Complex variable function and model construction method for solving stress field of rigid inclusion problem

Abstract

The present application relates to a complex variable function and model construction method for solving the stress field of rigid inclusion problems. First, for a given distal loading method and rigid inclusion shape, a corresponding conformal mapping function and distal boundary conditions are established; secondly, based on the boundary conditions of the matrix and the inclusion, a closed-form solution of the potential function that characterizes the stress field and displacement field in the matrix is ​​given with the help of the analytical properties of the potential function and the Cauchy integral; then, through the solved potential function, a numerical program for calculating the stress distribution on the interface between the matrix and the rigid inclusion is given; finally, for the stress at the corner point of the polygonal inclusion that has been obtained, an empirical formula for predicting the stress level using the curvature at the corner point is provided. It is convenient to calculate the stress distribution at the interface and analyze the influence of relevant parameters on the stress field; the research on the distribution of the stress field and the prediction of the stress level is relatively in-depth, and compared with the finite element method, it can greatly improve the calculation accuracy and speed.

Description

Technical Field

[0001] The present application relates to the field of rigid inclusion technology, and in particular to a complex variable function and model construction method for solving the stress field of rigid inclusion problems. Background Art

[0002] In flexible electronics, optical metasurfaces, and bioengineering, rigid inclusions (such as electronic components, optical devices, and inorganic materials) are often integrated into soft materials to achieve rich and unique functions. This soft and hard system has achieved remarkable achievements in the fields of information, energy, and medicine. In fact, due to the huge difference in the elastic properties of the matrix and the inclusions (the Young's modulus of soft materials in engineering is generally very small, usually one thousandth or even one ten-thousandth of the Young's modulus of electronic components, optical devices, inorganic materials, etc. integrated on its surface or inside), there is a more serious stress concentration phenomenon in such structures. Once an external load is applied, this contradiction often becomes the main factor of structural failure. Therefore, studying the problem of rigid inclusions in soft materials has very important theoretical significance for engineering design. Obtaining the stress distribution on the interface can also play a vital role in reducing stress concentration and preventing structural damage.

[0003] In the prior art, Muskhelishvili studied the problem of infinite plates containing elliptical rigid cores and derived explicit expressions of stress components in the matrix under uniform load at infinity. Based on Muskhelishvili's complex variable function method and Schwartz-Chr i stoffel integral equation, Chang studied the case of rigid rectangular inclusions and gave the relationship between the stress field at the interface of infinite elastic matrix and the aspect ratio of the inclusion shape. When considering the rigid body displacement of the inclusion relative to the matrix, Zou and He used the method of series expansion to obtain more accurate results for the problem of irregular shaped rigid inclusions (compared with previous literature). However, it is not difficult to find that the above results are either not widely applicable to the inclusion shape or the solution of the potential function is relatively complicated. In addition, they did not conduct in-depth research on the stress field distribution on the interface between the inclusion and the matrix, and in-depth research in this regard is missing.

[0004] A similar study to this one is the one conducted by Zou's group (Zou WN and He QC, International Journal of Engineering Science 126 (2018)). The method can be summarized as follows:

[0005] A general method for solving the planar problem of an elastic matrix containing a single arbitrarily shaped rigid inclusion under far-field uniform loads was established based on complex variable methods and conformal mapping. The method primarily utilizes the series method, whereby the potential function is hypothesized to be described in terms of a series of variables z (or ξ) based on its analytical properties in the corresponding region. Boundary conditions (taking into account the rigid body displacement of the inclusion relative to the matrix) are then substituted, and the corresponding set of equations is established based on the equality of coefficients of the same power of z (or ξ). Based on the obtained results, a numerical program was used to investigate the stress distribution at the boundary of the rhombus-shaped rigid inclusion.

[0006] With the rapid development of computer performance and the continuous development of corresponding software, the finite element method has attracted increasing attention from researchers. The finite element method has broken through the limitations of many theoretical methods and can numerically solve many problems that cannot be solved by theory. However, when the finite element method is applied to soft materials containing rigid inclusions, the following problems arise:

[0007] (1) In order to ensure the accuracy of the calculation results, the established geometric model should also have high accuracy. However, due to the complexity of the shape of general rigid inclusions, the accuracy of the established geometric model may not be guaranteed.

