A method for finding the shape of a plate thinning defect based on a few-frequency reflected ultrasonic guided wave

Abstract

The application provides a plate thinning defect shape finding method based on a few-frequency reflected ultrasonic guided wave, and comprises the following steps: S1: mode separation is performed on a total field of a reflected wave to obtain a reflection coefficient of a required guided wave mode; S2: a Green function in a defect-free plate is solved to obtain an approximate solution of a far field; S3: a reconstruction equation of a plate defect is established and preliminary reconstruction is performed; and S4: accurate determination of a plate defect shape is performed. The method is used for rapid quantitative detection of a plate structure thinning defect, and based on reflection data of the ultrasonic guided wave at a few frequency points, the shape of the defect can be found, the defect shape is accurately determined, a better detection effect is realized at a relatively low cost, and thus a convenient and reliable detection technology is provided for engineering detection.

Description

Technical Field

[0001] This invention relates to the field of nondestructive testing technology, and in particular to a method for finding defects in thinned flat plates based on low-frequency reflected ultrasonic guided waves. Background Technology

[0002] With the continuous development of science and technology, the demand for defect detection and assessment technology in industries such as aerospace, machinery manufacturing, and bridge construction is increasing. It is no longer enough to just determine the location and vague shape of defects, but to be able to quantitatively provide the accurate location of defects and determine their specific shape.

[0003] Conventional non-destructive testing (NDT) and evaluation methods in industry include magnetic particle testing, radiographic testing, eddy current testing, and ultrasonic testing, among which ultrasonic NDT is a widely used method. Due to the high frequency and short wavelength of ultrasound, it can interact with minute features in a structure, such as defects, cracks, and delamination, thus exhibiting high detection sensitivity and adaptability. However, traditional ultrasonic testing techniques mostly utilize ultrasonic transducers placed on the surface of the structure to transmit and receive volume waves, detecting defects within the material or near the contact surface, resulting in extremely limited coverage. For large components, a complete grid scan is often required to obtain comprehensive information, which is time-consuming and labor-intensive, and it is generally impossible to inspect areas that cannot be directly contacted. Summary of the Invention

[0004] This invention provides a method for finding thinning defects in flat plate structures based on ultrasonic guided waves reflected at a limited number of frequencies, enabling rapid quantitative detection of these defects. By relying solely on ultrasonic guided wave reflection data at a small number of frequencies, the method achieves accurate shape determination of the defect, resulting in good detection performance at a relatively low cost. This provides a convenient and reliable detection technology for engineering inspection.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A method for finding defects in plate thinning based on low-frequency reflected ultrasonic guided waves includes the following steps:

[0007] S1: Perform mode separation on the total field of the reflected wave to obtain the reflection coefficient of the desired guided wave mode; specifically: utilize the orthogonality between different modes during the propagation of the elastic wave in the waveguide material to perform mode separation on the total field of the reflected wave to obtain the amplitude of the reflected wave displacement corresponding to the mode of the incident waveguide wave. The ratio of this reflected wave amplitude to the incident wave amplitude is the reflection coefficient.

[0008] S2: Solve for the Green's function in a defect-free plate to obtain an approximate solution for the far field; specifically: combine the free boundary conditions of the traction force at the upper and lower boundaries of the plate and the wave equation, and utilize the orthogonality between different modes of guided waves to solve for an approximate solution for the far field of the Green's function in a defect-free plate.

[0009] S3: Establish the reconstruction equation of the plate defect and perform preliminary reconstruction; specifically: construct the boundary integral equation by using the dynamic reciprocity theorem and the approximate solution of the far field, and derive the reconstruction equation connecting the defect shape and the reflection coefficient. Based on a small amount of reflection coefficient data at discrete frequency points, combine the inverse Fourier transform and the trapezoidal integral rule to perform preliminary reconstruction of the defect and obtain the defect location and range.

[0010] S4: Accurately determine the shape of the plate defect; specifically: based on the defect location and range obtained from the preliminary reconstruction, assume the defect shape as a combination of basis functions with different amplitudes and substitute them into the reflection data at different frequencies to form a set of discrete equations for defect reconstruction; then, based on the obtained set of discrete equations for reconstruction, solve them using regularization correlation methods to obtain the amplitude of each basis function, and complete the precise quantitative reconstruction of the defect shape.

[0011] In some embodiments, in step S1, for the SH wave in the isotropic flat plate medium, the displacement of the particle is zero in the planes containing the x1 and x2 directions, and the displacement u3 in the x3 direction is not zero, and the displacement u3 satisfies the following equation of motion: Where c T The shear wave velocity of the material. Let x1 and x2 represent the second partial derivatives of the variable with respect to the x1 and x2 directions, respectively. This represents the second partial derivative of the variable with respect to time, where x1 is the axial direction of the plate, x2 is the thickness direction of the plate, and x3 is the direction of the antiplane.

[0012] Combining the free boundary conditions of the upper and lower boundary surfaces, we obtain the time-omitted harmonic term e. iωt nth order SH wave displacement The expression is Where h is the plate thickness, ω is the waveguide frequency, and c T For material shear wave velocity, As an intermediate parameter, A n The undetermined coefficients representing the amplitude, Let be the wave number of the nth mode SH wave along the x1 direction; The function is undetermined. When n = 0, 2, 4..., it is called a symmetric mode. In this case, f n =cos(β) n x2); n = 1, 3, 5... is called an antisymmetric mode, where f n =sin(β)n x2);

[0013] Consider two modes of the SH guided wave in the plate, such as the nth and mth modes, and denote the displacement, surface force, and wave number in the x1 direction on its cross-section as follows: and Based on the orthogonality of different waveguide modes, we can conclude that: Where * is the conjugate symbol; when detecting defects, the sensor receives total field information: total field displacement. Overall field strength Where, ∑ n The symbol represents summation over variable n, where n = n1, n2, ..., n N The order of all modes present in the guided wave at the current frequency;

[0014] For total field data containing various modes that are difficult to use directly, the amplitudes of the desired modes are extracted based on modal orthogonality: The incident wave is selected as a single-mode guided wave: displacement field in The amplitude of the incident wave is represented by: The reflected wave in the far field is: This represents the amplitude of the reflected wave; therefore, the reflection coefficient is defined as: It is a function of wavenumber / frequency.

[0015] In some embodiments, in step S1, for the Lamb wave in the isotropic flat plate medium and the Lamb-like wave in the orthotropic laminate structure, the displacement u of the particle along the x1 and x2 directions satisfies the wave equation: Where ρ is the material density, C ijlk This is the fourth-order elastic modulus tensor of the material;

[0016] For homogeneous and isotropic materials, C ijlk It can be represented as: C ijkl =λδ ij δ kl +μ(δ ik δ jl +δ il δ jk ); where λ is Lamé's constant, μ is the shear modulus, and δ ij The Kronecker symbol is used to determine whether two subscripts are the same; that is, it equals 1 when the subscripts are the same and 0 when they are different.

