Method for piezoelectric double curved plate wave propagation analysis of functionally graded sandwich

Abstract

The application discloses a kind of functionally graded sandwich piezoelectric double curved plate wave propagation analysis method, based on plane bending hypothesis, the reduction constitutive equation of piezoelectric layer and functionally graded sandwich considering piezoelectric effect and thermal effect is constructed;In the framework of first-order shear deformation theory, the wave transmission characteristic control equation set is derived by Hamilton variation principle;By extending harmonic method, the dispersion relation of piezoelectric double curved plate of functionally graded sandwich is obtained analytically, and the relationship between the first three order phase velocities and wave numbers is further numerically solved.The analysis method provided by the application overcomes the problem of multi-physical field coupling and has strong applicability.The method extends the harmonic method for piezoelectric effect, accurately obtains the first three order phase velocities of piezoelectric double curved plate corresponding to piezoelectric effect and other design variables, and reveals the influence mechanism of piezoelectric-based synergistic effect, effectively providing technical reference for the structure design of piezoelectric double curved plate in multi-physical field.

Description

Technical Field

[0001] This application belongs to the field of multiphysics dynamics technology of composite material structures, specifically relating to a wave propagation analysis method for a functionally graded sandwich piezoelectric hyperbolic plate. Background Technology

[0002] Hyperbolic plates are widely used in aircraft structural systems. In the flight environment, hyperbolic plates may experience damage. Wave propagation studies can be used for structural health monitoring. Currently, the fundamental theories surrounding hyperbolic plates are constantly being refined, and static and dynamic problems and phenomena are being continuously revealed. A series of analytical methods have been proposed to explore the static behavior of hyperbolic plates, showing that their bending behavior is influenced by geometric characteristics. Furthermore, the stability of hyperbolic plates is often significantly affected by multiphysics fields. Through the analysis of the static performance and dynamic characteristics of hyperbolic plates, the necessity of wave propagation analysis in hyperbolic plates is increasingly being highlighted. With the development of intelligent structural systems, piezoelectric materials are gradually being applied to basic aircraft components.

[0003] Piezoelectric materials, due to their electromechanical coupling effect, have inspired extensive interdisciplinary research and are continuously expanding their applications into emerging fields such as intelligent structural systems and electromechanical systems. The ongoing development of piezoelectric structures highlights their broad application prospects in the field of aircraft. The piezoelectric layer influences structural performance through internal energy conversion. Therefore, its impact on the mechanical properties of hyperboloids, particularly wave propagation characteristics, is significant. Currently, the mechanism by which the piezoelectric effect affects mechanical behavior has attracted considerable attention. In the static problem of piezoelectric hyperboloids, the application of an external voltage directly leads to internal deformation, further resulting in changes in lateral deflection under lateral load conditions. The mechanical properties of piezoelectric structures in thermal environments are significantly affected by temperature. Furthermore, the piezoelectric layer exhibits electromechanical coupling characteristics, and the degree of coupling is influenced by external factors. Therefore, analyzing the wave propagation problem of piezoelectric structures, especially the influence of the piezoelectric effect on wave propagation characteristics under different external factors, is beneficial for understanding their dynamic characteristics and failure mechanisms, providing reliable technical references for the intelligent development of aircraft structures. However, the current proposed piezoelectric structure dynamic analysis methods do not involve wave propagation of functionally graded sandwich piezoelectric hyperboloids. The synergistic effect and influence mechanism of piezoelectric effect, thermal effect and other factors on wave propagation characteristics need further analysis. Summary of the Invention

[0004] This invention discloses a wave propagation analysis method for a functionally graded sandwich piezoelectric hyperboloid plate, which can effectively solve at least one technical problem involved in the background art.

