High-elongation steel cord stress-strain response fitting method and application thereof

Abstract

The invention discloses a fitting method for stress-strain response of a high-elongation steel cord and application of the fitting method. The fitting method comprises the steps that stress-strain data points are obtained; the number N of turning points is determined based on a local curvature mutation detection method, the fitting curve is divided into N + 1 fitting line segments, a starting point is an original point when the first fitting line segment is drawn, and the starting point is located in data points with curvature mutation when other fitting line segments are drawn; the optimal slope of each fitting line segment, which maximizes the value, can be calculated by performing derivation on a formula for determining the coefficient about the slope, and a ray is drawn according to the calculated optimal slope of each fitting line segment, so that the fitting line segment of the stress-strain response of the high-elongation steel cord is obtained. According to the method, the number of turning points is determined based on a curvature mutation detection mode by utilizing the switching characteristics of a multi-segment linear function and a max function, the optimal slope is calculated through a formula, local high-precision fitting of each segment and corresponding experimental data is ensured, and the overall goodness of fit is greatly improved.

Description

Technical Field

[0001] This invention belongs to the field of materials mechanics testing technology, and particularly relates to a fitting method for the stress-strain response of high elongation steel cord and its application. Background Technology

[0002] The stress-strain test data and their fitting curves of steel cord play a crucial role in finite element simulation, and their accuracy directly determines the reliability of the simulation results and the effectiveness of engineering predictions.

[0003] Currently, the common practice in engineering is to directly input test data into finite element software, relying on the software's built-in algorithms to automatically identify constitutive relations, or for engineers to manually select a constitutive model based on experience, and then fit the test data. However, these methods have significant limitations when dealing with high-elongation steel cords. Because high-elongation steel cords exhibit highly nonlinear mechanical behavior and have low linear correlation, traditional constitutive models struggle to accurately capture their deformation mechanisms. Therefore, there is often a large deviation between the fitted curve and the test data, and the coefficient of determination is low. The values ​​are generally too low, and in some cases below 0.9, which fails to meet the requirements of high-precision simulation. This fitting bias further introduces constitutive error, resulting in poor accuracy of finite element simulation results.

[0004] To improve simulation accuracy, existing technologies include higher-order polynomial fitting and spline interpolation. Higher-order polynomial fitting uses a single higher-order polynomial curve for fitting, but it is prone to overfitting or underfitting when dealing with the complex nonlinear behavior of high-elongation steel cords, resulting in poor fitting stability and difficulty in guaranteeing overall and local goodness of fit (e.g., ...). While spline interpolation (e.g., using cubic spline interpolation) can produce smooth stress-strain curves, it may be sensitive to data fluctuations, cannot guarantee physical monotonicity, and its fitting results focus more on smoothness than on the deterministic coefficients of the original data. Therefore, it carries a higher risk of introducing non-physical oscillations into finite element simulations. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the technical problem this invention aims to solve is that, due to the highly nonlinear mechanical behavior exhibited by high-elongation steel cords, there is often a large deviation between the fitting curve of the traditional constitutive model and the test data. This invention proposes a fitting method for the stress-strain response of high-elongation steel cords with high accuracy, high reliability, and high linear correlation, and its application.

[0006] To solve the aforementioned technical problem, the technical solution adopted by the present invention is as follows: This invention provides a method for fitting the stress-strain response of high-elongation steel cord, comprising the following steps: S1: Obtain stress-strain data points for high elongation steel cord; S2: Analyze the stress and strain data, determine the number of inflection points N based on the method of detecting local curvature abrupt changes, divide the fitted curve into N+1 fitted line segments, and draw the first fitted line segment with the origin (0,0) as the starting point. When drawing the remaining fitted line segments, the starting point is located in the data points where curvature abrupt changes occur. S3: By analyzing the coefficient of determination The formula for slope By taking the derivative and setting it to zero, we can calculate the product that makes the derivative equal to zero. The optimal slope of each fitted line segment whose value is maximized.

[0007] S4: Based on the calculated optimal slope of each fitted line segment Rays are drawn with the origin (0,0) and the starting point determined in step S2 respectively. Except for the ray drawn from the origin (0,0), the remaining rays are extended in the opposite direction. The intersection of the extended rays is taken as the turning point. The extra rays outside the intersection point are deleted to obtain the fitted line segment of the stress-strain response of the high elongation steel cord.

