Method for calculating anisotropic stress of composite laminated plate based on isogeometric analysis

Abstract

The invention belongs to the field of composite material simulation, and provides a method for calculating the anisotropic stress of a composite material laminated plate based on isogeometric analysis, which comprises the following steps of: establishing a mapping relation between a physical model and a spline surface based on the size of the composite material laminated plate, establishing a displacement strain relation of the composite material laminated plate through a high-order shear deformation function, and calculating the stress of the composite material laminated plate. The stress-strain relation of the composite laminated plate is established through a constitutive equation, the displacement of control points on the composite laminated plate is solved and obtained by applying boundary conditions and adopting an isogeometric analysis method, a virtual work principle and a discrete equilibrium equation, and the stress-strain relation, the constitutive equation and a high-order shear deformation function are used for calculating the displacement of the control points on the composite laminated plate. Stress distribution of the composite material laminated plate in the plate thickness direction is obtained, and stress calculation of the composite material laminated plate is completed. According to the method, the solving efficiency is high, the obtained stress value is accurate, the simulation calculation amount is small, and the method has important significance for evaluating the structural strength of the composite material.

Description

Technical Field

[0001] This invention belongs to the field of composite material simulation, and relates to a method for calculating the stress of composite laminates, and more particularly to a method for calculating the stress of composite laminates in all directions based on isogeometric analysis. Background Technology

[0002] Composite materials possess high specific strength, specific stiffness, excellent fatigue strength, and corrosion resistance, along with high design flexibility. Composite laminates are widely used in engineering fields such as aerospace, automotive, and sports equipment. Accurately predicting the mechanical properties of composite laminates under various load conditions is a prerequisite for structural design. Interlaminar stress in composite laminates is one of the main factors causing structural failure. However, calculating interlaminar stress requires a three-dimensional model, resulting in large computational loads and high costs. Therefore, it is necessary to develop more efficient models and corresponding simulation methods to achieve accurate prediction of interlaminar stress, thereby improving the simulation efficiency of composite laminate structures.

[0003] CN201810095947.3 converts the deformation of composite material sheets into uncoupled tension, bending, and shear, and finally calculates the total stiffness and total load of the composite material sheets, and analyzes the interlaminar stress, but it is not used to solve the technical problem of this invention. Summary of the Invention

[0004] To address the aforementioned problems, this invention provides a method for calculating the anisotropic stress of composite laminates based on isogeometric analysis, particularly enabling accurate prediction of interlaminar stress.

[0005] The technical solution of the present invention is as follows: A method for calculating the anisotropic stress of composite laminates based on isogeometric analysis includes the following steps: S1. Based on the dimensions of the composite laminate, establish the mapping relationship between the physical model and the spline surface; S2, by using higher-order shear deformation functions, establish the displacement-strain relationship of composite laminates; S3, through constitutive equations, establish the stress-strain relationship of composite laminates; S4. By applying boundary conditions, using the isogeometric analysis method and the principle of virtual work, the equilibrium equations are discretized, and the displacements of the control points on the composite laminate are solved and obtained. S5. By using displacement-strain relationship, constitutive equation, and higher-order shear deformation function, the stress distribution in the thickness direction of the composite laminate is obtained, and the stress calculation of the composite laminate is completed.

[0006] Furthermore, in S1, the dimensions of the composite laminate include the layup angle, thickness, and length and width dimensions of each layer.

[0007] Furthermore, in S1, the physical model of the composite laminate is transformed into a non-uniform rational B-spline surface model. A mapping relationship between the physical model and the spline surface is established, and the mapping relationship is represented by the product of the control point coordinates and the non-uniform rational B-spline basis functions. (1) Where S is a two-dimensional non-uniform rational B-spline surface; where These are the coordinates of the control points. For a non-uniform rational B-spline surface, the subscripts i and j indicate the control point numbers; p and q are the orders of the non-uniform rational B-spline curve. and Represents different directions in the natural coordinate system; The spline surface mapping relationship is represented by the product of the control point coordinates and the non-uniform rational B-spline basis functions:

[0008] M and N are expressions for non-uniform rational B-spline functions; and In the natural coordinate system and The non-uniform rational B-spline basis functions corresponding to the direction. These are the weighting coefficients for the corresponding control points.