[0005] To achieve the above objectives, the technical solution of the present invention is as follows:

[0006] A method for wave propagation analysis of a functionally graded sandwich piezoelectric hyperboloid includes the following steps:

[0007] Step S1: Based on the structural characteristics, viscoelastic basis, and external voltage loading form of the piezoelectric hyperbolic plate, establish a first-order shear displacement field model, a reduced constitutive model, and an electric potential field model;

[0008] Step S2: Based on Hamilton's variational principle, establish the dynamic governing equations of the piezoelectric hyperboloid plate expressed in terms of generalized displacement and potential;

[0009] Step S3: Using the harmonic solution method, the shape functions of the generalized displacement and potential that satisfy the boundary conditions are proposed. The mode discretization of each order of the dynamic control equation system is realized by combining the Galerkin method. The dispersion relationship of the piezoelectric hyperboloid is obtained by the matching eigenvalue method. Then, the relationship between each order of phase velocity and wave number is solved numerically.

[0010] Step S4: Conduct model verification to determine the correctness of the displacement field model and the reduced constitutive model;

[0011] Step S5: Conduct wave propagation characteristic analysis of piezoelectric hyperboloid plates in multiphysics fields, and establish a design variable database based on piezoelectricity and wave propagation characteristic database of piezoelectric hyperboloid plates. The design variables include temperature field, piezoelectric layer to core layer thickness ratio, core layer functional gradient index, and viscoelastic foundation. The wave propagation characteristic database includes bending wave phase velocity and tensile wave phase velocity.

[0012] As a preferred improvement of the present invention, the piezoelectric hyperboloid plate includes a core layer and a piezoelectric layer; the core layer is composed of a functionally graded material and the reinforcing phase is symmetrically distributed in a power function form along the thickness direction; the piezoelectric layer is disposed on the upper and lower sides of the core layer to carry external voltage.

[0013] As a preferred improvement of the present invention, step S1 specifically includes:

[0014] Step S11: Establish the first-order shear displacement field and electric potential field based on the structural characteristics of the piezoelectric hyperboloid plate, and construct the strain field based on this.

[0015] Step S12: Based on the planar bending assumption, construct reduced constitutive relations for the core layer and the piezoelectric layer respectively.

[0016] As a preferred improvement of the present invention, step S3 includes:

[0017] Step S31: Using the harmonic solution method, based on the multi-physics field, the shape functions of the displacement field and electric potential field are proposed. The expressions for the displacement field and electric potential field in harmonic form are as follows:

[0018]

[0019]

[0020]

[0021] φ β =φ β * exp(iαk α +iβk β -iωt)

[0022] w = w * exp(iαk α +iβk β -iωt)

[0023] ψ=ψ * exp(iαk α +iβk β -iωt)

[0024] in, For planar displacement, φ α φ β For the cross-sectional angle, ω is the amplitude, k is the wave number in the α and β directions, and ω is the angular frequency;

[0025] Step S32: By discretizing the dynamic governing equations using the harmonic form solution, the dispersion relation of the piezoelectric hyperboloid is obtained:

[0026] {[K]+ω[C]+ω 2 [M]}{X}=0

[0027] Where K is the stiffness matrix, C is the damping matrix, M is the mass matrix, and X is the generalized displacement field and electric potential field;

[0028] Step S33: Solve the relationship between phase velocity and wave number for each order by numerical calculation, where the first order corresponds to bending wave and the second and third orders correspond to stretching wave.

[0029] As a preferred improvement of the present invention, step S4, namely, conducting model verification, includes:

[0030] Calculate the first-order dimensionless frequency of a piezoelectric hyperboloid with different power-law exponents;

[0031] Verify the modal frequencies of the piezoelectric hyperboloid plate;

[0032] Determine the correctness of the displacement field model, the reduced constitutive model, and the calculation using the harmonic solution method.

[0033] As a preferred improvement of the present invention, the analysis in step S5 includes: the synergistic effects of piezoelectricity on the temperature field, the thickness ratio of the piezoelectric layer to the core layer, the functional gradient index of the core layer, and the viscoelastic foundation on the first three phase velocities of the piezoelectric hyperboloid plate.

[0034] The beneficial effects of this invention are as follows:

[0035] 1. The method proposed in this invention overcomes the problem of multi-physics coupling, extends the harmonic method for the piezoelectric effect, and combines it with the Galerkin method to achieve mode separation and obtain the dispersion relation equations of each order after discretization.