[0008] (2) Experimental data and theoretical research results show that the most serious stress concentration often occurs at the corners of inclusions. Therefore, in order to obtain a more accurate stress distribution, the mesh quality of the inclusion interface (especially the corners) must be high.

[0009] (3) It is necessary to prepare multiple grid models with different densities and masses, and obtain converged results by calculating the results of these models separately. This will consume a lot of manpower and computing time compared with theoretical calculations.

[0010] Therefore, although the finite element method is an effective method for dealing with rigid inclusion problems in soft materials and obtaining stress field distribution at the interface, the manpower and computing power consumed to obtain high precision are enormous and it is not the optimal solution. Summary of the Invention

[0011] In view of this, the purpose of this application is to provide a complex variable function and model construction method for solving the stress field of rigid inclusion problems, so as to solve the problems existing in the prior art.

[0012] According to a first aspect of an embodiment of the present application, a method for constructing a complex variable function model for solving a stress field of a rigid inclusion problem is provided, comprising:

[0013] Obtaining a rigid inclusion shape to be solved, calculating a conformal mapping function corresponding to the rigid inclusion shape to be solved, and calculating the curvature at the corner point based on the conformal mapping function;

[0014] The equation of the target potential function to be solved is established using the preset loading condition at infinity and the boundary condition at the interface between the matrix and the inclusion; wherein the loading condition at infinity is a far-field uniform load;

[0015] Based on the conformal mapping function and the analytical properties of the target potential function to be solved in the matrix region, assuming a series expression of the target potential function to be solved;

[0016] Obtaining a first formula of the target potential function to be solved by using the series expression of the target potential function to be solved, the conformal mapping function, and the boundary conditions to be satisfied, analyzing the number of preset terms in the first formula of the target potential function to be solved, and clarifying its analytical properties in the relevant area;

[0017] Performing Cauchy integration on the first formula of the target potential function to be solved and its conjugate to obtain a linear equation system of the first equation of the target potential function to be solved and the undetermined coefficients and their conjugates in the series expression of the target potential function to be solved;

[0018] Solving the system of linear equations and the preset supplementary equations using a preset program to determine the complex potential function within the matrix;

[0019] The stress field on the interface between the matrix and the inclusion is calculated using the preset Muskhe li shvi li complex variable function theory and the complex potential function in the matrix.

[0020] Furthermore, the calculating of the conformal mapping function corresponding to the rigid inclusion shape to be solved includes:

[0021] Calculating a conformal mapping function corresponding to the rigid inclusion shape to be solved using a preset Schwarz-Christoffel mapping and preset engineering calculation software, and calculating the curvature at the corner point based on the mapping function;

[0022] The expression of the conformal mapping function is as follows:

[0023]

[0024] Among them, z corresponds to the physical plane, ξ corresponds to the mapping plane, and c j To find the coefficient, N represents the number of terms.

[0025] Furthermore, the equation of the target potential function to be solved includes:

[0026]

[0027] where Γ and Γ′ are quantities related to the load at infinity. and ψ0(z) are complex potential functions analyzed in the matrix domain.

[0028] Furthermore, the series expression of the target potential function to be solved includes:

[0029]

[0030] Among them, α j (j=1,2,...) and β j (j=1,2,...) is a series of unknown coefficients to be solved.

[0031] Furthermore, the first formula of the target potential function to be solved is obtained using the series expression of the target potential function to be solved, the conformal mapping function, and the boundary conditions to be satisfied. The number of preset terms in the first formula of the target potential function to be solved is analyzed, and its analytical properties in the relevant area are clarified, including:

[0032] Substituting the series expression of the target potential function to be solved and the conformal mapping function into the equation of the target potential function to be solved to obtain a first formula;

[0033] Analyze the number of preset terms in the first formula and clarify its analytical properties in the relevant area.

[0034] Furthermore, the first formula of the target potential function to be solved and its conjugate are subjected to Cauchy integration to obtain a linear equation system of the first equation of the target potential function to be solved and the undetermined coefficients and their conjugates in the series expression of the target potential function to be solved, including:

[0035] Performing Cauchy integration on the first formula and its conjugate, a first equation for the target potential function is obtained; wherein the first equation is as shown below:

[0036]

[0037] Setting the coefficients specified on both sides of the first equation of the target potential function to be equal, thereby obtaining a system of linear equations regarding the undetermined coefficients and their conjugates in the potential function series expression;

[0038]

[0039] Among them, c j (j=1, 2, .., N) are the known coefficients in the above conformal mapping function, G represents the shear modulus of the matrix, ε represents the rotation angle of the rigid inclusion, and κ is a material constant related to the Poisson's ratio of the matrix.

