Dynamic effective property prediction method of piezoelectric composite material

Abstract

The invention relates to a method for predicting dynamic effective properties of a piezoelectric composite material based on an effective medium theory of micromechanics. The dynamic effective properties refer to effective material parameters and combination parameters thereof, which are macroscopically shown by the piezoelectric composite material under the action of elastic waves and change along with frequency. The prediction method comprises the following five steps: step 1, obtaining material parameters of each component phase; 2, preparing or obtaining a piezoelectric composite material sample; 3, establishing a dynamic effective property prediction model-self-consistent equation set; 4, calculating dynamic effective parameters of the composite material; and 5, drawing a prediction curve of the dynamic effective property and verifying a prediction model. According to the method, the influence of the alignment degree, the radius, the slenderness ratio, the volume fraction, the frequency and the like of the inclusion on the macroscopic effective performance of the piezoelectric composite material is mainly considered, and a prediction model is established. The model is well matched with an experimental result. The method has obvious advantages in the aspects of efficiency and economy, and can overcome the defects of long period and high cost of a dynamic characteristic test of the piezoelectric composite material.

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 dynamic effective properties of piezoelectric composite materials. Background Art

[0002] Piezoelectric materials can realize the mutual conversion of mechanical energy and electrical energy on a macro scale. This effect is called piezoelectric effect, which is divided into positive piezoelectric effect and inverse piezoelectric effect. The positive piezoelectric effect converts mechanical quantity (deformation or force) into electrical signal; the inverse piezoelectric effect converts electrical signal into mechanical quantity (deformation or force). Based on the positive and inverse piezoelectric effects, piezoelectric materials are widely used in various fields. With the continuous breakthrough of technology, higher requirements are also put forward for the performance of piezoelectric materials. Conventional piezoelectric monomer materials, such as piezoelectric single crystals, piezoelectric ceramics, etc., can no longer meet the demand.

[0003] Piezoelectric composite materials are generally composed of a piezoelectric inclusion phase and a polymer matrix phase (see Appendix Figure 1 , with or without piezoelectricity) are compounded according to certain connectivity, volume or weight ratio and spatial geometric distribution. This information constitutes the microstructural information of piezoelectric composite materials and plays a vital role in their macroscopic effective properties. The piezoelectric phase is used to produce the piezoelectric effect, while the polymer phase is mainly used to reduce the density and permittivity of the material and increase the elastic compliance constant of the material. This design can multiply the macroscopic piezoelectric performance of the material and also has some excellent properties that piezoelectric monomer materials do not have.

[0004] Based on the classical elastic micromechanics theory, people have established a micromechanics model of static piezoelectric composites, and predicted the static macroscopic effective properties of piezoelectric composites based on microstructural information, thereby optimizing the macroscopic behavior of composites by rationally designing microstructures. Under the action of elastic waves, the dynamic macroscopic effective properties of piezoelectric composites are no longer constants, but change depending on the frequency. At present, there are few theoretical prediction methods for the dynamic effective properties of piezoelectric composites, which to a certain extent restricts the widespread application of piezoelectric composites 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 YP, Wan YP, Theoretical study on dynamic effective electroelastic properties of random piezoelectric composites with alignedinhomogeneities.Applied Mathematics and Mechanics(English Edition),2023,44(4),525-546

[0007] The above two documents 1-2 are papers published by the inventors in the past, which respectively provide prediction methods for piezoelectric composites containing aligned ellipsoidal inclusions and semi-ellipsoidal inclusions. However, the above studies do not consider the influence of the distribution of inclusion directions on the macroscopic dynamic effective properties of piezoelectric composites.

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

[0009] [3]Gururaja TR, Cross LE, 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 MG, Gururaja TR, 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 VK,Dispersive wave propagation in random particulatecomposites.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 above-mentioned documents 3-5 have the problems of difficulty, long cycle and high cost in experimental testing of dynamic effective properties of piezoelectric composite materials. This is because piezoelectric composite materials often work within a frequency range, which requires that the designed piezoelectric composite materials need to show the best dynamic effective properties within the working frequency range. In fact, there are many parameters that affect the dynamic effective properties of piezoelectric composite materials, 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, and the inclusion size, and these parameters work together with the frequency to affect the macroscopic effective properties of the composite material. 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 each sample needs to be tested multiple times within the frequency range. Therefore, this traditional experimental method has the problems of difficulty, long cycle and high cost. Summary of the invention

[0013] The present invention aims at the deficiencies of the prior art and establishes a method for predicting the dynamic effective properties of a piezoelectric composite material.

[0014] The comprehensive influence of the microstructure information and frequency of inclusions in the composite material on the macroscopic dynamic effective properties will be given priority consideration in the present invention. These comprehensive influencing factors include: the distribution of the orientation angle 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 composite material. The present invention solves the problem that the research method of the macroscopic dynamic effective performance of the piezoelectric composite material in the prior art is high in cost and difficult to fully explore the mechanism of the comprehensive influence effect of microstructure information and frequency.

