposition and tensor inversion operations respectively; the tensor I 2nd represents the second-order unit tensor; the five tensors in formulas (63) and (68)-(71) and The tensor given by formulas (31)-(35) in substep 3.6 is and According to the coordinate transformation rules given in sub-step 3.3, the local coordinate system is transformed into the global coordinate system; the tensor C 0 , e 0 , κ 0 and ρ 0 Represent the dynamic effective material parameters of the piezoelectric composite material to be determined; the tensor C n′ , e n′ , κ n′ and ρ n′ Represent the material properties C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , according to the coordinate transformation rules of sub-step 3.3, it is converted to the tensor in the global coordinate system; the material properties C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , obtained in step 1; 3.8 Establishing a self-consistent set of equations for the dynamic effective properties of piezoelectric composites Based on the effective medium theory of micromechanics and using the above expressions, a self-consistent set of equations for the dynamic effective material parameters of piezoelectric composites can be established in the global coordinate system: C 0 =C m +c n <V (1) (C 0 ,e 0 ,k 0 ,r 0 ,θ,φ,ω,α,β)> (76) e 0 =e m +c n <V (2) (C 0 ,e 0 ,k 0 ,r 0 ,θ,φ,ω,α,β)> (77) k 0 =k m +c n <V (3) (C 0 ,e 0 ,k 0 ,r 0 ,θ,φ,ω,α,β)> (78) r 0 =ρ m +c n <V (4) (C 0 ,e 0 ,k 0 ,r 0 ,θ,φ,ω,α,β)> (79) Among them C 0 , e 0 , κ 0 and ρ 0 They represent the dynamic effective properties of the piezoelectric composite material to be determined, namely the effective elastic tensor, effective piezoelectric tensor, effective dielectric tensor, and effective density tensor; C 0 There are 5 independent non-zero components to be determined, which can be expressed as and e 0 There are three independent non-zero components to be determined, which can be expressed as and κ 0 There are two independent non-zero components to be determined, which can be expressed as and ρ 0 There are two independent non-zero components to be determined, which can be expressed as and C m , e m , κ m and ρ m Represent the elastic tensor, piezoelectric tensor, dielectric tensor and density tensor of the matrix phase, which have been obtained in step 1; c n represents the volume fraction of the inclusion phase, which has been obtained in step 2; 4 combined parameter tensor function V (1) , V (2) , V (3) and V (4) , given in substep 3.7; the angle brackets <> in equations (76)-(79) represent the function V averaged over the direction angle, expressed as follows: The function P λ (α) See sub-step 3.2; the parameter λ has been obtained in step 2; The system of equations formed by formulas (76)-(79) is the final system of equations to be solved; since both sides of the equal sign of the system of equations contain dynamic effective parameters to be solved, the system of equations is self-consistent and can be numerically solved by an iterative method.

6. The method for predicting dynamic effective properties of a piezoelectric composite material according to claim 5, characterized in that: The calculation and extraction of the dynamic effective parameters of the piezoelectric composite material in step 4 is carried out according to the following sub-steps: 4.1 Using the material parameters C of the inclusion phase obtained in step 1 n , e n , κ n and ρ n , the material parameter C of the matrix phase m , e m , κ m and ρ m , and the volume fraction of the inclusion phase obtained in step 2 is c n , the inclusion radius a, the cross-sectional coefficient δ and the parameter λ of the inclusion alignment are substituted into the self-consistent equations (76)-(79) in step 3, and then combined with the propagation direction angles θ and φ of the applied elastic wave, as well as the frequency f, the dynamic effective material parameter C of the piezoelectric composite material at the specified frequency f is numerically calculated by the iterative method. 0 , e 0 , κ 0 and ρ 0 ; In the first iteration, the dynamic effective properties C of the piezoelectric composite material involved in all functions on the right side of the equal sign are 0 , e 0 , κ 0 and ρ 0 The parameter C corresponding to the matrix can be taken m , e m , κ m and ρ m , and then a temporary C 0 , e 0 , κ 0 and ρ 0 , continue to substitute into the right side of the equal sign to calculate, repeat in sequence until convergence, and the final predicted dynamic effective parameter C of the piezoelectric composite material is obtained. 0 , e 0 , κ 0 and ρ 0 ; 4.2 Calculate the effective wave propagation characteristics of the piezoelectric composite material; use the effective parameter C obtained in substep 4.1 0 , e 0 , κ 0 and ρ 0 Substitute into formula (8)-(10) to obtain the effective phase velocity; then substitute into formula (29), and according to the attenuation formula Im(k N *a), get the attenuation; then according to the acoustic impedance formula The effective acoustic impedance is calculated; all the effective properties that can be predicted by this prediction model have been obtained.

7. The method for predicting the dynamic effective properties of a piezoelectric composite material according to claim 6, characterized in that: In step 5, the prediction curve of the dynamic effective properties and the verification of the prediction model are carried out according to the following scheme: 5.1 Substitute the determined volume fraction of the inclusion phase into the established model, taking into account different frequencies, to obtain a complete prediction curve of the dynamic effective performance of the piezoelectric composite material as a function of frequency. Compare this prediction curve with the effective phase velocity or effective acoustic impedance under the action of elastic waves obtained in step 2 to verify the prediction model; 5.2 Considering different volume fractions of the included phase, draw another prediction curve of the dynamic effective performance of the piezoelectric composite material as a function of frequency, and compare the prediction curve with the effective phase velocity or effective acoustic impedance under the action of elastic waves obtained in step 2 to verify the prediction model.

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