A finite element method for solving generalized axisymmetric plane problems using stress as the fundamental variable.

Abstract

This invention provides a finite element method for solving generalized axisymmetric plane problems using stress as the fundamental variable, relating to the field of engineering analysis. The method includes: S1. Selecting basis functions that satisfy the equilibrium equations; S2. Applying the complementary energy principle to establish the element equations for each element; S3. Assembling all element equations to form the overall stiffness equation; S4. Applying forced boundary conditions to the stiffness equation; S5. Solving the stiffness equation; S6. Obtaining all stress components at any point within the element through basis function interpolation; S7. Repeating S1 to S6 to obtain the element stress distribution under each basic load condition; S8. Obtaining the stress state under any composite load based on the principle of linear superposition; S9. Evaluating the strength of the stressed body according to material failure criteria. This invention does not require users to possess professional modeling skills; through simple parameter input, the stress solution of complex axisymmetric problems can be obtained quickly and accurately, and strength evaluation and performance characterization can be performed accordingly.

Description

Technical Field

[0001] This invention relates to the fields of engineering analysis, computer-aided engineering (CAE) and materials characterization, and specifically to a finite element method for solving generalized axisymmetric plane problems using stress as the basic variable. Background Technology

[0002] In engineering practice, there are numerous structural components with axisymmetric or quasi-axisymmetric geometric shapes and stress conditions, such as rotating disks, thick-walled cylinders, and unit cell models of unidirectional fiber-reinforced composite materials. Utilizing their axisymmetry, three-dimensional problems can be simplified to two-dimensional or even one-dimensional problems, significantly reducing the difficulty of solving them. Classical elasticity mechanics can provide exact solutions for such problems, especially when limited to homogeneous, isotropic materials under simple loads.

[0003] However, with the emergence of new materials and structures, the aforementioned classical analytical methods face challenges. For example, the turbine disks of modern aero engines are usually made of high-temperature alloys or composite materials, with complex geometries (variable thickness) and material properties (anisotropy, multiphase composite), and are subjected to combined loads such as gradient temperature change fields and centrifugal inertial forces; thick-walled composite pipes are composed of single-layer plates with different winding angles, exhibiting anisotropy, and their wall thickness may far exceed the applicability of classical laminate theory; the micromechanical analysis of fiber-reinforced composite materials requires consideration of the complex microstructure of fibers, interface layers, and the matrix.

[0004] Currently, solving such complex problems mainly relies on general-purpose commercial finite element analysis software. These software programs typically use displacement as the basic variable and possess strong versatility. However, they have a high learning curve, demanding significant expertise in modeling, mesh generation, and physical understanding from the user. Even minor deviations in the modeling process can lead to distorted results or even mislead engineering designs. For routine problems requiring rapid iteration in the early stages of engineering design, the cumbersome operation steps and time-consuming calculations of general-purpose software fail to provide the most convenient, fast, and reliable solutions.

[0005] Therefore, there is an urgent need for a targeted, easy-to-use, and reliable analysis method to meet the needs of engineering designers for rapid stress analysis and performance evaluation of generalized axisymmetric problems at both macroscopic and microscopic scales. Summary of the Invention

[0006] In view of this, embodiments of this application provide a finite element method for solving generalized axisymmetric plane problems using stress as the fundamental variable. This method does not require users to have professional modeling skills; by simply inputting parameters, it can quickly and accurately obtain the stress solution of generalized axisymmetric plane problems under arbitrary combined loads, and thereby perform strength evaluation and performance characterization, thus providing an effective tool for rapid iteration in the early stages of engineering design.

[0007] This application provides the following technical solution: a finite element method for solving generalized axisymmetric plane problems using stress as the basic variable, comprising the following steps:

[0008] S1. Select the stress basis functions that satisfy the equilibrium equations based on the problem type and material properties of the body to be analyzed;

[0009] S2. Divide the stressed body into several elements, apply the principle of minimum complementary energy, and establish the element equations for each element. The element equations include the stiffness matrix and the load vector; each element includes several nodes.

[0010] S3. Assemble the element equations of all elements according to the nodal degrees of freedom of each node in the element to form the overall stiffness equation;

[0011] S4. Apply forced boundary conditions to the overall stiffness equation, including surface force boundary conditions and stress finite value conditions at the center of the solid body;

[0012] S5. Solve the overall stiffness equation to obtain the stress values ​​of all nodal points in the structure;

[0013] S6. Based on the nodal stress values ​​of the element, the stress components at any point within the element are obtained by interpolation using the stress basis function, thus obtaining the element stress distribution;

[0014] S7. Based on the theoretical model of each load condition, repeat steps S1 to S6 to solve for the element stress distribution under multiple basic load conditions.