[0008] Preferably, the expression for the fitted line segment of the stress-strain response of the high-elongation steel cord is as follows: ; in, The strain value, This corresponds to the stress value; This represents the slope of the i-th line segment. Let n be the strain value at the (i-1)th inflection point, and n be the total number of segments. When x≤x i-1 When, max(xx) i-1 If (,0)=0, then the contribution of this segment is zero. When x>x i-1 When, max(xx) i-1 ,0)=xx i-1 The segment was activated.

[0009] Preferably, in step S1, obtaining the stress-strain data of the high-elongation steel cord includes the following steps: Test the force-displacement data of high-elongation steel cords, and calculate the stress and strain values ​​according to the following formulas: The stress value of high elongation steel cord is calculated using the following formula: ; Where F is the tension acting on the high elongation steel cord, i.e., the force in Table 1, and A is the cross-sectional area of ​​the high elongation steel cord. The strain value of high elongation steel cord is calculated using the following formula: ; in L0 is the displacement of the high-elongation steel cord after being subjected to force, and L0 is the original length of the high-elongation steel cord.

[0010] Preferably, in step S2, the data is moved sequentially along the horizontal axis with a step size of 1, and 3 adjacent data points are taken as a group each time to determine whether there is a curvature change in the grouped data. The method for determining whether there is a curvature abrupt change in grouped data is as follows: Calculate the curvature of the circle defined by the three data points in the group, and then calculate the curvature C of the circle in the current group. j The curvature C of the circle in the previous group j-1 and the curvature C of the next grouped circle j+1 The relative rate of change, if >0.8 and If the value is greater than 0.8, then the data in this group is determined to have a curvature abrupt change.

[0011] Preferably, if there are two or more consecutive grouped circles C j C j+1 , ..., C j+(n+1) Simultaneously satisfy >0.8 and If the value is greater than 0.8, then the group of circles with the smallest curvature is determined to have a curvature abrupt change.

[0012] As a preferred option, except for the first line segment, the starting point of each fitted line segment is the midpoint among the three data points with abrupt curvature changes.

[0013] Preferably, in step S3, the determination coefficient is calculated according to the following formula. The slope of each fitted line segment when the maximum value is taken. : ; in, The strain value at the i-th point obtained in step S1. The stress value at the i-th point obtained in step S1.

[0014] As a preferred method, the coefficient of determination is calculated. The slope of each fitted line segment when the maximum value is taken. The derivation method of the formula is as follows: 1) Give the coefficient of determination The calculation formula is as follows: ; in, It is the sum of squared residuals, calculated using the following formula:

[0015] It is the total sum of squares, calculated using the following formula:

[0016] in, These are actual observed values. It is the slope of each fitted line segment. These are model predictions. It is the average of the observed values. It is the sample size; 2) To Differentiate and expand the squared terms:

[0017] 3) To Differentiate:

[0018] Set it to zero:

[0019] Then when the coefficient of determination The slope of each fitted line segment when the maximum value is taken. The calculation formula is: .

[0020] In another aspect, the present invention also provides the application of the above-mentioned fitting method in the stress-strain response fitting of metal wire products.

[0021] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. Improved high-precision fitting and simulation reliability: Existing methods do not fit the nonlinear stress-strain response of high-elongation steel cords (whose elongation at break is typically at least 5%) well, and the coefficient of determination is low. The coefficient of determination is often below 0.9, leading to significant constitutive errors in finite element simulations. This invention determines the number of inflection points and creates a piecewise linear function based on curvature abrupt change detection, ensuring that the coefficient of determination of each segment corresponds to the experimental data. By taking the maximum value, the overall goodness of fit was greatly improved, which directly enhanced the accuracy of the finite element simulation results and the reliability of engineering predictions. 2. Adaptive nonlinear characterization capability: High elongation steel cord exhibits high elastic modulus and complex nonlinear mechanical behavior, which traditional constitutive models cannot accurately capture. This invention utilizes the switching characteristics of multi-segment linear functions and max functions to effectively decompose and accurately fit the nonlinear stage in stress-strain data, better characterize the material properties in different strain ranges, and solve the problem of relatively low fitting accuracy of traditional models. 3. Objectivity and repeatability: Existing methods rely on engineers' experience and the fitting results of finite element software. This invention determines the number of inflection points and creates piecewise linear functions by using mathematical rules (such as calculating curvature at three points and detecting curvature abrupt changes), which improves the repeatability and consistency of the fitting process and reduces errors caused by reliance on software and insufficient experience. 4. Enhanced Engineering Applicability: By improving the accuracy of stress-strain fitting, this invention directly contributes to improving the accuracy of finite element analysis, thereby optimizing product design. In addition, the parameters set in this method (such as the 80% curvature change threshold) are based on empirical values ​​from a large number of experiments, achieving a good balance between capturing abrupt changes in material properties and avoiding overfitting caused by small fluctuations in data. Attached Figure Description