[0036] 2. The method proposed in this invention has strong applicability. When effectively combined with the eigenvalue method, it can accurately obtain the phase velocities of the piezoelectric hyperboloid plate corresponding to the piezoelectric effect and other design variables, and reveal the synergistic effect mechanism based on piezoelectricity. The analysis accuracy meets the requirements of engineering applications.

[0037] 3. A database of design parameters and wave propagation characteristics was established through design variable analysis, covering the relationship between the first three phase velocities and wave numbers. Appropriate design parameters can be selected according to actual engineering design requirements, effectively promoting the parametric design of hyperboloids in multiphysics fields. Attached Figure Description

[0038] Figure 1 A flowchart illustrating the wave propagation analysis method for a functionally graded sandwich piezoelectric hyperboloid plate provided in this application embodiment;

[0039] Figure 2 A schematic diagram of the structure of a functionally graded sandwich piezoelectric hyperboloid plate provided in an embodiment of this application;

[0040] Figure 3 The diagram shows the synergistic effect of piezoelectricity and temperature on the relationship between the third-order phase velocity and wavenumber of a piezoelectric hyperboloid, provided in the embodiments of this application. In the diagram, (a-1) is the first-order phase velocity-wavenumber diagram, (a-2) is the second-order phase velocity-wavenumber diagram, and (a-3) is the third-order phase velocity-wavenumber diagram. Detailed Implementation

[0041] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0042] It should be noted that all directional indications (such as up, down, left, right, front, back, etc.) in the embodiments of the present invention are only used to explain the relative positional relationship and movement of each component in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indication will also change accordingly.

[0043] Furthermore, in this invention, descriptions involving "first," "second," etc., are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0044] In this invention, unless otherwise explicitly specified and limited, the terms "connection," "fixed," etc., should be interpreted broadly. For example, "fixed" can mean a fixed connection, a detachable connection, or an integral part; it can mean a mechanical connection or an electrical connection; it can mean a direct connection or an indirect connection through an intermediate medium; it can mean the internal communication of two components or the interaction between two components, unless otherwise explicitly limited. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0045] Furthermore, the technical solutions of the various embodiments of the present invention can be combined with each other, but only if they are feasible for those skilled in the art. If the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0046] Please see Figure 1 As shown in the figure, this application provides a method for wave propagation analysis of a functionally graded sandwich piezoelectric hyperboloid plate, including the following steps:

[0047] Step S1: Based on the structural characteristics, viscoelastic basis, and external voltage loading form of the piezoelectric hyperbolic plate, establish a first-order shear displacement field model, a reduced constitutive model, and an electric potential field model.

[0048] like Figure 2 As shown, the piezoelectric hyperboloid plate includes a core layer 1 and a piezoelectric layer 2. The core layer 1 is made of a functionally graded material and is symmetrically distributed in a power-law manner along the thickness direction. The piezoelectric layers 2 are disposed on the upper and lower sides of the core layer 1 to bear external voltage. The piezoelectric hyperboloid plate is assembled on a viscous Pasternak base.

[0049] Specifically, step S1 includes:

[0050] Step S11: Establish the first-order shear deformation displacement field Electric potential field [Eα,Eβ,E z ], and based on this, construct the strain field [ε αα ,ε ββ ,γ βz ,γ αz ,γ αβ];

[0051] The displacement field is constructed using first-order shear deformation theory, and the expression is as follows:

[0052]

[0053] in, Let represent the in-plane displacement and lateral displacement of each point on the hyperboloid. φ and w represent the displacements of points in the plane in the directions α, β, and z, respectively. α and φ β R is the rotation angle of the transverse normal in the directions α and β. α and R β Let be the radius of curvature of the hyperboloid in the α and β directions, and (α,β,z,t) be the spatial coordinates and time point.

