A method for detecting a defect-containing piezoelectric structure based on non-local analysis

Abstract

The application discloses a kind of based on non-local analysis's detection method of piezoelectric structure containing defect, belong to nondestructive testing technical field, including constructing piezoelectric structure geometric model;Establish the local differential expression of Gibbs free energy density of piezoelectric structure;The local differential expression of Gibbs free energy density is converted into non-local integral form;Numerical dispersion is carried out to non-local integral form, and numerical dispersion expression is established;The numerical dispersion expression is expressed as matrix form, and the variational principle is applied to matrix form, and implicit numerical solution format is established;The dynamic control equation of piezoelectric structure is obtained, and free vibration equation is constructed;Solve free vibration equation, obtain the reference natural circular frequency and current natural circular frequency of piezoelectric structure, and the difference between reference natural circular frequency and current natural circular frequency is compared to identify and detect defect;The application effectively improves the accuracy of inherent frequency prediction and the reliability of defect detection of piezoelectric structure containing defect under force-electric coupling condition.

Description

Technical Field

[0001] This invention relates to the field of nondestructive testing technology, and in particular to a method for testing defective piezoelectric structures based on nonlocal analysis. Background Technology

[0002] With the rapid development of precision manufacturing, smart materials, and microelectromechanical systems (MEMS), piezoelectric materials, due to their unique electromechanical coupling characteristics, are widely used in core functional devices such as sensors and actuators. However, during the fabrication and service process, defects such as cracks, pores, and inclusions inevitably occur within the material. These defects severely disrupt the local stress and electric field distribution of the piezoelectric structure, causing stress concentration and electric field concentration, which in turn leads to device performance degradation, signal distortion, and even sudden failure.

[0003] Vibration-based defect detection methods identify defects by monitoring changes in the natural frequencies of a structure, offering advantages such as global applicability and non-destructive nature. However, for piezoelectric structures containing defects, the displacement field is discontinuous at the defect location. Traditional numerical methods based on local differential theory (such as the finite element method) face difficulties in solving these discontinuities due to the undefined differential operators, and the electromechanical coupling effect further increases the complexity of defect characterization. Therefore, existing methods lack sufficient accuracy in predicting the natural frequencies of defective piezoelectric structures under electromechanical coupling conditions, affecting the accuracy and reliability of defect detection.

[0004] In recent years, near-field dynamics theory has provided a new approach for the mechanical analysis of structures with defects by using nonlocal integral operators instead of spatial differential operators, thus maintaining good numerical adaptability at displacement field discontinuities. However, there are currently no mature solutions for extending near-field dynamics methods to piezoelectric structures with electromechanical coupling and constructing a free vibration analysis framework suitable for defect detection. Therefore, developing a piezoelectric structure detection method that can effectively characterize the effects of defects and is applicable to electromechanical coupling conditions has significant engineering value. Summary of the Invention

[0005] The purpose of this invention is to provide a method for detecting defective piezoelectric structures based on nonlocal analysis. By predicting the natural circular frequency and displacement mode of the piezoelectric structure and based on the difference in natural frequency, the method can accurately detect whether there are defects in the piezoelectric structure, thus solving the problem that existing detection methods are difficult to efficiently and accurately identify defects in piezoelectric structures.

[0006] To achieve the above objectives, this invention provides a method for detecting defective piezoelectric structures based on nonlocal analysis, comprising the following steps: S1. Determine the dimensional parameters and electromechanical coupling material properties of the piezoelectric structure to be tested, and construct the geometric model of the piezoelectric structure; S2. Establish the local differential expression for the Gibbs free energy density of the piezoelectric structure. The local differential expression is a function including displacement components and electric potential. S3. The local differential expression of Gibbs free energy density is transformed into a nonlocal integral form using the near-field dynamics operator method. S4. Based on the piezoelectric structure geometric model, the discrete spacing of material points is set, and the nonlocal integral form is numerically discretized to establish a numerical discrete expression for the Gibbs free energy density. S5. Express the numerical discrete expression in matrix form, apply the variational principle to the matrix form, and establish an implicit numerical solution scheme. S6. Introduce an inertial term into the implicit numerical solution scheme to obtain the dynamic control equation of the piezoelectric structure, and further construct the free vibration equation. S7. Solve the free vibration equation to obtain the reference natural circular frequency and the current natural circular frequency of the piezoelectric structure. By comparing the difference between the reference natural circular frequency and the current natural circular frequency, defects can be identified and detected.

[0007] Preferably, the electromechanical coupling material properties in S1 include elastic constant, piezoelectric constant, dielectric constant, and material density.