[0015] S8. Based on the principle of linear superposition, the stress distribution of the units under the multiple basic load conditions is combined according to the proportion of the actual composite load to obtain the stress state under any composite load.

[0016] S9. Based on the obtained stress state, perform strength assessment and material characterization on the stressed body according to the material failure criteria.

[0017] According to one embodiment of this application, the stress basis function in step S1 is selected as follows: first, a function that can obtain the exact solution mode of a single-layer material under the condition of no gradient temperature change field is selected as the basis function. Then, based on the basis function, a function that is different from but close to the exact solution mode is introduced to expand it in order to handle gradient temperature change field or complex load conditions.

[0018] According to one embodiment of this application, in step S2, the principle of minimum redundant energy is applied to establish the element equations for each element, including: expressing the stress field within the element as a function of the nodal stress, substituting it into the expression for the total redundant potential energy, and through variation of the nodal stress, ensuring that the set stress field satisfies the deformation compatibility condition in the sense of minimum redundant potential energy, thereby obtaining the algebraic equations for all nodal stresses of the structure.

[0019] According to one embodiment of this application, in step S4, the surface force boundary condition is applied as a forced boundary condition, which is achieved by directly specifying the surface force value at the boundary node; the stress finite value condition at the center of the solid body is used to determine the integral constant of the central element or constrain the degree of freedom of the central node.

[0020] According to one embodiment of this application, the method further includes: for quasi-axisymmetric problems, first reducing the order by the separation of variables method, and then solving according to steps S1-S6.

[0021] According to one embodiment of this application, in step S8, obtaining the stress state under any composite load specifically includes: firstly obtaining the basic stress solutions for uniaxial tension / compression along the principal axis of the material and pure shear in the principal plane of the material, and then linearly combining the obtained basic stress solutions according to the proportion of the actual composite load.

[0022] According to one embodiment of this application, in step S9, the strength assessment is to judge the stress state of the calculated stress body by applying a failure criterion applicable to orthotropic materials; the material characterization is applicable to orthotropic materials.

[0023] According to one embodiment of this application, the finite element method is applied to stress analysis of rotating turbine disks and thick-walled composite material pipes at the macroscopic scale, or to equivalent performance prediction and micro-stress analysis of unidirectional fiber-reinforced composite materials at the microscopic scale.

[0024] Compared with the prior art, the beneficial effects that can be achieved by at least one of the above-mentioned technical solutions adopted in the embodiments of this specification include: the embodiments of this invention encapsulate complex physical models and mathematical solution processes within the software. Users only need to input a small amount of data such as geometric dimensions, material parameters, and load conditions, and can quickly obtain results without any modeling and mesh generation experience. It is convenient to use and can obtain high-precision approximate solutions, which has strong engineering practical value. Attached Figure Description

[0025] To more clearly illustrate the technical solutions of the embodiments of this application, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0026] Figure 1 This is a flowchart illustrating the finite element method for solving generalized axisymmetric plane problems using stress as the basic variable, according to the present invention.

[0027] Figure 2 This is a schematic diagram of the column model according to an embodiment of the present invention;

[0028] Figure 3 This is a schematic diagram illustrating the application of this invention to the analysis of thick-walled multilayer composite material pipes at a macroscopic scale.

[0029] Figure 4 This is a schematic diagram illustrating the application of an embodiment of the present invention to the analysis of a rotating turbine disk with varying thickness at a macroscopic scale;

[0030] Figure 5 This is a schematic diagram of the bending analysis of a multilayer composite material thick-walled pipe based on a quasi-axisymmetric stress analysis problem according to an embodiment of the present invention. Detailed Implementation

[0031] The embodiments of this application will now be described in detail with reference to the accompanying drawings.

