A method for dynamic analysis of a functionally graded material cracked plate structure

Abstract

The application discloses a kind of functional gradient material crack plate structure dynamics analysis methods, comprising the following steps: obtaining the structure parameters and physical property parameters of crack plate, constructs the physical property parameter distribution function of functional gradient material;Based on first-order shear deformation theory and penalty function, crack plate energy functional is established;Based on the displacement of normal plate is extended to the improved fourier series, constructs the displacement function of normal plate;Establish polar coordinates on crack plate, based on polar coordinates, crack complementary function is constructed;The displacement function of normal plate and crack complementary function are added to obtain the overall displacement function of crack plate;Based on the expansion coefficient vector of overall displacement function, the characteristic equation of functional gradient crack plate structure is constructed;Based on crack plate energy functional, functional gradient crack plate structure characteristic equation is solved, and the dynamic response of functional gradient crack plate is solved by iteration.The application has less limitation to plate thickness, and most of crack plate modeling requirements can be realized according to actual demand.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of structural dynamics, and particularly relates to a functional gradient material cracked plate structure dynamics analysis method. BACKGROUND

[0002] The functional gradient material is a new type of multiphase composite material, and its main feature is that the macroscopic material characteristics are continuously changed in space by continuously changing the volume fraction of each component phase. Since the interface between each component is weakened, the functional gradient material can effectively avoid or reduce stress concentration caused by differences in material performance, and overcome the shortcomings of easy interface separation and interlayer cracking of traditional composite materials. The plate structure is one of the most common components in engineering applications, and is widely used in the fields of ships, aviation, aerospace and the like. During manufacturing or use, the plate may be cracked due to corrosion, fatigue or external load impact, and the existence of the crack changes the vibration characteristics of the plate structure and may cause complex nonlinear dynamics behavior, thereby seriously affecting the reliability and performance of the structure.

[0003] In the traditional modeling for homogeneous thin cracked plates, the crack effect is replaced by a virtual external load mainly according to the deformation equivalence principle, and the elastic plate motion equation coupled with the crack term is established through the mechanical equilibrium principle, and the crack term expression is derived based on the Rice and Levy stress relationship. Since this method is based on the resonance fatigue test of typical homogeneous thin plates, its empirical formula is not applicable to functional gradient materials, and therefore cannot meet the modeling requirements of functional gradient cracked plates. At present, most of the existing researches are focused on the modeling and analysis of cracked beams or cracked thin plates, and there are few researches on the modeling and analysis of cracked medium-thick plates and functional gradient medium-thick plates. Therefore, it is of important application value and research value to break through the limitations of the existing modeling theory and method, and to research and establish a modeling theory of functional gradient cracked plates which is less restricted by plate thickness and boundary conditions.

[0004] The theoretical research on the functional gradient cracked plate structure can analyze the influence of crack parameters on the natural frequency, mode and dynamics response of the plate structure, provide a theoretical basis for online damage detection technology, and enrich the data reference required by damage detection. However, the existence of the crack greatly increases the difficulty of theoretical modeling and analysis of the functional gradient cracked plate structure. Compared with the research results obtained by the finite element method and the experimental method, the theoretical research is still lacking. Therefore, it is necessary to establish an accurate dynamics analysis method of the functional gradient cracked plate structure. SUMMARY

[0005] The purpose of the present application is to provide a functional gradient material cracked plate structure dynamics analysis method to solve the problems existing in the prior art.

[0006] To achieve the above object, the application provides a functional gradient material crack plate structure dynamics analysis method, comprising the following steps:

[0007] Obtaining the structure parameters and physical property parameters of the crack plate, and constructing a physical property parameter distribution function of the functional gradient material;

[0008] Based on the physical property parameter distribution function, a first-order shear deformation theory and a penalty function are used to establish an energy functional of the crack plate;

[0009] Based on the improved Fourier series, the displacement of the normal plate is extended, and a displacement function of the normal plate is constructed;

[0010] Polar coordinates are established on the crack plate, and a crack complementary function is constructed based on the polar coordinates;

[0011] The displacement function of the normal plate and the crack complementary function are added to obtain the overall displacement function of the crack plate;

[0012] Based on the expansion coefficient vector of the overall displacement function, a characteristic equation of the functional gradient crack plate structure is constructed;

[0013] Based on the structure characteristic equation, the dynamic response of the functional gradient crack plate is solved by iteration.