[0015] The technical solution of the present invention is: a method for predicting the dynamic effective properties of a piezoelectric composite material, the piezoelectric composite material containing an inclusion material and a matrix material, characterized in that: the prediction method comprises the following five steps;

[0016] Step 1: Determine the inclusion material and matrix material to 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 2: Determine the inclusion material and matrix material, and prepare or obtain the piezoelectric composite material sample

[0019] 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 degree parameter λ of the inclusion, the section coefficient δ of the inclusion, and the radius a of the inclusion; then the effective phase velocity and the effective acoustic impedance under the action of elastic waves are obtained;

[0020] Step 3: Dynamic effective property prediction model-establishment of self-consistent equations

[0021] 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;

[0022] Step 4: Calculate the dynamic effective properties of the composite material

[0023] Substituting the material parameters of the inclusion phase and the matrix phase obtained in step 1 and the experimental data parameters obtained in step 2 into the prediction model established in step 3, i.e., the self-consistent equation group, the dynamic effective material parameters are obtained through numerical calculation;

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

[0025] Step 5: Draw the prediction curve of dynamic effective properties and verify the prediction model

[0026] Substitute different frequencies into the established model to obtain a complete prediction curve of the dynamic effective properties of the piezoelectric composite material changing with frequency, compare the prediction curve with the effective phase velocity or effective acoustic impedance under the action of elastic waves obtained in step 2, and verify the prediction model;

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

[0028] Preferably, the material parameters of each component phase are obtained in step 1 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 , the superscript x=n represents the inclusion phase parameter, and the superscript x=m represents the matrix phase parameter; 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 , they represent the elastic modulus in different directions and are 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 , the superscript x=n represents the inclusion phase parameter, and the 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 , they represent the piezoelectric coefficients in different directions, which are obtained by measurement or table lookup; if a phase material does not have piezoelectricity, its piezoelectric coefficient is set to zero;

[0031] 1.3 The second-order dielectric tensor of the inclusion phase and the matrix phase is κ x , the superscript x=n represents the inclusion phase parameter, the 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 , they represent the dielectric constant in different directions and are 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 , the superscript x=n represents the inclusion phase parameter, the 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 is obtained by measurement or table lookup.

[0033] Preferably, the experimental data parameters in step 2 are obtained by measurement or by referring to existing technical data, and 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.

[0034] Preferably, step 2 includes the following sub-steps:

[0035] 2.1 Preparation of piezoelectric composite samples with an inclusion phase volume fraction of c n It is known that the volume fraction of the matrix phase c m =1-c n ;

[0036] If the obtained 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 ;

[0037] 2.2 The prepared piezoelectric composite material, based on the SEM image of the sample, measured the radius a of the inclusion and the cross-sectional coefficient δ;

[0038] If the obtained piezoelectric composite material sample is used, the radius a and the cross-sectional coefficient δ of the inclusion are obtained by consulting the existing technical data;

[0039] 2.3 The piezoelectric composite material prepared was analyzed according to the SEM image of the sample, and the probability of different values ​​of the inclusion direction angle α was calculated according to the following formula: The probability distribution function P of the inclusion direction λ (α) Inversely calculate the parameter λ of the degree of inclusion alignment;

[0040]

[0041] 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;

[0042] 2.4 The piezoelectric composite material prepared is used to measure the effective wave propagation characteristics of the piezoelectric composite material sample under the action of elastic waves, including at least the effective phase velocity and the effective acoustic impedance;

[0043] If a piezoelectric composite material sample is obtained, the macroscopic effective wave propagation characteristics of the sample are obtained by consulting existing technical data.

[0044] 5. The method for predicting the dynamic effective properties of a piezoelectric composite material according to claim 1, characterized in that: the dynamic effective property prediction model in step 3 - the self-consistent equation group, is established according to the following sub-steps:

[0045] 3.1 In step 3, a prediction model for the dynamic effective properties of piezoelectric composite materials containing arbitrary directional inclusions will be established, that is, a set of self-consistent equations about macroscopic dynamic effective parameters; it is divided into 7 parts;

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

[0047] In 3.3, set the coordinate system transformation rules of the tensor;

[0048] In 3.4, determine the expression for the effective phase velocity of the piezoelectric composite material;

[0049] In 3.5, determine the expression for the structural coefficient h;

[0050] In 3.6, determine the expression for the volume average tensor of the five effective Green operators of the piezoelectric composite;

[0051] In 3.7, determine the expressions for the four combined parameter tensors in the self-consistent system of equations;

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

[0053] 3.2 Setting the microstructure information parameters of piezoelectric composite materials

[0054] In the overall coordinate system o-x1x2x3, the piezoelectric composite material exhibits transverse isotropy in the macroscopic view, and the symmetry axis is the x3 axis; the piezoelectric composite material is composed of a matrix phase and an inclusion phase, in which 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

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

[0056] The positions of inclusions in the matrix are randomly distributed but do not overlap; the shape of the inclusion is a rotating ellipsoid with the rotation axis being the X3 axis. In the local coordinate system, the equation of the rotating ellipsoid is:

[0057]

[0058] 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;

[0059] The directions of inclusions in the matrix are randomly distributed. The direction angle of the X3 axis of the inclusion in the overall 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 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 2; λ = 0 means that the orientation of the inclusions is completely randomly distributed; λ > 0 means that the orientation of the inclusions begins to show a certain directional preference; the larger the value of λ, the more obvious the directional preference of the inclusions, that is, the higher the degree of alignment; λ → ∞ means that the rotation axis of the inclusions is completely aligned with the x3 axis;

[0062] 3.3 Setting the coordinate system transformation rules for tensors

[0063] When the tensor expressed in the local coordinate system is converted to the global coordinate system, the conversion tensor Q required is

[0064]

[0065] Among them, α and β are the direction angles mixed in the global coordinate system. There is no need to calculate the specific values ​​of α and β, as they will be integrated in the subsequent calculations. If it is assumed that the fourth-order tensor F, the third-order tensor T, and the 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

[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 subscripts in formula 5-7 can be 1, 2, or 3, and the Einstein summation convention is implemented;

[0070] 3.4 Expression for determining the effective phase velocity of piezoelectric composites

[0071] 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;

[0072]

[0073] Among them, the following combination parameters are used:

[0074]

[0075] M2=A1sin 4 θ+A2cos 4 θ+A3cos 2 θsin 2 θ (12)

[0076]

[0077]

[0078] 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 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 to be determined are obtained by solving the self-consistent equations in sub-step 3.8;