[0008] Preferably, the displacement component in S2 includes displacement Directional components and displacement Directional components The potential component is Gibbs free energy density of piezoelectric structures The local differential expression is: ; in, for right Partial derivatives of coordinates, To Differential operators for coordinate differentiation for right Partial derivatives of coordinates, To Differential operators for coordinate differentiation for right Partial derivatives of coordinate derivatives , and Intermediate variables used for formula conciseness have no specific meaning; their specific expression is: ; ; ; in, , , , Where is the elastic constant. , , It is the piezoelectric constant. , is the dielectric constant.

[0009] Preferably, the Gibbs free energy density in the nonlocal integral form of S3 for: ; in, , , They are respectively , , The nonlocal integral form corresponding to the partial derivative term is expressed as follows: ; ; ; , and Intermediate variables used for formula conciseness have no specific meaning; their specific expression is: ; ; ; in, As a matter point The non-local near-field range centered on the object. for A point of matter within, , , They are material points Displacement at point Components, displacement Components, electric potential , , They are material points Displacement at point Components, displacement Components, electric potential For matter points The area of ​​the micro element, , , , and This is the near-field dynamics function.

[0010] Preferably, the discrete spacing of the material points in S4 is 0.01 mm, which is suitable for any material point. The Gibbs free energy density after numerical discretization is : ; in, , , They are respectively , , At matter point The discrete form of the partial derivative term at point is specifically expressed as: ; ; ; , and Intermediate variables used for formula conciseness have no specific meaning; their specific expression is: ; ; ; in, As a matter point The non-local near-field range centered on the object. for A point of matter within, 、 、 They are material points Displacement at point Components, displacement Components, electric potential 、 、 They are material points Displacement at point Components, displacement Components, electric potential For matter points The area; 、 、 It is a nonlocal difference operator matrix. 、 、 It is a matrix of nonlocal integral operators; Indicates transpose. For matter points Near field range The displacement vectors of all material points within the space. For matter points Near field range The electric potential vector of all material points within the vector; subscript Represents the peri-field dynamics function at the material point and matter points It can take values ​​between these ranges.

[0011] Preferably, the Gibbs free energy density in S5 The matrix form is as follows: ; in, For matter points Near field range The displacement vectors of all material points within the space. For matter points Near field range The electric potential vector of all material points within the interior; For matter points The mechanical stiffness matrix, For matter points The mechanical-electric coupling stiffness matrix, For matter points The dielectric stiffness matrix is ​​specifically expressed as: ; ; ; Applying the variational principle to the matrix form of all matter points and assembling the matrix, we obtain the implicit numerical solution formula: ; ; in, This is the global displacement vector. The overall electric potential vector. Given the mechanical load vector, Given the electric load vector, For the overall mechanical stiffness matrix, The overall electromechanical coupling stiffness matrix is... This represents the overall dielectric stiffness matrix.

[0012] Preferably, in S6, the dynamic control equation obtained after introducing the inertial term is: ; ; in, For the overall quality matrix, For piezoelectric structure density, The global displacement vector The second derivative with respect to time; By focusing our efforts, we solve the second equation and substitute it into the first equation. Then, setting both the mechanical load vector and the electric load vector to zero, we obtain the free vibration equation: ; in, The natural angular frequency, This is the displacement mode vector.

[0013] Preferably, the implicit algorithm in S7 is the subspace iteration method or the Lanczos method, and the extracted intrinsic circular frequencies are the first 6 intrinsic circular frequencies; defect identification is achieved by comparing the offset or offset pattern of each order frequency in the first 6 intrinsic circular frequencies of the structure to be detected and the defect-free structure.

[0014] Therefore, the present invention employs the above-mentioned method for detecting defective piezoelectric structures based on nonlocal analysis, which has the following beneficial effects: (1) The present invention adopts the near-field dynamics operator method to transform the local differential form of Gibbs free energy density into a non-local integral form. Since the integral operator is still defined at the discontinuity of the displacement field, it can directly perform free vibration analysis on discontinuous structures with defects, thus overcoming the solution difficulties caused by the lack of definition of the differential operator at the defect in the traditional local differential method. (2) By establishing the dynamic control equation of the piezoelectric structure and performing static condensation, this invention can effectively solve the natural circular frequency and displacement mode distribution of the piezoelectric structure under the combined action of mechanical and electrical loads; (3) The present invention has a stronger applicability to piezoelectric structures with defects. By comparing the inherent frequency difference between the structure to be tested and the defect-free structure, structural defects can be effectively identified and detected.