[0040] According to a second aspect of an embodiment of the present application, a complex function method for solving the stress field of a rigid inclusion problem is provided, the method comprising:

[0041] Obtaining a rigid inclusion shape to be solved, calculating a conformal mapping function corresponding to the rigid inclusion shape to be solved using a preset Schwarz-Christoffel mapping and preset engineering calculation software, and calculating the curvature at the corner point based on the mapping function;

[0042] Taking multiple sets of values ​​for the number of terms in the conformal mapping function (i.e., the value of N therein) and obtaining different curvature values ​​corresponding to the same corner point;

[0043] Inputting different values ​​of N into the complex variable function model for solving the stress field of the rigid inclusion problem to solve the stress field in the rigid inclusion problem and obtain the stress distribution at the corner point;

[0044] Using the curvature and stress distribution at the corner points, an empirical formula for predicting stress levels using curvature is obtained through curve fitting, including:

[0045] σ=a·(K·R) b , (6)

[0046] Where σ represents the stress level, K represents the curvature, R is the coefficient in the conformal mapping function, and a and b are parameters obtained by curve fitting.

[0047] The technical solutions provided by the embodiments of the present application may have the following beneficial effects:

[0048] It can be understood that the technical solution provided by the present application obtains the shape of the rigid inclusion to be solved, calculates the corresponding conformal mapping function of the rigid inclusion shape to be solved, and calculates the curvature at the corner point based on the mapping function; uses the preset loading condition at infinity and the boundary condition at the interface between the matrix and the inclusion to establish the equation of the target potential function to be solved; wherein, the loading condition at infinity is a far-field uniform load; then, based on the conformal mapping function and the analytical properties of the target potential function to be solved in the matrix area, assumes the series expression of the target potential function to be solved; uses the The first formula of the target potential function to be solved is obtained from the series expression, the conformal mapping function, and the boundary conditions to be satisfied. The preset number of terms in the first formula of the target potential function to be solved is analyzed, and its analytical properties in the relevant area are clarified; the first formula of the target potential function to be solved and its conjugate are subjected to Cauchy integration to obtain the first equation of the target potential function to be solved and a linear equation system of the undetermined coefficients and their conjugates in the series expression of the target potential function to be solved; the linear equation system and the preset supplementary equation are solved by using a preset program to determine the complex potential function in the matrix; finally, the stress field on the interface between the matrix and the inclusion is calculated by using the preset Muskhe li shv ili complex variable function theory and the complex potential function in the matrix. Take multiple sets of values ​​for the number of terms in the conformal mapping function (i.e., the value of N therein) and obtain different curvature values ​​at the corresponding same corner point; use the complex function method to solve the stress field in the rigid inclusion problem for different values ​​of N to obtain the stress distribution at the corner point; use the curvature and stress distribution at the corner point to obtain an empirical formula for predicting the stress level using curvature through curve fitting. It can be understood that the technical solution provided by this application concisely gives the analytical solution of the potential function that characterizes the stress field in the matrix, providing a theoretical basis for in-depth research on stress distribution. In addition, the numerical program provided can easily give the stress distribution on the interface between the matrix and the inclusion, and the empirical formula provided can reliably predict the stress level at the corner point of the polygonal inclusion through the curvature, which can provide support for structural design.

[0049] It should be understood that the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the present application. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the present application and, together with the description, serve to explain the principles of the present application.

