Ballastless track fatigue damage semi-analytical method based on tension-compression anisotropy

Abstract

The invention relates to the technical field of ballastless tracks, and provides a ballastless track fatigue damage semi-analytical method based on tension-compression anisotropy, which comprises the following steps of: 1, expressing a material damage state by using equivalent free energy for a complex stress state and a simple stress state, and performing unified measurement on the complex stress state and the simple stress state to obtain a material damage state; the stress values are equivalent to the stress values of single-axis tension and compression under the condition of single-axis load; step 2, on the basis of free energy defined by effective tension and compression stress, by combining material fatigue three-stage damage characteristics, using a natural constant as a base and a damage variable as an index, constructing a function reflecting fatigue damage three-stage change; step 3, establishing an analytic expression of fatigue loading cycle times and damage through integration of tension and compression damage in a load cycle time to obtain a discrete semi-analytic expression; and 4, solving the semi-analytical expression through a numerical integration method, and drawing a calculation result graph. According to the method, ballastless track fatigue damage semi-analysis can be well carried out.

Description

Technical Field

[0001] The invention relates to the technical field of ballastless track, in particular to a semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy. Background Art

[0002] Ballastless track is the main structural type of high-speed railway. During its design service life, it can experience tens of millions or even hundreds of millions of train loads, which is a typical high-cycle fatigue load. Under the excitation of continuous and repeated high-cycle train loads, the track will suffer damage such as track plate cracking and block falling.

[0003] Track slab cracking will gradually degrade the mechanical properties of ballastless track. As the service time increases, if these damages are not discovered in time, they will often quickly develop into larger damages, further weakening the durability and service life of the track structure. In order to ensure the safe operation of the line, the track status must be accurately grasped, which requires scientific and reasonable damage prediction technology.

[0004] Usually, the fatigue damage constitutive model of concrete materials is composed of control equations and damage evolution functions. In order to establish the fatigue damage relationship of concrete, researchers have gradually carried out the construction of constitutive equations under different assumptions. Common methods include linear damage accumulation method, microscopic plane method, and competition mechanism method based on damage energy release rate. Among them, the concrete fatigue damage constitutive model proposed by the competition mechanism method based on damage energy release rate is the most representative. This model drives the development of concrete damage with damage energy release rate, decomposes the effective stress into tensile and compressive stresses through the fourth-order projection tensor, and establishes tensile damage energy release rate function and compressive damage energy release rate function respectively. The tensile damage energy release rate function is independent of plasticity, and the compressive damage energy release rate function is divided into elastic part and plastic part. At the same time, its damage function is selected based on the concept of competition between the two mechanisms of self-healing and damage evolution, and the tensile and compressive fatigue damage functions have the same form. And the numerical calculation is realized through the stress update algorithm of elastic prediction-plastic correction-damage correction.

[0005] However, the model algorithm is complex. For the fatigue problem of ballastless track concrete subjected to high-cycle train loads, the plastic behavior is far less than the yield function because the stress level is far less than the yield function, and the plastic deformation is generally not reflected. The model can be further simplified. The fatigue damage equation adopts a semi-empirical formula, and its applicability to concrete fatigue problems with different load characteristics needs to be recalibrated and selected. Due to the complexity of the model algorithm and the high degree of nonlinearity, the model calculation efficiency is low during parameter calibration, and there is no suitable analytical formula to quickly feedback the relationship and value pattern between material parameters. Summary of the invention

[0006] The present invention provides a semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy, which can better perform semi-analytical analysis of fatigue damage of ballastless track.

[0007] According to the present invention, a semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy comprises the following steps: Step 1: Use equivalent free energy to express the material damage state of complex stress state and simple stress state, and use it for unified measurement of complex stress and simple stress state, so that it is equivalent to the uniaxial tensile and compressive stress values ​​under uniaxial loading conditions; Step 2: Based on the free energy defined by the effective stress of tension and compression, combined with the three-stage damage characteristics of material fatigue, a function reflecting the three-stage changes is constructed using the natural constant as the base and the damage variable as the exponent; Step 3, obtain a semi-analytical expression by analytically expressing the tensile and compressive damage and the number of fatigue loadings; Step 4: Solve the semi-analytical expression through numerical integration method and draw a graph of the calculation results.

[0008] Preferably, in step 1, the complex stress state includes Simple stress states include uniaxial tension and uniaxial compression.

[0009] Preferably, in step 1, the effective stress after tension and compression decomposition is used to define the tensile and compressive damage free energy, the tensile and compressive stress decomposition is realized by the fourth-order projection tensor, and the tensile and compressive fatigue damage evolution is described by the stress state respectively, as shown in formula (1): ; In the formula, pull is +, press is -, is the effective stress tensor in tension and compression The component at the position, the multiplied variable in the expression is repeated twice as a repeated index, For repeated indices, all repeated indices need to be traversed and summed within their indicator definition domain, that is, they obey the Einstein summation convention.

