Method and apparatus for analyzing impact response of piezoelectric structure of functionally graded sandwich

Abstract

The application discloses a kind of functional gradient sandwich piezoelectric structure impact response analysis method and equipment, based on plane bending hypothesis, the constitutive model considering electromechanical coupling effect is revised, and displacement field framework is constructed using revised high-order shear beam theory;In order to obtain the post-buckling equilibrium path caused by external voltage, an external voltage perturbation scheme is proposed;The dynamic control equation of pre-buckling and post-buckling piezoelectric nanobeam is derived, and the two-step perturbation technique is combined with high-order Galerkin integration to develop an approximate solution for impact response.The analysis method provided by the application overcomes the problems of multi-physical field and nonlinearity, and improves the calculation accuracy.The method takes the impact response of pre-buckling and post-buckling piezoelectric beam as the measurement index of piezoelectric beam impact resistance, effectively providing technical reference for the structural design of piezoelectric beam 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 method and equipment for analyzing the impact response of a functionally graded sandwich piezoelectric structure. Background Technology

[0002] Piezoelectric materials have attracted widespread attention across various fields due to their unique electromechanical coupling effect, showing promising application prospects in emerging fields such as electromechanical systems. Structurally, the concept of functional graded materials (FJTs) has driven the development of lightweight structures, and the combined application of FJTs with sandwich structure designs has significantly improved the dynamic performance of structures. The continuously developing piezoelectric sandwich structures have important value in vibration reduction, vibration energy harvesting technology, health monitoring, and other applications in aerospace.

[0003] The applied voltage determines the overall dynamic performance of the structure, significantly affecting the stability of functionally graded sandwich piezoelectric structures and even leading to post-buckling configurations. Furthermore, piezoelectric systems are highly susceptible to low-velocity impacts in their operating environment. The dynamic behavior and stress field induced by these low-velocity impacts pose a serious threat to the normal operation of the entire piezoelectric system. Particularly under post-buckling conditions induced by an applied electric field, the dynamic phenomena exhibited by the piezoelectric structure under impact are more complex, potentially exhibiting bistable oscillatory dynamic behavior, and the stress field is also affected. Currently, no systematic analytical method has been proposed for the impact response of piezoelectric sandwich structures, addressing the nonlinear characteristics arising from impact problems and post-buckling configurations, as well as the multiphysics issues in the electromechanical coupling effect.

[0004] Therefore, it is necessary to provide a method for analyzing the impact response of functionally graded sandwich piezoelectric structures, so as to provide technical support for the design of functionally graded sandwich piezoelectric structures under impact conditions. Summary of the Invention

[0005] The purpose of this application is to provide a method and equipment for analyzing the impact response of functionally graded sandwich piezoelectric structures. This method can determine the contact force, displacement response, and stress field of functionally graded sandwich piezoelectric structures under low-speed impact, thus laying the foundation for the impact-resistant design of piezoelectric structures.

[0006] To solve the above-mentioned technical problems, this application is implemented as follows:

[0007] This application provides a method for analyzing the impact response of a functionally graded sandwich piezoelectric structure, including:

[0008] Based on the geometric characteristics and impact loading form of the piezoelectric sandwich beam with functionally graded core, a displacement field model, a material model, and an impact model are established.

[0009] Based on Hamilton's variational principle, the impact dynamics control equations of piezoelectric sandwich beams are established and dimensionless.

[0010] Using the quadratic perturbation method, a perturbation scheme for generalized displacement and potential is proposed to obtain the buckling equilibrium path induced by applied voltage. This is used as the initial configuration for impact analysis. Then, by extending and combining the quadratic perturbation technique and higher-order Galerkin integrals, the closed-form solution of the impact response is obtained.

[0011] Conduct model validation and convergence analysis to determine the effectiveness of the adopted model and the convergence requirements of the approximate analytical calculation;

[0012] Design variables were selected, and a numerical analysis of low-velocity impact response was carried out using the fourth-order Runge-Kutta method. A database of design variables and low-velocity impact response was established. The design variables include applied voltage, piezoelectric constant, and functional gradient index of the sandwich layer. The low-velocity impact response database includes the contact force, displacement response, and stress field of the piezoelectric sandwich beam.