[0032] The following specific examples illustrate the implementation of this application. Those skilled in the art can easily understand other advantages and effects of this application from the content disclosed in this specification. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. This application can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of this application. It should be noted that, in the absence of conflict, the following embodiments and features in the embodiments can be combined with each other. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0033] like Figure 1As shown, this embodiment of the invention provides a finite element method for solving generalized axisymmetric plane problems using stress as the fundamental variable. In this embodiment, a "generalized axisymmetric plane problem" refers to a three-dimensional load-bearing body whose length along the axis of symmetry is much greater or less than the dimensions of its cross-section. Under certain load conditions, its stress analysis problem can be simplified into a two-dimensional mathematical problem defined within the plane of its cross-section. All stress analysis problems satisfying this condition are generally referred to as generalized plane problems. "Generalized axisymmetric" includes the usual "axisymmetric problem" as one category; it also includes another category called "quasi-axisymmetric problem," in which the load-bearing body is geometrically axisymmetric, but its load condition varies along the circumferential direction according to a cosine function. Therefore, the stress function as the solution to the problem can be expressed as the product of a function about the radial coordinates (axisymmetric) and a cosine function (non-axisymmetric).

[0034] The finite element method includes the following steps:

[0035] S1. Select the stress basis functions that satisfy the equilibrium equations based on the problem type and material properties of the body to be analyzed;

[0036] S2. Divide the stressed body into several elements, and apply the principle of minimum complementary energy to establish the element equations for each element. The element equations include the stiffness matrix and the load vector. Each element includes several nodes. (Here, "stiffness" is a common term in the finite element method; in physics, its property is actually flexibility.)

[0037] S3. Assemble the element equations of all elements according to the nodal degrees of freedom of each node in the element to form the overall stiffness equation;

[0038] S4. Apply forced boundary conditions to the overall stiffness equation, including surface force boundary conditions and stress finite value conditions at the center of the solid body;

[0039] S5. Solve the overall stiffness equation to obtain the stress values ​​of all nodal points in the structure;

[0040] S6. Based on the element nodal stress values, all stress components at any point within the element are obtained through interpolation of the stress basis function, thus obtaining the element stress distribution; the above basis function is used as post-processing in the finite element method, and the stress within the element can be represented by nodal stress.

[0041] S7. Based on the theoretical model of each load condition, repeat steps S1 to S6 to solve for the element stress distribution under multiple basic load conditions (i.e., the stress state under uniaxial tension, compression and pure shear in the principal axis direction of the material).

[0042] S8. Based on the principle of linear superposition, the stress distribution of the units under the multiple basic load conditions is combined according to the proportion of the actual composite load to obtain the stress state under any composite load.

[0043] S9. Based on the obtained stress state, the strength of the stressed body is evaluated according to the material failure criteria to determine the strength of the stressed body material at macroscopic and microscopic scales; and material characterization applicable to orthotropic materials is performed.

[0044] This invention aims to address stress analysis problems in special load-bearing bodies such as cylinders or disks at macroscopic or microscopic scales. Users can obtain solutions and assess strength without modeling, using convenient, fast, and reliable application software. Simultaneously, this invention can also solve related problems such as heat conduction and electrical conduction in these cylindrical or disk-shaped load-bearing bodies. The characteristic of this analysis method is that it uses stress as the fundamental variable and establishes a finite element method for axisymmetric or quasi-axisymmetric problems. Most current commercial finite element software uses displacement as the fundamental variable and requires strong modeling skills. This invention addresses the limitations of commercial finite element software from both theoretical and practical perspectives, overcoming the problem of users without professional modeling training being unable to operate it correctly.

[0045] The method is applied to stress analysis of rotating turbine disks and thick-walled composite pipes at the macroscopic scale, or to the prediction of equivalent properties and micro-stress analysis of unidirectional fiber-reinforced composite materials at the microscopic scale. Specifically, the stress analysis problems covered by this invention include, but are not limited to:

[0046] 1. Stress analysis and strength assessment of a rotating turbine disk under a gradient temperature change field;

[0047] 2. Stress analysis and strength assessment of thick-walled pipes laminated with composite materials under uniform internal / external pressure, axial tension / compression, and torsion around the axial direction, and under gradient temperature change fields, especially when the geometric dimensions far exceed the applicable range of classical laminated plate and shell theory (such as the ratio of wall thickness to radius exceeding 1:5 or even higher).

[0048] 3. Characterize the equivalent elastic properties, equivalent strength properties, equivalent coefficient of thermal expansion, equivalent thermal conductivity, and equivalent electrical conductivity of unidirectional composite materials (especially ceramic matrix composites) made of fibers with multiple layers of different coatings;

[0049] 4. By analyzing the microscopic stress of unidirectional composite materials made of fibers with multiple layers of different coatings, a strength assessment suitable for macroscopic applications is made for the material, and the material is characterized.