[0015] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0016] Figure 1 This is a flowchart of a defective piezoelectric structure detection method based on nonlocal analysis according to the present invention; Figure 2 This is a schematic diagram of the geometric model and boundary conditions of the piezoelectric sensor according to an embodiment of the present invention; Figure 3 This is a cloud diagram of the first displacement mode of the piezoelectric sensor under electrical coupling conditions according to an embodiment of the present invention. Figure 4 This is a cloud diagram of the second-order displacement mode of the piezoelectric sensor under electrical coupling conditions according to an embodiment of the present invention. Figure 5 This is a cloud diagram of the third displacement mode of the piezoelectric sensor under electrical coupling conditions according to an embodiment of the present invention. Figure 6 This is a cloud diagram of the fourth displacement mode of the piezoelectric sensor under electrical coupling conditions according to an embodiment of the present invention. Figure 7 This is a cloud diagram of the fifth displacement mode of the piezoelectric sensor under electrical coupling conditions according to an embodiment of the present invention. Figure 8 This is a cloud diagram of the sixth displacement mode of a piezoelectric sensor under electrical coupling conditions according to an embodiment of the present invention. Detailed Implementation

[0017] The following detailed description of embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely illustrates selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0018] Example like Figure 1 As shown, this invention provides a method for detecting defective piezoelectric structures based on nonlocal analysis, comprising the following steps: S1. Determine the dimensional parameters and electromechanical coupling material properties of the piezoelectric structure to be tested, and construct a geometric model of the piezoelectric structure; the electromechanical coupling material properties include elastic constant, piezoelectric constant, dielectric constant and material density.

[0019] In this embodiment, a piezoelectric sensor is selected as the piezoelectric structure, and the geometric parameters and boundary conditions are as follows: Figure 2 As shown. The piezoelectric sensor is related to... y The axis is symmetrical, and the specific geometric parameters are as follows: , , , , , The piezoelectric material chosen is PZT-4, with the polarization direction along... y Shaft, density 7500 kg / m³ 3 The material properties are shown in Table 1.

[0020] Table 1 Material Properties of PZT-4

[0021] S2. Establish the local differential expression for the Gibbs free energy density of the piezoelectric structure. The local differential expression is a function including displacement components and electric potential; the displacement components include displacement... x Directional components and displacement y Directional components The electric potential is .

[0022] based on Figure 2 The geometric model shown and the material properties listed in Table 1 indicate the Gibbs free energy density of the piezoelectric structure. The local differential expression is: ; in, for right x Partial derivatives of coordinates, To x Differential operators for coordinate differentiation for right y Partial derivatives of coordinates, To y Differential operators for coordinate differentiation for right y Partial derivatives of coordinate derivatives , and Intermediate variables used for formula conciseness have no specific meaning; their specific expression is: ; ; ; in, , , , Where is the elastic constant. , , It is the piezoelectric constant. , is the dielectric constant.

[0023] S3. Using the near-field dynamics operator method, the local differential expression of Gibbs free energy density is transformed into a non-local integral form.

[0024] for Figure 2The piezoelectric structure with defects shown exhibits a displacement field discontinuity at the defect location, where the local differential operator is undefined. Therefore, a near-field dynamics operator method is employed to transform the local differential form of the Gibbs free energy density in step S2 into a nonlocal integral form. After the transformation, the nonlocal integral form of the Gibbs free energy density... for: ; in, , , They are respectively , , The nonlocal integral form corresponding to the partial derivative term is expressed as follows: ; ; ; , and Intermediate variables used for formula conciseness have no specific meaning; their specific expression is: ; ; ; in, As a matter point The non-local near-field range centered on the object. for A point of matter within, , , They are material points Displacement at point Components, displacement Components, electric potential , , They are material points Displacement at point Components, displacement Components, electric potential For matter points The area of ​​the micro element, , , , and Here is the near-field dynamics function, expressed as follows: ; ; in, and For the weight function, For matter points With matter point Relative positions between x Quantity, For relative positions y Quantity.

[0025] S4. Based on the piezoelectric structure geometric model, the discrete spacing of material points is set, and the nonlocal integral form is numerically discretized to establish a numerical discrete expression for the Gibbs free energy density.

[0026] based on Figure 2 The geometric model of the piezoelectric structure shown is set with a discrete spacing of 0.1 mm. Figure 2 The two-dimensional planar region shown is discretized into material points. For any material point... The Gibbs free energy density after numerical discretization is : ; in, , , They are respectively , , At matter point The discrete form of the partial derivative term at point is specifically expressed as: ; ; ; , and Intermediate variables used for formula conciseness have no specific meaning; their specific expression is: ; ; ; in, As a matter point The non-local near-field range centered on the object. for A point of matter within, 、 、 They are material points Displacement at point Components, displacement Components, electric potential 、 、 They are material points Displacement at point Components, displacement Components, electric potential For matter points The area; 、 、 It is a nonlocal difference operator matrix. 、 、 It is a matrix of nonlocal integral operators; Indicates transpose. For matter points Near field range The displacement vectors of all material points within the space. For matter points Near field range The electric potential vector of all material points within the vector; subscript Represents the peri-field dynamics function at the material point and matter points It can take values ​​between these ranges.