[0017] Based on the orthogonality of different modes of guided waves in the wave field, taking any two modes of the Lamb wave in the plate, we record the displacement, surface force, and wave number in the x1 direction on its cross-section as follows: Based on the orthogonality of different waveguide modes, we can conclude that:

[0018] Where * is the conjugate symbol;

[0019] When detecting defects, the sensor receives total field information: total field displacement. Overall field strength

[0020] For total field data containing various modes that are difficult to use directly, the amplitudes of the desired modes should be extracted based on modal orthogonality.

[0021] The incident wave is selected as a single-mode guided wave: in Indicates the amplitude of the incident wave;

[0022] The reflected wave in the far field is: Indicates the amplitude of the reflected wave;

[0023] The reflection coefficient can be expressed as: It is a function of wavenumber / frequency. For the same guided wave and the same mode, the wavenumber and frequency are in one-to-one correspondence. Therefore, the reflection coefficient can also be said to be a function of frequency.

[0024] In some embodiments, in step S2, a time-harmonic concentrated load δ is applied at a source point y at any location within the plate. in δ(xy)e iωt The displacement response is obtained at the signal receiving location, field point x. For the SH wave in the flat plate, the wave equation corresponding to its Green's function state is: The upper and lower boundaries of the flat plate are subject to the free boundary conditions of traction force: Among them, c T ω is the shear wave velocity of the material, h is the plate thickness, and μ is the shear modulus.

[0025] The Green's function of the SH wave is solved using Fourier transform and residue theorem, and its far-field approximation is taken. The final expression is: in, ξ n Let be the wave number of the nth order SH wave along the x1 direction.

[0026] In some embodiments, for Lamb waves and Lamb-like waves in a flat plate, the wave equation corresponding to their Green's function states is:

[0027]

[0028] Among them, Cijlk is the fourth-order elastic modulus tensor, ρ is the material density, ω is the guided wave frequency, represents the displacement response in the x n direction caused by applying a unit external body force in the x i direction;

[0029] When x1 > y1, the far-field Green's function displacement:

[0030]

[0031] When x1 < y1, the far-field Green's function displacement:

[0032]

[0033] where i, n = 1, 2; ξ m is the wave number of the m-th order guided wave in the x1 direction, f i represents the component of the applied body force in the x i direction; σ i1 represents the stress in the cross-section of the plate caused by the guided wave, respectively represent the displacement and stress fields corresponding to the m-th order guided wave mode propagating along the positive x1 direction, respectively represent the function terms that only vary with x2 after removing the term; the negative sign in the superscripts and subscripts (such as u -i , σ -i1 ) represents the guided wave propagating along the negative x1 direction.

[0034] In some embodiments, in step S3, the relationship between the boundary integral and the scattered field can be constructed through the dynamic reciprocity theorem:

[0035]

[0036] where the subscripts i, j, k, l, n = 1, 2; is the scattered field displacement, is the total field displacement, U ln (x, y) is the displacement in the Green's function state, S represents the boundary of the defect region V, x = (x1, x2) represents the axial and thickness direction coordinates of the source point, and y = (y1, y2) represents the axial and thickness direction coordinates of the field point;

[0037] The Green's function state surface force

[0038] satisfies the traction-free condition on the defect boundary S in the total field:​​​​​

[0040] Using the Born approximation, the total field over the defect is approximated as the incident field. The boundary integral equation is written as:

[0041]

[0042] Applying Gauss's theorem to transform the boundary integral S(x) into an integral V(x) over the entire defect, we can obtain:

[0043]

[0044] In some embodiments, for SH waves in a flat plate, substituting the incident field and the far-field approximation of the Green's function, the incident wave is a symmetric mode:

[0045] in, For intermediate parameters, Let ξ be the amplitude of the p-th order guided wave of the incident wave. p Let x1 be the wave number of the p-th guided wave; This represents the amplitude of the m-th order guided wave in the scattered wave;

[0046] Using the orthogonality of modes, we can obtain:

[0047]

[0048] The expression for the reflection coefficient is:

[0049] Using the Gaussian integral formula and Taylor expansion, the reflection coefficient can be approximated as:

[0050]

[0051] Where, ξ p Let x be the wave number in the x1 direction of the p-th guided wave. a ,x b d(x1) represents the range of defects; d(x1) is the expression for the defect depth, and d(x1) = 0 when there are no defects.

[0052] In some embodiments, for Lamb waves and lamb-like waves in a flat plate, the incident field and the far-field approximation of the Green's function are substituted, and the incident field is a p-th order modal guided wave:

[0053] Scattered field:

[0054]

[0055] Among them, g m This is an intermediate parameter representing the m-th mode, and its value is:

[0056]

[0057] Let x and y represent the displacement and stress caused by the m-th order guided wave propagating along the positive x1 direction, respectively. They represent Remove The term following the term is a function term that changes only with respect to x²; u -i σ -i1 These represent guided waves propagating along the negative x1 direction;

[0058] Using the orthogonality of modes, we can obtain:

[0059] The expression for the reflection coefficient is:

[0060] Using the Gaussian integral formula, the reflection coefficient can be approximated as:

[0061] Where N(x2,ξ) p ) is an intermediate variable, the specific form of which is obtained by separating variables after summing the Gaussian integral;

[0062] Since the interval [h0, h0+d(x1)] is very small, the integrand is expanded using Taylor at h0:

[0063] N(x2,ξ p )=N(h0,ξ p )+N'(h0,ξ p (x2-h)+O(x2-h0) 2 ;

[0064] When the integrand retains only the first term:

[0065] Wherein G(ξ) p )=N(h0,ξ p ), ξ p Let x be the wave number in the x1 direction of the p-th guided wave. a ,x b d(x1) represents the range of defects; d(x1) is the expression for the defect depth, and d(x1) = 0 when there are no defects.

[0066] In some embodiments, for SH waves, Lamb waves, and lamb-like waves, rearranging terms yields the defect reconstruction equation as follows:

[0067]

[0068] Among them, for SH wave:

[0069] Lamb waves and lamb-like waves:

[0070] in, There is a Fourier transform-like relationship between d(x1) and d(x1). Given some discrete data points, the defects are initially reconstructed by combining the inverse Fourier transform and the trapezoidal integral rule.

[0071] Given the p-th mode, the wavenumber point ξ = ξ corresponding to the reflected data point. 1 ,ξ 2 ,ξ 3 …ξ N ξ omitted p If the subscript is used, the preliminary reconstructed equation is:

[0072]

[0073] Based on this formula, a preliminary reconstruction of the defect is achieved, obtaining the defect location and extent (x). a ,x b ).