[0050] In one embodiment of the present invention, the stress basis function in step S1 is selected as follows: First, a function that yields an exact solution mode for a single-layer material under no gradient temperature change field is selected as the basis function. Then, based on this basis function, a function that is different from but close to the exact solution mode is introduced to expand the basis function, in order to handle complex load conditions with gradient temperature change fields. In this embodiment, the basis function is selected according to the mode of the exact solution that can be obtained from a single layer under the condition of no gradient temperature change field, ensuring that an exact solution can also be obtained for multi-layer structures under the same conditions. Then, a function that is similar to but different from the mode of the exact solution is selected to expand the basis function, ensuring that the approximate solution has sufficiently high accuracy under the condition of temperature change gradient field.

[0051] In one embodiment of the present invention, in step S2, the principle of minimum complementary energy is applied to establish the element equations of each element, including: expressing the stress field in the element as a function of the nodal stress, substituting it into the expression of the total complementary potential energy, and through variation of the nodal stress, making the set stress field satisfy the deformation compatibility condition in the sense of minimum complementary potential energy, thereby obtaining the algebraic equations about the nodal stress.

[0052] In one embodiment of the present invention, in step S4, the surface force boundary condition is applied as a forced boundary condition and is achieved by directly specifying the surface force value at the boundary node; the stress finite value condition at the center of the solid body is used to determine the integral constant of the central element or constrain the degree of freedom of the central node.

[0053] In this embodiment, the surface force boundary conditions must be applied as mandatory boundary conditions to ensure they are precisely satisfied. For solid bodies, the method for handling the boundary conditions at the center point is: the boundary condition at the center of the solid body is that all stresses at that point must be finite values. This cannot directly determine any stress at the center node, but it can help determine the relevant integration constants. Solving the stiffness equation after applying all boundary conditions directly yields the nodal stresses.

[0054] In one embodiment of the present invention, for quasi-axisymmetric problems, the order is first reduced using the separation of variables method, and then the solution is obtained according to steps S1-S6. The quasi-axisymmetric problems include in-plane shear and out-of-plane shear. For quasi-axisymmetric problems such as in-plane and out-of-plane shear, the analysis process first uses the separation of variables method to express the stress as the product of an unknown function only about the radial coordinate and a cosine function about the circumferential coordinate, obtaining the governing equation of the unknown function, thus reducing the problem to a one-dimensional problem, and then performing finite element analysis.

[0055] In one embodiment of the present invention, step S8, obtaining the stress state under arbitrary composite load, specifically includes: firstly, obtaining the basic stress solutions for uniaxial tension / compression along the principal axis of the material and pure shear in the principal plane of the material; then, linearly combining the obtained basic stress solutions according to the proportion of the actual composite load. This embodiment utilizes the superposition principle; after obtaining the stress state under uniaxial tension or compression along any principal axis of the material and shear in any principal plane of the material, the stress state under composite load conditions is linearly combined according to the applied composite load ratio.

[0056] In one embodiment of the present invention, in step S9, the strength assessment is to judge the stress state of the calculated stress body by applying a failure criterion applicable to orthotropic materials.

[0057] This invention is applicable to several classes of real-world engineering problems that are physically approximated. Its theoretical basis is elasticity, but it can be applied at both microscopic and macroscopic scales. Through mathematical analogy, it can also be applied to characterize the thermal and electrical conductivity of materials. Existing solutions in elasticity are only applicable to homogeneous, isotropic materials. When the cylindrical material to be analyzed is anisotropic and radially inhomogeneous, no readily available solution exists. Without using existing general-purpose finite element software, independent solutions are unavoidable. This invention addresses such problems by establishing independent solution methods under essentially the same assumptions and solution approaches. These methods are implemented on the Matlab platform using self-developed software, yielding solutions to the required analysis problems with minimal and simple steps and data input. The resulting software can serve as an application tool for engineers in related fields.

[0058] The problems that this invention can analyze and solve include the following types:

[0059] 1. Microscopic (applicable to unidirectional fiber-reinforced resin matrix composites and ceramic matrix composites): equivalent elasticity, strength constant; equivalent thermal conductivity, electrical conductivity; microscopic stress analysis, strength assessment;

[0060] 2. Macroscopic: Stress analysis and strength assessment of composite material thick-walled pipes; stress analysis and strength assessment of composite material turbine disks.