[0027] S5. Express the numerical discrete expression in matrix form, apply the variational principle to the matrix form, and establish an implicit numerical solution scheme.

[0028] Gibbs free energy density The matrix form is as follows: ; in, For matter points Near field range The displacement vectors of all material points within the space. For matter points Near field range The electric potential vector of all material points within the interior; For matter points The mechanical stiffness matrix, For matter points The mechanical-electric coupling stiffness matrix, For matter points The dielectric stiffness matrix is ​​specifically expressed as: ; ; ; Applying the variational principle to the matrix form of all matter points and assembling the matrix, we obtain the implicit numerical solution formula: ; ; in, This is the global displacement vector. The overall electric potential vector. Given the mechanical load vector, Given the electric load vector, For the overall mechanical stiffness matrix, The overall electromechanical coupling stiffness matrix is... This represents the overall dielectric stiffness matrix.

[0029] S6. Introduce an inertial term into the implicit numerical solution scheme to obtain the dynamic control equation of the piezoelectric structure, and further construct the free vibration equation.

[0030] Based on the implicit numerical solution scheme in step S5, a method related to material density is introduced. The relevant inertial terms lead to the dynamic governing equations of the piezoelectric structure: ; ; in, For the overall quality matrix, For piezoelectric structure density, The global displacement vector The second derivative with respect to time; By focusing our efforts, we solve the second equation and substitute it into the first equation. Then, setting both the mechanical load vector and the electric load vector to zero, we obtain the free vibration equation: ; in, The natural angular frequency, This is the displacement mode vector.

[0031] S7. Solve the free vibration equation to obtain the reference natural circular frequency and the current natural circular frequency of the piezoelectric structure. By comparing the difference between the reference natural circular frequency and the current natural circular frequency, defects can be identified and detected.

[0032] The free vibration equation in step S6 is solved using the subspace iteration method, and the first 6 reference natural circular frequencies of the piezoelectric structure in the defect-free state and the first 6 current natural circular frequencies in the test state are extracted respectively.

[0033] The first six natural frequencies of a defect-free piezoelectric sensor under force-electric coupling conditions are shown in Table 2.

[0034] Table 2 Natural Frequency of Piezoelectric Sensors

[0035] Figures 3 to 8 The first six displacement modes of the piezoelectric sensor under the combined action of mechanical and electrical loads are presented.

[0036] Repeat steps S1 to S6 and the solution process described above for the piezoelectric structure under test to obtain the first six natural angular frequencies under the test state. Defects in the piezoelectric structure are identified and detected by comparing the offset or offset pattern of the current natural angular frequencies with the reference natural frequencies listed in Table 2. If a significant offset occurs in one or more natural frequencies, a defect is determined to exist in the structure. The numerical distribution characteristics of the offset can further reflect the location and severity of the defect.

[0037] The numerical results show that this embodiment can effectively predict the natural frequency and displacement mode distribution of piezoelectric ceramic structures.

[0038] Therefore, the present invention adopts the above-mentioned nonlocal analysis-based method for detecting defective piezoelectric structures. It uses a near-field dynamic nonlocal integral operator to replace the traditional local differential operator, which can directly perform free vibration analysis in defective regions where the displacement field is discontinuous. This overcomes the shortcomings of traditional methods, which are difficult to solve at defects due to the lack of defined differential operators. It effectively improves the accuracy of predicting the natural frequency of defective piezoelectric structures under force-electric coupling conditions and the reliability of defect detection.

[0039] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for detecting defective piezoelectric structures based on nonlocal analysis, characterized in that, Includes the following steps: S1. Determine the dimensional parameters and electromechanical coupling material properties of the piezoelectric structure to be tested, and construct the geometric model of the piezoelectric structure; S2. Establish the local differential expression for the Gibbs free energy density of the piezoelectric structure. The local differential expression is a function including displacement components and electric potential. S3. The local differential expression of Gibbs free energy density is transformed into a nonlocal integral form using the near-field dynamics operator method. S4. Based on the piezoelectric structure geometric model, the discrete spacing of material points is set, and the nonlocal integral form is numerically discretized to establish a numerical discrete expression for the Gibbs free energy density. S5. Express the numerical discrete expression in matrix form, apply the variational principle to the matrix form, and establish an implicit numerical solution scheme. S6. Introduce an inertial term into the implicit numerical solution scheme to obtain the dynamic control equation of the piezoelectric structure, and further construct the free vibration equation. S7. Solve the free vibration equation to obtain the reference natural circular frequency and the current natural circular frequency of the piezoelectric structure. By comparing the difference between the reference natural circular frequency and the current natural circular frequency, defects can be identified and detected.