[0051] Figure 1 This is a flow chart of a method for constructing a complex variable function model for solving the stress field of a rigid inclusion problem according to an exemplary embodiment;

[0052] Figure 2 is a structural diagram of an infinite planar system including a rigid inclusion according to an exemplary embodiment;

[0053] Figure 3 is a diagram showing stress distribution on an equilateral triangle rigid inclusion interface when N is different according to an exemplary embodiment;

[0054] Figure 4 is a graph showing an exponential relationship fitting curve of normal stress at corner point 1 of an equilateral triangle rigid inclusion according to an exemplary embodiment;

[0055] Figure 5 is a graph showing an exponential relationship fitting curve of the hoop stress at corner point 1 of an equilateral triangle rigid inclusion according to an exemplary embodiment;

[0056] Figure 6 FIG. 1 is a diagram showing stress distribution on a square rigid inclusion interface when N is different according to an exemplary embodiment. DETAILED DESCRIPTION

[0057] Exemplary embodiments will be described in detail herein, with examples illustrated in the accompanying drawings. In the following description, when referring to the drawings, identical numerals in different figures represent identical or similar elements, unless otherwise indicated. The embodiments described in the following exemplary embodiments are not intended to represent all embodiments consistent with the present application. Rather, they are merely examples of apparatus and methods consistent with certain aspects of the present application, as detailed in the appended claims.

[0058] Example 1

[0059] See also Figure 1 , Figure 1 The present invention is a flow chart showing a method for constructing a complex variable function model for solving a stress field of a rigid inclusion problem according to an exemplary embodiment. The method includes:

[0060] S1. Obtaining a rigid inclusion shape to be solved, calculating a conformal mapping function corresponding to the rigid inclusion shape to be solved, and calculating the curvature at the corner point based on the mapping function;

[0061] S2. The target potential function equation to be solved is established by using the preset loading conditions at infinity and the boundary conditions at the matrix and inclusion interface: wherein the loading condition at infinity is the far-field average load;

[0062] S3. Assume a series expression of the target potential function to be solved based on the analytical properties of the target potential function to be solved in the matrix region;

[0063] S4. deriving a first formula for the target potential function to be solved using the series expression of the target potential function to be solved, the conformal mapping function, and the boundary conditions to be satisfied, analyzing the preset terms in the first formula for the target potential function to be solved, and clarifying their analytical properties in the relevant region;

[0064] S5. Performing Cauchy integration on the first formula of the target potential function to be solved and its conjugate to obtain a linear equation system of the first equation of the target potential function to be solved and the undetermined coefficients and their conjugates in the series expression of the target potential function to be solved;

[0065] S6. Solve the system of linear equations and the preset supplementary equation using a preset program to determine the potential function within the matrix;

[0066] S7. Calculate the stress field on the interface between the matrix and the inclusion using the known complex variable function theory of Muskhe li shvi li and the potential function within the matrix;

[0067] S8. For the conformal mapping function in S1, change the number of terms it contains and obtain different curvature values ​​at the same corner point corresponding to the change, then repeat the above S2-S7 process to calculate the stress field on the interface between the matrix and the inclusion, and then obtain an empirical formula for predicting the stress level using curvature through curve fitting.

[0068] For specific implementation, please refer to Figure 2 As described in step S1, the step of calculating the conformal mapping function corresponding to the rigid inclusion shape to be solved includes:

[0069] Calculating the conformal mapping function corresponding to the rigid inclusion shape to be solved using a preset Schwarz-Christoffel mapping and preset engineering calculation software;

[0070] It should be noted that Figure 2 Here, S1 represents the area occupied by the matrix, and S2 represents the area occupied by inclusions.

[0071] Specifically, according to the given rigid inclusion shape, the Schwarz-Christoffel mapping is used and the corresponding conformal mapping function is calculated with the help of Maple software to calculate the curvature K at the corresponding inclusion corner point.

[0072] The expression of the conformal mapping function is as follows:

[0073]

[0074] Among them, z corresponds to the physical plane, ξ corresponds to the mapping plane, and c jTo find the coefficient, N represents the number of terms.

[0075] In the specific implementation, as described in step S2, the equation of the target potential function to be solved is established using the preset loading condition at infinity and the boundary condition at the interface between the matrix and the inclusion; wherein, the loading condition at infinity is a far-field uniform load; the boundary condition at the interface between the matrix and the inclusion is that the two are perfectly connected, and the rigid inclusion has only one rigid body displacement (rotation and translation) relative to infinity.

[0076] Furthermore, the equation of the target potential function to be solved includes:

[0077]

[0078] where Γ and Γ′ are quantities related to the load at infinity. and ψ0(z) are complex potential functions analyzed in the matrix domain.

[0079] In specific implementation, as described in step S3, the series expression of the target potential function to be solved includes:

[0080]

[0081] Among them, α j (j=1,2,...) and β j (j=1,2,...) is a series of unknown coefficients to be solved.