[0009] Furthermore, the non-uniform rational B-spline basis functions in the natural coordinate system are expressed as: when p When = 0,

[0010] for p ≥1,

[0011] in, It is the first i 1 node n It is the number of basis functions. p It is the order of the non-uniform rational B-spline curve.

[0012] Furthermore, the specific method of S2 is as follows: The higher-order shear deformation function is: (3) in, (4) Where h is the thickness of the composite laminate. Let z represent a dimensionless constant, and z in formulas (3) and (4) represent the coordinates of the composite laminate in the thickness direction; Any point of the composite laminate relative to x , y , z Displacement in direction U=[ UVW ] is represented as: (5) in, u , v , w For the physical plane correspondence x , y , z Displacement in three directions α , β for y , x Rotation of direction.

[0013] Based on equation (5), the strain ε at any point of the composite laminate is obtained.

[0014] Furthermore, S3 specifically refers to: The constitutive equation, i.e., the stress-strain relationship, of the composite laminate is: (7) (8) ε is the strain at any point in the composite laminate. k The first layer of composite laminate represents the composite laminate. k layer; It is the equivalent material constant.

[0015] Furthermore, in S4, the isogeometric analysis method used is as follows: The displacement of each point on the surface of the composite laminate can be expressed as the product of a non-uniform rational B-spline surface and control points: (10) In the formula, u represents the magnitude of the displacement at any point; ij For the first i , j The magnitude of the displacement corresponding to each control point It is a non-uniform rational B-spline surface; Based on the control point displacements, the strain coefficient matrix of the composite laminate is obtained and expressed in element form: (11) Where L is the differential operator; R is the combination matrix of non-uniform rational B-spline basis functions.

[0016] Furthermore, according to the principle of virtual work, the work done by the external force is equal to the deformation energy of the object, and the energy relationship in any unit is: (12) in, D This refers to the mid-surface region of the composite laminate. Ds The area under the influence of external forces. Let δ be the external load, w be the lateral displacement of the composite laminate, assemble the element stiffness matrix into the global stiffness matrix, eliminate the illusory displacement, and obtain the discretized equilibrium equation. Then solve the discretized equilibrium equation to obtain the displacement d of all control points.

[0017] Furthermore, S5 specifically refers to: The control point displacement is obtained by discretizing the equilibrium equation in S4, the strain is obtained by stress-strain relationship in S2, and the stress of each ply of the composite laminate is obtained by stress-strain constitutive equation of composite laminate in S3.

[0018] Furthermore, after S5, the stress of each ply of the composite laminate is output along the thickness direction, and the stress variation curve is plotted.

[0019] Technical effects of the present invention: 1. This invention provides a method for calculating anisotropic stress in composite laminates based on isogeometric analysis. Compared with the traditional finite element method, this method directly performs mesh generation and calculation based on the geometric model, achieving integration from modeling to simulation. It directly discretizes the solution domain through the geometric model and accurately describes the physical model using control point coordinates and non-uniform rational B-spline basis functions, resulting in higher solution efficiency and more accurate stress values. Compared with the common Reddy model, the calculated normal stress error is reduced by 2% and 1.1%, and the transverse shear stress error is significantly reduced by 16% and 9%, respectively.

[0020] 2. This invention uses a shear deformation function to describe the stress variation law of composite laminates along the thickness direction, avoiding the model architecture that requires a three-dimensional model to obtain interlaminar stress, especially interlaminar shear stress. By using a shear deformation function to describe the stress variation law of composite laminates along the thickness direction, this invention avoids the model architecture that requires a three-dimensional model to obtain interlaminar stress, especially interlaminar shear stress. This results in a smaller simulation computation and higher simulation efficiency; the computational scale is only one-quarter of that of a three-dimensional model.