[0022] Figure 1 This is a schematic diagram illustrating the fitting of the first ray based on stress-strain test data points from high-elongation steel cord A; Figure 2 This is a schematic diagram illustrating the fitting of the first two rays based on the stress-strain test data points of high-elongation steel cord A; Figure 3 This is a schematic diagram illustrating the fitting of the first three rays based on the stress-strain test data points of high-elongation steel cord A; Figure 4 This is a schematic diagram illustrating the fitting of the first four rays based on the stress-strain test data points of high-elongation steel cord A; Figure 5 This is a schematic diagram illustrating the determination of the turning point by extending the second, third, and fourth rays in the reverse direction after fitting the complete ray from the stress-strain test data points of high-elongation steel cord A. Figure 6 The piecewise linear graph obtained by fitting the stress-strain test data of high elongation steel cord A using the method of this embodiment of the invention; Figure 7 The curves are obtained by fitting the stress-strain test data of high elongation steel cord A using finite element software. Detailed Implementation

[0023] The technical solutions in specific embodiments of the present invention will now be described in detail and completely with reference to the accompanying drawings. Obviously, the described embodiments are merely some specific implementations of the overall technical solution of the present invention, and not all implementations. Based on the overall concept of the present invention, all other embodiments obtained by those skilled in the art fall within the protection scope of the present invention.

[0024] This invention provides a method for fitting the stress-strain response of high-elongation steel cord, comprising the following steps: S1: Obtain stress-strain data points for high elongation steel cord; S2: Analyze the stress and strain data, determine the number of inflection points N based on the method of detecting local curvature abrupt changes, divide the fitted curve into N+1 fitted line segments, and draw the first fitted line segment with the starting point (0,0). When drawing the remaining fitted line segments, the starting point is located in the data points where curvature abrupt changes occur. S3: By analyzing the coefficient of determination The formula for slope By taking the derivative and setting it to zero, we can calculate the product that makes the derivative equal to zero. The optimal slope of each fitted line segment whose value is maximized.

[0025] S4: Based on the calculated optimal slope of each fitted line segment Rays are drawn with the origin (0,0) and the starting point determined in step S2 respectively. Except for the ray drawn from the origin (0,0), the remaining rays are extended in the opposite direction. The intersection of the extended rays is taken as the turning point. The extra rays outside the intersection point are deleted to obtain the fitted line segment of the stress-strain response of the high elongation steel cord.

[0026] It should be noted that high-elongation steel cords (typically with an elongation at break of at least 5%) exhibit high elastic modulus and complex nonlinear mechanical behavior. Traditional constitutive models struggle to accurately capture their deformation mechanisms, and existing methods do not fit the nonlinear stress-strain response of high-elongation steel cords well. Currently, the common practice in engineering is to directly input test data into finite element software, relying on the software's built-in algorithm to automatically identify constitutive relations, or for engineers to manually select a constitutive model based on experience, and then fit the test data. However, these methods have significant limitations when dealing with high-elongation steel cords. Due to the highly nonlinear mechanical behavior of high-elongation steel cords, their linear correlation is low, and traditional constitutive models struggle to accurately capture their deformation mechanisms. Therefore, there is often a large deviation between the fitted curve and the test data, and the coefficient of determination is low. The values ​​are generally too low, and in some cases below 0.9, which fails to meet the requirements of high-precision simulation. This fitting bias further introduces constitutive error, resulting in poor accuracy of finite element simulation results.

[0027] To address the aforementioned issues, this invention first tests the force-displacement data of the steel cord and converts it into stress-strain data. Then, based on curvature abrupt change detection, it determines the number of inflection points and creates a multi-segment linear function. This decomposes the complex nonlinear mechanical properties into multiple continuous linear stages, and the number of inflection points and the optimal slope for each segment are determined using formulas. This enables high-precision fitting of the nonlinear stress-strain response of steel cord, thereby helping to improve the reliability of its results in engineering analyses such as finite element simulation.

[0028] In a preferred embodiment, the expression for the fitted line segment of the stress-strain response of the high-elongation steel cord is: ; in, The strain value, This corresponds to the stress value; This represents the slope of the i-th line segment. Let n be the strain value at the (i-1)th inflection point, and n be the total number of segments. When x≤x i-1 When, max(xx) i-1 If (,0)=0, then the contribution of this segment is zero. When x>x i-1 When, max(xx) i-1 ,0)=xx i-1 The segment was activated.