2. The method for detecting defective piezoelectric structures based on nonlocal analysis according to claim 1, characterized in that: The electromechanical coupling material properties in S1 include elastic constant, piezoelectric constant, dielectric constant, and material density.

3. The method for detecting defective piezoelectric structures based on nonlocal analysis according to claim 1, characterized in that, The displacement components in S2 include displacement Directional components and displacement Directional components The potential component is Gibbs free energy density of piezoelectric structures The local differential expression is: ； in, for right Partial derivatives of coordinates, To Differential operators for coordinate differentiation for right Partial derivatives of coordinates, To Differential operators for coordinate differentiation for right Partial derivatives of coordinate derivatives , and As an intermediate variable, the specific expression is: ； ； ； in, , , , Where is the elastic constant. , , It is the piezoelectric constant. , is the dielectric constant.

4. The method for detecting defective piezoelectric structures based on nonlocal analysis according to claim 3, characterized in that, Gibbs free energy density in nonlocal integral form of S3 for: ； in, , , They are respectively , , The nonlocal integral form corresponding to the partial derivative term is expressed as follows: ； ； ； , and As an intermediate variable, the specific expression is: ； ； ； in, As a matter point The non-local near-field range centered on the object. for A point of matter within, , , They are material points Displacement at point Components, displacement Components, electric potential , , They are material points Displacement at point Components, displacement Components, electric potential For matter points The area of ​​the micro element, , , , and This is the near-field dynamics function.

5. The method for detecting defective piezoelectric structures based on nonlocal analysis according to claim 4, characterized in that, The discrete spacing of the material points in S4 is 0.01 mm. For any material point... The Gibbs free energy density after numerical discretization is : ； in, , , They are respectively , , At matter point The discrete form of the partial derivative term at point is specifically expressed as: ； ； ； , and As an intermediate variable, the specific expression is: ； ； ； in, As a matter point The non-local near-field range centered on the object. for A point of matter within, 、 、 They are material points Displacement at point Components, displacement Components, electric potential 、 、 They are material points Displacement at point Components, displacement Components, electric potential For matter points The area; 、 、 It is a nonlocal difference operator matrix. 、 、 It is a matrix of nonlocal integral operators; Indicates transpose. For matter points Near field range The displacement vectors of all material points within the space. For matter points Near field range The electric potential vector of all material points within the vector; subscript Represents the peri-field dynamics function at the material point and matter points It can take values ​​between these ranges.

6. The method for detecting defective piezoelectric structures based on nonlocal analysis according to claim 5, characterized in that, Gibbs free energy density in S5 The matrix form is as follows: ； in, For matter points Near field range The displacement vectors of all material points within the space. For matter points Near field range The electric potential vector of all material points within the interior; For matter points The mechanical stiffness matrix, For matter points The mechanical-electric coupling stiffness matrix, For matter points The dielectric stiffness matrix is ​​specifically expressed as: ； ； ； Applying the variational principle to the matrix form of all matter points and assembling the matrix, we obtain the implicit numerical solution formula: ； ； in, This is the global displacement vector. The overall electric potential vector. Given the mechanical load vector, Given the electric load vector, For the overall mechanical stiffness matrix, The overall electromechanical coupling stiffness matrix is... This represents the overall dielectric stiffness matrix.

7. The method for detecting defective piezoelectric structures based on nonlocal analysis according to claim 6, characterized in that, In S6, the dynamic control equations obtained after introducing the inertial term are: ； ； in, For the overall quality matrix, For piezoelectric structure density, The global displacement vector The second derivative with respect to time; By focusing our efforts, we solve the second equation and substitute it into the first equation. Then, setting both the mechanical load vector and the electric load vector to zero, we obtain the free vibration equation: ； in, The natural angular frequency, This is the displacement mode vector.

8. The method for detecting defective piezoelectric structures based on nonlocal analysis according to claim 1, characterized in that: The implicit algorithm in S7 is either the subspace iteration method or the Lanczos method, and the extracted natural circular frequencies are the first 6 natural circular frequencies. Defect identification is achieved by comparing the offset or offset pattern of each order of the first 6 natural circular frequencies of the structure to be detected and the defect-free structure.

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