[0054] The electric potential field expression that satisfies Maxwell's equations is as follows:

[0055]

[0056] Where ψ is the electric potential field on the mid-surface of the plate, ψ0 is the external voltage, and h p Z represents the thickness of the piezoelectric layer. p The potential distribution along the thickness direction, z P The expression is as follows:

[0057]

[0058] Where h is the thickness of the sandwich layer and z is the thickness direction variable.

[0059] The expression for the strain field is as follows:

[0060]

[0061]

[0062]

[0063]

[0064]

[0065] Step S12: Based on the planar bending assumption, construct reduced constitutive relations for the core layer and the piezoelectric layer respectively.

[0066] Specifically, the reduced constitutive relation of the sandwich layer is constructed under the assumption of planar bending, considering the stress-strain relationship of thermal effects:

[0067]

[0068] Where ΔT is the change in temperature field. and The coefficient of thermal expansion of the sandwich layer is _____. K represents the reduced stiffness factor of the sandwich layer. F The core layer is represented as follows:

[0069]

[0070]

[0071]

[0072] in, These are the elastic modulus and Poisson's ratio of the core layer as they vary along the thickness direction, and the variation pattern along the thickness direction is as follows:

[0073]

[0074]

[0075]

[0076]

[0077]

[0078] Where V is the volume fraction of each part of the composite material, ρ is the density, and the subscripts g and m represent the reinforcing phase and the matrix phase, respectively.

[0079] Specifically, for the piezoelectric layer, the constitutive equation considering both piezoelectric and thermal effects is expressed in detail as follows:

[0080]

[0081]

[0082] in, The reduced stiffness factor of the piezoelectric layer is represented by the superscript k. P Indicates a piezoelectric layer α and a β The coefficient of thermal expansion of the piezoelectric layer is... To reduce the voltage constant, Let be the dielectric constant. The detailed expression for the reduced stiffness coefficient is:

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] The reduced dielectric constant and dielectric constant are shown below:

[0089]

[0090]

[0091]

[0092]

[0093] Step S2: Based on Hamilton's variational principle, establish the dynamic governing equations of the piezoelectric hyperboloid plate expressed in terms of generalized displacement and potential. The specific expressions are as follows:

[0094]

[0095] Where, χ ij Ξ represents the generalized stiffness coefficient. ij ,S ij ,F ij H ij I represents the generalized electric coefficient. i Represents generalized density. This represents the planar displacement of points in the plate, with subscripts α and β representing the α and β directions respectively, and φ. α ,φ β N represents the cross-sectional rotation angle. E This represents the axial internal forces in the α and β directions caused by the piezoelectric effect.

[0096] Step S3: Using the harmonic solution method, shape functions of the generalized displacement and potential satisfying the boundary conditions are proposed. Combined with the Galerkin method, the modes of each order of the dynamic governing equations are discretized. The dispersion relation of the piezoelectric hyperboloid is obtained using the matched eigenvalue method. Then, the relationship between each order of phase velocity and wavenumber is numerically solved. Specifically, this includes:

[0097] Step S31: By employing the harmonic solution method, shape functions satisfying the boundary conditions for the displacement field and electric potential field are proposed around the multi-physics field. The expressions for the displacement field and electric potential field in harmonic form are as follows:

[0098]

[0099] φ α =φ α * exp(iαk α +iβk β -iωt)

[0100]

[0101] φ β =φ β * exp(iαk α +iβk β -iωt)

[0102] w = w * exp(iαk α +iβk β -iωt)

[0103] ψ=ψ * exp(iαkα+iβk β -iωt) (12)

[0104] in, ω is the amplitude, k is the wave number in the α and β directions, and ω is the angular frequency;

[0105] Step S32: By discretizing the dynamic governing equations using the harmonic form solution, the dispersion relation of the piezoelectric hyperboloid is obtained:

[0106] {[K]+ω[C]+ω 2 [M]}{X}=0 (13)

[0107] In the formula, K is the stiffness matrix, C is the damping matrix, M is the mass matrix, and X is the generalized displacement field and electric potential field; the detailed expressions for K, C, and M are as follows:

[0108]

[0109]

[0110]

[0111] Where, N Ta N Tβ This is the axial force due to thermal expansion.