[0021] 3. This invention can accurately calculate the anisotropic stress value at any point in a composite laminate, as well as the stress distribution between layers. This is of great significance for evaluating the structural strength of composite materials, ensuring quality and usability. It is a major innovation in simulation methods for interlaminar stress in composite laminates, and has significant social and economic benefits. Attached Figure Description

[0022] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the embodiments of the present invention will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0023] Figure 1 Schematic diagram of control point distribution and coordinates for composite laminate. Figure 1 .

[0024] Figure 2 Schematic diagram of control point distribution and coordinates for composite laminate. Figure 2 .

[0025] Figure 3 Schematic diagram of control point distribution and coordinates for composite laminate. Figure 3 .

[0026] Figure 4 Schematic diagram of control point distribution and coordinates for composite laminate. Figure 3 .

[0027] Figure 5 This is a schematic diagram showing the geometry and layup sequence of the composite laminate.

[0028] Figure 6 Schematic diagram of interlaminar stress distribution in composite laminate. Figure 1 .

[0029] Figure 7 Schematic diagram of interlaminar stress distribution in composite laminate. Figure 2 .

[0030] Figure 8 Schematic diagram of interlaminar stress distribution in composite laminate. Figure 3 .

[0031] Figure 9 Schematic diagram of interlaminar stress distribution in composite laminate. Figure 4 .

[0032] Figure 10 Schematic diagram of interlaminar stress distribution in composite laminate. Figure 5 .

[0033] in, Figure 1 The geometric mesh and control point coordinates are given, where p = 3, n = 5, and the node vectors are... .

[0034] Figure 2 The geometric mesh and control point coordinates are given, where p = 3, n = 6, and the node vectors are... .

[0035] Figure 3 The geometric mesh and control point coordinates are given, where p = 3, n = 8, and the node vectors are... .

[0036] Figure 4 The geometric mesh and control point coordinates are given, where p = 3, n = 12, and the node vectors are... .

[0037] Figure 6 For point Stress at location Distribution curve along the thickness direction, layup sequence [0°, 90°] s ; Figure 7 For point Stress at location Distribution curve along the thickness direction, layup sequence [0°, 90°] s ; Figure 8 For point Stress at location Distribution curve along the thickness direction, layup sequence [0°, 90°] s ; Figure 9 For point Stress at location Distribution curve along the thickness direction, layup sequence [0°, 90°] s ; Figure 10 For point Stress at location Distribution curve along the thickness direction, layup sequence [0°, 90°] s a and b are the length and width dimensions of the composite laminate, respectively. Detailed Implementation

[0038] This section describes embodiments of the present invention, used to explain and illustrate the technical solutions of the present invention. Unless otherwise specified, the embodiments and features described herein can be combined with each other.

[0039] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicating directions or positional relationships, are given in the accompanying drawings and are used only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or device referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implying the number of indicated technical features. Thus, features defined with "first," "second," etc., may explicitly or implicitly include more than one of those features. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0040] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to fixed connections, detachable connections, or integrated connections; they can refer to mechanical connections or point connections; they can refer to direct connections or indirect connections through an intermediate medium; and they can refer to the internal communication between two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0041] Example 1: A method for calculating the anisotropic stress of composite laminates based on isogeometric analysis includes the following steps: S1. Based on the dimensions of the composite laminate, establish the mapping relationship between the physical model and the spline surface; S2, by using higher-order shear deformation functions, establish the displacement-strain relationship of composite laminates; S3, through constitutive equations, establish the stress-strain relationship of composite laminates; S4. By applying boundary conditions, using the isogeometric analysis method and the principle of virtual work, the equilibrium equations are discretized, and the displacements of the control points on the composite laminate are solved and obtained. S5. By using displacement-strain relationship, constitutive equation, and higher-order shear deformation function, the stress distribution in the thickness direction of the composite laminate is obtained, and the stress calculation of the composite laminate is completed.