[0061] Because the complexity of the problem exceeds the scope of typical analytical solutions in elasticity mechanics, this invention employs a finite element method (FEM) with stress as the fundamental variable, based on the principle of minimum total complementary energy. While the FEM with stress as the fundamental variable lacks general applicability, it offers unparalleled convenience, speed, and reliability as a specific solution for a particular problem, even if general applicability is not a primary concern. For a specific problem, the analysis itself essentially defines the physical and mathematical models, eliminating the need for users to independently model the problem. This avoids stringent requirements on the user's professional modeling skills and experience, and its use is straightforward, providing immediate results. Its convenience and speed are akin to using a calculator for addition, subtraction, multiplication, and division, making it extremely suitable for the rapid iterations required in the early stages of engineering design. Its conception, derivation, and implementation all demonstrate significant originality.

[0062] This invention addresses several specific problems with practical engineering value, deriving unique solutions that allow designers to quickly and easily obtain solutions without the need for modeling and simulation on large, general-purpose design platforms, thus meeting engineering design requirements. Specifically, for axisymmetric or quasi-axisymmetric problems under plane stress, generalized plane stress, or plane strain and generalized plane strain conditions, this invention proposes a finite element method based on the principle of minimum total complementary energy, using stress as the fundamental variable. This method can accurately (under the condition of no gradient in the temperature field, ignoring unavoidable truncation errors in the computer) or approximately solve the problem. Because the problem is one-dimensional, the mesh is merely an interval on a number line, requiring no special skills for partitioning. Furthermore, due to the reasonable selection of the approximate solution mode, the mesh requires almost no further subdivision to obtain a sufficiently accurate solution.

[0063] The so-called quasi-axisymmetric problem in this invention is actually a non-axisymmetric problem, but it exhibits a simple regularity along the circumferential direction. Therefore, the solution can be expressed as the product of an undetermined function that depends only on the radial coordinate and a sine or cosine function that depends only on the circumferential coordinate using the method of separation of variables, and then solved. The focus of this invention is on how to determine the undetermined coefficients in the aforementioned undetermined function using the finite element method with stress as the basic variable after applying the method of separation of variables.

[0064] The so-called generalized plane stress in this invention refers to the assumption that all other plane stresses are satisfied. However, the thickness of the sheet elastic body involved is not uniform. In this case, the in-plane membrane force (in-plane stress multiplied by the local sheet thickness) can be used to replace the in-plane stress, and then the mathematical problem can be established.

[0065] The so-called generalized plane strain in this invention refers to satisfying all other plane strain assumptions. However, the columnar elastic body involved can simultaneously withstand tension and compression along the length direction and torsion about the length axis. In this case, the mathematical problem can still be simplified into a two-dimensional problem defined in the cross-section of the column, but it can still involve strain in the third direction.

[0066] like Figure 2 As shown, Figure 2 This is a schematic diagram of the columnar model in an embodiment of the present invention, used to illustrate the physical meaning of the model at different scales (macro / micro) and different problems (plane stress / strain, generalized / narrowly defined). The dashed circles represent multi-layered composites along the thickness direction. External loads can be any combination of the components shown in the diagram. ΔT represents a temperature-varying field applied to the model, which may or may not have a radial gradient, used to simulate non-uniform temperature loads. The arrows on the cubes in the diagram represent all stress components in a macroscopic sense, which can all be applied to the model as loads. When its length is much larger than its cross-sectional diameter, it is described by plane strain or generalized plane strain problems; when its length is much smaller than its cross-sectional diameter, it resembles a disk, and its length is the disk thickness, which can be described by plane stress; when the disk thickness is non-uniform (but still axisymmetric), it is described by generalized plane stress problems. At the microscale, this columnar model approximates a unit cell of a unidirectional fiber-reinforced resin-based or ceramic-based composite material, which can be used to characterize and analyze such materials. The model is a continuum consisting of a coaxial solid cylinder (located at the center) and multiple hollow cylinders (excluding the central layer), with no limit on the number of layers (represented by the dashed lines in the figure). It is applicable to composite materials formed by fibers with multiple interfaces and an outer matrix.