[0014] Optionally, the process of constructing the physical property parameter distribution function of the functional gradient material comprises: representing the physical property parameters of the metal material based on the Young's modulus, Poisson's ratio and density of the metal material; representing the physical property parameters of the ceramic material based on the Young's modulus, Poisson's ratio and density of the ceramic material; and constructing the physical property parameter distribution function of the functional gradient material based on the physical property parameters of the metal material, the physical property parameters of the ceramic material, the thickness of the crack plate and the gradient index.

[0015] The physical property parameter distribution function of the functional gradient material comprises a Young's modulus distribution function of the functional gradient material, a Poisson's ratio distribution function of the functional gradient material and a density distribution function of the functional gradient material.

[0016] Optionally, the process of establishing the energy functional of the crack plate comprises: representing the potential energy of the functional gradient crack plate structure based on the strain vector, the transpose and the material stiffness coefficient matrix; representing the kinetic energy of the functional gradient crack plate structure based on the components of the displacement in the length direction, the width direction and the normal direction of the functional gradient crack plate, the rotation angles in the length direction and the width direction, the time amount and the inertia amount; representing the boundary functional of the functional gradient crack plate structure based on the boundary coefficients of the length end and the width end; representing the external force work of the functional gradient crack plate structure based on the components of the external force in the length direction, the width direction and the normal direction; and obtaining the energy functional of the crack plate based on the potential energy, the kinetic energy, the boundary functional and the external force work of the functional gradient crack plate structure.

[0017] Optionally, the process of constructing the displacement function of the normal plate comprises: obtaining the displacement function of the normal plate based on the displacement expansion order, the expansion coefficient of the normal plate displacement field, the first improved Fourier series term, the second improved Fourier series term and the time-dependent generalized coordinate position coefficient.

[0018] Optionally, the process of constructing the crack complementary function comprises: obtaining the polar radius complementary series based on the displacement expansion order of the polar radius and polar angle complementary function of any point on the cracked plate in the polar coordinate system; obtaining the polar angle complementary series based on the polar angle of any point on the cracked plate in the polar coordinate system; and obtaining the crack complementary function based on the polar radius complementary series, the polar angle complementary series, the expansion coefficient of the complementary function displacement field, the time-dependent generalized coordinate position coefficient, the displacement expansion order of the complementary function and the displacement expansion order of the polar angle complementary function.

[0019] Optionally, the process of establishing the polar coordinate system comprises: establishing the polar coordinate system with the crack tip as the pole and the crack and its extension line as the polar axis on the cracked plate.

[0020] Optionally, the process of constructing the structure characteristic equation of the functionally graded cracked plate comprises: substituting the integral displacement function into the energy functional, and deriving the displacement expansion coefficient of the integral displacement function to obtain the structure characteristic equation of the functionally graded cracked plate.

[0021] Optionally, the process of iteratively solving the dynamic response of the functionally graded cracked plate comprises: solving the eigenvalue of the structure characteristic equation, wherein the eigenvalue is the natural frequency of the cracked plate; bringing the eigenvector corresponding to the eigenvalue into the integral displacement function of the cracked plate to obtain the displacement of vibration, and further obtaining the dynamic response of the functionally graded cracked plate.