[0013] Optionally, the piezoelectric sandwich beam consists of a piezoelectric layer and a functionally graded layer;

[0014] The piezoelectric layer and functionally graded layer include:

[0015] The core layer is composed of functionally graded materials and is symmetrically distributed along the neutral layer in a power-law manner.

[0016] The piezoelectric layers are disposed on the upper and lower sides of the sandwich layer to carry external voltage.

[0017] Optionally, the step of establishing a displacement field model, a material model, and an impact model based on the geometric characteristics and impact loading form of the piezoelectric sandwich beam with functionally graded core includes:

[0018] Establish the displacement field and electric potential field of the high-order shear beam, and construct the nonlinear strain field based on these.

[0019] Within the planar bending assumption framework, constitutive relations considering the piezoelectric effect are constructed for the piezoelectric layer; for the functionally graded layer, a sandwich layer with a power-law exponential distribution is designed, and its constitutive relation is constructed.

[0020] The modified Hertzian model was used to evaluate the contact forces during the loading and unloading phases.

[0021] Optionally, the dimensionless transformation includes:

[0022] Dimensionless parameters are introduced to make the impact dynamics control equations dimensionless.

[0023] Optionally, the method employs a quadratic perturbation approach to propose a perturbation scheme for the generalized displacement and potential, thereby obtaining the buckling equilibrium path induced by the applied voltage, including:

[0024] The dynamic control equations are reduced to static equations. Based on the external voltage-buckling deflection solution, a perturbation scheme that satisfies the boundary conditions is proposed to obtain the first three approximate solutions of the buckling equilibrium path.

[0025] Optionally, using this as the initial configuration for impact analysis, and then obtaining the closed-form solution of the impact response by extending and combining the second-order perturbation technique and higher-order Galerkin integrals, includes:

[0026] Using the buckling configuration as the initial configuration for impact analysis, a higher-order perturbation scheme is proposed around the impact load-displacement solution. Combined with the application of the higher-order Galerkin integral method, the time-domain equations of impact dynamics of piezoelectric sandwich beams are obtained.

[0027] We extend the fourth-order Runge-Kutta method to obtain numerical solutions for the impact problems of piezoelectric sandwich beams with pre- and post-buckling.

[0028] Optionally, the model validation process includes:

[0029] The material model, displacement field model, and calculation method were validated.

[0030] The convergence analysis includes:

[0031] Convergence analysis is performed to calculate multiple high-order truncation solutions of contact force, displacement, and stress field, and convergence requirements are evaluated.

[0032] Optionally, the numerical analysis of low-velocity impact response using the fourth-order Runge-Kutta method includes:

[0033] A database of contact force, displacement, and stress field under pre- and post-buckling states was constructed, with applied voltage, piezoelectric constant, and functional gradient index of the sandwich layer as the core design variables.

[0034] This invention also provides an apparatus including a processor, a memory, and a program or instructions stored in the memory and executable on the processor. When the program or instructions are executed by the processor, they implement the steps of the functionally graded sandwich piezoelectric structure impact response analysis method.

[0035] This invention also provides a readable storage medium storing a program or instructions, which, when executed by a processor, implement the steps of the functionally graded sandwich piezoelectric structure impact response analysis method.

[0036] Compared with related technologies, the beneficial effects of this application are as follows:

[0037] The method proposed in this application can overcome the nonlinear deformation problem under multi-physics coupling and in the impact process, and extend the high-order Galerkin integral method—the second perturbation method;

[0038] The method proposed in this application has strong applicability and can accurately obtain the nonlinear impact response under pre- and post-buckling states, with analysis accuracy meeting the requirements of engineering applications.

[0039] A database of design parameters and impact responses was established through design variable analysis, allowing for the selection of appropriate design parameters based on actual engineering design requirements. Attached Figure Description

[0040] Figure 1 A flowchart illustrating the impact response analysis method for a functionally graded sandwich piezoelectric structure provided in this application embodiment;

[0041] Figure 2 This is a schematic diagram of the structure of the piezoelectric beam of the functionally graded sandwich structure provided in the embodiments of this application;

[0042] Figure 3 A configuration diagram of the piezoelectric beam in the pre-buckling state provided in the embodiments of this application;

[0043] Figure 4 This is a configuration diagram of the piezoelectric beam in the post-buckling state provided in an embodiment of this application;

[0044] Figure 5 (a)-(c) are dynamic response curves under different applied voltages in the embodiments of this application, where (a) is the contact force response diagram, (b) is the displacement response diagram of the mid-span point, and (c) is the stress distribution diagram of the mid-span point.