[0074] In some embodiments, in step S4, a typical trigonometric function basis function is used to set the defect shape as follows:

[0075]

[0076] in, a n ,b n Unknown undetermined coefficients;

[0077] Substitute different wavenumber data

[0078] For the sake of simplicity, the real and imaginary parts are separated:

[0079]

[0080]

[0081] The system of equations is obtained as follows:

[0082]

[0083] in:

[0084]

[0085]

[0086]

[0087] The above system of equations can be denoted in vector form: b = AX;

[0088] To solve the above system of equations, the frequency point selected must satisfy the relationship between the possible range of defects. or When matrix A is full rank, the equation can be solved directly;

[0089] When matrix A is not full rank, solving the system of equations becomes a problem of solving an ill-conditioned system of equations.

[0090] For solving ill-conditioned equation systems, the Tikhonov regularization method is used, and the final solution formula can be expressed as:

[0091] Wherein, λ is the regularization parameter, which can be obtained through the L-curve method;

[0092] After solving the system of equations, the unknown undetermined coefficient 'a' is obtained. n ,b n After obtaining the value, substitute it into the defect shape expression:

[0093] The specific location and shape of the defect are obtained, and the defect is accurately quantified and reconstructed.

[0094] In summary, the present invention has at least the following beneficial effects:

[0095] In this invention, the defect reconstruction expression is directly obtained based on the understanding of the material and plate structure properties. By analyzing the transmitted and received signals through sensors, and using guided wave reflection data at a small number of frequency points, the shape of the defect can be accurately determined, enabling rapid quantitative detection. Furthermore, this invention can use various types of guided waves for detection, such as SH waves and Lamb waves. For orthotropic plates, Lamb-like waves can also be used, making it more adaptable to experimental or engineering testing. The detection process can achieve simultaneous detection of defects over a large area of ​​components, providing an efficient and accurate solution for quantitative ultrasonic guided wave detection, which has significant application value in engineering.

[0096] The feasibility and accuracy of this paper are demonstrated through numerical simulation: In the examples, the lowest-order symmetric mode of the SH wave in an isotropic single-layer plate and the reflection coefficient of the lowest-order mode of the Lamb wave in an isotropic single-layer plate and an isotropic anisotropic double-layer plate are used to reconstruct defects of different sizes and shapes. It is found that relatively accurate results can be obtained in these examples. This method can reconstruct multiple defects with relatively accurate results, which can meet the accuracy requirements of engineering health detection. It also has the following advantages: (1) It can detect invisible parts without the need for complex rotation and travel devices; (2) It can use a variety of different types of guided waves; (3) It is adaptable to a variety of different plate structures; (4) It has high sensitivity and accuracy to defects; (5) It has low energy consumption and economy. Attached Figure Description

[0097] Figure 1 This is a schematic diagram illustrating the steps of a plate thinning defect finding method based on low-frequency reflected ultrasonic guided waves.

[0098] Figure 2 This is a schematic diagram of a single-layer flat plate model and wave field involving arbitrary thinning defects, as described in this invention.

[0099] Figure 3 This is a schematic diagram illustrating the solution of the Green's function in the flat plate structure involved in this invention.

[0100] Figure 4 This is a schematic diagram of the triangular defect shape involved in this invention.

[0101] Figure 5 This is a data diagram of the reflection coefficient of a triangular defect based on SH wave reconstruction involved in this invention.

[0102] Figure 6 This is a schematic diagram of the L-curve method for reconstructing triangular defects based on SH waves in this invention.

[0103] Figures 7a-7b The images show the initial and final reconstruction results of the triangular defect based on the SH wave.

[0104] Figures 8a-8d This is a diagram showing the reconstruction results of triangular defects of various sizes based on SH waves in this invention.

[0105] Figures 9a-9b This is a diagram showing the reconstruction results of various shape defects based on SH waves involved in this invention.

[0106] Figures 10a-10b This is a diagram showing the reconstruction results of various shape defects based on Lamb waves in this invention.

[0107] Figure 11 This is a schematic diagram of a multilayer flat plate model and wave field containing arbitrary thinning defects in this invention.

[0108] Figures 12a-12b This is a diagram showing the reconstruction result of a double elliptical defect based on a Lamb-like wave in this invention. Detailed Implementation

[0109] In the following description, only certain exemplary embodiments are presented briefly. As those skilled in the art will recognize, the described embodiments can be modified in various ways without departing from the spirit or scope of the embodiments of the invention. Therefore, the drawings and description are considered to be exemplary in nature and not restrictive.

[0110] The following disclosure provides many different implementations or examples for carrying out different structures of the embodiments of the present invention. To simplify the disclosure of the embodiments of the present invention, specific examples of components and arrangements are described below. Of course, these are merely examples and are not intended to limit the embodiments of the present invention. Furthermore, reference numerals and / or reference letters may be repeated in different examples of the embodiments of the present invention; such repetition is for simplification and clarity and does not in itself indicate a relationship between the various implementations and / or arrangements discussed.

[0111] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0112] like Figure 1 As shown, this embodiment provides a method for finding defects in plate thinning based on low-frequency reflected ultrasonic guided waves, including the following steps:

[0113] S1: Perform mode separation on the total field of the reflected wave to obtain the reflection coefficient of the desired guided wave mode;

[0114] Specifically, by utilizing the orthogonality between different modes during the propagation of elastic waves in the waveguide material, mode separation is performed on the total field of the reflected wave to obtain the amplitude of the reflected wave displacement corresponding to the mode of the incident waveguide wave. The ratio of this reflected wave amplitude to the incident wave amplitude is the reflection coefficient.

[0115] In some embodiments, in step S1, for the SH wave in the isotropic flat plate medium, the displacement of the particle is zero in the planes containing the x1 direction (plane axis) and the x2 direction (plane thickness direction), while the displacement u3 in the opposite plane direction, i.e., the x3 direction, is not zero, and the displacement u3 satisfies the following equation of motion: Where c T The shear wave velocity of the material. Let x1 and x2 represent the second partial derivatives of the variable with respect to the x1 and x2 directions, respectively. This represents the second partial derivative of the variable with respect to time.

[0116] Combining the free boundary conditions of the upper and lower boundary surfaces, the displacement of the nth-order SH wave can be obtained. The expression (omitting the time harmonic term e) iωt )for:

[0117] Where h is the plate thickness, ω is the waveguide frequency, and c T For material shear wave velocity, As an intermediate parameter, A n The undetermined coefficients representing the amplitude, Let be the wave number of the nth mode SH wave along the x1 direction; The function is undetermined. When n = 0, 2, 4..., it is called a symmetric mode. In this case, f n =cos(β) n x2); n = 1, 3, 5... is called an antisymmetric mode, where f n =sin(β) n x2).

[0118] Consider two modes of the SH guided wave in the flat plate, such as the nth and mth modes, and denote their displacement and stress, respectively. The wave number in the x1 direction is: and

[0119] Based on the orthogonality of different waveguide modes, we can conclude that: Where * is the conjugate symbol;

[0120] When detecting defects, the sensor receives total field information, including the total field displacement. Overall performance: Where ∑ n The symbol represents summation over variable n, where n = n1, n2, ..., n N The order of all modes present in the guided wave at the current frequency.