[0022] The technical effects of the present application are as follows:

[0023] In the dynamic analysis of the functionally graded material cracked plate structure, the transverse shear deformation of the plate is considered, so that the limitation on the thickness of the plate is less, and most of the modeling requirements of the cracked plate can be realized according to the actual requirements; in addition, for different boundary conditions and different functionally graded materials, only the boundary constraint application and the functionally graded material configuration need to be completed by changing the boundary parameter and material property parameter setting, and reprogramming is not needed, which can greatly save the calculation cost and greatly improve the calculation efficiency. BRIEF DESCRIPTION OF DRAWINGS

[0024] The accompanying drawings, which form a part of the present application, are included to provide a further understanding of the application and are incorporated herein for explanation by reference. In the drawings:

[0025] Figure 1This is a schematic diagram of the dynamic analysis method for a functionally graded material cracked plate structure in an embodiment of the present invention;

[0026] Figure 2 This is a schematic diagram of the rectangular functionally graded crack plate structure in an embodiment of the present invention;

[0027] Figure 3 This is a distribution diagram of the functionally graded material along the plate thickness in an embodiment of the present invention;

[0028] Figure 4 This is a schematic diagram of the plate partitioning and additional polar coordinate system in an embodiment of the present invention. Detailed Implementation

[0029] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0030] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0031] Example 1

[0032] like Figure 1 As shown, this embodiment provides a method for dynamic analysis of a functionally graded material cracked plate structure, including the following steps:

[0033] (1) Extract the plate structure shape parameters, crack shape parameters, material and boundary parameters; its length is L, width is b, and thickness is h; functionally graded materials are generally composed of a combination of metallic and ceramic materials, and their physical properties change in a gradient. The distribution function is expressed as follows:

[0034] P(z)=(P1-P2)(z / h+1 / 2) γ +P2

[0035] Where P(z) represents the distribution function of the physical properties of the functionally graded material [E(z), υ(z), ρ(z)]; P1 represents the physical properties of the metallic material [E1, υ1, ρ1]; P2 represents the physical properties of the ceramic material [E2, υ2, ρ2]; E1 / E2 represents the Young's modulus of the metallic / ceramic material; υ1 / υ2 represents the Poisson's ratio of the metallic / ceramic material; ρ1 / ρ2 represents the density of the metallic / ceramic material; E(z) represents the distribution function of the Young's modulus of the functionally graded material; υ(z) represents the distribution function of the Poisson's ratio of the functionally graded material; ρ(z) represents the distribution function of the density of the functionally graded material; γ represents the gradient exponent, ranging from (0, +∞).

[0036] (2) The energy functional of the cracked plate is established based on the first-order shear deformation theory and the penalty function

[0037]

[0038]

[0039]

[0040]

[0041] Π = T s +W s -U k -U p

[0042] U p , T s , U k and W s are the potential energy, kinetic energy, boundary functional and work of external forces of the functionally graded cracked plate structure, respectively; Π is the energy functional of the cracked plate structure; ε is the strain vector; T represents transposition; D represents the material stiffness coefficient matrix; I0, I1 and I2 are the inertial quantities; x and y represent the coordinates of the functionally graded plate in the length and width directions, respectively, and the coordinate ranges are (0, L) and (0, b), respectively; u, v and w are the displacement components in the x direction, y direction and normal direction, respectively; φ x and φ y represent the angular displacements in the x direction and y direction; t represents the time quantity; k uxi , k vxi , k wxi , k φxxi , k φyxi represent the boundary coefficients of the x i end, i = 0, l; k uyi , k vyi , k wyi , k φxyi , k φyyi represent the boundary coefficients of the y i end, i = 0, l; f u , f v and f w are the components of the external force in the x direction, y direction and normal direction.

[0043] (3) The displacement function of the cracked plate is composed of the displacement function of the normal plate and the crack complementary function. First, the displacement function of the normal plate is constructed. The displacement function of the normal plate is extended by using the improved Fourier series to extend the displacement of the plate, and the specific expression is as follows:

[0044]

[0045]

[0046]

[0047] where D p represents the displacement of the normal plate, M and N are the displacement expansion orders, D pmn represents the expansion coefficients of the normal plate displacement field; F m (x) and Q n (y) represent the mth and nth order improved Fourier series terms, respectively; q(t) represents the time-dependent generalized coordinate position coefficient.