[0079] 3.5 Determine the expression for the structural coefficient h

[0080] The structural coefficient h will be used in subsequent calculations, and its expression is

[0081]

[0082] The expression of parameter b is as follows:

[0083]

[0084] Among them, 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, 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 overall coordinate system, and there is no need to calculate the specific values ​​of α and β; k is the wave number, and its expression is:

[0085]

[0086] ω=2πf (30)

[0087] 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, when subscript N=1, it is a quasi-P wave, when subscript N=2, it is a quasi-SR wave, when subscript N=3, it is 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;

[0088] 3.6 Determine the expression of the volume average tensor of the five effective Green operators of piezoelectric composites

[0089] 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:

[0090]

[0091] 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;

[0092] n=(sinθcosφ,sinθsinφ,cosθ) (36)

[0093] Here, θ and φ do not need specific values; is the polarization vector U N The component of N is 1, 2, or 3;

[0094]

[0095]

[0096] Q1=B4 sin 4 θ+B5 sin 2 θcos 2 θ (46)

[0097] Q2=-B1 sin 4 θ+B2 sin 2 θcos 2 θ+B3 cos 4 θ (47)

[0098] Where M1 is shown in formula (11); α 下标 is the component of the vector α

[0099]

[0100] The remaining parameters in formulas (31)-(35) are as follows:

[0101]

[0102] F N =T[1-χ(z N )] (52)

[0103]

[0104] Among them, v in formula (51) NSee 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;

[0105] From this step, we know that the volume average tensor of the effective Green's 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 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 to be determined will be obtained by solving the self-consistent equations in sub-step 3.8;

[0106] 3.7 Determine the expressions of the four combined parameter tensors in the self-consistent equations

[0107] Four combined parameter tensor functions V will be used in the self-consistent equations (1) , V (2) , V (3) and V (4) , their expressions are:

[0108] V (1) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)=h 2 [(C n′ -C m ):J1-(e n′ -e m ) T ·J2] (55)

[0109] V (2) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)=h 2 [(en′ -e m ):J1+(κ n′ -κ m )·J2] (56)

[0110] V (3) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)=h 2 [(κ n′ -κ m )·J4-(e n′ -e m ):J3] (57)

[0111] V (4) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)=h 2 (ρ n′ -ρ m )·J5 (58)

[0112] The expression of the structural coefficient h is shown 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) All are 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 , the 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 , the dielectric tensor κ n and the density tensor ρ n It will be obtained through step 1; the expression of tensor J1-J5 is:

[0113] J1=B1:R+F1·W T (59)

[0114] J2=A1·W T -F1 T :R (60)

[0115] J3 = B1:W - F1·Z (61)

[0116] J4 = F1 T :W + A1·Z (62)

[0117]

[0118]

[0119] Wherein:

[0120]

[0121] A1 = [(e n′ - e 0 ):(C n′ - C 0 ) -1 :(e n′ - e 0 ) T +(κ n′ - κ 0 )] -1 (72)

[0122] 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)

[0123] F1 = (C n′ - C 0 ) -1 :(e n′ - e 0 ) T ·A1 (74)

[0124] F1 T = A1·(e n′ - e 0 ):(C n′ - C 0 ) -1 (75)

[0125] In formulas (55)-(75), the operation 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 Respectively 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 of the inclusion phase in the local coordinate system C n , e n , κ n and ρ n , according to the coordinate transformation rule of sub-step 3.3, it is converted to a tensor in the global coordinate system; the material property C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , obtained in step 1;

[0126] 3.8 Establishment of a self-consistent set of equations for the dynamic effective properties of piezoelectric composites

[0127] Based on the effective medium theory of micromechanics, 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:

[0128] C 0 =C m +c n <V (1) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (76)

[0129] e 0 =e m +c n <V (2) (C 0 ,e 0 ,κ 0 ,ρ0 ,θ,φ,ω,α,β)> (77)

[0130] κ 0 =κ m +c n <V (3) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (78)

[0131] ρ 0 =ρ m +c n <V (4) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (79)

[0132] 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, the effective piezoelectric tensor, the effective dielectric tensor, and the 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;

[0133] 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:

[0134]

[0135] The function P λ (α) See sub-step 3.2; parameter λ has been obtained in step 2;

[0136] 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 determined, the system of equations is self-consistent and can be numerically solved by an iterative method.

[0137] Preferably, the calculation and extraction of the dynamic effective parameters of the piezoelectric composite material in step 4 are performed according to the following sub-steps:

[0138] 4.1 Using the material parameters C of the inclusion phase obtained in step 1 n , e n , κ n and ρ n , material parameters of the matrix phase C m , e m , κ m and ρ m , and the volume fraction of the inclusion phase obtained in step 2 is c n , the radius a of the inclusion, the cross-sectional coefficient δ and the parameter λ of the degree of 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 iteration 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 the temporary C 0 , e 0 , κ 0 and ρ 0 , continue to substitute into the right side of the equal sign for calculation, repeat in sequence until convergence, and the final predicted dynamic effective parameter C of the piezoelectric composite material is obtained. 0 , e 0 , κ0 and ρ 0 ;

[0139] 4.2 Calculate the effective wave propagation characteristics of the piezoelectric composite material; Substitute the effective parameter C obtained in substep 4.1 0 , e 0 , κ 0 and ρ 0 Substitute into formulas (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.

[0140] Preferably, the drawing of the prediction curve of the dynamic effective property and the verification of the prediction model in step 5 are performed according to the following scheme:

[0141] 5.1 Substitute the determined volume fraction of the inclusion phase into the established model considering different frequencies to obtain a complete prediction curve of the dynamic effective performance of the piezoelectric composite material changing with 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;

[0142] 5.2 Considering different volume fractions of the included phase, draw another prediction curve of the dynamic effective performance of the piezoelectric composite material changing with 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.