[0045] Figure 6 This is a structural framework diagram of the equipment in an embodiment of this application. Detailed Implementation

[0046] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0047] The terms "first," "second," etc., used in the specification and claims of this application are used to distinguish similar objects and not to describe a specific order or sequence. It should be understood that such use of data can be interchanged where appropriate so that embodiments of this application can be implemented in orders other than those illustrated or described herein, and the objects distinguished by "first," "second," etc., are generally of the same class and the number of objects is not limited; for example, a first object can be one or more. Furthermore, in the specification and claims, "and / or" indicates at least one of the connected objects, and the character " / " generally indicates that the preceding and following objects are in an "or" relationship.

[0048] The impact response analysis method for functionally graded sandwich piezoelectric structures provided in this application will be described in detail below with reference to the accompanying drawings, through specific embodiments and application scenarios.

[0049] Please see Figure 1 This application provides a method for analyzing the impact response of a functionally graded sandwich piezoelectric structure, including:

[0050] S101. Based on the geometric characteristics and impact loading form of the piezoelectric sandwich beam with functionally graded core, establish the displacement field model, material model and impact model.

[0051] S102 establishes and dimensionlessly renders the impact dynamics control equations of piezoelectric sandwich beams based on Hamilton's variational principle.

[0052] S103. Using the second perturbation method, a perturbation scheme for generalized displacement and potential is proposed to obtain the buckling equilibrium path induced by applied voltage. This is used as the initial configuration for impact analysis. Then, by extending and combining the second perturbation technique and higher-order Galerkin integral, the closed-form solution of the impact response is obtained.

[0053] S104. Conduct model validation and convergence analysis to determine the effectiveness of the adopted model and the convergence requirements of the approximate analytical calculation.

[0054] S105. Select design variables and conduct numerical analysis of low-speed impact response using the fourth-order Runge-Kutta method. Establish a database of design variables and low-speed impact response. The design variables include applied voltage, piezoelectric constant, and functional gradient index of the sandwich layer. The low-speed impact response database includes contact force, displacement response, and stress field of the piezoelectric sandwich beam.

[0055] In step S101, the piezoelectric sandwich beam is as follows: Figure 2 As shown, it consists of piezoelectric functionally graded layers;

[0056] The piezoelectric functional gradient layer includes:

[0057] The core layer 1 is composed of functionally graded materials and is distributed symmetrically in a power-law manner along the neutral layer;

[0058] The piezoelectric layer 2 is disposed on the upper and lower sides of the sandwich layer 1 and bears the external voltage.

[0059] The piezoelectric sandwich beam is subjected to an external initial voltage φ0 and is subjected to a radius of r. i and initial velocity v i The impact of a rigid sphere 3 at both ends of a piezoelectric sandwich beam has simple and immovable boundary conditions, as shown in the following figure. Figure 3 and 4 As shown.

[0060] The displacement field model, material model, and impact model are established based on the geometric characteristics and impact loading form of the piezoelectric sandwich beam with functionally graded core, including:

[0061] Establish the displacement field and electric potential field of the high-order shear beam, and construct the nonlinear strain field based on these.

[0062] To accurately reflect the influence of the post-buckling configuration on the displacement field, a modified higher-order shear beam theory was used to construct the displacement field of the piezoelectric sandwich beam, as follows:

[0063]

[0064]

[0065] in and These are the axial and lateral displacements of any point, respectively, and u1 and u3 are defined as generalized displacements of points on the neutral axis, u3 * and Let f(z) represent the buckling deflection and the cross-sectional rotation, respectively. f(z) and g(z) are higher-order displacement functions, given by the following equation:

[0066]

[0067] For a piezoelectric layer, the electric potential field that satisfies Maxwell's equations is expressed in the following format:

[0068]

[0069]

[0070] V0 is the applied voltage, and V(x,t) is the piezoelectric layer potential distribution. Therefore, the electric field is expressed as E. x and E z The specific expression is as follows:

[0071]

[0072] Within the framework of the von Kármán nonlinear assumption, the strain components are derived as follows:

[0073]

[0074]

[0075] The generalized strain components in the displacement field equations are written as:

[0076]

[0077] Within the planar bending assumption framework, constitutive relations considering the piezoelectric effect are constructed for the piezoelectric layer; for the functionally graded layer, a sandwich layer with a power-law exponential distribution is designed, and its constitutive relation is constructed.