[0042] In S1, the dimensions of the composite laminate include the layup angle, thickness, and length and width of each layer.

[0043] In S1, the physical model of the composite laminate is transformed into a non-uniform rational B-spline surface model. A mapping relationship between the physical model and the spline surface is established, and the mapping relationship is represented by the product of the control point coordinates and the non-uniform rational B-spline basis functions. (1) Where S is a two-dimensional non-uniform rational B-spline surface; where These are the coordinates of the control points. For a non-uniform rational B-spline surface, the subscripts i and j indicate the control point numbers; p and q are the orders of the non-uniform rational B-spline curve. and Represents different directions in the natural coordinate system; The spline surface mapping relationship is represented by the product of the control point coordinates and the non-uniform rational B-spline basis functions:

[0044] M and N are expressions for non-uniform rational B-spline functions; and In the natural coordinate system and The non-uniform rational B-spline basis functions corresponding to the direction. These are the weighting coefficients for the corresponding control points.

[0045] The non-uniform rational B-spline basis functions in the natural coordinate system are expressed as follows: when p When = 0,

[0046] for p ≥1,

[0047] in, It is the first i 1 node n It is the number of basis functions. p It is the order of the non-uniform rational B-spline curve.

[0048] The specific method for S2 is as follows: The higher-order shear deformation function is: (3) in, (4) Where h is the thickness of the composite laminate. Let z represent a dimensionless constant, and z in formulas (3) and (4) represent the coordinates of the composite laminate in the thickness direction; Any point of the composite laminate relative to x , y, z Displacement in direction U=[ UVW ] is represented as: (5) in, u , v , w For the physical plane correspondence x , y , z Displacement in three directions α , β for y , x Rotation of direction.

[0049] Based on equation (5), the strain ε at any point of the composite laminate is obtained.

[0050] S3 specifically refers to: The constitutive equation, i.e., the stress-strain relationship, of the composite laminate is: (7) (8) ε is the strain at any point in the composite laminate. k The first layer of composite laminate represents the composite laminate. k layer; It is the equivalent material constant.

[0051] In S4, the specific isogeometric analysis method used is as follows: The displacement of each point on the surface of the composite laminate can be expressed as the product of a non-uniform rational B-spline surface and control points: (10) In the formula, u represents the magnitude of the displacement at any point; ij For the first i , j The magnitude of the displacement corresponding to each control point It is a non-uniform rational B-spline surface; Based on the control point displacements, the strain coefficient matrix of the composite laminate is obtained and expressed in element form: (11) Where L is the differential operator; R is the combination matrix of non-uniform rational B-spline basis functions.

[0052] According to the principle of virtual work, the work done by external forces is equal to the deformation energy of the object. The energy relationship in any unit is as follows: (12) in, D This refers to the mid-surface region of the composite laminate. DsThe area under the influence of external forces. Let δ be the external load, w be the lateral displacement of the composite laminate, assemble the element stiffness matrix into the global stiffness matrix, eliminate the illusory displacement, and obtain the discretized equilibrium equation. Then solve the discretized equilibrium equation to obtain the displacement d of all control points.

[0053] S5 specifically refers to: The control point displacement is obtained by discretizing the equilibrium equation in S4, the strain is obtained by stress-strain relationship in S2, and the stress of each ply of the composite laminate is obtained by stress-strain constitutive equation of composite laminate in S3.

[0054] After S5, the stress of each ply of the composite laminate is output along the thickness direction, and the stress change curve is plotted.