[0082] Further, as described in step S4, the first formula of the target potential function to be solved is obtained by using the series expression of the target potential function to be solved and the conformal mapping function, and the preset number of terms in the first formula of the target potential function to be solved is analyzed, and its analytical properties in the relevant area are clarified, including:

[0083] Substituting the series expression of the target potential function to be solved and the conformal mapping function into the equation of the target potential function to be solved to obtain a first formula;

[0084] The number of preset terms in the first formula is analyzed, and its analytical properties in the relevant area are clarified.

[0085] Furthermore, the first formula of the target potential function to be solved and its conjugate are subjected to Cauchy integration to obtain a linear equation system of the first equation of the target potential function to be solved and the undetermined coefficients and their conjugates in the series expression of the target potential function to be solved, including:

[0086] Performing Cauchy integration on the first formula and its conjugate, a first equation for the target potential function is obtained; wherein the first equation is as shown below:

[0087]

[0088] Setting the coefficients specified on both sides of the first equation pair of the target potential function to be equal, thereby obtaining a linear equation system of the undetermined coefficients and their conjugates in the potential function series expression;

[0089] In the specific implementation, let both sides of the equation be about ξ i The coefficients of (i=1,2,...,N) are equal, thus obtaining a series of linear equations about the unknown coefficients in the potential function series expression and their conjugates, and the supplementary equations are obtained by the inclusion of moment balance;

[0090]

[0091] Among them, c j (j=1, 2, .., N) are the known coefficients in the above conformal mapping function, G represents the shear modulus of the matrix, ε represents the rotation angle of the rigid inclusion, and κ is a material constant related to the Poisson's ratio of the matrix.

[0092] In specific implementation, by writing a corresponding calculation program, the linear equations are solved to obtain the undetermined complex coefficients, thereby determining the complex potential function in the matrix.

[0093] According to Muskhe li shvi li's complex variable function theory, the stress field on the interface between the matrix and the inclusion is calculated based on the known complex potential function in the matrix.

[0094] According to a second aspect of an embodiment of the present application, an empirical formula for predicting the stress field of a rigid inclusion problem is provided, the method comprising:

[0095] Obtaining a rigid inclusion shape to be solved, and calculating a conformal mapping function corresponding to the rigid inclusion shape to be solved using a preset Schwarz-Christoffel mapping and preset engineering calculation software;

[0096] Taking multiple sets of values ​​for the number of terms in the conformal mapping function (i.e., the value of N therein), and obtaining different curvature values ​​corresponding to the same corner point;

[0097] For different values ​​of N, the complex function method described in claim 1 is used to solve the stress field in the rigid inclusion problem to obtain the stress distribution at the corner point;

[0098] By using the curvature and stress distribution at the corner points, an empirical formula for predicting the stress level using curvature was obtained through curve fitting, and its validity and reliability were verified by increasing N.

[0099] The complex variable function method implemented in this application includes an angle-preserving mapping function module, a complex linear equation solving module, a matrix complex potential function expression module, an interface stress data output module, and a curve fitting module for corresponding data. DETAILED DESCRIPTION

[0101] Example 1:

[0102] This embodiment is an infinite matrix material (such as Figure 3 The entire system is subjected to a unidirectional uniform tensile load in the y-axis direction. The material constants of the matrix are as follows:

[0103] G=7.38MPa, μ=0.49. (6)

[0104] 1. Using the Schwarz-Christoffel mapping and the Map le software, the corresponding conformal mapping function is calculated as:

[0105]

[0106] In fact, the conformal mapping function corresponding to a strictly equilateral triangle is an infinite series, but due to the limitation of the solution, we need to choose the truncated form (that is, the highest negative power of ξ is -N), which is also the default conformal mapping function form in the following steps. Then the curvature at the corner point is obtained through a preset procedure.

[0107] 2. Based on the given loading conditions at infinity (unidirectional uniform stretching in the y-axis direction) and the boundary conditions at the interface between the matrix and the inclusion, an equation for the unknown potential function is established.

[0108]

[0109] where Γ and Γ′ are quantities related to the load at infinity. and ψ0(z) are complex potential functions analyzed in the matrix domain, and

[0110]

[0111] Where L is the boundary on the physical plane.