[0143] Beneficial effects of the present invention:

[0144] 1. The present invention 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 properties of the piezoelectric composite material, and has a clear physical meaning. The macroscopic dynamic effective parameters of the piezoelectric composite material can be predicted by using the experimental data parameters obtained by experiments or consulting the existing technical materials, and the effective wave propagation characteristics when the elastic wave propagates in the piezoelectric composite material, such as attenuation, effective phase velocity, effective acoustic impedance, etc.

[0145] 2. The method for predicting the dynamic effective properties of piezoelectric composite materials based on the effective medium theory of micromechanics of the present invention has obvious advantages in efficiency and economy, and can solve the shortcomings of long cycle and high cost of dynamic characteristic test experiments of piezoelectric composite materials.

[0146] 3. The present invention is not only applicable to piezoelectric composite materials, but also to the prediction of dynamic effective properties of pure elastic composite materials, and it is only necessary to degenerate the piezoelectric coefficient in the model to zero.

[0147] 4. The present invention is applicable to the situations where the inclusions are completely aligned, the inclusions are completely random, and the inclusion directions are arbitrary and satisfy certain direction preferences, and therefore has the characteristics of a wide range of applications.

[0148] 5. There is currently no dynamic homogenization theory based on material microstructure information that links the distribution of inclusion direction angles with the macroscopic dynamic effective properties of piezoelectric composites. The present invention introduces a function of the distribution of inclusion direction angles to characterize the degree of alignment of inclusions. Inclusions can be completely aligned, or they can be misaligned but have a certain directional preference, or they can be completely random in direction. The present invention uses a self-consistent method to establish a prediction model for the macroscopic dynamic effective properties of piezoelectric composites containing inclusions in arbitrary directions, thereby obtaining a continuous change relationship between dynamic effective properties and frequency. BRIEF DESCRIPTION OF THE DRAWINGS

[0149] Figure 1 Schematic diagram of the geometric configuration of a piezoelectric composite material containing inclusions in arbitrary directions;

[0150] Figure 2 For the general Figure 1 Schematic diagram of the geometric configuration transformed into a single inclusion scattering problem in an equivalent medium.

[0151] Figure 3 The figure compares the predicted value of the dynamic effective acoustic impedance of the PZT501A / Spurrs epoxy 1-3 piezoelectric composite material with the experimental value based on the literature [3].

[0152] Figure 4 The figure is a comparison between the predicted value of the dynamic effective phase velocity of the pure elastic composite material of lead / EPON828Z epoxy 0-3 with a volume fraction of 0.05 and the experimental value based on the literature [5].

[0153] Figure 5 The figure is a comparison between the predicted value of the dynamic effective phase velocity of the pure elastic composite material of lead / EPON828Z epoxy 0-3 with a volume fraction of 0.15 and the experimental value based on the literature [5]. DETAILED DESCRIPTION

[0154] Embodiment 1: This embodiment provides 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 comprises the following five steps;

[0155] Step 1: Determine the inclusion material and matrix material to obtain the material parameters of each component phase

[0156] 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;

[0157] The material parameters of each component phase in step 1 are obtained according to the following sub-steps:

[0158] 1.1 The fourth-order elastic tensor of the inclusion phase and the matrix phase is C x , the superscript x=n represents the inclusion phase parameter, and the 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 , they represent the elastic modulus in different directions, obtained by measurement;

[0159] 1.2 The third-order piezoelectric tensor of the inclusion phase and the matrix phase is e x , the superscript x=n represents the inclusion phase parameter, and the 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 , they represent the piezoelectric coefficients in different directions, which are obtained by measurement; if a phase material does not have piezoelectricity, its piezoelectric coefficient is set to zero;

[0160] 1.3 The second-order dielectric tensor of the inclusion phase and the matrix phase is κ x , the superscript x=n represents the inclusion phase parameter, the 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;

[0161] 1.4 The second-order dielectric density tensor of the inclusion phase and the matrix phase is ρ x , the superscript x=n represents the inclusion phase parameter, the 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 is obtained by measurement;

[0162] Step 2: Determine the inclusion material and matrix material according to step 1, and prepare or obtain the piezoelectric composite material sample

[0163] 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 degree parameter λ of the inclusion, the section coefficient δ of the inclusion, and the inclusion radius a; then the effective phase velocity and the effective acoustic impedance under the action of elastic waves are obtained;

[0164] The experimental data parameters in step 2 are obtained by measurement, and the effective phase velocity and effective acoustic impedance under the action of elastic waves in step 2 are obtained by measurement;

[0165] The step 2 includes the following sub-steps:

[0166] 2.1 Preparation of piezoelectric composite samples with an inclusion phase volume fraction of c n It is known that the volume fraction of the matrix phase c m =1-c n ;

[0167] 2.2 The prepared piezoelectric composite material, based on the SEM image of the sample, measured the radius a of the inclusion and the cross-sectional coefficient δ;

[0168] 2.3 The piezoelectric composite material prepared was analyzed according to the SEM image of the sample, and the probability of different values ​​of the inclusion direction angle α was calculated according to the following formula: The probability distribution function P of the inclusion direction λ (α) Inversely calculate the parameter λ of the degree of inclusion alignment;

[0169]

[0170] 2.4 The piezoelectric composite material prepared is used to measure the effective wave propagation characteristics of the piezoelectric composite material sample under the action of elastic waves, including at least the effective phase velocity and the effective acoustic impedance;

[0171] Step 3: Dynamic effective property prediction model-establishment of self-consistent equations