[0112] Step S33: Obtain the relationship between the angular frequency ω and the wave number k by numerically solving the eigenvalues ​​of the dispersion relation, while determining the phase velocity of the piezoelectric hyperboloid plate. Therefore, the relationship between the third-order phase velocity and the wave number can be further obtained, where the first order corresponds to the bending wave, and the second and third orders correspond to the stretching wave.

[0113] Step S4: Conduct model verification, calculate the first dimensionless frequency of the piezoelectric hyperboloid with different power-law exponents, verify the modal frequencies of the piezoelectric hyperboloid, ensure the correctness of the displacement field model, the reduced constitutive model and the calculation method used in step S3, and ensure that the existing theoretical and calculation models can provide feasibility.

[0114] Step S5: Conduct wave propagation characteristics analysis of piezoelectric hyperboloids in multiphysics, establish a database of design variables based on piezoelectricity and wave propagation characteristics of piezoelectric hyperboloids, and effectively promote the parametric design of hyperboloids in multiphysics.

[0115] The design variables include the temperature field ΔT and the thickness ratio h between the piezoelectric layer and the core layer. p / h, functional gradient index N of the sandwich layer, and viscoelastic foundation k1,k2,C d The wave propagation characteristic database includes the phase velocity of bending waves c1 and the phase velocities of tensile waves c2 and c3.

[0116] The wave propagation analysis method of a functionally graded sandwich piezoelectric hyperboloid plate provided in this application will be described in detail below with reference to a specific embodiment.

[0117] Example 1

[0118] In this embodiment, the synergistic effect of piezoelectricity and temperature field on the wave propagation characteristics of the functionally graded piezoelectric hyperboloid is considered. The curvature of the hyperboloid in the α and β directions, the thickness of the functionally graded core layer, and the thickness of the piezoelectric layer are respectively taken as: R α =R β =1m;h P =h=0.02m. The functionally graded sandwich layer consists of ZrO2 and Ti-6Al-4V, with a power-law exponent (N) of 3 by default. Additionally, the piezoelectric layer is made of PZT-5A.

[0119] The embodiments described analyze the effect of temperature on the first three phase velocities of a hyperboloid, focusing on the coupling effect of thermal and piezoelectric effects. Figure 3 As shown in (a-1), increasing temperature leads to a decrease in the first-order phase velocity across the entire wavenumber range, with the sensitivity of the first-order phase velocity initially weakening and then increasing. This is because bending stiffness decreases with increasing temperature. On the other hand, the effect of the piezoelectric effect on the first-order phase velocity is modulated by temperature changes. Specifically, the increase in the first-order phase velocity (corresponding to k = 110) due to the piezoelectric effect is more pronounced at ΔT = 400 K, which is attributed to the deepening of the piezoelectric layer polarization. Figure 3 (a-2) shows the relationship between the second-order phase velocity and wavenumber of the hyperbolic plate. It can be observed that the phase velocity decreases with increasing temperature across the entire wavenumber range. Similarly, the tensile stiffness of the hyperbolic plate decreases directly depending on temperature. Figure 3In (a-3), the third-order phase velocity corresponding to the smaller wavenumber is very sensitive to temperature.

[0120] Therefore, it can be concluded that the analysis method of the present invention is effective for wave propagation analysis of functionally graded sandwich piezoelectric hyperboloids, and the design parameters and wave propagation characteristic database of functionally graded sandwich piezoelectric hyperboloids provided by the present invention have reference value for the structural design of piezoelectric hyperboloids in multiphysics fields.

[0121] The beneficial effects of this invention are as follows:

[0122] 1. The method proposed in this invention overcomes the problem of multi-physics coupling, extends the harmonic method for the piezoelectric effect, and combines it with the Galerkin method to achieve mode separation and obtain the dispersion relation equations of each order after discretization.

[0123] 2. The method proposed in this invention has strong applicability. When effectively combined with the eigenvalue method, it can accurately obtain the phase velocities of the piezoelectric hyperboloid plate corresponding to the piezoelectric effect and other design variables, and reveal the synergistic effect mechanism based on piezoelectricity. The analysis accuracy meets the requirements of engineering applications.