[0048] (4) The whole plate is divided into four regions by the crack and its extension line and a straight line passing through the crack tip and parallel to the y-axis. The angle between the crack and the x-direction is α, the coordinates of the crack tip P are (x0, y0), and the height of the crack is c. The crack and its extension line can be expressed as y = c - L tan α + x tan α in the coordinate system. The ranges of the four regions are shown in Table 1:

[0049] Table 1

[0050] (y = c - L tan a + x tan a, y = b) Region Two Region Four (y = 0, y = c - L tan a + x tan a) Region One Region Three (x = 0, x = x0) (x = x0, x = L)

[0051] (5) An polar coordinate system is established with the crack tip as the pole and the crack and its extension line as the polar axis. The crack extension line is taken as the starting point of the polar angle 0°, and the polar angle ranges are (0, π) and (0, -π), respectively. On the basis of the polar coordinates, a crack supplementary function is introduced to simulate the effect of the existence of the crack. The specific expression is as follows:

[0052]

[0053] R(x, y) = r (2k-1) / 2

[0054] where D c represents the supplementary displacement of the crack, J is the displacement expansion order of the supplementary function, D ickl represents the expansion coefficients of the supplementary function displacement field. u, v, and w are the components of the supplementary displacement in the axial, circumferential, and normal directions, respectively, φ x and φ y represent the rotation displacement in the x and y directions; r represents the polar radius of any point on the cracked plate in the polar coordinate system; θ represents the polar angle of any point on the cracked plate in the polar coordinate system; R(x, y) is defined as the polar radius supplementary series, Ψ i (x, y) is defined as the polar angle supplementary series, and k is the order of the displacement expansion function of the polar angle supplementary function.

[0055] (6) The expression of the overall displacement function of the cracked plate obtained by adding the displacement function of the normal plate and the crack supplementary function is as follows:

[0056] D = D p + D c (D = u, v, w, φ x , φ y )

[0057] (7) Calculate the eigen-equation of the functionally graded cracked plate structure by the energy functional constructed in step (2), substitute the overall displacement function into the energy functional, and derive the displacement expansion coefficients of the overall displacement function respectively to obtain the eigen-equation of the functionally graded cracked plate structure. The specific expression is as follows:

[0058] (K-ω 2 M)H = F

[0059] wherein K is a stiffness matrix; ω is a frequency parameter; M is a mass matrix; H is an expansion coefficient vector of the overall displacement function; and F is a load vector;

[0060] (8) Solve the dynamic response of the functionally graded cracked plate by iteration. Specifically, according to the obtained eigen-equation of the structure, solve the eigenvalues of the eigen-equation, the eigenvalues are the natural frequencies of the cracked plate, and the eigenvectors corresponding to the eigenvalues are vectors composed of the displacement expansion coefficients in the overall displacement function, which are brought into the overall displacement function of the plate to obtain the displacement of vibration, and then the dynamic response of the functionally graded cracked plate is obtained.

[0061] Example Two

[0062] Consider a rectangular functionally graded cracked plate structure, the top view of which is shown in Figure 2 , L = 0.4m, b = 0.2m, thickness = 0.01; crack length L c = 0.4L, c = 0.1, α = 15°; the composition phases of the functionally graded material are ZrO2 and Al, and the material properties are E1 = 168GPa, υ1 = 0.3, ρ1 = 5700kg / m 3 and E2 = 70GPa, υ2 = 0.3, ρ2 = 2707kg / m 3 , the boundary condition is S-S-S-S (four edges simply supported); the gradient index γ = 5, and the material distribution along the plate thickness is shown in Figure 3 .