[0172] Based on the effective medium theory of micromechanics, a self-consistent equation group about the dynamic effective parameters of the piezoelectric composite material is established; the dynamic effective property prediction model in step 3 - the self-consistent equation group is established according to the following sub-steps:

[0173] 3.1 In step 3, a prediction model for the dynamic effective properties of piezoelectric composite materials containing arbitrary directional inclusions will be established, that is, a set of self-consistent equations about macroscopic dynamic effective parameters; it is divided into 7 parts;

[0174] In 3.2, set the microstructure information parameters of the piezoelectric composite material;

[0175] In 3.3, set the coordinate system transformation rules of the tensor;

[0176] In 3.4, determine the expression for the effective phase velocity of the piezoelectric composite material;

[0177] In 3.5, determine the expression for the structural coefficient h;

[0178] In 3.6, determine the expression for the volume average tensor of the five effective Green operators of the piezoelectric composite;

[0179] In 3.7, determine the expressions for the four combined parameter tensors in the self-consistent system of equations;

[0180] In 3.8, a self-consistent set of equations for predicting the dynamic effective properties of piezoelectric composites is established;

[0181] 3.2 Setting microstructure information parameters of piezoelectric composite materials

[0182] In the overall coordinate system o-x1x2x3, the piezoelectric composite material exhibits transverse isotropy in the macroscopic view, and the symmetry axis is the x3 axis; the piezoelectric composite material is composed of a matrix phase and an inclusion phase, in which 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; can be expressed as

[0183] c m +c n =1 (1)

[0184] The positions of inclusions in the matrix are randomly distributed but do not overlap; the shape of the inclusion is a rotating ellipsoid with the rotation axis being the X3 axis. In the local coordinate system, the equation of the rotating ellipsoid is:

[0185]

[0186] 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;

[0187] The directions of inclusions in the matrix are randomly distributed. The direction angle of the X3 axis of the inclusion in the overall 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 direction angle β has no effect on the macroscopic properties. The probability distribution function of the inclusion direction angle α is expressed as

[0188]

[0189] Where λ and α are obtained in step 2; λ = 0 means that the orientation of the inclusions is completely randomly distributed; λ > 0 means that the orientation of the inclusions begins to show a certain directional preference; the larger the value of λ, the more obvious the directional preference of the inclusions, that is, the higher the degree of alignment; λ → ∞ means that the rotation axis of the inclusions is completely aligned with the x3 axis;

[0190] 3.3 Setting the coordinate system transformation rules for tensors

[0191] When the tensor expressed in the local coordinate system is converted to the global coordinate system, the conversion tensor Q required is

[0192]

[0193] Among them, α and β are the direction angles mixed in the global coordinate system. There is no need to calculate the specific values ​​of α and β, as they will be integrated in the subsequent calculations. If it is assumed that the fourth-order tensor F, the third-order tensor T, and the 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

[0194] F ij ' kl =Q pi Q qj Q rk Q sl F pqrs (5)

[0195] T ij ' k =Q pi Q qj Q rk T pqr (6)

[0196] S i ' j =Q pi Q qj S pq (7)

[0197] All subscripts in formula 5-7 can be 1, 2, or 3, and the Einstein summation convention is implemented;

[0198] 3.4 Expression for determining the effective phase velocity of piezoelectric composites

[0199] 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;

[0200]

[0201] Among them, the following combination parameters are used:

[0202]

[0203] M2=A1 sin 4 θ+A2cos 4 θ+A3 cos 2 θsin 2 θ (12)

[0204]

[0205]

[0206] 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 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 to be determined are obtained by solving the self-consistent equations in sub-step 3.8;

[0207] 3.5 Determine the expression for the structural coefficient h

[0208] The structural coefficient h will be used in subsequent calculations, and its expression is

[0209]

[0210] The expression of parameter b is as follows:

[0211]

[0212] Among them, 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, 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 overall coordinate system, and there is no need to calculate the specific values ​​of α and β; k is the wave number, and its expression is:

[0213]

[0214] ω=2πf (30)

[0215] 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, when subscript N=1, it is a quasi-P wave, when subscript N=2, it is a quasi-SR wave, when subscript N=3, it is 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;

[0216] 3.6 Determine the expression of the volume average tensor of the five effective Green operators of piezoelectric composites

[0217] 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:

[0218]

[0219] 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;

[0220] n=(sinθcosφ,sinθsinφ,cosθ) (36)

[0221] Here, θ and φ do not need specific values; U 下 N 标 is the polarization vector U N The component of N is 1, 2, or 3;

[0222]

[0223] Q1=B4 sin 4 θ+B5 sin 2 θcos 2 θ (46)

[0224] Q2=-B1 sin 4 θ+B2 sin 2 θcos 2 θ+B3 cos 4 θ (47)

[0225] Where M1 is shown in formula (11); α 下标 is the component of the vector α

[0226]

[0227] The remaining parameters in formulas (31)-(35) are as follows:

[0228]

[0229] F N =T[1-χ(z N )] (52)

[0230]

[0231] 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;

[0232] From this step, we know that the volume average tensor of the effective Green's 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 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 to be determined will be obtained by solving the self-consistent equations in sub-step 3.8;

[0233] 3.7 Determine the expressions of the four combined parameter tensors in the self-consistent equations

[0234] Four combined parameter tensor functions V will be used in the self-consistent equations (1) , V (2) , V (3) and V (4) , their expressions are:

[0235] V (1) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)=h2 [(C n′ -C m ):J1-(e n′ -e m ) T ·J2] (55)

[0236] V (2) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)=h 2 [(e n′ -e m ):J1+(κ n′ -κ m )·J2] (56)

[0237] V (3) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)=h 2 [(κ n′ -κ m )·J4-(e n′ -e m ):J3] (57)

[0238] V( 4) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)=h 2 (ρ n′ -ρ m )·J5 (58)