[0078] For a functionally graded sandwich made of isotropic material, the constitutive equation is as follows:

[0079]

[0080] In the formula, and It is the reduced stiffness coefficient of the functionally graded layer based on the plane bending assumption, and its expression is as follows:

[0081]

[0082] Among them, v k (z) is the Poisson's ratio of the functionally graded layer, E k (z) is Young's modulus, ρ k (z). v k (z) and E k (z) all follow a power-law function distributed along the thickness direction, and can be written as the following expression:

[0083]

[0084]

[0085]

[0086]

[0087] The volume fraction function in equation (15) takes z = 0 as the plane of symmetry, where “(1)” and “(2)” represent the upper and lower sides of the z = 0 plane, respectively. N is the power-law exponent. In addition, the superscripts “r” and “m” represent the reinforcing phase and the matrix phase, respectively.

[0088] For piezoelectric layers, the reduced constitutive relation is written as:

[0089]

[0090]

[0091] in, and It is the reduced stiffness factor of the piezoelectric layer, and its detailed expression is:

[0092]

[0093]

[0094] In addition, the reduced voltage constant ( and ) and reduced dielectric constant ( and As shown below:

[0095]

[0096]

[0097] The modified Hertzian model was used to evaluate the contact forces during the loading and unloading phases;

[0098] The modified Hertz model is as follows:

[0099]

[0100] In the formula, and These represent the displacements at the contact points on the impactor and the beam, respectively; η is the indentation, η max This is the maximum indentation calculated during the loading process. It is assumed that the entire impact process is in the elastic phase, i.e., η0 = 0. (F i ) max It is the maximum contact force; while K i It refers to contact stiffness, expressed as follows:

[0101]

[0102] In equation (20), E i ,μ i and r i These are the impactor's elastic modulus, Poisson's ratio, and impactor radius, respectively. And E... 33 It is the transverse elastic modulus at the contact point and depends on the properties of the upper piezoelectric layer material.

[0103] Step S102 specifically includes:

[0104] Based on Hamilton's variational principle, the governing equations for the impact dynamics of piezoelectric sandwich beams are established. By substituting constitutive relations and nonlinear strain fields, the dynamic equations expressed in terms of external forces are transformed into governing equations expressed in terms of generalized displacement and potential. The specific expressions are as follows:

[0105]

[0106]

[0107]

[0108] In the formula, χ0,χ1,χ2,χ3,χ4 are generalized stiffness coefficients, and Ξ gz ,Ξ f ,Ξ g , G f , G g G e , f is the generalized electric coefficient. i ρ represents the contact force. a ,ρ b ,ρ c ,ρ d For generalized density, N x This is the axial force. The specific expression is as follows:

[0109]

[0110]

[0111]

[0112]

[0113]

[0114]

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121] S22: Introduce dimensionless parameters to make the dynamic control equations dimensionless;

[0122] The dimensionless parameters are as follows:

[0123]

[0124]

[0125]

[0126]

[0127]

[0128] In the above steps, the specific dimensionless dynamic equations are as follows:

[0129]

[0130]

[0131]

[0132] In addition, in step S102, the method of second-order perturbation is used to propose a perturbation scheme for the generalized displacement and potential, and to obtain the buckling equilibrium path induced by the applied voltage, including:

[0133] The dynamic control equations are reduced to static equations. Based on the external voltage-buckling deflection solution, a perturbation scheme that satisfies the boundary conditions is proposed to obtain the first three approximate solutions of the buckling equilibrium path.

[0134] The process of reducing the dynamic governing equations to a static equations includes:

[0135] By removing the dynamic terms, the dynamic governing equations degenerate into a static equation system, which is expressed as follows:

[0136]

[0137]

[0138]

[0139] To determine the relationship between the applied voltage and the first-order maximum lateral deflection during buckling, the second-order perturbation method is extended.