[0063] (1) Extract the shape parameters of the plate, the shape of the crack, the parameters of the material and the boundary parameters; the expression of the physical property parameters is as follows:

[0064] E(z) = (E1-E2)(z / h+1 / 2) γ +E2

[0065] υ(z) = (υ1-υ2)(z / h+1 / 2) γ+ υ2

[0066] ρ(z) = (ρ1-ρ2)(z / h+1 / 2) γ + ρ2

[0067] (2) The energy functional of the cracked plate is established based on the first-order shear deformation theory and the penalty function

[0068]

[0069]

[0070]

[0071]

[0072] Π = T s + W s - U k - U p

[0073] U s , T s , U p and W s are the potential energy, kinetic energy, boundary functional and external work of the functionally graded cracked plate structure, respectively; Π is the energy functional of the cracked plate structure; ε is the strain vector; T represents the transpose; D represents the material stiffness coefficient matrix; I0, I1 and I2 are the moments of inertia; x and y are the coordinates of the functionally graded plate in the length and width directions, respectively, and the coordinate ranges are (0, L) and (0, b), respectively; u, v and w are the displacement components in the x direction, y direction and normal direction, respectively; φ x and φ y are the angular displacements in the x direction and y direction; t represents the time; k uxi , k vxi , k wxi , k φxxi , k φyxi are the boundary coefficients of the x i end, i = 0, l; k uyi , k vyi , k wyi , k φxyi , k φyyi are the boundary coefficients of the y i end, i = 0, l; f u , f v and f w are the components of the external force in the x direction, y direction and normal direction, respectively.

[0074] (3) The displacement function of the normal plate is constructed. The expansion order is taken as M = N = 8, and the specific expression is as follows:

[0075]

[0076] where D p represents the displacement of the normal plate; D pmn represents the expansion coefficient of the normal plate displacement field; F m (x) and Q n (y) represent the mth and nth order improved Fourier series terms, respectively; q(t) represents the time-dependent generalized coordinate position coefficient.

[0077] (4) As shown in Figure 4 , the whole plate is divided into four areas by the crack and its extension line and the straight line passing through the crack tip and parallel to the y-axis. The angle between the crack and the x direction is 0°, the crack tip P coordinate is (0.24, 0.1), the height of the crack is 0.1 m, and the crack and its extension line can be expressed as y = 0.1 in the coordinate system. The ranges of the four areas are shown in Table 2:

[0078] Table 2

[0079] (y = 0.1, y = 0.2) Region Two Region Four (y = 0, y = 0.1) Region One Region Three (x = 0, x = 0.24) (x = 0.24, x = 0.4)

[0080] (5) And taking the crack tip as the pole and the crack and its extension line as the polar axis to establish a polar coordinate system, a crack supplementary function is introduced to simulate the effect of the existence of the crack on the basis of the polar coordinate, and the expansion order is taken as J = 6. The specific expression is as follows:

[0081]

[0082] where D c represents the supplementary displacement of the crack supplementary function; D ickl represents the expansion coefficient of the supplementary function displacement field; u, v and w are the components of the supplementary displacement in the axial, circumferential and normal directions, respectively, and φ x , φ y represent the angular displacements in the x and y directions.

[0083] (6) The expression of the overall displacement function of the cracked plate obtained by adding the displacement function of the normal plate and the crack supplementary function is as follows:

[0084] D = D p + D c (D = u, v, w, φ x , φ y )

[0085] (7) The function gradient cracked plate structure characteristic equation is calculated by step (2), and the specific expression is as follows:

[0086] (K-ω 2 M)H = F

[0087] wherein K is a stiffness matrix; ω is a frequency parameter; M is a mass matrix; H is an expansion coefficient vector of the overall displacement function; and F is a load vector.

[0088] (8) solving the dynamic response of the functionally graded cracked plate by iteration.