[0239] The expression of the structural coefficient h is shown 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) All are 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 , the 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 , the dielectric tensor κ n and the density tensor ρ n It will be obtained through step 1; the expression of tensor J1-J5 is:

[0240] J1=B1:R+F1·W T (59)

[0241] J2=A1·W T -F1 T :R (60)

[0242] J3=B1:W-F1·Z (61)

[0243] J4=F1 T :W+A1·Z (62)

[0244]

[0245]

[0246] in:

[0247]

[0248] A1=[(e n′ -e 0 ):(C n′ -C 0 ) -1 :(e n′ -e 0 ) T +(κ n′ -κ 0 )] -1 (72)

[0249] 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)

[0250] F1=(C n′-C 0 ) -1 :(e n′ -e 0 ) T ·A1 (74)

[0251] F1 T =A1·(e n′ -e 0 ):(C n′ -C 0 ) -1 (75)

[0252] In formulas (55)-(75), the operation 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 Respectively 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 of the inclusion phase in the local coordinate system C n , e n , κ n and ρ n , according to the coordinate transformation rule of sub-step 3.3, it is converted to a tensor in the global coordinate system; the material property C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , obtained in step 1;

[0253] 3.8 Establishment of a self-consistent set of equations for the dynamic effective properties of piezoelectric composites

[0254] Based on the effective medium theory of micromechanics, 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:

[0255] C 0 =C m +cn <V (1) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (76)

[0256] e 0 =e m +c n <V (2) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (77)

[0257] κ 0 =κ m +c n <V (3) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (78)

[0258] ρ 0 =ρ m +c n <V (4) (C 0 ,e 0 ,κ 0 ,ρ 0 ,θ,φ,ω,α,β)> (79)

[0259] 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, the effective piezoelectric tensor, the effective dielectric tensor, and the 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;

[0260] 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:

[0261]

[0262] The function P λ (α) See sub-step 3.2; parameter λ has been obtained in step 2;

[0263] 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 determined, the system of equations is a self-consistent system of equations and is numerically solved by an iterative method.

[0264] Step 4: Calculate the dynamic effective properties of the composite material

[0265] Substituting the material parameters of the inclusion phase and the matrix phase obtained in step 1 and the experimental data parameters obtained in step 2 into the prediction model established in step 3, i.e., the self-consistent equation group, the dynamic effective material parameters are obtained through numerical calculation;

[0266] Then the effective elastic wave propagation characteristics are calculated and a complete prediction model is obtained;

[0267] 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:

[0268] 4.1 Using the material parameters C of the inclusion phase obtained in step 1 n , e n , κ n and ρ n , material parameters of the matrix phase C m , e m , κ m and ρ m , and the volume fraction of the inclusion phase obtained in step 2 is c n, the radius a of the inclusion, the cross-sectional coefficient δ and the parameter λ of the degree of 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 iteration 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 the temporary C 0 , e 0 , κ 0 and ρ 0 , continue to substitute into the right side of the equal sign for calculation, 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 ;

[0269] 4.2 Calculate the effective wave propagation characteristics of the piezoelectric composite material; Substitute the effective parameter C obtained in substep 4.1 0 , e 0 , κ 0 and ρ 0 Substitute into formulas (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.

[0270] The steps one and two described are not in any particular order.

[0271] Step 5: Draw the prediction curve of dynamic effective properties and verify the prediction model

[0272] Substituting different frequencies into the established model, a prediction curve of the dynamic effective properties of the complete piezoelectric composite material changing with 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.

[0273] The drawing of the prediction curve of the dynamic effective property and the verification of the prediction model in step 5 are carried out according to the following scheme:

[0274] 5.1 Substitute the determined volume fraction of the inclusion phase into the established model considering different frequencies to obtain a complete prediction curve of the dynamic effective performance of the piezoelectric composite material changing with 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;

[0275] 5.2 Considering different volume fractions of the included phase, draw another prediction curve of the dynamic effective performance of the piezoelectric composite material changing with frequency, compare the prediction curve with the effective phase velocity or effective acoustic impedance under the action of elastic waves obtained in step 2, and verify the prediction model.

[0276] Embodiment 2: Embodiment 2 is basically the same as Embodiment 1, and the similarities are not repeated here, and the differences are as follows:

[0277] The material parameters of each component phase are obtained in step 1 of embodiment 2 according to the following sub-steps:

[0278] 1.1 The fourth-order elastic tensor of the inclusion phase and the matrix phase is C x , the superscript x=n represents the inclusion phase parameter, and the 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 looking up the table;

[0279] 1.2 The third-order piezoelectric tensor of the inclusion phase and the matrix phase is e x , the superscript x=n represents the inclusion phase parameter, and the 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 , they represent the piezoelectric coefficients in different directions, which are obtained by looking up the table; if a phase material does not have piezoelectricity, its piezoelectric coefficient is set to zero;

[0280] 1.3 The second-order dielectric tensor of the inclusion phase and the matrix phase is κ x , the superscript x=n represents the inclusion phase parameter, the 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 constants in different directions and are obtained by looking up the table;

[0281] 1.4 The second-order dielectric density tensor of the inclusion phase and the matrix phase is ρ x , the superscript x=n represents the inclusion phase parameter, the 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 is obtained by looking up the table;

[0282] Step 2 in Embodiment 2: The experimental data parameters in step 2 are obtained by consulting the prior art data, and the effective phase velocity and effective acoustic impedance under the action of elastic waves in step 2 are obtained by consulting the prior art data;

[0283] The step 2 includes the following sub-steps:

[0284] 2.1 Obtained piezoelectric composite material samples, by consulting existing technical data to obtain the volume fraction of the inclusion phase c of the existing piezoelectric composite material n ;

[0285] 2.2 Obtain the piezoelectric composite material sample, and obtain the radius a and section coefficient δ of the inclusion by consulting the existing technical data;

[0286] 2.3 The obtained piezoelectric composite material sample obtains the alignment degree parameter λ of the inclusion by consulting the existing technical data;

[0287] 2.4 Obtain the piezoelectric composite material sample, and obtain the macroscopic effective wave propagation characteristics of the sample by consulting the existing technical data, including at least the effective phase velocity and effective acoustic impedance.