[0029] The above technical solution uses a mathematical expression based on the switching characteristics of the max function to create a piecewise linear fitting function. The core of this model lies in its ability to decompose complex nonlinear mechanical properties into multiple continuous linear stages through superposition, and to calculate the optimal slope of each segment using a formula. This enables high-precision fitting.

[0030] In a preferred embodiment, step S1, obtaining stress-strain data of high-elongation steel cord includes the following steps: Test the force-displacement data of high-elongation steel cords, and calculate the stress and strain values ​​according to the following formulas: The stress value of high elongation steel cord is calculated using the following formula: ; Where F is the tension acting on the high elongation steel cord, i.e., the force in Table 1, and A is the cross-sectional area of ​​the high elongation steel cord. The strain value of high elongation steel cord is calculated using the following formula: ; in L0 is the displacement of the high-elongation steel cord after being subjected to force, and L0 is the original length of the high-elongation steel cord.

[0031] In a preferred embodiment, in step S2, the data points are moved sequentially along the horizontal axis with a step size of 1, and three adjacent data points are taken as a group each time to determine whether there is a sudden change in curvature in the circle formed by them.

[0032] The reason why the above technical solution groups three adjacent data points as a group is that, according to the three-point theorem, the curvature C of the circle that uniquely determines it can be calculated. j Then, the number of inflection points can be determined by whether there is a curvature abrupt change.

[0033] In a preferred embodiment, in step S2, the method for determining whether there is a curvature abrupt change in the grouped data is as follows: Calculate the curvature of the circle defined by the three data points in the group, and then calculate the curvature C of the circle in the current group. j The curvature C of the circle in the previous group j-1 and the curvature C of the next grouped circle j+1 The relative rate of change, if >0.8 and If the value is greater than 0.8, then the circle in the group is determined to have a sudden change in curvature.

[0034] It should be noted that, since the curvature of adjacent points cannot be obtained simultaneously at the beginning and end of the data grouping, the first 3 points and the last 3 points are not involved in the above mutation judgment.

[0035] If there are two or more consecutive grouped circles, such as C j C j+1 , ..., C j+(n+1) Simultaneously satisfy >0.8 and If the value is greater than 0.8, then the group of circles with the smallest curvature is determined to have a curvature abrupt change.

[0036] It should be noted that when determining whether there is a sudden change in curvature in a group of circles, the first step is to determine whether the three points are collinear. If they are collinear, it indicates that the region has high linearity, and the curvature calculation for this group is skipped; if they are not collinear, the relative rate of change of curvature between this group and the preceding and following groups is calculated. The method for determining collinearity of three points in a Cartesian coordinate system is the area method.

[0037] For point A(x) A ,y A ),B(x B ,y B ) and C(x C ,y C The calculation formula is as follows: ; If S = 0, then the area is zero and the three points are collinear. If S ≠ 0, then the area is not zero, the three points are not collinear, and they form a triangle.

[0038] If they are not collinear, then further determine whether there is a sudden change in curvature. If the curvature C of the current grouped circle is... j The curvature C of the circle in the previous group j-1 The curvature C of the circle and the next group of circles j+1 The relative rates of change all satisfy >0.8 and If the value is greater than 0.8, it is determined that there is a curvature change in the region where the three points are located, that is, a turning point is identified. Three non-collinear points A(x) in a rectangular coordinate system A ,y A ),B(x B ,y B ) and C(x C ,y C The curvature C of the circle is determined by j The calculation formula is as follows: ; Where S is the area of ​​the triangle formed by points A, B, and C, which can be calculated using the formula for S above; ; ; .

[0039] In a preferred embodiment, except for the first line segment, the starting point of each fitted line segment is the midpoint among the three data points with abrupt curvature changes.

[0040] In a preferred embodiment, in step S3, the determination coefficient is calculated according to the following formula. The slope of each fitted line segment when the maximum value is taken. : ; in, The strain value at the i-th point obtained in step S1. The stress value at the i-th point obtained in step S1.

[0041] In a preferred embodiment, the determination coefficient is calculated. The slope of each fitted line segment when the maximum value is taken. The derivation method of the formula is as follows: 1) Give the coefficient of determination The calculation formula is as follows: ; in, It is the sum of squared residuals, calculated using the following formula:

[0042] It is the total sum of squares, calculated using the following formula:

[0043] in, These are actual observed values. It is the slope of each fitted line segment. These are model predictions. It is the average of the observed values. It is the sample size; 2) To Differentiate and expand the squared terms:

[0044] 3) To Differentiate:

[0045] Set it to zero:

[0046] Then when the coefficient of determination The slope of each fitted line segment when the maximum value is taken. The calculation formula is: .