[0140] For electric potential and generalized displacement, the following perturbation scheme is proposed:

[0141]

[0142]

[0143] Here, ε is the small perturbation parameter. The equations are discretized by substituting the static equations of the force-electric coupling satisfying the boundary conditions into the dimensionless dynamic equations. Thus, the first three asymptotic solutions are determined:

[0144]

[0145]

[0146]

[0147]

[0148]

[0149] in, The specific expression is:

[0150]

[0151]

[0152] Furthermore, the critical buckling voltage was determined as follows:

[0153]

[0154] Using this as the initial configuration for impact analysis, and then by extending and combining the second-order perturbation technique and higher-order Galerkin integrals, the closed-form solution of the impact response is obtained, including:

[0155] Using the buckling configuration as the initial configuration for impact analysis, a higher-order perturbation scheme is proposed around the impact load-displacement solution. Combined with the application of the higher-order Galerkin integral method, the time-domain equations of impact dynamics of piezoelectric sandwich beams are obtained.

[0156] Combining dimensionless parameters and the dimensionless dynamic equations, the dynamic governing equations can be optimized to the following form:

[0157]

[0158]

[0159]

[0160] Lateral loads, generalized displacements and potentials satisfying boundary conditions are extended to the following perturbation scheme:

[0161]

[0162]

[0163] The optimized dynamic control equations are substituted into the critical buckling voltage equation, and by introducing... The first three solutions were obtained:

[0164]

[0165]

[0166]

[0167] in,

[0168]

[0169]

[0170]

[0171]

[0172] Substituting the perturbation scheme equations into the first three order solution equations, and applying higher-order Galerkin integrals to obtain higher-order time-domain equations, as follows:

[0173]

[0174] The coefficients of each order of dynamic terms are expressed as follows:

[0175]

[0176]

[0177]

[0178] For the impactor, the dimensionless kinetic equations can be expressed as:

[0179]

[0180] in, The dimensionless impact stiffness is expressed as follows:

[0181]

[0182] By combining equations By combining the coefficient equations of each order of dynamic terms, the fourth-order Runge-Kutta method is extended to obtain numerical solutions to the impact problems of piezoelectric sandwich beams with pre- and post-buckling.

[0183] In step S104, the model validation includes:

[0184] The material model, displacement field model, and calculation method were validated.

[0185] Specifically, regarding the structural and material models, the critical buckling external voltage of the piezoelectric sandwich beam needs to be verified. On the other hand, considering the nonlinear characteristics of the impact problem, the accuracy of the contact model and calculation methods needs further verification. This is to ensure that existing theoretical and computational models provide feasibility and correctness.

[0186] The convergence analysis includes:

[0187] Convergence analysis is performed to calculate multiple high-order truncation solutions of contact force, displacement, and stress field, and convergence requirements are evaluated.

[0188] Specifically, before conducting a detailed study on the impact response of piezoelectric nanobeams with symmetric functionally graded material cores, convergence analysis of higher-order truncation solutions is necessary to ensure computational accuracy. During the analysis, since the numerical results of the stress field have high requirements for truncation, the focus is on comparing the numerical results of stress field truncation.

[0189] In step S105, the numerical analysis of low-velocity impact response using the fourth-order Runge-Kutta method includes:

[0190] A database of contact force, displacement, and stress field under pre- and post-buckling states was constructed, with applied voltage, piezoelectric constant, and functional gradient index of the sandwich layer as the core design variables.

[0191] The impact response analysis method of a functionally graded sandwich piezoelectric structure provided in this application will be described in detail below with reference to specific embodiment 1.

[0192] Example 1

[0193] In this embodiment, the influence of applied voltage on the impact response of the functionally graded piezoelectric sandwich beam is considered. The length of the piezoelectric beam, the thickness of the functionally graded layer, and the thickness of the piezoelectric layer are taken as: L = 300 mm, h = 7 mm, hp = 1.5 mm, respectively. The functionally graded material layer is composed of Al and Al2O3, and the power-law exponent (N) is assumed to be 3. In addition, the piezoelectric layer is made of BaTiO3 (50%) and CoFe2O4 (50%). The impact velocity is 17 m / s, and the applied voltage Φ0 is selected as [-2Φ cr ,Φ cr ,1.01Φ cr 1.03Φ cr ].