[0288] In Example 2, the material parameters of each component phase are obtained by looking up a table, and the experimental data parameters, effective phase velocity, and effective acoustic impedance of the composite material are obtained by referring to the existing technical information. Although they are different from those measured by experiments in Example 1, they are also data obtained by technicians in this field in their respective research and experiments before the technical solution of the present invention is proposed. They are also a kind of experimental data and can be borrowed and used directly.

[0289] Verification Example 1 based on Example 2:

[0290] In order to facilitate the understanding of the present invention, the present invention will be fully described below in conjunction with an example of a PZT501A / Spurrs epoxy 1-3 type piezoelectric composite material.

[0291] 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:

[0292] 1. According to the literature [3], the inclusion phase is PZT501A piezoelectric material, and its elastic tensor C n The non-zero components of and Its piezoelectric tensor e n The non-zero components of and Its dielectric tensor κ n The non-zero components of and κ0=8.85*10 -12 C 2 / Nm 2 ; Its density tensor ρ n The non-zero components of

[0293] 2. According to the literature [3], the matrix phase is Spurrs epoxy, and its elastic tensor C m The non-zero components of and Its dielectric tensor κ m The non-zero components of Its density tensor ρ m The non-zero components of

[0294] 3. According to the record in reference [3], the radius of the inclusion is a = 0.225 mm, the cross-sectional coefficient is δ = 4.4; the parameter λ of the inclusion alignment degree is 10; Reference [3] experimentally measured samples with inclusion volume fractions of 10% and 20%, so the inclusion phase volume fraction c is taken in the theoretical model respectively. n = 0.1 and c n =0.2;

[0295] 4. Reference [3] experimentally measured the effective acoustic impedance of PZT501A / Spurrsepoxy 1-3 piezoelectric composites with inclusion volume fractions of 10% and 20% 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 triangles and hollow circles in .

[0296] 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 equation group. First, the effective material parameter C is numerically calculated by the iteration method. 0 , e 0 , κ 0 and ρ 0Continuous 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. 2; 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 = 0.2 The curve of the predicted data, such as Figure 3 shown.

[0297] 6. If Figure 3 As shown, the curve of the predicted data of this prediction model is consistent with the effective acoustic impedance measured experimentally.

[0298] Verification Example 2 based on Example 2:

[0299] In order to facilitate the understanding of the present invention, the present invention will be fully described below in conjunction with an example of a PZT501A / Spurrs epoxy 0-3 type piezoelectric composite material.

[0300] 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:

[0301] 1. According to the literature [4], the inclusion phase is PZT501A piezoelectric material, and its elastic tensor C n The non-zero components of and Its piezoelectric tensor e n The non-zero components of and Its dielectric tensor κ n The non-zero components of and κ0=8.85*10 -12 C 2 / Nm 2 ; Its density tensor ρ n The non-zero components of

[0302] 2. According to the literature [4], the matrix phase is Spurrs epoxy, and its elastic tensor C m The non-zero components of and Its dielectric tensor κ m The non-zero components of Density tensor ρ mThe non-zero components of

[0303] 3. Reference [4] experimentally measured the inclusion radius a = 0.75 μm and the inclusion volume fraction c when a 5 MHz elastic longitudinal wave propagated along the direction n = (0, 0, 1) (i.e., θ = 0). n = 0.1 and 0.2, the measured values ​​are shown in Table 1 below.

[0304] 4. According to the literature [4], the inclusion radius a = 0.75 μm, the cross-sectional coefficient δ = 1; the inclusion alignment parameter λ is taken as 0;

[0305] Substituting the above microstructure information and frequency parameters into the prediction model established by the present invention, i.e., the self-consistent equations, the final predicted effective acoustic impedance data is shown in Table 1. It can be seen that the predicted data of the prediction model is consistent with the experimental data.

[0306] Table 1 Comparison of acoustic impedance of 0-3 type piezoelectric composites

[0307]

[0308] Verification Example 3 based on Example 2:

[0309] In order to facilitate the understanding of the present invention, the present invention will be fully described below in conjunction with an example of the lead / EPON828Z epoxy 0-3 type pure elastic composite material.

[0310] 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:

[0311] 1. According to the literature [5], the inclusion phase is lead particle material, and its elastic tensor C n The non-zero components of and Density tensor ρ n The non-zero components of

[0312] 2. According to the literature [5], the matrix phase is EPON828Z epoxy, and its elastic tensor C m The non-zero components of and Density tensor ρ m The non-zero components of

[0313] 3. Reference [5] experimentally measured the variation of effective longitudinal wave velocity v1 in the frequency range of 0-1.5MHz. Inclusion radius a = 660μm, inclusion volume fraction c n = 0.05 and 0.15 for the effective longitudinal wave velocity v1 Figure 4 and Figure 5 The hollow circle in .

[0314] 4. According to the literature [5], the radius of the inclusion is a = 660 μm, the cross-sectional coefficient is δ = 1; the parameter λ of the inclusion alignment degree is 0; the above microstructure information and frequency parameters are substituted into the prediction model established by the present invention, i.e., the self-consistent equation group. The fitting curve of the final predicted effective longitudinal wave velocity v1 is shown in Figure 4 and Figure 5 The solid line in .