[0089] The above description is merely the preferred specific embodiments of the present application, and the protection scope of the present application is not limited to this. Any changes or replacements within the technical scope disclosed by the present application can be easily thought of by those skilled in the art, and should be covered within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method for dynamic analysis of a functionally graded material cracked plate structure, characterized by, The method comprises the following steps: obtaining the structural parameters and physical property parameters of the cracked plate, and constructing a physical property parameter distribution function of the functionally graded material; based on the physical property parameter distribution function, establishing a cracked plate energy functional by using a first-order shear deformation theory and a penalty function; based on the improved Fourier series, extending the displacement of the normal plate to construct a displacement function of the normal plate; establishing a polar coordinate on the cracked plate, and based on the polar coordinate, constructing a crack complementary function; adding the displacement function of the normal plate and the crack complementary function to obtain the overall displacement function of the cracked plate; based on the expansion coefficient vector of the overall displacement function, constructing a functionally graded cracked plate structure characteristic equation; based on the structure characteristic equation, solving the dynamic response of the functionally graded cracked plate by iteration; the displacement function of the normal plate extends the displacement of the plate by using the improved Fourier series, and the specific expression is as follows: where D p represents the displacement of the normal plate, M and N are the displacement expansion orders, D pmn represents the expansion coefficients of the normal plate displacement field; F m (x) and Q n (y) represent the mth and nth order improved Fourier series terms, respectively; q(t) represents the time-dependent generalized coordinate position coefficient; u, v, and w are the displacement components in the x direction, y direction, and normal direction, respectively, and φ x , φ y represent the angular displacements in the x direction and y direction; x, y represent the length and width coordinates of the functionally graded plate, respectively; the length is L, and the width is b. the construction process of the crack complementary function includes: based on the polar radius and polar angle of any point on the cracked plate in the polar coordinate system, obtaining the polar radius complementary series; based on the polar angle of any point on the cracked plate in the polar coordinate system, obtaining the polar angle complementary series; based on the polar radius complementary series, the polar angle complementary series, the expansion coefficient of the complementary function displacement field, the time-dependent generalized coordinate position coefficient, and the complementary function displacement expansion order and the displacement expansion order of the polar angle complementary function, the crack complementary function is obtained.

2. The functionally graded material cracked plate structure dynamic analysis method according to claim 1, wherein the process of constructing the physical property parameter distribution function of the functionally graded material includes: representing the physical property parameters of the metal material based on the Young's modulus, Poisson's ratio and density of the metal material; representing the physical property parameters of the ceramic material based on the Young's modulus, Poisson's ratio and density of the ceramic material; based on the physical property parameters of the metal material, the physical property parameters of the ceramic material, the thickness of the cracked plate and the gradient index, the physical property parameter distribution function of the functionally graded material is constructed; wherein the physical property parameter distribution function of the functionally graded material includes the Young's modulus distribution function of the functionally graded material, the Poisson's ratio distribution function of the functionally graded material and the density distribution function of the functionally graded material.

3. The functionally graded material cracked plate structure dynamic analysis method according to claim 1, wherein the process of establishing the cracked plate energy functional includes: representing the potential energy of the functionally graded cracked plate structure based on the strain vector, the transpose and the material stiffness coefficient matrix; representing the kinetic energy of the functionally graded cracked plate structure based on the displacement components in the length direction, width direction and normal direction of the functionally graded cracked plate, the angular displacement in the length direction and width direction, the time amount and the inertia amount; representing the boundary functional of the functionally graded cracked plate structure based on the boundary coefficients of the length end and the width end; representing the external force work of the functionally graded cracked plate structure based on the components of the external force in the length direction, width direction and normal direction; and obtaining the cracked plate energy functional based on the potential energy, kinetic energy, boundary functional and external force work of the functionally graded cracked plate structure.

4. The functionally graded material cracked plate structure dynamic analysis method according to claim 1, wherein The polar coordinate system establishment process comprises: establishing a polar coordinate system with the crack tip as a polar point and the crack and its extension line as polar axes on the crack plate. 5.The functional gradient material crack plate structure dynamic analysis method according to claim 1, characterized in that, The functional gradient crack plate structure characteristic equation construction process comprises: substituting the integral displacement function into the energy functional, and respectively deriving displacement expansion coefficients of the integral displacement function to obtain the functional gradient crack plate structure characteristic equation. 6.The functional gradient material crack plate structure dynamic analysis method according to claim 5, characterized in that, The iterative solution process of the functional gradient crack plate dynamic response comprises: solving eigenvalues of the structure characteristic equation, the eigenvalues being the natural frequencies of the crack plate; bringing the eigenvectors corresponding to the eigenvalues into the integral displacement function of the crack plate to obtain the vibration displacement, and further obtaining the functional gradient crack plate dynamic response.

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