[0067] like Figure 3 As shown, Figure 3 This is a schematic diagram illustrating the application of this invention to the analysis of multilayer composite thick-walled pipes on a macroscopic scale. The dashed circles represent multilayer composites along the thickness direction. ΔT represents a temperature-varying field applied to the model, which may or may not have a radial gradient, used to simulate non-uniform temperature loads. Arrows on the inner circle of the end face represent internal pressure, and arrows on the outer circle represent external pressure. On the pipe body, straight arrows represent axial tension (compression when negative), and curved arrows represent torsion, which can also be subject to temperature-varying loads. The column model, applied on a macroscopic scale, is used to analyze multilayer thick-walled pipes. The pipe is hollow, and each layer can be an axisymmetric anisotropic material, including but not limited to composite materials wound at an arbitrary angle by unidirectional composite materials. The number of layers is unlimited (represented by dashed lines in the diagram). The pipe can withstand uniform internal pressure, uniform external pressure, axial tension or compression, torsion around the axial direction, and temperature variations distributed arbitrarily along the radial direction.

[0068] like Figure 4 As shown, Figure 4This is a schematic diagram illustrating the application of this invention to the analysis of a variable-thickness rotating turbine disk at a macroscopic scale. The dashed circles represent multiple segments of the disk made of different materials or of different thicknesses; the arrows represent rotational angular velocities. The columnar model, applied at the macroscopic scale to the analysis of a variable-thickness rotating disk, allows for segmental definition of thickness (steps are indicated by dashed lines; multiple steps are possible, but they must be symmetrical about the mid-plane), stiffness, and strength. The material can be anisotropic, but axisymmetric, and can have arbitrarily distributed temperature variations along the radial direction.

[0069] The embodiments of this invention use an interpolation basis function selected with reference to the exact solution of a simpler problem, and a finite element method obtained with stress as the basic variable. The theoretical basis for the implementation of the claimed method is as follows:

[0070] Taking a case with 5 nodes per element (including the two endpoints) as an example, the number of nodes may be more or less. The nodes within the element can be, but are not limited to, equally spaced nodes. The Airy stress function is set as follows:

[0071] (1)

[0072] Where r is the radial coordinate, log is the natural logarithm, a is a characteristic length, and the subsequent related expressions are dimensionless without loss of generality. It can be the radius of a fiber, pipe, or impeller, etc., and C1 to C5 are undetermined integration constants. The element of the first matrix factor on the right-hand side of equation (1) is the selected basis function. The general solution of the exact solution of the stress of an isotropic material under the condition of no temperature gradient can be obtained from the first and third elements in the basis function matrix. The form of the general solution may change due to the anisotropy of the material. Therefore, equation (1) does not exclude the possibility of using other suitable basis functions. The fifth element in the basis function matrix corresponds to the particular solution of the exact solution of the stress under the action of rotational inertial force. The particular solution corresponding to the temperature gradient generally does not have an analytical form. Therefore, when constructing the basis function of the approximate solution, other functions besides the above functions must be added. The second and fourth elements in the basis function matrix are for this purpose, and at the same time, they supplement the default intermediate order. Applications show that, because these basis functions cover the true solution space sufficiently well, a sufficiently accurate approximation can generally be obtained by simulating a region that has to be divided due to physical reasons using a single element. Subdividing the elements can sometimes yield a small improvement in accuracy, but the improvement is usually not significant.

[0073] From equation (1) and the definition of the Airy stress function, the stress field can be obtained as follows:

[0074] (2)

[0075] in:

[0076] (3)

[0077] θ is the circumferential coordinate; g r g θ These are terms corresponding to rotational inertia forces; in problems without rotational inertia forces, they are all zero.

[0078] For a solid object, the Airy stress function corresponding to equation (1) for the element at the center is set as:

[0079] (4)

[0080] Accordingly,

[0081] (5)

[0082] The equation (2) representing the stress field is derived from the Airy stress function, and thus the equilibrium equation of the problem is satisfied.

[0083] C1 to C5 are expressed using the radial stress at the nodes of the elements:

[0084] (6)

[0085] Where σ1 to σ5 represent the radial stresses at the five nodes of the element, and also the degrees of freedom of all nodes in the element, and G1 to G5 represent the stresses at the five nodes of the element. r The values ​​at the 5 nodes of the cell;

[0086] (7)

[0087] For units that do not contain a center:

[0088] (8)

[0089] For the unit containing the center:

[0090] (9)

[0091] Where r1 to r5 are the radial coordinates of the five nodes of the element.

[0092] In equations (2) and (6), G1 to G5 and g r g θ These only appear in the rotating bladed disk problem and represent the contribution of inertial forces; in problems without inertial forces, they are all zero.

[0093] The continuity of normal surface forces between elements is guaranteed by the sharing of nodal degrees of freedom between intersecting elements.