[0315] Will Figure 4 and 5 Compared with the fitting curve, the hollow circle in the figure shows that the effective longitudinal wave velocity v1 predicted by this prediction model is consistent with the effective longitudinal wave velocity v1 measured in the experiment.

Claims

1. A method for predicting 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 to obtain the material parameters of each component phase 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; 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 degree parameter λ of the inclusion, the section coefficient δ of the inclusion, and the radius a of the inclusion; then the effective phase velocity and the 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 Substituting the material parameters of the inclusion phase and the matrix phase obtained in step 1 and the experimental data parameters obtained in step 2 into the prediction model established in step 3, i.e., the self-consistent equation group, 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 Substitute different frequencies into the established model to obtain a complete prediction curve of the dynamic effective properties of the piezoelectric composite material changing with 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; The steps one and two described are not in any particular order.

2. The method for predicting dynamic effective properties of a piezoelectric composite material according to claim 1, characterized in that: 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 , the superscript x=n represents the inclusion phase parameter, and the 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 , they 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 , the superscript x=n represents the inclusion phase parameter, and the 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 , they represent the piezoelectric coefficients in different directions, which are 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 , the superscript x=n represents the inclusion phase parameter, the 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 , they 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 , the superscript x=n represents the inclusion phase parameter, the 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 is obtained by measurement or table lookup.

3. The method for predicting dynamic effective properties of a piezoelectric composite material according to claim 1, characterized in that: The experimental data parameters in step 2 are obtained by measurement or by referring to existing technical data, and 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, characterized in that: Step 2 includes the following sub-steps: 2.1 Preparation of piezoelectric composite samples with an inclusion phase volume fraction of c n It is known that the volume fraction of the matrix phase c m =1-c n ; If the obtained 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 The prepared piezoelectric composite material, based on the SEM image of the sample, measured the radius a of the inclusion and the cross-sectional coefficient δ; If the obtained piezoelectric composite material sample is used, the radius a and the cross-sectional coefficient δ of the inclusion are obtained by consulting the existing technical data; 2.3 The piezoelectric composite material prepared was analyzed according to the SEM image of the sample, and the probability of different values ​​of the inclusion direction angle α was calculated according to the following formula: The probability distribution function P of the inclusion direction 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 The piezoelectric composite material prepared is used to measure the effective wave propagation characteristics of the piezoelectric composite material sample under the action of elastic waves, including at least the effective phase velocity and the 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, characterized in that: 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 directional inclusions will be established, that is, a set of self-consistent equations about 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 average tensor of the five effective Green operators of the piezoelectric composite; 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 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, and the symmetry axis is the x3 axis; the piezoelectric composite material is composed of a matrix phase and an inclusion phase, in which 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; 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 shape of the inclusion 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 direction angle of the X3 axis of the inclusion in the overall 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 direction angle β has no effect on the macroscopic properties. The probability distribution function of the inclusion direction angle α is expressed as Where λ and α are obtained in step 2; λ = 0 means that the orientation of the inclusions is completely randomly distributed; λ > 0 means that the orientation of the inclusions begins to show a certain directional preference; the larger the value of λ, the more obvious the directional preference of the inclusions, that is, the higher the degree of alignment; λ → ∞ means that the rotation axis of the inclusions is completely aligned with the x3 axis; 3.3 Setting the coordinate system transformation rules for tensors When the tensor expressed in the local coordinate system is converted to the global coordinate system, the conversion tensor Q required is Among them, α and β are the direction angles mixed in the global coordinate system. There is no need to calculate the specific values ​​of α and β, as they will be integrated in the subsequent calculations. If it is assumed that the fourth-order tensor F, the third-order tensor T, and the 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 formula 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 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 to be determined 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: Among them, 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, 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 overall coordinate system, and there is no need to calculate the specific values ​​of α and β; 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, when subscript N=1, it is a quasi-P wave, when subscript N=2, it is a quasi-SR wave, when subscript N=3, it is 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 piezoelectric composites 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, or 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 the 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 step, we know that the volume average tensor of the effective Green's 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 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 to be determined 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 equations Four combined parameter tensor functions V will be used in the self-consistent equations (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 substep 3.5; the fourth-order tensor V (1) , the third-order tensor V (2) , the second-order tensor V (3) and V (4) All are 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 , the 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 , the 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 operation 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 Respectively 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 of the inclusion phase in the local coordinate system C n , e n , κ n and ρ n , according to the coordinate transformation rule of sub-step 3.3, it is converted to a tensor in the global coordinate system; the material property C of the inclusion phase in the local coordinate system n , e n , κ n and ρ n , obtained in step 1; 3.8 Establishment of a self-consistent set of equations for the dynamic effective properties of piezoelectric composites Based on the effective medium theory of micromechanics, 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, the effective piezoelectric tensor, the effective dielectric tensor, and the 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; 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 determined, 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 , material parameters of the matrix phase C m , e m , κ m and ρ m , and the volume fraction of the inclusion phase obtained in step 2 is c n , the radius a of the inclusion, the cross-sectional coefficient δ and the parameter λ of the degree of 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 iteration 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 the temporary C 0 , e 0 , κ 0 and ρ 0 , continue to substitute into the right side of the equal sign for calculation, 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; Substitute the effective parameter C obtained in substep 4.1 0 , e 0 , κ 0 and ρ 0 Substitute into formulas (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: The prediction curve of the dynamic effective property and the verification of the prediction model in step 5 are drawn according to the following scheme: 5.1 Substitute the determined volume fraction of the inclusion phase into the established model considering different frequencies to obtain a complete prediction curve of the dynamic effective performance of the piezoelectric composite material changing with 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; 5.2 Considering different volume fractions of the included phase, draw another prediction curve of the dynamic effective performance of the piezoelectric composite material changing with 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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