[0121] For total field data containing various modes that are difficult to use directly, the amplitudes of the desired modes are extracted based on modal orthogonality:

[0122] The incident wave is chosen as a single-mode guided wave, which we can denote as the Pth-order guided wave. Its displacement field in Indicates the amplitude of the incident wave;

[0123] Displacement field of reflected waves in the far field Indicates the amplitude of the reflected wave;

[0124] Therefore, the reflection coefficient is defined. It is a function of wavenumber / frequency.

[0125] In some embodiments, in step S1, for Lamb waves in isotropic flat plate media and quasi-lamb waves in orthotropic laminated plate structures, the displacement u1 of the particles in the x1 direction (plane axis) and the displacement u2 in the x2 direction (plane thickness direction) are not zero, while the displacement u3 in the antiplane direction, i.e., the x3 direction, is zero, and the displacement field satisfies the wave equation: Among them, C ijlk Let ρ be the fourth elastic modulus tensor of the material, and ρ be the material density.

[0126] For a homogeneous and isotropic material, the fourth elastic modulus tensor C ijkl =λδ ij δ kl +μ(δ ik δ jl +δ il δ jk ); where λ is Lamé's constant, μ is the shear modulus, and δ ij The Kronecker symbol is used to determine whether two subscripts are the same; that is, it equals 1 when the subscripts are the same and 0 when they are different.

[0127] Consider any two modes of the Lamb wave in the flat plate, and denote their displacement, stress, and wave number in the x1 direction as follows:

[0128] Based on the orthogonality of different waveguide modes, we can conclude that: Where * is the conjugate symbol;

[0129] When detecting defects, the sensor receives total field information, specifically the total field displacement. Overall field strength

[0130] For total field data containing various modes that are difficult to use directly, the amplitudes of the desired modes should be extracted based on modal orthogonality.

[0131] The incident wave is chosen as a single-mode guided wave, which we can denote as the Pth-order guided wave, and its corresponding displacement is... in Indicates the amplitude of the incident wave;

[0132] Displacement of reflected waves in the far field Indicates the amplitude of the reflected wave;

[0133] Define reflection coefficient It is a function of wavenumber, and since the wavenumber and frequency are in one-to-one correspondence for the same mode of the same guided wave, the reflection coefficient can also be considered as a function of frequency.

[0134] S2: Solve for the Green's function in a defect-free plate to obtain an approximate solution for the far field;

[0135] Specifically, by combining the free boundary conditions of the traction force at the upper and lower boundaries of the plate with the wave equation, and utilizing the orthogonality between different modes of guided waves, an approximate solution for the far field of the Green's function in a defect-free plate is obtained.

[0136] In step S2, because the fundamental solution of the Green's function for elastic waves is complex and inconvenient to directly use to construct the boundary integral equation, and because the incident wave and the defect in actual detection produce rapidly attenuating volume waves and unattenuated guided waves, and the distance between the defect and the sensor is relatively far, the sensor only receives the unattenuated portion. Therefore, the far-field approximation solution can be used for the far-field Green's function. The Green's function in a flat plate refers to: (e.g., ...) Figure 3 As shown, a time-harmonic concentrated load δ is applied at an arbitrary source point y within the flat plate. in δ(xy)e iωt The displacement response is obtained at the signal receiving location, field point x.

[0137] For an SH wave in a flat plate, the displacement in the antiplane direction, i.e., the x3 direction, is in its Green's function state. The wave equation that is satisfied is:

[0138] The upper and lower boundaries of the flat plate are free boundary conditions for traction force, from which the traction force can be obtained.

[0139] Among them, c T Let ω be the shear wave velocity of the material, h be the plate thickness, and μ be the shear modulus.

[0140] The Green's function of the SH wave is solved using Fourier transform and residue theorem, and the far-field approximation is taken. Finally, the expression for the displacement field is obtained as follows:

[0141]

[0142] in, ξ n Let be the wave number of the nth order SH wave along the x1 direction.

[0143] In some embodiments, for Lamb waves and Lamb-like waves in a flat plate, the wave equation corresponding to their Green's function states is: Among them, C ijlk The fourth-order elastic modulus tensor is given by ρ, where ρ is the material density and ω is the guided wave frequency. Indicates in x n When a unit external force is applied in the x direction iDisplacement response caused by direction.

[0144] According to the roots of the dispersion equation, at a given frequency, solving the dispersion equation yields a finite number of real roots (denoted as M), which correspond to guided wave modes that do not decay. In contrast, infinitely many complex roots are obtained because they contain imaginary parts, corresponding to dissipative wave modes that decay rapidly with the propagation distance. Therefore, it can be assumed that the far-field Green's function only contains non-decayed guided wave modes.

[0145] For single-layer structures, dispersion equations can be established based on the free boundary conditions of the upper and lower boundaries of the plate as traction forces, which is relatively simple. For more complex multi-layer plate structures, dispersion equations can also be established by combining the transfer matrix method or the global matrix method with the boundary conditions. By solving the dispersion equations, the propagation direction, wavenumber, displacement mode, and other properties of the guided waves of each mode propagating in the structure can be obtained.

[0146] Given the orthogonality between different guided wave modes, in order to calculate the Green's function in the plate, we further consider two states, one of which is the Green's function state in the plate:

[0147] f i =δ in δ(xy); f i =δ in δ(xy); i = 1, 2;

[0148] Another state is that of an incident wave (denoted as the Pth-order guided wave) incident along the positive x1 axis: f i =0; i=1,2; where c m Let ξ be the amplitude of the m-th order guided wave. m Let f be the wave number of the m-th guided wave in the x1 direction. i Indicates additional physical effort; u i This indicates the x caused by the guided wave propagating along the positive x1 direction. i Directional displacement, σ i1 This represents the stress on the flat plate cross-section caused by the guided wave. Let x and y represent the displacement and stress caused by the m-th order guided wave propagating along the positive x1 direction, respectively. They represent Remove The term following the term is a function term that changes only with respect to x²; the subscripts and superscripts contain negative signs (e.g., u). -i , σ -i1 ) represents the guided wave propagating along the negative x1 direction;

[0149] Substituting the two states into the reciprocal work theorem in elastic dynamics: where the integration region is V (enclosed by the boundary S), respectively represent the body force, displacement, and surface force corresponding to state 1, respectively represent the body force, displacement, and surface force corresponding to state 1; where, the relationship between the surface force and the stress is: For the plane where the cross-section is located, the projection value n1 of its normal vector on the x1 axis is n1 = 1; it can be obtained that:

[0150]

[0151]

[0152] After simplification, it can be obtained that:

[0153]

[0154]

[0155] Using the previous modal orthogonality, discuss separately when m ≠ p and m = p, and finally simplify to obtain:

[0156] where the amplitude of the reflected wave

[0157] Similarly, considering another state is an incident wave propagating along the negative axis, it can also be obtained that:

[0158]

[0159] After simplification, it is obtained that:

[0160] where the amplitude of the reflected wave

[0161] Therefore, it can be concluded that when x1 > y1,

[0162] ;

[0163] when x1 < y1,

[0164]

[0165] The above derivation process is valid for both Lamb waves in isotropic media and Lamb-like waves in orthotropic media. The difference is that for Lamb waves propagating in different materials and different layer structures, the modal characteristics of the propagating guided waves are different, that is, in the displacement function the form is different, and the wave number ξ m in the x1 direction is also different.