[0094] The energy (total residual potential energy) is in the following form:

[0095] (10)

[0096] Where [S] is the compliance matrix of the material under the corresponding generalized plane problem, ν rz ν θz These are the Poisson's ratios in the radial and circumferential directions, respectively, α rr α θθ α zz These are the coefficients of thermal expansion along the three coordinate axes, and ΔT is the temperature change. It is a constant strain along the axial direction. The expression of the total complementary energy principle implies that the constitutive equations are satisfied. Applying the variational principle, that is:

[0097] (11)

[0098] This ensures the satisfaction of the compatibility equations (corresponding to the application of variational principles in the conventional finite element method to satisfy the equilibrium equations). Substituting equation (6) into equation (10) expressing the residual potential energy, we can obtain the problem's "stiffness" (actually flexibility) matrix and the corresponding "loads" through equation (11):

[0099] (12)

[0100] Where m is the number of elements and n is the number of nodes, satisfying:

[0101] n = 4 × m + 1 (13)

[0102] The superscript of the elements of the stiffness matrix is ​​the element number, and the superscript of the elements of the nodal stress and nodal load is the node number. The steps of generating the overall "stiffness" equation of the assembled structure involved in the generation of equation (12) are similar to those of the usual finite element method, and will not be described in detail here.

[0103] Regarding boundary conditions, there are two significant differences compared to the typical finite element method that uses displacement as the fundamental variable:

[0104] First, the surface force applied on the boundary in a physics problem is not a load, but a forced boundary condition, which needs to be imposed through nodal stress at the boundary.

[0105] Second, for a solid cylinder / disc, the center is a special boundary, and the boundary conditions on it are also special, that is, all stresses must be finite values, the effect of which is that C1 and C2 in equation (1) must be zero.

[0106] For a hollow object, the boundary conditions are:

[0107] (14)

[0108] in , These are the radial stresses applied to the inner and outer boundaries, respectively. Applying boundary conditions results in:

[0109] (15)

[0110] For solid objects, there is no However, the first and second rows and columns of the structural stiffness matrix are all zero. In order to prevent the structural stiffness matrix from becoming singular due to the selection of basis functions in equation (4), a positive value is artificially assigned to the two diagonal elements of the structural stiffness matrix. In order to avoid unnecessary differences in the values ​​between the elements of the structural stiffness matrix and thus cause ill-conditioning, the positive value is taken from the third diagonal element.

[0111] (16)

[0112] This treatment of the first and second degrees of freedom means that:

[0113] (17)

[0114] This is only to make the structural stiffness equation solvable, but the stresses at the first and second nodes obtained are not the true solutions, and they need to be recalculated according to equation (4) during post-processing.

[0115] The above outlines the main steps of axisymmetric problems. Quasi-axisymmetric problems have a relatively narrow application scope, primarily serving as a tool for characterizing unidirectional fiber-reinforced composite materials, especially ceramic matrix composites reinforced with multi-layered coated ceramic fibers, through micromechanics. Furthermore, they only address the shear within and outside the cross-section of the material, without needing to consider temperature changes, gradients, or inertial forces. In this case, two nodes per element are sufficient. Examples of quasi-axisymmetric stress analysis problems include microscopic bending of multi-layered coated ceramic fibers and macroscopic bending of multi-layered thick-walled pipes, such as... Figure 5 As shown, the dashed circles represent multi-layered composite structures along the thickness direction, and the arrows in the figure represent bending loads. Bending loads can be bending moments acting on the cross-section of the column, or the curvature of the column's axis. This load condition is one of the load conditions that can be analyzed in column-column models.

[0116] For shear problems within a cross section, the Airy stress function can be defined based on the analytical solution obtainable for a single layer, as follows:

[0117] (18)

[0118] The stress within the element can be obtained as:

[0119] (19)

[0120] Where C1 to C4 are undetermined integration constants, which can be expressed by the stress at the nodes, and thus obtained:

[0121] (20)

[0122] The nodal degrees of freedom of element σ1, σ2, τ1, and τ2 are the radial stress and shear stress at the nodes, respectively. The nodal degrees of freedom of the entire structure are:

[0123] (twenty one)

[0124] Because this analysis is only a part of the characterization and stress analysis of unidirectional fiber reinforced composite materials, the corresponding physical model is solid, and the center is implemented according to the approach of equation (16).