[0055] Example 2: This invention discloses a method for calculating the anisotropic stress of composite laminates based on isogeometric analysis. Isogeometric analysis employs a non-uniform rational B-spline surface, establishing a precise mapping relationship between the physical model and the computational model through control point coordinates. Combined with a higher-order shear deformation theory model of the composite laminate, the anisotropic stress components are calculated, achieving an integrated process from modeling to computational simulation of the composite laminate. The steps of this invention are as follows: Based on the length and width dimensions of the composite laminate, establish the mapping relationship between the physical model and the spline surface; by selecting a suitable higher-order shear deformation function, establish the displacement-strain relationship of the laminate; through constitutive equations, establish the stress-strain relationship of the laminate; by applying boundary conditions, using isogeometric analysis and the principle of virtual work, discretize the equilibrium equations, solve for and obtain the displacement of the control points; through the displacement-strain relationship, constitutive equations, and higher-order shear deformation function, obtain the stress distribution at any location, realizing the computational simulation of the anisotropic stress of the composite laminate.

[0056] A method for calculating the anisotropic stress of composite laminates based on isogeometric analysis is specifically implemented as follows: Without loss of generality, the examples used in this invention employ dimensionless geometric parameters and material properties, as follows: Geometric information: a = b = 10 h The layup sequence is [0°, 90°]. s .

[0057] Material properties: E 1 = 25 E 2, G 12 = G 13 = 0.5 E 2, G 23= 0.2 E 2, ν 12 = ν 21 = 0.25.

[0058] The boundary conditions are as follows: Displacement boundary conditions: simply supported on four sides, i.e., in place, ;exist place, .

[0059] Boundary conditions of force: subjected to sinusoidal distributed load in the Z direction ,in .

[0060] The stress is dimensionless as follows:

[0061] Step 1, based on the length dimension a and width dimension b of the composite laminate (e.g., ... Figure 1 Establish the mapping relationship between the physical model and the spline surface, and define this mapping relationship using the coordinates of control points (such as...). Figure 1 ) and the product representation of non-uniform rational B-spline basis functions.

[0062] (1) in These are the coordinates of the control points. It is a non-uniform rational B-spline surface, where p and q are the orders of the basis functions. Figure 1 This represents the distribution of control points and their corresponding coordinate values ​​when p=q=3.

[0063] For composite laminates, the mapping relationship of spline surfaces is determined by the following equation: (2) in and In the natural coordinate system and The non-uniform rational B-spline basis functions corresponding to the direction. These are the corresponding weighting coefficients.

[0064] The non-uniform rational B-spline basis functions in the natural coordinate system are expressed as follows: when p When = 0,

[0065] for p ≥1,

[0066] in,

[0067] in, It is the first i 1 node n It is the number of basis functions. p It is the order of the non-uniform rational B-spline curve.

[0068] Special, for Figure 1 , p = 3, n = 5, node vector is Mesh refinement can be achieved by inserting node coordinates into the node vector.

[0069] For example, for Figure 2 p = 3, n = 6, node vector is ;for Figure 3 p = 3, n = 8, node vector is .

[0070] for Figure 4 , p = 3, n = 12, node vector is The coordinates of the corresponding control points are respectively in Figures 1 to 4 The information is provided in the text.

[0071] By inserting nodes into the node vector, control points can be automatically added and generated, thereby refining the spline surface and improving simulation accuracy.

[0072] Step 2: By selecting a suitable higher-order shear deformation function, the displacement-strain relationship of the laminate is established, as follows: The higher-order shear deformation function is: (3) in, (4) in, h Let be the thickness of the composite laminate. This shear deformation function satisfies the condition that the shear stress on the upper and lower surfaces of the composite laminate is zero, and ensures that the shear stress in each layer is continuously distributed along the thickness direction.

[0073] Based on equation (3), any point of the composite laminate relative to... x , y , z Displacement in direction U=[ UVW ] is represented as: (5) in, u , v , w In the physical plane, the corresponding global coordinate systemx , y , z Displacement in three directions α , β for y , x Rotation of direction.