[0166] S3: Establish the reconstruction equation for the plate defect and perform preliminary reconstruction;

[0167] Specifically, the boundary integral equation is constructed using the reciprocity theorem of dynamics and the approximate solution of the far field, and the reconstruction equation connecting the defect shape and the reflection coefficient is derived. Based on the reflection coefficient data at discrete frequency points, the defect is initially reconstructed by combining the inverse Fourier transform and the trapezoidal integral rule to obtain the defect location and range.

[0168] In some embodiments, in step S3, the relationship between the boundary integral and the scattering field can be constructed using the dynamic reciprocity theorem:

[0169] Where the subscripts i,j,k,l,n = 1, 2; It is the displacement of the scattered field. It is the total field displacement, U ln (x,y) represents the displacement of the Green's function state, S represents the boundary of the defect range V, where x=(x1,x2) represents the axial and thickness coordinates of the source point, and y=(y1,y2) represents the axial and thickness coordinates of the field point.

[0170] Surface forces in Green's function state

[0171] The traction free condition is satisfied on the defect boundary S in the total field:

[0172] At this point, the boundary integral equation can be simplified to:

[0173] Because the defect size is very small compared to the sample itself in ultrasonic guided wave nondestructive testing, the scattered field is very weak relative to the incident field; using the Born approximation, the total field on the defect is approximated as the incident field. The boundary integral equation is now expressed as:

[0174] Applying Gauss's theorem to transform the boundary integral S(x) into an integral V(x) over the entire defect, we can obtain:

[0175] In some embodiments, for SH waves in a flat plate, the incident field and the far-field approximation of the Green's function are substituted, and the incident wave is a symmetric mode (p-th order): in For intermediate parameters, Let ξ be the amplitude of the p-th order guided wave of the incident wave. p Let x1 be the wave number of the p-th guided wave; This represents the amplitude of the m-th order guided wave in the scattered wave;

[0176] Using the orthogonality of modes, we can obtain:

[0177]

[0178] Define reflection coefficient

[0179] Using the Gaussian integral formula and Taylor expansion, the reflection coefficient can be approximated as:

[0180] Where, ξ p Let x be the wave number in the x1 direction of the p-th guided wave. a ,x b d(x1) represents the range of defects; d(x1) is the expression for the defect depth, and d(x1) = 0 when there are no defects.

[0181] In some embodiments, for Lamb waves and lamb-like waves in a flat plate, the incident field and the far-field approximation of the Green's function are substituted, and the incident field is a p-th order modal guided wave:

[0182] Scattered field: Among them, g m It is an intermediate parameter representing the m-th mode.

[0183] Except g m The definitions of other variables are the same as in S2: that is... Let x and y represent the displacement and stress caused by the m-th order guided wave propagating along the positive x1 direction, respectively. They represent Remove The term following the term is a function term that changes only with respect to x²; the subscripts and superscripts contain negative signs (e.g., u). -i , σ -i1 () represents the guided wave propagating along the negative x1 direction; using the orthogonality of modes, we can obtain:

[0184]

[0185] The expression for the reflection coefficient is:

[0186]

[0187] Using the Gaussian integral formula, the reflection coefficient can be approximated as:

[0188] Where N(x2,ξ) p ) is an intermediate variable, the specific form of which is obtained by separating variables after summing the Gaussian integral.

[0189] Since the interval [h0, h0+d(x1)] is very small, the integrand is expanded in Taylor series at h0 as: N(x2, ξ) p )=N(h0,ξ p )+N'(h0,ξ p (x2-h)+O(x2-h0) 2 ;

[0190] When the integrand retains only the first term: Among them, the intermediate variable G(ξ) p )=N(h0,ξ p ), x a ,x b d(x1) represents the range of defects; d(x1) is the expression for the defect depth, and d(x1) = 0 when there are no defects.

[0191] In some embodiments, for SH waves, Lamb waves, and lamb-like waves, rearranging terms yields the defect reconstruction equation as follows: in These are variables related to the reflection coefficient and wavenumber.

[0192] For SH wave: Lamb waves and lamb-like waves: in, There is a Fourier transform-like relationship between d(x1) and d(x1). Given some discrete data points, the defects are initially reconstructed by combining the inverse Fourier transform and the trapezoidal integral rule.

[0193] Given several reflection data points when the incident wave is in the p-th mode, where the wavenumber points corresponding to the reflection data points are ξ = ξ 1 ,ξ 2 ,ξ 3 …ξ N ξ is the same as ξ in S3. p For the sake of simplicity, ξ will be omitted in the following text. p The subscript of the equation is used to initially reconstruct the equation as follows:

[0194] Based on this formula, a preliminary reconstruction of the defect is achieved, obtaining the approximate location and possible extent of the defect (x). a ,x b ).

[0195] S4: Accurately determine the shape of defects in the flat plate;

[0196] Specifically, based on the defect location and range obtained from the initial reconstruction, the defect shape is assumed to be a combination of basis functions with different amplitudes, and the reflection data at different frequencies are substituted to form a set of discrete equations for defect reconstruction. Then, based on the obtained set of discrete equations for reconstruction, the amplitude of each basis function is obtained by solving the regularization correlation method, thus completing the precise quantitative reconstruction of the defect shape.

[0197] In some embodiments, in step S4, the approximate location of the defect obtained from step S3 and its possible range are (x... a ,x b To further achieve precise physical reconstruction of the defect, the defect shape within the possible range is assumed to be a combination of basis functions with different amplitudes. Using typical trigonometric basis functions, the defect shape is defined as follows:

[0198] in, a n ,b n Unknown undetermined coefficients;

[0199] Substitute different wavenumber data

[0200] For simplicity of solution, separating the real and imaginary parts of the equation yields: Real part virtual part

[0201] The system of equations is obtained as follows:

[0202]

[0203] in:

[0204]

[0205]

[0206]

[0207]

[0208] The above system of equations can also be written in vector form: b = AX;

[0209] Among the measured values Demand

[0210] coefficient matrix

[0211] To solve the above system of equations, the frequency point selected must satisfy the relationship between the possible range of defects. or When matrix A is full rank, the equations can be solved directly; however, in most cases, it is impossible to make the selected wavenumber points satisfy this condition. In this case, matrix A is not full rank, and solving the system of equations becomes a problem of solving an ill-conditioned system of equations.