[0010] Preferably, in step 2, the slope of the function is large in the first stage, and the damage accumulation is correspondingly fast, while the second stage is relatively gentle, the accumulation speed is slow, and it is the stage that accounts for the largest proportion of the three stages; The damage speed in the first stage can be controlled. The larger the damage speed, the faster it decays, and the smaller the damage speed, the slower it decays. Controls the damage speed in the third stage. The larger the damage, the faster the damage accumulates. The smaller the damage, the slower the damage accumulates. Optimize the cumulative effect of the first and second stages of the damage curve under different loading stress levels, and use The distance between the damage evolution relationships under different stress levels is reduced, and the parameters related to the damage function are calibrated through test data. The tensile and compressive fatigue damage functions of ballastless track concrete are reorganized through fatigue tests, as shown in formulas (2) and (3), where represents the tensile and compressive damage variables, , is the characteristic parameter of fatigue damage function; ; .

[0011] Preferably, in step 3, under the action of 1-dimensional uniaxial tensile and compressive fatigue loads, the tensile and compressive damage free energy is Simplified to effective tensile and compressive stress ; Therefore, the left sides of equations (2) and (3) remain unchanged, and the right sides of the equations are replaced by the effective tensile and compressive stresses, respectively. Then, the terms on the right side of the equations related to the damage variable are replaced by By moving them to the left side of the equation and merging similar terms, we can obtain the damage expression after separating the variables, and derive the analytical expression based on this. The maximum and minimum values ​​of train loads, the mean value of frequency characteristics, the variance and the amplitude range of the confidence interval of three times the mean square error are obtained through the statistical characteristics analysis of train loads, which provides the load premise for the actual track structure loading conditions. The characteristic value is used as the key parameter of the loading function to calculate the maximum compressive stress of the track structure components. Based on this as the loading condition, the nominal stress of the loading is According to the sine function expression, as shown in formula (4): ; In the formula are the load average nominal stress and nominal stress amplitude, respectively. is the load frequency, and three load characteristic parameters are used to define the load variation. These characteristics are obtained through train load statistics and stress calculation based on track structure; effective stress is the nominal stress. times, so the effective stress Divide the nominal stress by To express; The effective stress of load versus time Derivative, where time The relevant variables are and , by partial derivation, we can get formula (5): ; The analytical solution of the result is found at the expense of a certain accuracy to improve the efficiency of parameter calibration; thus, formula (5) Ignore, only the first term remains on the right side of equation (5); To make the expression concise, let , substituting the simplified formula (5) into the formula (2) after combining similar terms, we can obtain the damage evolution function related to the load average nominal stress and nominal stress amplitude and load frequency; where the loading period is the inverse of the frequency , so in one cycle, the left side of the equation is about time The derivative of will be transformed into the derivative with respect to the number of cycles , integrate the right side of the equation within one cycle, and get the integral of the damage with respect to the number of cycles as shown in equation (6): ; Considering that the damage does not increase during unloading, When unloading, it automatically becomes 0. The integral term on the right side of equation (6) is calculated according to a cycle. Segmented means that only in the interval and is not equal to 0, and the integral term on the right side of the final equation becomes ; Substituting the integral result into equation (6), we get the differential expression of the number of cycles and damage as equation (7): ; Since the right side of equation (7) cannot be directly integrated, the numerical integration method is used to construct multiple sets of equations with equal spacing to obtain a discrete equation format. The discrete equation can adjust the equal spacing value by itself. Take the spacing equal to 1 as an example, as shown in equation (8): ; Belongs to the natural number set, represented by 0, 1, 2…; Similarly, according to the uniaxial tension case, the semi-analytical expression of tensile damage and fatigue loading times is as follows: ; in .

[0012] Preferably, in step 4, specifically: (1) Define function variables; The variables that need to be defined in equations (14) and (15) are: , ; (2) Set the initial values ​​of the variables and adjust each initial value as needed; Get the constant, under pressure: ; Tension: ; (3) Establish an iterative format; (3.1) Set the maximum number of cycles, and define the value yourself; (3.2) Set the initial value of the iteration count variable; (3.3) Set a two-dimensional empty set, the row dimension of which is the maximum number of loops N , the column dimension is 2, which are used to store the cycle ratio n / N and damage variables d , use for loop to construct iterative calculation format about damage; (4) Draw a graph of the calculation results; The normalized cycle ratio of the loading times is n / N As the horizontal axis, the damage variable is used as the vertical axis to draw a graph, including setting the graph title, setting the legend, and setting the horizontal and vertical axis names of damage and number of cycles.

[0013] The beneficial effects of the present invention are as follows: The present invention sets the tensile and compressive fatigue elastic damage free energy of ballastless track concrete, filters the influence of material plasticity on fatigue damage, reduces complexity, and improves fatigue damage calculation efficiency.