[0047] In another aspect, the present invention provides the application of the above-described fitting method in the finite element analysis of metal wire products.

[0048] The key to this invention lies in: (1) Piecewise linear function model: The mathematical expression based on the switching characteristics of the max function is adopted. The model constructs multiple linearly fitted line segments. Its core lies in its ability to decompose complex nonlinear mechanical properties into multiple continuous linear stages through superposition. It determines the number of inflection points and the starting point of each fitted line segment through curvature abrupt change detection, and calculates the coefficient of determination using a formula. The slope of each fitted line segment is taken at its maximum value to ensure high fitting accuracy between each segment and the corresponding experimental data.

[0049] (2) Method for determining the number of inflection points: A method based on the detection of sudden changes in local curvature is proposed to determine the number of inflection points. Specifically, this includes: calculating the local curvature using a sliding window (3 points per group), and setting a threshold of 80% relative change rate of curvature as the number of inflection points. This method is the cornerstone for achieving high-precision segmentation.

[0050] (3) Strict control of segment quality: When constructing each line segment, according to the formula The optimal slope value for each line segment is rigorously calculated, which ensures the accuracy of the fitting for each segment and even the whole.

[0051] (4) Specialization for high elongation steel cords: The method is specifically designed for the high nonlinear characteristics of stress-strain response of high elongation steel cords, solving its specific problems in finite element simulation.

[0052] In another aspect, the present invention also provides the application of the above-mentioned fitting method in the stress-strain response fitting of metal wire products.

[0053] To more clearly and in detail introduce the fitting method for stress-strain response of high elongation steel cord provided by the embodiments of the present invention and its application, the following description will be based on specific embodiments.

[0054] Example 1 1. Acquisition of scatter points for stress-strain test data The force-displacement data of steel cord were tested according to the test procedures specified in 6.1 Breaking force and breaking elongation and 6.2 Elongation between specified forces in GB / T33159-2016 Test Methods for Steel Cord. The stress calculation formula for steel cord is Equation 1: σ = F / A (1); Where F is the tension acting on the steel cord, i.e., the force in Table 1, and A is the cross-sectional area of ​​the steel cord; The strain calculation formula for steel cord is Equation 2: (2); Where ΔL is the change in length after being subjected to force, i.e., the displacement in Table 1, and L0 is the original length of the object, i.e., the test distance of the fixture.

[0055] 2. Determining the number of inflection points based on local curvature abrupt change detection: Move along the horizontal axis in sequence with a step size of 1, and take 3 adjacent data points as a group each time to determine whether there is a sudden change in curvature in the grouped circle.

[0056] First, determine whether the three points are collinear. In a rectangular coordinate system, the method to determine whether three points are collinear is the area method.

[0057] For point A(x) A ,y A ),B(x B ,y B ) and C(x C ,y C The calculation formula is as follows: (3); If S = 0, then the area is zero and the three points are collinear. If S ≠ 0, then the area is not zero, the three points are not collinear, and they form a triangle.

[0058] If the points are collinear, it indicates high linearity in the region, and the curvature calculation for this group is skipped. If they are not collinear, the relative rate of change of curvature between this group and the preceding and following groups is calculated. After determining whether the coordinate points in the coordinate system are collinear using the above formula, the curvature of the circle can be used to determine how many inflection points the fitted line segment contains. Find three non-collinear points A(x) in a rectangular coordinate system. A ,y A ),B(x B ,y B ) and C(x C ,y C The formula for the curvature of a circle is as follows: (4); Where S is the area of ​​the triangle formed by points A, B, and C, which can be calculated using formula 3 above. ; ; .

[0059] Calculate the curvature of the circle defined by the three data points in the group, and then calculate the curvature C of the circle in the current group. j The curvature C of the circle in the previous group j-1 and the curvature C of the next grouped circle j+1 The relative rate of change, if >0.8 and If the value is greater than 0.8, then the data in this group is determined to have a curvature abrupt change.

[0060] If there are two or more consecutive grouped circles, such as C j C j+1 , ..., C j+(n+1) Simultaneously satisfy >0.8 and If the value is greater than 0.8, then the group of circles with the smallest curvature is determined to have a curvature abrupt change.

[0061] Using the above method, if there are N curvature abrupt changes, there are N inflection points. The fitted curve is divided into N+1 fitted line segments. The rule for drawing the starting point of each fitted line segment is that the starting point of the first line segment is (0,0), and the starting point of subsequent line segments is the midpoint of the three data points in the curvature abrupt change group.