[0112] 3. According to the conformal mapping and potential function and ψ0(z) should be analytical in the matrix region. We can assume the series expression of the potential function

[0113]

[0114] 4. Substitute the potential function series expression (10) and the mapping function (7) into the equation established in step 2 to obtain

[0115]

[0116] Where L′ is the boundary on the mapping plane. For the complex terms in Eq. The analysis can clarify its analytical properties in the relevant area and then expand it to obtain

[0117]

[0118] Where f1 represents a function containing only positive power terms and constant terms.

[0119]

[0120] b i (i=1,2,...,N) can be calculated based on Expanding at ξ=0, we get

[0121]

[0122] 5. Perform Cauchy integration on the obtained equation (11) and its conjugate to obtain the equation for the unknown potential function:

[0123]

[0124] and

[0125]

[0126] Let both sides of equation (15) be about ξ -j The coefficients of (j=1,2,...,N) are equal, thus obtaining a series of linear equations about the unknown coefficients in the potential function series expression and their conjugates:

[0127]

[0128] And through the inclusion of moment balance, we get the supplementary equation

[0129]

[0130] 6. By writing a corresponding calculation program, solving the linear equations composed of equations (17) and (18) can obtain the unknown complex coefficients in equation (10), thereby determining the complex potential function in the matrix.

[0131] 7. According to Muskheli Shvili's theory of complex functions:

[0132]

[0133] We can write a program to calculate the stress field on the interface between the matrix and the inclusion. Figure 3 The stress distribution on the interface is shown when different N is taken.

[0134] 8. Targeting By taking multiple sets of values ​​of N in the mapping function, the corresponding curvature K is obtained for different N, and steps 2-7 are repeated to obtain the stress distribution at the corner point. The empirical formula (exponential type) for predicting stress level using curvature is obtained by curve fitting.

[0135] σ=a·(K·R) b , (20)

[0136] (like Figure 4 and 5 As shown in Figure 2, we also increased N to verify its effectiveness and reliability.

[0137] Example 2:

[0138] This embodiment is an infinite matrix material containing a square rigid inclusion (such as Figure 6 As shown in Figure 2), the entire system is subjected to a unidirectional uniform tensile load in the y-axis direction. At this time, the conformal mapping function is

[0139]

[0140] The specific solution process is similar to that of Example 1. First, an equation for the unknown potential function is established. Then, based on the conformal mapping and analytical assumptions, the series expression of the potential function is substituted into the obtained loading conditions and boundary conditions and the relevant terms are analyzed. Then, the obtained equation is Cauchy integrated to obtain a system of linear equations for the undetermined complex coefficients in the potential function. All the undetermined coefficients are solved to determine the expression of the potential function in the matrix. The stress distribution on the boundary can be further calculated. Finally, multiple groups of N values ​​are calculated to obtain the relevant empirical formula.

[0141] In one embodiment, linear elasticity theory is commonly used to solve the problem of rigid inclusions in soft materials, and the validity of its results has been verified by photoelastic experiments (performed by Misseron). Complex functions and conformal mapping are powerful mathematical tools for solving noncircular inclusion problems. However, the introduction of conformal mapping functions increases the difficulty of deriving analytical solutions to potential functions that characterize the stress and displacement fields within the matrix. Existing studies have mostly employed series methods, which are relatively cumbersome and result in complex potential functions. This has hindered further research into stress fields within the matrix, particularly at the boundary with the inclusion.

[0142] This application is based on complex functions and combines Cauchy integrals to provide a simple method for solving the problem of an infinite matrix containing arbitrary-shaped rigid inclusions under a uniform distal load. First, the corresponding conformal mapping function and distal boundary conditions are established for a given distal loading method and rigid inclusion shape; secondly, based on the boundary conditions of the matrix and the inclusion, the closed-form solution of the potential function that characterizes the stress field and displacement field in the matrix is ​​given with the help of the analytical properties of the potential function and the Cauchy integral; then, through the solved potential function, a numerical program for calculating the stress distribution on the interface between the matrix and the rigid inclusion is given; finally, for the stress at the corner point of the polygonal inclusion that has been obtained, an empirical formula for predicting the stress level using the curvature at the corner point is provided. In summary, this application is applicable to rigid inclusions of arbitrary shapes, and the results are relatively simple, which facilitates the calculation of the stress distribution at the interface and the analysis of the influence of relevant parameters on the stress field; the research on the distribution of the stress field and the prediction of the stress level is relatively in-depth, and compared with the finite element method, it can greatly improve the calculation accuracy and speed.