[0074] Based on equation (5), the strain at any point in the composite laminate is expressed as: (6) Step 3: Establish the stress-strain relationship of the laminated plate using the constitutive equations, as follows: The constitutive equation, i.e., the stress-strain relationship, of the composite laminate is: (7) (8) in, k The first layer of composite laminate represents the composite laminate. k layer; It is an equivalent material constant, derived from the elastic modulus, Poisson's ratio, and ply angle of a composite single-layer plate. The result is as follows:

[0075] in:

[0076] Step 4: By applying boundary conditions, using the isogeometric analysis method and the principle of virtual work, the equilibrium equations are discretized, and the displacements of the control points are solved, as follows: The boundary conditions include force boundary conditions (Neumann conditions) and displacement boundary conditions (Dirichlet conditions). By applying external nodal forces to the nodes and constraining the displacements, a unique displacement solution can be obtained.

[0077] The specific isogeometric analysis method used is as follows: The displacement of each point on the surface of the composite laminate can be expressed as the product of a non-uniform rational B-spline surface and control points: (10) In the formula, u represents the magnitude of the displacement at any point; ij For the first i , j The magnitude of the displacement corresponding to each control point.

[0078] Based on the control point displacement, the strain coefficient matrix of the laminate can be obtained from equations (5) and (6), and expressed in element form: (11) Where L is the differential operator; R is the combination matrix of non-uniform rational B-spline basis functions.

[0079] According to the principle of virtual work, the work done by external forces is equal to the deformation energy of the object. The energy relationship in any unit is as follows: (12) in, D This represents the area of ​​the mid-surface region of the laminate. Ds The area of ​​the region where the external force acts. For external loads. Assemble the element stiffness matrices into a global stiffness matrix, eliminate virtual displacements, and obtain the discrete equations: (13) Solving the above equation yields the displacement d of all control points.

[0080] Step 5: Using displacement-strain relationships, constitutive equations, and higher-order shear deformation functions, the stress distribution along the thickness of the plate is obtained, realizing the interlaminar stress analysis and simulation of the composite laminate, as detailed below: The method for determining the interlaminar stress of the composite laminate is as follows: The control point displacement d obtained from equation (13) is used to obtain the strain through equation (6), and the stress of each ply is obtained through equation (7).

[0081] Finally, the stress of each layer is output along the thickness direction, and the stress variation curve is plotted to complete the simulation. Table 1 shows a comparison of the calculation scale, stress results, and errors relative to the three-dimensional model obtained using the model in this invention, the Reddy theory model, and the three-dimensional model. It can be seen that although this invention still has some errors compared to the three-dimensional model, they are all within an acceptable range.

[0082] Compared to the Reddy theoretical model, the stress results obtained by this invention show a significant improvement in accuracy, especially the transverse shear stress results, with an error of less than 6% compared to the three-dimensional model. This provides highly reliable simulation results for predicting the interlaminar strength of composite materials. Furthermore, in terms of computational scale, this invention's computational scale is only one-quarter that of the three-dimensional model, greatly improving computational efficiency.

[0083] Table 1. Comparison of stress results and errors obtained from different models

[0084] Figure 5 This refers to the configuration of a composite laminate. Figures 6-10 The layup sequence is [0°, 90°]. s The stress distribution curve at a specified location on the composite material plate.

[0085] The present invention provides a simple method for calculating the anisotropic stress of composite laminates based on isogeometric analysis, enabling accurate prediction of stress at any location and between layers.

[0086] The above description is merely a specific embodiment of the present invention, providing a detailed description of the invention. Parts not covered herein are conventional techniques. However, the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention. The scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for calculating each direction stress of a composite laminate based on isogeometric analysis, characterized by, The method comprises the following steps: S1, based on the size of the composite laminated plate, a physical model and a spline surface mapping relationship are established; S2, a displacement-strain relationship of the composite laminated plate is established through a high-order shear deformation function; S3, a stress-strain relationship of the composite laminated plate is established through a constitutive equation; S4, through the application of boundary conditions, an equal geometry analysis method and a virtual work principle are adopted to discretize the balance equation, and the displacement of the control points on the composite laminated plate is solved and obtained; S5, the stress distribution of the composite laminated plate in the thickness direction is obtained through the displacement-strain relationship, the constitutive equation and the high-order shear deformation function, and the stress calculation of the composite laminated plate is completed.