[0062] 3. Calculate the coefficient of determination using the formula. The slope of each fitted line segment when the maximum value is taken. The coefficient of determination, also known as the factor of determination or the factor of judgment, is used to determine the outcome of a decision. The coefficient of determination for linearity without intercept is expressed as follows. As the core evaluation index for goodness of fit, its value ranges from [0,1]. The closer its value is to 1, the stronger the explanatory power of the fitted curve for the variation of the original data, and the better the fitting effect. Since the starting point of the fitted line segment is (0,0), the formula for the coefficient of determination for linearity without intercept is shown in Equation 5.

[0063] Calculate the coefficient of determination The slope of each fitted line segment when the maximum value is taken. The derivation method of the formula is as follows: 1) Give the coefficient of determination The calculation formula is as follows: (5); in, It is the sum of squared residuals, calculated using the following formula:

[0064] It is the total sum of squares, calculated using the following formula:

[0065] in, These are actual observed values. It is the slope of each line segment. These are model predictions. It is the average of the observed values. It is the sample size; 2) Because With the slope of each line segment Irrelevant, seeking The maximum value is equivalent to finding The minimum value. For Differentiate and expand the squared terms:

[0066] 3) To Differentiate:

[0067] Set it to zero:

[0068] Then when the coefficient of determination The slope of each fitted line segment when the maximum value is taken. The calculation formula is: (6).

[0069] 4. Using the origin (0,0) and the starting point determined in step 2 as the starting points for drawing each fitted line segment, and based on the calculated slope of each segment... Draw rays, except for the ray originating from the origin (0,0), extend the remaining rays in the opposite direction, and take the intersection of the extended rays as the turning point. Delete the redundant rays outside the intersection point to obtain the fitted line segment of the stress-strain response of the high-elongation steel cord: The expression for the fitted line segment of the stress-strain response of the high-elongation steel cord is: ; in, The strain value, This corresponds to the stress value; This represents the slope of the i-th line segment. Let n be the strain value at the (i-1)th inflection point, and n be the total number of segments. When x≤x i-1 When, max(xx) i-1 If (,0)=0, then the contribution of this segment is zero. When x>x i-1 When, max(xx) i-1 ,0)=xx i-1 The segment was activated.

[0070] Application Example 1 1. Acquisition of scatter points for stress-strain test data The force-displacement data of the steel cord were tested according to the test procedures specified in 6.1 Breaking force and breaking elongation and 6.2 Elongation between specified forces in GB / T33159-2016 Test Methods for Steel Cord; (Table 1); The stress calculation formula for steel cord is Equation 1: σ = F / A (1); Where F is the tension acting on the steel cord, i.e., the force in Table 1, and A is the cross-sectional area of ​​the steel cord (A=1.4103mm²). 2 ); The strain calculation formula for steel cord is Equation 2: (2); Where ΔL is the change in length after being subjected to force, i.e., the displacement in Table 1, and L0 is the original length of the object, i.e., the test distance of the fixture (L0=1000mm). The stress-strain data of steel cord A can be obtained by calculation as shown in Table 2.

[0071] Table 1A Steel Cord Force-Displacement Data

[0072] Table 2 Stress-strain data for steel curtain A

[0073] 2. Determining the location region of inflection points based on local curvature abrupt change detection: Move along the horizontal axis in sequence with a step size of 1, and take 3 adjacent data points as a group each time to determine whether there is a sudden change in curvature in the grouped circle.

[0074] First, determine whether the three points are collinear. In a rectangular coordinate system, the method to determine whether three points are collinear is the area method.

[0075] For three points A(x) A ,y A ),B(x B ,y B ) and C(x C ,y C The calculation formula is as follows: (3); If S = 0, then the area is zero and the three points are collinear. If S ≠ 0, then the area is not zero, the three points are not collinear, and they form a triangle.

[0076] By substituting the data from Table 2 into Formula 3, it can be seen that there are no three collinear points. Therefore, the curvature of each group of circles is calculated. Find the three non-collinear points A(x) in the rectangular coordinate system. A ,y A ),B(x B ,y B ) and C(x C ,y C The formula for the curvature of a circle is as follows: (4); Where S is the area of ​​the triangle formed by points A, B, and C, which can be calculated using formula 3 above. ; ; .

[0077] Calculate the curvature of the circle defined by the three data points in the group, and then calculate the curvature C of the circle in the current group. j The curvature C of the circle in the previous group j-1 and the curvature C of the next grouped circle j+1 The relative rate of change, if >0.8 and If the value is greater than 0.8, then the data in this group is determined to have a curvature abrupt change.