[0143] The key points of this application are as follows:

[0144] By introducing conformal mapping, the shape of rigid inclusions can be arbitrary, which has wide applicability;

[0145] By using complex variable functions and Cauchy integrals, the solution process is clear and the obtained potential function is relatively simple, which is convenient for the implementation of numerical programs and the study of the influence of relevant parameters on the stress field at the interface and in the matrix.

[0146] Based on a large number of examples, the proposed empirical formula for predicting the stress magnitude at the corner points of polygonal inclusions has high accuracy and reliability.

[0147] In comparison, the advantages of this application are mainly the following two points:

[0148] 1. Most technologies use series methods to solve potential functions, which is a relatively tedious and complicated process. This application uses Cauchy integration, which is relatively simple and clear.

[0149] 2. The existing technology focuses on solving the potential function and does not conduct in-depth research on the stress distribution in the matrix, especially on the interface. The present application provides a numerical program corresponding to the analytical solution to provide relevant images of the stress distribution, and provides a reliable empirical formula to predict the value of the stress level at a higher level (at the corner points of the polygon) through curvature.

[0150] It can be understood that the same or similar parts of the above embodiments can be referenced to each other, and the contents not described in detail in some embodiments can refer to the same or similar contents in other embodiments.

[0151] It should be noted that, in the description of this application, the terms "first", "second", etc. are used for descriptive purposes only and should not be understood as indicating or implying relative importance. In addition, in the description of this application, unless otherwise specified, the meaning of "plurality" refers to at least two.

[0152] Any process or method description in a flowchart or otherwise described herein may be understood to represent a module, segment or portion of code comprising one or more executable instructions for implementing the steps of a specific logical function or process, and the scope of the preferred embodiments of the present application includes alternative implementations in which functions may be performed out of the order shown or discussed, including performing functions in a substantially simultaneous manner or in the reverse order depending on the functions involved, which should be understood by those skilled in the art to which the embodiments of the present application belong.

[0153] It should be understood that various parts of the present application can be implemented using hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented using software or firmware stored in a memory and executed by a suitable instruction execution system. For example, if implemented using hardware, as in another embodiment, any one of the following technologies known in the art or a combination thereof can be used to implement: a discrete logic circuit having a logic gate circuit for implementing a logic function on a data signal, an application-specific integrated circuit having a suitable combination of logic gate circuits, a programmable gate array (PGA), a field programmable gate array (FPGA), etc.

[0154] Those skilled in the art will understand that all or part of the steps in the method of the above embodiment can be completed by instructing related hardware through a program, and the program can be stored in a computer-readable storage medium. When the program is executed, it includes one or a combination of the steps of the method embodiment.

[0155] In addition, the functional units in the various embodiments of the present application may be integrated into a processing module, or each unit may exist physically separately, or two or more units may be integrated into a module. The above-mentioned integrated module may be implemented in the form of hardware or in the form of a software functional module. If the integrated module is implemented in the form of a software functional module and sold or used as an independent product, it may also be stored in a computer-readable storage medium.

[0156] The storage medium mentioned above can be a read-only memory, a magnetic disk or an optical disk, etc.

[0157] Throughout this specification, reference to terms such as "one embodiment," "some embodiments," "examples," "specific examples," or "some examples" means that a specific feature, structure, material, or characteristic described in conjunction with that embodiment or example is included in at least one embodiment or example of the present application. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

[0158] Although the embodiments of the present application have been shown and described above, it can be understood that the above embodiments are exemplary and cannot be understood as limitations on the present application. Ordinary technicians in this field can change, modify, replace and modify the above embodiments within the scope of the present application.