[0212] For solving ill-conditioned equation systems, the Tikhonov regularization method is used, and the final solution formula can be expressed as: in, To perform a regularization of X in the original equation, I is related to matrix A. T A is an identity matrix of the same order; λ is a regularization parameter, which can be obtained through the L-curve method;

[0213] After solving the system of equations, the unknown undetermined coefficient 'a' is obtained. n ,b n After obtaining the value, substitute it into the defect shape expression:

[0214] The specific location and shape of the defect are obtained, and the defect is accurately quantified and reconstructed.

[0215] The inventive concept of this invention is as follows:

[0216] Taking the reconstruction of surface thinning defects in a monolayer isotropic structure using SH waves as an example, such as... Figure 2 As shown.

[0217] Process overview: such as Figure 2 The figure shows a plate with a thinning defect, where h is the plate thickness, S and S′ are the virtual boundaries of the defect surface boundary and the defect portion of the plate surface, respectively, and the defect range V is bounded by boundaries S and S′. d(x1) is the expression for the defect to be determined, with the horizontal coordinate x1 located at half the plate thickness. Figure 2 As shown, the vertical axis x2 points downwards, the incident wave is along the positive x1 direction, and the interaction with the defect produces a reflected wave u. ref and transmitted wave u tra .

[0218] 1. Perform mode separation on the total field to obtain the reflection coefficient of the desired SH wave mode.

[0219] Based on the orthogonality of different guided modes, the amplitude of the guided mode can be extracted from the SH wave in the flat plate.

[0220] The incident wave is selected as a single-mode guided wave: The reflected wave in the far field is:

[0221] The reflection coefficient can be expressed as: It is a function of wavenumber, and for the same guided wave and the same mode, the wavenumber and frequency are in one-to-one correspondence, so the reflection coefficient can also be called a function of frequency.

[0222] Using the method described in S3, the following is given: Figure 4 As shown, when the dimensionless width L of the defect max / h = 1.0; Maximum dimensionless depth d max / h=0.05, sampling wavenumber points (with respect to the frequency-thickness product hω / c) T The reflection coefficient data of the lowest mode of the SH wave are obtained from 10 points that are uniformly distributed in the range of 0.5-5 (normalized). Figure 5 As shown.

[0223] 2. Solving for the far-field Green's function in a defect-free flat plate.

[0224] The Green's function in a flat plate refers to the time-harmonic concentrated load δ applied at any point within the plate (source point Y). in δ(xy)e iωt Displacement response obtained at the signal receiving location (referred to as field point X)

[0225] For the SH-wave plate, using the method described in S2, its far-field Green's function can be obtained as follows:

[0226] in ξ n Let c be the wave number of the nth order SH wave along the x1 direction. T Let h be the shear wave velocity of the material, h be the plate thickness, and μ be the shear modulus.

[0227] 3. Derive the defect equation based on the reciprocity theorem of dynamics, and perform preliminary reconstruction of the defect by combining the inverse Fourier transform and the trapezoidal integral rule.

[0228] Using the lowest-order SH mode guided wave as the incident guided wave, and employing the method described in S3, we can obtain:

[0229] Where ξ is the wave number of the lowest-order guided wave in the x1 direction, x a ,x b d(x1) represents the extent of the defect; d(x1) represents the defect depth expression.

[0230] Given the wavenumber points ξ = ξ corresponding to the reflection data points. 1 ,ξ 2 ,ξ 3 …ξ N The initial reconstructed equation is as follows:

[0231] Based on this formula, a preliminary reconstruction of the defect can be achieved, and the approximate location and possible extent of the defect (x) can be obtained. a ,x b ),like Figure 7a As shown.

[0232] 4. Based on the preliminary reconstruction results, the defect shape within the possible range is assumed to be a combination of a set of basis functions with different amplitudes. A discrete system of equations is established and solved to finally obtain the accurate reconstruction results of the defect.

[0233] Based on the scope of defects obtained from the initial reconstruction, such as Figure 7a As shown, x a =-2; x b =2, within this range, the defect shape is represented as a combination of a set of basis functions with different amplitudes:

[0234] in a n ,b n Unknown undetermined coefficients;

[0235] Substitute different wavenumber data

[0236] Simultaneously, the boundary conditions of the basis function set will be... Substituting the values, we obtain the system of equations:

[0237]

[0238] The solution to this ill-conditioned system of equations employs the Tikhonov regularization method, and the final solution formula can be expressed as: Where λ is the regularization parameter, which can be obtained through the L-curve method, such as... Figure 6 As shown. Substituting the final solution back into the defect shape expression, the final reconstructed defect shape can be obtained as follows. Figure 7b As shown.

[0239] Analysis of Reconstruction Results and Influencing Factors: Apart from the dimensional defects in the above calculation example, Figures 8a-8d The results of reconstructing triangular defects of various sizes using SH waves are also shown. It should be noted that the reflection coefficient of the lowest order guided wave mode is used for defect reconstruction.

[0240] The impact of changes in defect depth on defect reconstruction results: such as Figures 8a-8d Mid-sampling wavenumber (with respect to the frequency-thickness product hω / c) T (Normalized) represents 10 points uniformly distributed in the range of 0.5-5, (hω / c T It is a dimensionless wavenumber, where h is the plate thickness and c is the wavenumber.T (This refers to the shear wave velocity). When the maximum defect depth changes... Figure 8a Maximum dimensionless depth d max / h = 0.1, maximum dimensionless width L max / h = 1.0; Figure 8b Maximum dimensionless depth d max / h = 0.15, maximum dimensionless width L max / h=1.0; it can be observed that for shallower (d) max For triangular defects of varying depths (h <= 0.1), good reconstruction results can be obtained. However, as the defect depth continues to increase, the assumption of small defects / shallow defects in the theoretical derivation will no longer be satisfied, leading to a decrease in reconstruction accuracy.

[0241] The impact of varying defect width on defect reconstruction results: This section examines the effect of changes in defect surface width on the reconstruction results. Previously... Figure 7b Mid-sampling wavenumber (with respect to the frequency-thickness product hω / c) T The normalized depth d is 10 points uniformly distributed in the range of 0.5-5. max / h = 0.05, maximum dimensionless width L max / h=1.0, Figure 8c The sampling wavenumber points are the same, but the maximum dimensionless depth d of the defect is different. max / h = 0.05, maximum dimensionless width L max / h=0.6; it can be found that using the same wavenumber, the reconstruction effect becomes less than ideal for narrower triangular defects.

[0242] However, when using reflection coefficient data containing higher frequencies, such as Figure 8d Sampling 20 points with a wavenumber (normalized with respect to the frequency-thickness product hω / cT) uniformly distributed in the range of 0.5-10 will yield good reconstruction results. Therefore, when the defect is very narrow, more high-frequency reflection coefficients are needed for defect reconstruction, which can reduce the error of defect reconstruction and meet the needs of engineering inspection.