[0194] The embodiment described above studies the effect of an applied voltage Φ0 on the impact response of a piezoelectric beam with a symmetrical FGM sandwich core. According to... Figure 5(a) The contact force is almost the same for different applied voltages. This is mainly because the applied voltage has almost no direct effect on the contact stiffness or indentation.

[0195] exist Figure 5 (b) plots the mid-span displacement-time history curves under different applied voltages. From the pre-buckling state to the critical buckling state, the applied electric field increases along the polarization direction of the piezoelectric layer. Correspondingly, the displacement amplitude increases and the frequency decreases, indicating that the stiffness and stability of the piezoelectric beam in the pre-buckling state weaken with increasing external voltage. In particular, when Φ0 = Φ cr At that time, the restoring force of the piezoelectric beam is very small.

[0196] Under impact loads, the piezoelectric beam cannot return to its initial configuration. During the post-buckling stage, the buckling phenomenon first appears and then disappears with further increase in external voltage. This means that the post-buckling configuration caused by the piezoelectric inverse effect effectively improves the overall stiffness of the piezoelectric sandwich beam and suppresses vibration response. During the buckling process, the piezoelectric sandwich beam undergoes a transition from a positive stiffness region to a negative stiffness region. When the piezoelectric sandwich beam is at the critical bifurcation point, the higher-order modes become more pronounced.

[0197] Figure 5 (c) Highlighting the effect of applied voltage on the bending normal stress field at the moment of maximum contact force. The difference in bending stress field between the pre-buckling and critical buckling states is not significant. This is mainly because it is insensitive to contact force. Conversely, in the post-buckling state, the bending stress field is extremely sensitive to applied voltage. This is mainly attributed to the contribution of the post-buckling configuration to the initial stress field in a nonlinear system. Although the bending tensile stress increases in the post-buckling state compared to the pre-buckling state, the magnitude of the compressive stress decreases significantly. Therefore, it can be considered that, within the tensile strength range, the post-buckling piezoelectric sandwich beam exerts its mechanical properties to a greater extent.

[0198] Therefore, it can be concluded that the analysis method of the present invention is effective for the impact response analysis of piezoelectric beams with pre- and post-buckling, and the design parameters and impact response database of functionally graded sandwich piezoelectric beams provided by the present invention are of reference value for impact design.

[0199] like Figure 6 As shown, this application embodiment also provides an apparatus 100, which includes a processor 101, a memory 102, and a program or instructions stored in the memory 102 and executable on the processor 101. When the program or instructions are executed by the processor 101, they implement the various processes of the above-described embodiment of the piezoelectric structure impact response analysis method for functionally graded sandwich cores and achieve the same technical effect. To avoid repetition, they will not be described again here.

[0200] It should be noted that the equipment in the embodiments of this application includes mobile equipment and non-mobile equipment.

[0201] This application also provides a readable storage medium storing a program or instructions. When the program or instructions are executed by a processor, they implement the various processes of the above-described embodiment of the piezoelectric structure impact response analysis method for functionally graded sandwich cores and achieve the same technical effect. To avoid repetition, they will not be described again here.

[0202] The processor is the processor in the equipment described in the above embodiments. The readable storage medium includes computer-readable storage media, such as computer read-only memory (ROM), random access memory (RAM), magnetic disk, or optical disk.

[0203] This application embodiment also provides a chip, which includes a processor and a communication interface. The communication interface and the processor are coupled. The processor is used to run programs or instructions to implement the various processes of the above-described embodiment of the piezoelectric structure impact response analysis method for functionally graded sandwich cores, and can achieve the same technical effect. To avoid repetition, it will not be described again here.

[0204] It should be understood that the chip mentioned in the embodiments of this application may also be referred to as a system-on-a-chip, system chip, chip system, or system-on-a-chip, etc.

[0205] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element. Furthermore, it should be noted that the scope of the methods and apparatuses in the embodiments of this application is not limited to performing functions in the order shown or discussed, but may also include performing functions substantially simultaneously or in the reverse order, depending on the functions involved. For example, the described methods may be performed in a different order than described, and various steps may be added, omitted, or combined. Additionally, features described with reference to certain examples may be combined in other examples.