2. The method according to claim 1, wherein, In S1, the size of the composite laminated plate includes the laying angle, thickness and length-width size of each layer.

3. The method according to claim 1, wherein, In S1, the physical model of the composite laminated plate is converted into a non-uniform rational B-spline surface model, the mapping relationship between the physical model and the spline surface is established, and the physical model mapping relationship is expressed by the product of the control point coordinates and the non-uniform rational B-spline basis function: （1） Wherein, S is a two-dimensional non-uniform rational B-spline surface; wherein That is, the control point coordinates, is a non-uniform rational B-spline surface, and the subscripts i and j represent the control point number; p and q are the degrees of the non-uniform rational B-spline curve; and represent different directions in the natural coordinate system; And the spline surface mapping relationship is expressed by the product of the control point coordinates and the non-uniform rational B-spline basis function: M and N are the expressions of non-uniform rational B-spline function; and are the non-uniform rational B-spline basis functions corresponding to the directions of the natural coordinate system and are the non-uniform rational B-spline basis functions corresponding to the directions of the natural coordinate system are the weight coefficients of the corresponding control points.

4. The method according to claim 3, wherein, The non-uniform rational B-spline basis function in the natural coordinate system is expressed as: When p = 0, For p ≥1, in, It is the first i 1 node n It is the number of basis functions. p It is the order of the non-uniform rational B-spline curve.

5. The method according to claim 1, wherein, The method of S2 is as follows: The high-order shear deformation function is: （3） wherein (4) where h is the thickness of the composite laminate, where z is the coordinate in the thickness direction of the composite laminate, and z = 0 at the outer surface of the composite laminate. The displacement U of any point of the composite laminate with respect to x , y , z the direction of the displacement U=[ UVW ] is expressed as: （5） wherein u , v , w is a physical displacement in the direction of the surface normal x , y , z three directions, α , β is y , x a rotation in the direction Based on formula (5), the strain ε of any point of the composite laminated plate is obtained.

6. The method according to claim 1, wherein, S3 is specifically: The constitutive equation of the composite laminated plate, i.e., the stress-strain relationship, is: （7） （8） ε is the strain at any point of the composite laminate, k represents the first k layer of the composite laminate; is the equivalent material constant.

7. The method according to claim 1, wherein, In S4, the equal geometry analysis method is specifically: The displacement of each point on the surface of the composite laminated plate is expressed in the form of the product of the non-uniform rational B-spline surface and the control point: （10） where u represents the displacement size of any point; u ij is the first i , j displacement size corresponding to the control point; is a non-uniform rational B-spline surface; and represent different directions under the natural coordinate system; According to the displacement of the control points, the strain coefficient matrix of the composite laminated plate is obtained, which is expressed in the form of an element: （11） Wherein L is a differential operator; R is a non-uniform rational B-spline basis function combination matrix.

8. The method according to claim 7, wherein, According to the virtual work principle, the work done by the external force is equal to the deformation energy of the object, and the energy relationship in any element is: （12） wherein, D is the mid-plane region of the composite laminate, Ds is the out-of-plane region, is the external load, δ is the dummy symbol, w is the transverse displacement of the composite laminate, the element stiffness matrix is assembled into the global stiffness matrix, the dummy displacement is eliminated, and the discrete equilibrium equation is obtained. The discrete equilibrium equation is solved to obtain the displacement d of all control points.

9. The method of claim 1, wherein, S5 is specifically: The displacement of the control points is obtained through the discretized balance equation in S4, the strain is obtained through the stress-strain relationship in S2, and the stress of each layer of the composite laminated plate is obtained through the stress-strain relationship constitutive equation of the composite laminated plate in S3.

10. The method of claim 1, wherein, After S5, the stress of each layer of the composite laminated plate is output along the thickness direction, and the stress change curve is drawn.

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