Claims

1. A complex function model construction method for solving the stress field of rigid inclusion problems, characterized by: The method comprises: Obtain the rigid inclusion shape to be solved; Calculating a conformal mapping function corresponding to the rigid inclusion shape to be solved, and calculating the curvature at the corner point based on the conformal mapping function; The equation of the target potential function to be solved is established using the preset loading condition at infinity and the boundary condition at the interface between the matrix and the inclusion; wherein the loading condition at infinity is a far-field uniform load; Based on the conformal mapping function and the analytical properties of the target potential function to be solved in the matrix region, assuming a series expression of the target potential function to be solved; Obtaining a first formula of the target potential function to be solved by using the series expression of the target potential function to be solved, the conformal mapping function, and the boundary conditions to be satisfied, analyzing the preset terms in the first formula of the target potential function to be solved, and clarifying its analytical properties in the relevant area; Performing Cauchy integration on the first formula of the target potential function to be solved and its conjugate to obtain a linear equation system of the first equation of the target potential function to be solved and the undetermined coefficients and their conjugates in the series expression of the target potential function to be solved; Solving the system of linear equations and the preset supplementary equations using a preset program to determine the complex potential function within the matrix; The stress field on the interface between the matrix and the inclusion is calculated using the known Muskhelishvili's complex variable function theory and the complex potential function in the matrix.

2. The method according to claim 1, characterized in that The calculating of the conformal mapping function corresponding to the rigid inclusion shape to be solved includes: Calculating a conformal mapping function corresponding to the rigid inclusion shape to be solved using a preset Schwarz-Christoffel mapping and preset engineering calculation software, and calculating the curvature at the corner point based on the mapping function; The expression of the conformal mapping function is as follows: Among them, z corresponds to the physical plane, ξ corresponds to the mapping plane, and c j To find the coefficient, N represents the number of terms, and the curvature can be calculated according to a preset program.

3. The method according to claim 1, characterized in that The equation of the target potential function to be solved includes: ψ(z)=Γ′z+ψ0(z), (2) where Γ and Γ′ are quantities related to the load at infinity. and ψ0(z) are the analytical potential functions in the matrix domain; Γz is the product of Γ and Z, and Γ′z represents the product of Γ′ and Z.

4. The method according to claim 1, wherein The series expression of the target potential function to be solved includes: Among them, α j (j=1,2,...) and β j (j=1,2,...) is a series of unknown coefficients to be solved.

5. The method according to claim 1, wherein The first formula of the target potential function to be solved is obtained by using the series expression of the target potential function to be solved, the conformal mapping function, and the boundary conditions to be satisfied, and the number of preset terms in the first formula of the target potential function to be solved is analyzed, and its analytical properties in the relevant area are clarified, including: Substituting the series expression of the target potential function to be solved and the conformal mapping function into the boundary conditions satisfied by the target potential function to be solved to obtain a first formula; Analyze the number of preset terms in the first formula and clarify its analytical properties in the relevant area.

6. The method according to claim 1, characterized in that The first formula of the target potential function to be solved and its conjugate are subjected to Cauchy integration to obtain a linear equation system of the first equation of the target potential function to be solved and the undetermined coefficients and their conjugates in the series expression of the target potential function to be solved, including: Performing Cauchy integration on the first formula and its conjugate, a first equation for the target potential function is obtained; wherein the first equation is as shown below: Setting the coefficients specified on both sides of the first equation of the target potential function to be equal, thereby obtaining a system of linear equations regarding the undetermined coefficients and their conjugates in the potential function series expression; Among them, c j (j=1, 2, .., N) are the known coefficients in the above conformal mapping function, G represents the shear modulus of the matrix, ε represents the rotation angle of the rigid inclusion, and κ is a material constant related to the Poisson's ratio of the matrix.

7. The complex function method for solving the stress field of rigid inclusion problems is characterized by: The method comprises: Obtaining a rigid inclusion shape to be solved, calculating a conformal mapping function corresponding to the rigid inclusion shape to be solved using a preset Schwarz-Christoffel mapping and preset engineering calculation software, and calculating the curvature at the corner point based on the mapping function; Taking multiple sets of values ​​for the number of terms in the conformal mapping function (i.e., the value of N therein) and obtaining different curvature values ​​corresponding to the same corner point; Inputting different values ​​of N into the complex variable function model for solving the stress field of the rigid inclusion problem to solve the stress field in the rigid inclusion problem and obtain the stress distribution at the corner point; Using the curvature and stress distribution at the corner points, an empirical formula for predicting stress levels using curvature is obtained through curve fitting, including: σ=a·(K·R) b , (6) Where σ represents the stress level, K represents the curvature, R is the coefficient in the conformal mapping function, and a and b are parameters obtained by curve fitting; Wherein, the complex function model for solving the stress field of the rigid inclusion problem is constructed by using the method described in any one of claims 1 to 6.

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