[0206] Through the above description of the embodiments, those skilled in the art can clearly understand that the methods of the above embodiments can be implemented by means of software plus necessary general-purpose hardware platforms. Of course, they can also be implemented by hardware, but in many cases the former is a better implementation method. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product is stored in a storage medium (such as ROM / RAM, magnetic disk, optical disk) and includes several instructions to cause a terminal (which may be a mobile phone, computer, server, air conditioner, or network device, etc.) to execute the methods described in the various embodiments of this application.

[0207] 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 analyzing the impact response of a functionally graded sandwich piezoelectric structure, characterized in that, include: Based on the geometric characteristics and impact loading form of the piezoelectric sandwich beam with functionally graded core, a displacement field model, a material model, and an impact model are established. The piezoelectric sandwich beam is composed of a piezoelectric layer and a functionally graded layer. The functionally graded piezoelectric layer includes: a core layer, which is composed of functionally graded materials and is symmetrically distributed along the neutral layer in a power-law manner; and piezoelectric layers, which are disposed on the upper and lower sides of the core layer to bear external voltage. Based on Hamilton's variational principle, the impact dynamics control equations of piezoelectric sandwich beams are established and dimensionless. A quadratic perturbation method is employed to propose a perturbation scheme for generalized displacement and potential, yielding the buckling equilibrium path induced by applied voltage. This path serves as the initial configuration for impact analysis. Further extensions combining quadratic perturbation techniques and higher-order Galerkin integrals are used to obtain the closed-form solution of the impact response. Specifically, this involves: degenerating the dynamic governing equations into a static equation set; proposing a perturbation scheme satisfying boundary conditions around the external voltage-buckling deflection solution to obtain the first three approximate solutions of the buckling equilibrium path; using the buckling configuration as the initial configuration for impact analysis; proposing a higher-order perturbation scheme around the impact load-displacement solution; and combining this with the application of higher-order Galerkin integrals to obtain the time-domain equations of the impact dynamics of the piezoelectric sandwich beam; and extending the fourth-order Runge-Kutta method to obtain numerical solutions for the impact problems of the piezoelectric sandwich beam with and without buckling. Conduct model validation and convergence analysis to determine the effectiveness of the adopted model and the convergence requirements of the approximate analytical calculation; Design variables were selected, and a numerical analysis of low-velocity impact response was carried out using the fourth-order Runge-Kutta method. A database of design variables and low-velocity impact response was established. The design variables include applied voltage, piezoelectric constant, and functional gradient index of the sandwich layer. The low-velocity impact response database includes the contact force, displacement response, and stress field of the piezoelectric sandwich beam.

2. The method of claim 1, wherein, The displacement field model, material model, and impact model are established based on the geometric characteristics and impact loading form of the piezoelectric sandwich beam with functionally graded core, including: Establish the displacement field and electric potential field of the high-order shear beam, and construct the nonlinear strain field based on these. Within the planar bending assumption framework, constitutive relations considering the piezoelectric effect are constructed for the piezoelectric layer; for the functionally graded layer, a sandwich layer with a power-law exponential distribution is designed, and its constitutive relations are constructed. The modified Hertzian model was used to evaluate the contact forces during the loading and unloading phases.

3. The method of claim 2, wherein, The dimensionless transformation includes: Dimensionless parameters are introduced to make the impact dynamics control equations dimensionless.

4. The method of claim 1, wherein, The process of conducting model validation includes: The material model, displacement field model, and calculation method were validated. The convergence analysis includes: Convergence analysis is performed to calculate multiple high-order truncation solutions of contact force, displacement, and stress field, and convergence requirements are evaluated.

5. The method of claim 1, wherein, The numerical analysis of low-velocity impact response using the fourth-order Runge-Kutta method includes: A database of contact force, displacement, and stress field under pre- and post-buckling states was constructed, with applied voltage, piezoelectric constant, and functional gradient index of the sandwich layer as the core design variables.

6. An apparatus, comprising: It includes a processor, a memory, and a program or instructions stored in the memory and executable on the processor, wherein the program or instructions, when executed by the processor, implement the steps of the method for analyzing the impact response of a piezoelectric structure with a functionally graded sandwich as described in any one of claims 1 to 5.

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