[0078] If there are two or more consecutive grouped circles, such as C j Cj+1 , ..., C j+(n+1) Simultaneously satisfy >0.8 and If the value is greater than 0.8, then the group of circles with the smallest curvature is determined to have a curvature abrupt change.

[0079] Substituting the data from Table 2 into formula (4), we can obtain the grouping circles that satisfy the following conditions: >0.8 and >0.8; If there are two or more consecutive grouped circles, such as C j C j+1 , ..., C j+(n+1) Simultaneously satisfy >0.8 and If the value is greater than 0.8, then the group of circles with the smallest curvature is determined to have a curvature abrupt change.

[0080] Substituting the data from Table 2 into the above conditions, we can see that (7,8,9), (8,9,10), and (9,10,11) are consecutive and all satisfy the condition. >0.8 and For the three sets of circles with a curvature greater than 0.8, the point (9,10,11) has the smallest curvature, so it is determined to be the first point of curvature abrupt change. (18,19,20) and (23,24,25) are the second and third points of curvature abrupt changes, respectively. Therefore, the fitted line segment has three inflection points, meaning there are four fitted line segments.

[0081] 3. Calculate the coefficient of determination using the formula. The slope of each fitted line segment when the maximum value is taken. : Calculate the coefficient of determination The slope of each fitted line segment when the maximum value is taken. The derivation method of the formula is as follows: 1) Give the coefficient of determination The calculation formula is as follows: (5); in, It is the sum of squared residuals, calculated using the following formula:

[0082] It is the total sum of squares, calculated using the following formula:

[0083] in, These are actual observed values. It is the slope of each line segment. These are model predictions. It is the average of the observed values. It is the sample size; 2) Because With the slope of each line segment Irrelevant, seeking The maximum value is equivalent to finding The minimum value. For Differentiate and expand the squared terms:

[0084] 3) To Differentiate:

[0085] Set it to zero:

[0086] Then when the coefficient of determination The slope of each fitted line segment when the maximum value is taken. The calculation formula is: (6).

[0087] Based on the previous conclusions, the circles formed by the three sets of points (9, 10, 11), (18, 19, 20), and (23, 24, 25) in Table 2 meet the conditions. Therefore, the number of fitted line segments is 4. Let each fitted line segment be... , , and The data corresponding to each fitted line segment in Table 2 are shown in the following table: Table 3 shows the data for each fitted line segment in Table 2.

[0088] Substituting the coordinate values ​​of the data in Table 2 corresponding to each fitted line segment into Formula 6, we can obtain the optimal slope of each fitted line segment as shown in Table 4: Table 4 Optimal Slope Values ​​for Each Line Segment

[0089] 4. Methods for drawing fitted line segments and finding intersection points: Regarding the starting point for drawing each line segment, point 1 was chosen as the starting point for the first line segment because the starting point of the test data was (0,0), so the starting point of the fitted line segment should also be (0,0). Points 10, 19, and 24 are the midpoints within each curvature abrupt change group. Extensive experimental verification has shown that using the midpoint of each group as the starting point yields a relatively high coefficient of determination. .

[0090] (1) such as Figure 1As shown, taking the origin (0,0) (i.e., point 1) as the starting point, the slope is... =15.02 to make auxiliary ray R1.

[0091] (2) For example Figure 2 As shown, taking point 10 as the starting point, the slope is... =504.66 is used as auxiliary ray R2.

[0092] (3) such as Figure 3 As shown, taking point 19 as the starting point, the slope is... =178.74 as auxiliary ray R3.

[0093] (4) such as Figure 4 As shown, taking point 24 as the starting point, the slope is... =47.13 is used as auxiliary ray R4.

[0094] (5) such as Figure 5 As shown, extend rays R2, R3 and R4 in the opposite direction, and denote the intersection points as point A (1.8128, 27.2282), point B (3.5090, 883.0530) and point C (4.4260, 1046.9575), respectively.

[0095] (6) For example Figure 6 As shown, after removing the ray portion outside the intersection point, the four continuous line segments defined by the origin (0,0), point A, point B, and point C constitute the final multi-segment linear fitting model, which are as follows: segmented line segments (From origin to point A), segmented line segment (From point A to point B), segmented line segment (from point B to point C) and segmented line segments (After point C). This model and formula The results are consistent and accurately characterize the nonlinear stress-strain response of high-elongation steel cords.

[0096] (7) Based on the above, the fitted line segment representing steel cord A can be obtained as follows: ; The slopes of each line segment are shown in Table 4, and the x-coordinates of the turning points are shown in Table 5.