[0125] At the outer boundary, the boundary conditions are:

[0126] (twenty two)

[0127] Shear outside the cross section is mathematically a diffusion problem. The shear stress within the element can be obtained from the corresponding Prandtl stress function, based on the analytical solution for a single layer:

[0128] (twenty three)

[0129] Where τ1 and τ2 represent the nodal degrees of freedom of the element, i.e., the radial shear stress at the node. The nodal degrees of freedom of the entire structure are:

[0130] (twenty four)

[0131] At the center, the approach of equation (16) is followed, while at the outer boundary, the boundary conditions are:

[0132] (25)

[0133] The analysis method of this invention enables designers to obtain solutions to problems quickly and easily without having to model and simulate on large-scale general-purpose design tool platforms, thereby meeting the needs of engineering design.

[0134] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A finite element method for solving generalized axisymmetric plane problems using stress as the fundamental variable, characterized in that, Includes the following steps: S1. Select the stress basis functions that satisfy the equilibrium equations based on the problem type and material properties of the body to be analyzed; S2. Divide the stressed body into several elements, apply the principle of minimum complementary energy, and establish the element equations for each element. The element equations include the stiffness matrix and the load vector; each element includes several nodes. S3. Assemble the element equations of all elements according to the nodal degrees of freedom of each node in the element to form the overall stiffness equation; S4. Apply forced boundary conditions to the overall stiffness equation, including surface force boundary conditions and stress finite value conditions at the center of the solid body; S5. Solve the overall stiffness equation to obtain the stress values ​​of all nodal points in the structure; S6. Based on the stress values ​​at the element nodes, all stress components at any point within the element are obtained through interpolation using the stress basis function, thus obtaining the element stress distribution; S7. Based on the theoretical model of each load condition, repeat steps S1 to S6 to solve for the element stress distribution under multiple basic load conditions. S8. Based on the principle of linear superposition, the stress distribution of the units under the multiple basic load conditions is combined according to the proportion of the actual composite load to obtain the stress state under any composite load. S9. Based on the obtained stress state, perform strength assessment and material characterization on the stressed body according to the material failure criteria.

2. The finite element method for solving generalized axisymmetric plane problems with stress as the basic variable according to claim 1, characterized in that, The stress basis function in step S1 is selected as follows: First, a function that can obtain the exact solution mode of a single-layer material under the condition of no gradient temperature change field is selected as the basis function. Then, based on the basis function, a function different from the exact solution mode is introduced to expand it in order to handle complex load conditions with gradient temperature change field.

3. The finite element method for solving generalized axisymmetric plane problems with stress as the basic variable according to claim 1, characterized in that, In step S2, the principle of minimum complementary energy is applied to establish the element equations for each element, including: expressing the stress field within the element as a function of the radial stress at the nodes, substituting it into the expression for the total complementary potential energy, and through variation of the nodal stress, ensuring that the set stress field satisfies the deformation compatibility condition in the sense of minimum complementary potential energy, thereby obtaining the element algebraic equations for the radial stress at the nodes.

4. The finite element method for solving generalized axisymmetric plane problems with stress as the basic variable according to claim 1, characterized in that, In step S4, the surface force boundary condition is applied as a forced boundary condition by directly specifying the surface force value at the boundary node; the stress finite value condition at the center of the solid body is used to determine the integral constant of the central element or constrain the degree of freedom of the central node.

5. The finite element method for solving generalized axisymmetric plane problems with stress as the basic variable according to claim 1, characterized in that, Also includes: For quasi-axis symmetric problems, first reduce the order using the method of separation of variables, and then solve according to steps S1-S6.

6. The finite element method for solving generalized axisymmetric plane problems with stress as the basic variable according to claim 1, characterized in that, In step S8, the stress state under any composite load is obtained, specifically including: first, obtaining the basic stress solutions for uniaxial tension / compression along the principal axis of the material and pure shear in the principal plane of the material; then, linearly combining the obtained basic stress solutions according to the proportion of the actual composite load.

7. The finite element method for solving generalized axisymmetric plane problems with stress as the basic variable according to claim 1, characterized in that, In step S9, the strength assessment is to judge the stress state of the calculated load-bearing body by applying the failure criteria applicable to orthotropic materials; the material characterization is applicable to orthotropic materials.

8. The finite element method for solving generalized axisymmetric plane problems with stress as the basic variable according to any one of claims 1 to 7, characterized in that, The finite element method is applied to stress analysis of rotating turbine disks and thick-walled composite pipes at the macroscopic scale, or to the prediction of equivalent properties and micro-stress analysis of unidirectional fiber-reinforced composite materials at the microscopic scale.

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