[0124] 3. A database of design parameters and wave propagation characteristics was established through design variable analysis, covering the relationship between the first three phase velocities and wave numbers. Appropriate design parameters can be selected according to actual engineering design requirements, effectively promoting the parametric design of hyperboloids in multiphysics fields.

[0125] The embodiments of this application have been described above with reference to the accompanying drawings. However, this application is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of this application without departing from the spirit and scope of the claims, and all of these forms are within the protection scope of this application.

Claims

1. A method for wave propagation analysis of a functionally graded sandwich piezoelectric hyperboloid plate, characterized in that, Includes the following steps: Step S1: Based on the structural characteristics, viscoelastic basis, and external voltage loading form of the piezoelectric hyperbolic plate, establish a first-order shear displacement field model, a reduced constitutive model, and an electric potential field model; Step S2: Based on Hamilton's variational principle, establish the dynamic governing equations of the piezoelectric hyperboloid plate expressed in terms of generalized displacement and potential; Step S3: Using the harmonic solution method, shape functions of the generalized displacement and potential satisfying the boundary conditions are proposed. Combined with the Galerkin method, the modes of each order of the dynamic control equations are discretized. The dispersion relation of the piezoelectric hyperboloid is obtained using the matched eigenvalue method. Then, the relationship between each order of phase velocity and wavenumber is numerically solved; including: Step S31: Using the harmonic solution method, based on the multi-physics field, the shape functions of the displacement field and electric potential field are proposed. The expressions for the displacement field and electric potential field in harmonic form are as follows: in, , For planar displacement, , For the cross-sectional angle, For amplitude, for and Wave number in direction, It is the angular frequency; Step S32: By discretizing the dynamic governing equations using the harmonic form solution, the dispersion relation of the piezoelectric hyperboloid is obtained: Where K is the stiffness matrix, C is the damping matrix, M is the mass matrix, and X is the generalized displacement field and electric potential field; Step S33: Solve the relationship between phase velocity and wave number for each order numerically, where the first order corresponds to bending wave and the second and third orders correspond to stretching wave; Step S4: Conduct model verification to determine the correctness of the displacement field model and the reduced constitutive model; Step S5: Conduct wave propagation characteristic analysis of piezoelectric hyperboloid plates in multiphysics fields, and establish a design variable database based on piezoelectricity and wave propagation characteristic database of piezoelectric hyperboloid plates. The design variables include temperature field, piezoelectric layer to core layer thickness ratio, core layer functional gradient index, and viscoelastic foundation. The wave propagation characteristic database includes bending wave phase velocity and tensile wave phase velocity.

2. The method according to claim 1, characterized in that, The piezoelectric hyperboloid plate includes a core layer and a piezoelectric layer; the core layer is made of a functionally graded material and the reinforcing phase is symmetrically distributed in a power function form along the thickness direction; the piezoelectric layer is disposed on the upper and lower sides of the core layer to carry external voltage.

3. The method according to claim 2, characterized in that, Step S1 specifically includes: Step S11: Establish the first-order shear displacement field and electric potential field based on the structural characteristics of the piezoelectric hyperboloid plate, and construct the strain field based on this. Step S12: Based on the planar bending assumption, construct reduced constitutive relations for the core layer and the piezoelectric layer respectively.

4. The method according to claim 1, characterized in that, In step S4, the model validation includes: Calculate the first-order dimensionless frequency of a piezoelectric hyperboloid with different power-law exponents; Verify the modal frequencies of the piezoelectric hyperboloid plate; Determine the correctness of the displacement field model, the reduced constitutive model, and the calculation using the harmonic solution method.

5. The method according to claim 2, characterized in that, The analysis in step S5 includes: the synergistic effects of piezoelectricity on temperature field, piezoelectric layer to core layer thickness ratio, core layer functional gradient index, and viscoelastic foundation on the first three phase velocities of the piezoelectric hyperboloid plate.

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