[0097] Table 5. Axis coordinates of turning points

[0098] (8) Then, by using the determination coefficient formula (4), we can obtain the following: =0.9993.

[0099] Comparative Example 1: Fitting using finite element software For the stress-strain test data of high-elongation steel cord A, conventional finite element software was first used for fitting, and the resulting fitting curve is shown below. Figure 7 As shown, the coefficient of determination between it and the test data =0.8876. In comparison, the novel method proposed in this invention exhibits higher accuracy in stress-strain response fitting.

Claims

1. A method of fitting the stress-strain response of a high elongation steel cord, characterized in that, The method comprises the following steps: S1: obtaining stress-strain data points of the high-elongation steel cord; S2: analyzing the stress-strain data, determining the number N of turning points based on a local curvature mutation detection method, dividing the fitting curve into N+1 fitting line segments, and drawing the first fitting line segment with the starting point as the origin (0, 0), and drawing the remaining fitting line segments with the starting point located in the data points with curvature mutation; S3: The optimal slope of each fitted segment can be calculated by taking the derivative of the formula for the coefficient of determination with respect to the slope and setting the derivative equal to zero. ;​ S4: determining the optimal slope of each fitting line segment according to the calculation And respectively with the origin (0, 0) and the starting point determined in step S2 draw rays, in addition to the ray from the origin (0, 0), reverse extend the rest of the rays, take the intersection of the extended rays as the turning point, delete the excess rays outside the intersection, get the fitting line segment of the high elongation steel cord stress strain response.

2. The fitting method of claim 1, wherein, The expression of the fitting line segment of the stress-strain response of the high-elongation steel cord is: ； wherein is a strain value, is a corresponding stress value; This represents the slope of the i-th line segment. Let n be the strain value at the (i-1)th inflection point, and n be the total number of segments. When x < x i-1 , max(x-x i-1 , 0) = 0, the segment contributes zero. When x > x i-1 , max(x-x i-1 , 0) = x-x i-1 , the segment is activated.

3. The fitting method of claim 1, wherein, In step S1, obtaining the stress-strain data of the high-elongation steel cord comprises the following steps: Test the force-displacement data of the high-elongation steel cord, and calculate the stress value and the strain value according to the following formulas respectively: The stress value of the high-elongation steel cord is calculated by the following formula: ； Wherein, F is the tension on the high-elongation steel cord, that is, the force in Table 1, and A is the cross-sectional area of the high-elongation steel cord; The strain value of the high-elongation steel cord is calculated by the following formula: ； wherein is the displacement of the high elongation steel cord under load, L0is the original length of the high elongation steel cord.

4. The fitting method of claim 1, wherein, In step S2, along the horizontal coordinate axis in sequence, move with a step of 1, take 3 adjacent data points as a group each time, and judge whether the grouped data has curvature mutation.

5. The fitting method of claim 4, wherein, The method for judging whether the grouped data has curvature mutation is: According to the 3 data points in the group, the curvature of the circle determined by them is calculated, and the curvature C of the current group circle is calculated j The relative change rate of the curvature C of the current group circle and the curvature C of the previous group circle j-1 And the curvature C of the next group circle j+1 If > 0.8 and > 0.8, it is determined that there is a curvature mutation in the group data.

6. The fitting method of claim 5, wherein, If there are two or more consecutive group circles such as C j ,C j+1 ,..., C j+(n+1) that satisfy >0.8 and >0.8 at the same time, it is determined that the group circle with the smallest curvature has a curvature mutation.

7. The fitting method of claim 6, wherein, Except for the first line segment, the starting point of each fitting line segment is the middle point of the three data points with curvature mutation when drawing the fitting line segment.

8. The fitting method of claim 1, wherein, In step S3, the determination coefficient is calculated according to the following equation Slopes of the fitting line segments at the maximum : ； wherein is the strain value at the i-th point obtained in step S1, is the stress value at the i-th point obtained in step S1.

9. The fitting method of claim 8, wherein, Computing the determination coefficient Slope of each fitted line segment at the maximum The formula derivation method is: 1) The formula for calculating the coefficient of determination R2is given by: R2= 1 - (SSresidual / SStotal) ； wherein is the residual sum of squares, calculated as: is the total sum of squares, calculated as: wherein, is the actual observed value, is the slope of each fitted line segment, is the model predicted value, is the mean of the observed values, is the number of samples; 2) for derivation, expanding the square term: 3） Let it be zero: then the determination coefficient The slope of each fitted line segment at the maximum The formula for calculating the slope of each fitted line segment at the maximum is: 。 10. The application of the fitting method of any one of claims 1-9 in the stress-strain response fitting of metal wire products.

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