[0243] The impact of different defect shapes on the defect reconstruction results: Examining the impact of changes in defect shape on the reconstruction results. For example... Figures 9a-9b The defect dimensions are all of maximum dimensionless depth dmax / h = 0.05, maximum dimensionless width Lmax / h = 1.0, and the sampling wavenumbers (normalized with respect to the frequency-thickness product hω / cT) are 20 points uniformly distributed in the range of 0.5-10. Figure 9a The defect in the middle is semi-elliptical. Figure 9b The middle image shows a rectangular defect, and it can be observed that good reconstruction results can be obtained for defects of different shapes.

[0244] The impact of different types of guided waves on defect reconstruction results: In addition to SH waves, some examples of Lamb wave reconstruction were also obtained in this invention. For example... Figures 10a-10b As shown, a computational example of thinning defects in an isotropic monolayer plate using Lamb is presented. The defect dimensions are: maximum dimensionless depth dmax / h = 0.05, maximum dimensionless width Lmax / h = 1.0, and the sampling wavenumbers (dimensionless with respect to the frequency-thickness product hω / cT) are uniformly distributed at 10 points within the range of 0.5-3. Figure 10a The middle part is a triangular defect. Figure 10b The middle one is a rectangular defect. It can be seen that good reconstruction results can be obtained for defects of different shapes even with a small range of sampling number and sampling wave number.

[0245] Reconstruction of complex, multi-defect, multi-layered flat plate structures: for example Figure 11 The double-layer orthotropic anisotropic material plate structure shown is used for defect reconstruction using the reflection coefficient of the lowest-order mode-like Lamb wave. The reconstruction results are given as follows: Figures 12a-12b As shown in the figure. The thicknesses of the two plates are hA = hB = h, where h is the plate thickness and cT is the shear wave velocity of material A along the waveguide propagation direction. The maximum dimensionless depths of the two semi-elliptical defects are dmax / h = 0.03 and dmax / h = 0.05, respectively, and the maximum dimensionless widths are both Lmax / h = 0.6. The sampling wavenumber points (dimensionless with respect to the frequency-thickness product hω / cT) are 10 points uniformly distributed within the range of 0.5-3. Observation shows that the shapes of both defects can be reconstructed quite well. Therefore, this invention can directly locate and reconstruct defects, and the reconstruction of both defects is relatively accurate.

Claims

1. A method for finding defects in plate thinning based on low-frequency reflected ultrasonic guided waves, characterized in that, It includes the following steps: S1: Perform modal separation on the total field of the reflected wave to obtain the reflection coefficient of the desired guided wave mode; Specifically: Utilize the orthogonality between different modes during the propagation of elastic waves in the waveguide medium to perform modal separation on the total field of the reflected wave, and obtain the amplitude of the displacement of the reflected wave corresponding to the mode of the incident waveguide wave. The ratio of this reflected wave amplitude to the incident wave amplitude is the reflection coefficient; S2: Solve the Green's function in the defect-free plate to obtain an approximate solution in the far field; Specifically: Combine the traction-free boundary conditions on the upper and lower boundaries of the plate and the wave equation, and utilize the orthogonality between different mode guided waves to solve for the approximate solution of the Green's function in the far field of the defect-free plate; S3: Establish a reconstruction equation for the plate defect and perform preliminary reconstruction; Specifically: Construct a boundary integral equation from the dynamic reciprocity theorem and the approximate solution in the far field, and derive a reconstruction equation connecting the defect shape and the reflection coefficient. Based on the reflection coefficient data at a small number of discrete frequency points, combine the inverse Fourier transform and the trapezoidal integration rule to perform preliminary reconstruction on the defect, and obtain the defect position and range; S4: Accurately determine the shape of the plate defect; Specifically: Based on the defect position and range obtained in the preliminary reconstruction, assume the defect shape as a combination of a set of basis functions with different amplitudes and substitute them into the reflection data at different frequencies simultaneously to form a reconstructed discrete equation set for the defect; then, based on the obtained reconstructed discrete equation set, combine with regularization-related methods to solve and obtain the amplitude for each basis function, and complete the accurate quantitative reconstruction of the defect shape.

2. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 1, characterized in that, In step S1, for the SH wave in the isotropic flat plate medium, the displacement of the particle is zero in the planes containing the x1 and x2 directions, while the displacement u3 in the x3 direction is not zero, and the displacement u3 satisfies the following equation of motion: Where c T The shear wave velocity of the material. Let x1 and x2 represent the second partial derivatives of the variable with respect to the x1 and x2 directions, respectively. This represents the second partial derivative of the variable with respect to time, where x1 is the axial direction of the plate, x2 is the thickness direction of the plate, and x3 is the direction of the antiplane. Combining the free boundary conditions of the upper and lower boundary surfaces, we obtain the time-omitted harmonic term e. iωt nth order SH wave displacement The expression is Where h is the plate thickness, ω is the waveguide frequency, and c T For material shear wave velocity, As an intermediate parameter, A n The undetermined coefficients representing the amplitude, Let be the wave number of the nth mode SH wave along the x1 direction; The function is undetermined. When n = 0, 2, 4..., it is called a symmetric mode. In this case, f n =cos(β) n x2); n = 1, 3, 5... is called an antisymmetric mode, where f n =sin(β) n X2); Consider two modes of the SH guided wave in the plate, such as the nth and mth modes, and denote the displacement, surface force, and wave number in the x1 direction on its cross-section as follows: and According to the orthogonality of different modes of guided waves, it can be obtained that: where, * is the conjugate symbol; When detecting defects, the sensor receives the total field information: Total field displacement Overall field strength Where ∑ n The symbol represents summation over variable n, where n = n1, n2, ..., n N The order of all modes present in the guided wave at the current frequency; For total field data containing various modes that are difficult to use directly, the amplitudes of the desired modes are extracted based on modal orthogonality: The incident wave is selected as a single-mode guided wave: displacement field in Indicates the amplitude of the incident wave; The reflected wave in the far field is: Indicates the amplitude of the reflected wave; Therefore, the reflection coefficient is defined as: It is a function of wavenumber / frequency.

3. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 2, characterized in that, In step S1, for Lamb waves in an isotropic plate medium and Lamb-like waves in an orthotropic laminated plate structure, the displacement u of the particle along the x1 and x2 directions satisfies the wave equation: Where ρ is the material density, C ijlk This is the fourth-order elastic modulus tensor of the material; For homogeneous and isotropic materials, C ijlk It can be represented as: C ijkl =λδ ij d kl +μ(d ik d jl +d il d jk ); Where λ is Lamé constant, μ is shear modulus, and δ ij The Kronecker symbol is used to determine whether two subscripts are the same; that is, it equals 1 when the subscripts are the same and 0 when they are different. Based on the orthogonality of different modes of guided waves in the wave field, taking any two modes of the Lamb wave in the plate, we record the displacement, surface force, and wave number in the x1 direction on its cross-section as follows: According to the orthogonality of different modes of guided waves, it can be obtained that: where, * is the conjugate symbol; When detecting defects, the sensor receives the total field information: Total field displacement Overall field strength For total field data containing various modes that are difficult to use directly, the amplitudes of the desired modes should be extracted based on modal orthogonality. The incident wave is selected as a single-mode guided wave: in Indicates the amplitude of the incident wave; The reflected wave in the far field is: Indicates the amplitude of the reflected wave; The reflection coefficient can be expressed as: It is a function of wavenumber / frequency. For the same guided wave and the same mode, the wavenumber and frequency are in one-to-one correspondence. Therefore, the reflection coefficient can also be said to be a function of frequency.

4. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 1, characterized in that, In step S2, a time-harmonic concentrated load δ is applied at a source point y at any location within the plate. in δ(xy)e iωt ; At the signal receiving location point x, the displacement response is obtained. For SH waves in the plate, the wave equation corresponding to the Green's function state is: The upper and lower boundaries of the plate are traction-free boundary conditions: Among them, c T ω is the shear wave velocity of the material, h is the plate thickness, and μ is the shear modulus. Solve the Green's function of SH waves using the Fourier transform and the residue theorem, and take the far-field approximate form. Finally, its expression can be obtained as: in, ξ n Let be the wave number of the nth order SH wave along the x1 direction.

5. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 4, characterized in that, For Lamb waves and Lamb-like waves in the plate, the wave equation corresponding to the Green's function state is: Among them, C ijlk The fourth-order elastic modulus tensor is given by ρ, where ρ is the material density and ω is the guided wave frequency. Indicates in x n When a unit external force is applied in the x direction i Displacement response caused by direction; When x1 > y1, the displacement of the far-field Green's function: When x1 < y1, the displacement of the far-field Green's function: Where i,n=1,2; ξ m Let f be the wave number of the m-th guided wave in the x1 direction. i Indicates the additional physical effort at x i Directional component; σ i1 This represents the stress on the flat plate cross-section caused by the guided wave. These represent the displacement and stress fields corresponding to the m-th guided wave mode propagating along the positive x1 direction, respectively. They represent Remove The term following the term is a function term that changes only with respect to x²; the subscripts and superscripts contain negative signs (e.g., u). -i , σ -i1 ) represents the guided wave propagating along the negative x1 direction.

6. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 1, characterized in that, In step S3, through the dynamic reciprocity theorem, the relationship between the boundary integral and the scattered field can be constructed: Where the subscripts i,j,k,l,n = 1, 2; It is the displacement of the scattered field. It is the total field displacement, U ln (x,y) represents the displacement of the Green's function state, S represents the boundary of the defect range V, x = (x1,x2) represents the axial and thickness coordinates of the source point, and y = (y1,y2) represents the axial and thickness coordinates of the field point. Green's function state surface force The traction free condition is satisfied on the defect boundary S in the total field: The boundary integral equation simplifies to: Using the Born approximation, the total field over the defect is approximated as the incident field. The boundary integral equation is written as: Applying Gauss's theorem to transform the boundary integral S(x) into an integral V(x) over the entire defect, we can obtain:

7. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 6, characterized in that, For the SH wave in the flat plate, substituting the incident field and the far-field approximation of the Green's function, the incident wave has a symmetric mode: in, For intermediate parameters, Let ξ be the amplitude of the p-th order guided wave of the incident wave. p Let x1 be the wave number of the p-th guided wave; This represents the amplitude of the m-th order guided wave in the scattered wave; Utilize the orthogonality of the modes to obtain: The expression of the reflection coefficient is: Then, using the Gaussian integral formula and Taylor expansion, approximate the reflection coefficient as: Where, ξ p Let x be the wave number in the x1 direction of the p-th guided wave. a ,x b d(x1) represents the range of defects; d(x1) is the expression for the defect depth, and d(x1) = 0 when there are no defects.

8. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 7, characterized in that, For Lamb waves and lamb-like waves in a flat plate, substituting the incident field and the far-field approximation of the Green's function, the incident field is a p-th order mode guided wave: Scattered field: Among them, g m This is an intermediate parameter representing the m-th mode, and its value is: Let x and y represent the displacement and stress caused by the m-th order guided wave propagating along the positive x1 direction, respectively. They represent Remove The term following the term is a function term that changes only with respect to x²; u -i σ -i1 These represent guided waves propagating along the negative x1 direction; Utilize the orthogonality of the modes to obtain: The expression of the reflection coefficient is: Using the Gaussian integral formula, approximate the reflection coefficient as: Where N(x2, ξ) p ) is an intermediate variable, the specific form of which is obtained by separating variables after summing the Gaussian integral; Since the interval [h0, h0+d(x1)] is very small, the integrand is expanded in Taylor series at h0 as: N(x2, ξ) p )=N(h0,ξ p )+N′(h0,ξ p (x2-h)+O(x2-h0) 2 ; When the integrand retains only the first term: Wherein G(ξ) p )=N(h0,ξ p ), ξ p Let x be the wave number in the x1 direction of the p-th guided wave. a x b d(x1) represents the range of defects; d(x1) is the expression for the defect depth, and d(x1) = 0 when there are no defects.

9. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 8, characterized in that, For SH waves, Lamb waves and Lamb-like waves, after transposing, the defect reconstruction equation is: Among them, for SH wave: Lamb waves and lamb-like waves: in, There is a Fourier transform-like relationship between d(x1) and d(x1). Given some discrete data points, the defects are initially reconstructed by combining the inverse Fourier transform and the trapezoidal integral rule. Given the p-th mode, the wavenumber point ξ = ξ corresponding to the reflected data point. 1 ξ 2 ξ 3 …ξ N ξ omitted p If the subscript is used, the preliminary reconstructed equation is: Based on this formula, a preliminary reconstruction of the defect is achieved, obtaining the defect location and extent (x). a x b ).

10. The guided wave method for quantitative detection of thinning defects in flat plates according to claim 9, characterized in that, In step S4, adopt typical trigonometric basis functions and set the defect shape as: in, a n ,b n Unknown undetermined coefficients; Substitute different wavenumber data For the sake of simplicity in solving, separate the real part and the imaginary part: Obtain the equation set: where: The above system of equations can be denoted in vector form: b = AX; To solve the above system of equations, the frequency point selected must satisfy the relationship between the possible range of defects. or When matrix A is full rank, the equation can be solved directly; When matrix A is not full rank, solving the system of equations becomes a problem of solving an ill-conditioned system of equations. For solving ill-conditioned equation systems, the Tikhonov regularization method is used, and the final solution formula can be expressed as: Wherein, λ is the regularization parameter, which can be obtained through the L-curve method; After solving the system of equations, the unknown undetermined coefficient 'a' is obtained. n ,b n After obtaining the value, substitute it into the defect shape expression: The specific location and shape of the defect are obtained, and the defect is accurately quantified and reconstructed.

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