[0014] In order to make the tensile fatigue damage of ballastless track concrete more scientific and reasonable, the present invention takes the concrete fatigue test results with train load characteristics as a benchmark, proposes a new type of tensile fatigue damage evolution equation that is self-consistent with the fatigue test results, and more accurately describes the fatigue characteristics of ballastless track concrete.

[0015] In order to overcome the high complexity of the current model, the present invention proposes a semi-analytical method of fatigue damage and loading times based on the fatigue damage evolution equation, which, combined with a simple numerical integration method, effectively improves the data processing time and memory space requirements during parameter calibration. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a flow chart of a semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy in an embodiment; Figure 2 A comparison diagram of the semi-analytical solution and the finite element solution of compressive fatigue damage in the embodiment; Figure 3 It is a comparison diagram of the semi-analytical solution and the finite element solution of tensile fatigue damage in the embodiment. DETAILED DESCRIPTION

[0017] In order to further understand the content of the present invention, the present invention is described in detail in conjunction with the accompanying drawings and embodiments. It should be understood that the embodiments are only for explaining the present invention and are not intended to limit it.

[0018] Example like Figure 1As shown, this embodiment provides a semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy, which includes the following steps: Step 1: Use equivalent free energy to express the material damage state of complex stress state and simple stress state, which is used for unified measurement of complex stress and simple stress state, so that it is equivalent to the uniaxial tensile and compressive stress values ​​under uniaxial loading conditions.

[0019] In step 1, the complex stress state includes , simple stress states include uniaxial tension and uniaxial compression.

[0020] In step 1, the effective stress after tension and compression decomposition is used to define the tensile and compressive damage free energy. The tensile and compressive stress decomposition is realized by the fourth-order projection tensor. The evolution of tensile and compressive fatigue damage is described by the stress state respectively, as shown in formula (1): ; In the formula, pull is +, press is -, is the effective stress tensor in tension and compression The component at the position, the multiplied variable in the expression is repeated twice as a repeated index, For repeated indices, all repeated indices need to be traversed and summed within their indicator definition domain, that is, they obey the Einstein summation convention: , if 11 is the uniaxial loading direction, when the uniaxial pressure , when uniaxial tension .

[0021] Step 2: Based on the free energy defined by the effective stress of tension and compression, combined with the three-stage damage characteristics of material fatigue, a function reflecting the three-stage changes is constructed using the natural constant as the base and the damage variable as the exponent.

[0022] The function has a large slope in the first stage, and the damage accumulation is relatively fast. The second stage is relatively flat, and the accumulation speed is slow. It is the stage that accounts for the largest proportion of the three stages. The damage speed in the first stage can be controlled. The larger the damage speed, the faster it decays, and the smaller the damage speed, the slower it decays. Control the damage speed in the third stage. The larger the damage, the faster the damage accumulation speed. The smaller the damage, the slower the damage accumulation speed. According to the damage accumulation rate of the second stage of the bending and tension damage curve of concrete materials, it is larger than that of the compressive fatigue damage accumulation curve, and the constructed function It is impossible to balance the large difference in damage curves under different stress levels, resulting in large deviations between the tensile damage curves under different stress levels and the test results; therefore, Optimize the cumulative effect of the first and second stages of the damage curve under different loading stress levels, and use The distance between the damage evolution relationships under different stress levels is reduced, and the parameters related to the damage function are calibrated through test data. The tensile and compressive fatigue damage functions of ballastless track concrete are reorganized through fatigue tests, as shown in formulas (2) and (3), where represents the tensile and compressive damage variables, , is the characteristic parameter of fatigue damage function; ; .

[0023] Step 3: Analytically express the tensile and compressive damage and the number of fatigue loadings to obtain a semi-analytical expression.

[0024] During parameter analysis, test data obtained by simple loading is generally used for calibration. Considering that finite element calculation requires unit division, model establishment, material property allocation, material subroutine programming, overall stiffness matrix construction, and consistent tangent stiffness matrix algorithm compilation, the stress and strain in different coordinate axis directions, damage values ​​and displacements at different loading times, etc. are saved during result calculation and analysis. Both the calculation time and storage space bring redundant costs to parameter analysis under simple loading. At the same time, the model is a damage accumulation function that is directly related to time. Under the condition of equal amplitude loading, it is difficult to intuitively obtain the relationship between the number of loading times and damage accumulation. Therefore, an analytical expression of fatigue damage and loading times is sought to save calculation and storage costs and directly clarify the relationship between the number of cycles and damage. Under the action of one-dimensional uniaxial tension and compression fatigue loads, the tensile and compressive damage free energy can be Simplified to effective tensile and compressive stress Therefore, the left sides of equations (2) and (3) remain unchanged, and the right sides of the equations are replaced by the effective tensile and compressive stresses, respectively. Then, the terms on the right side of the equations related to the damage variable are replaced by By moving them to the left side of the equation and merging similar terms, we can obtain the damage expression after separating the variables, and derive the analytical expression based on this.

[0025] Since the derivation process of the analytical function for tension and compression is the same, the following takes uniaxial compression fatigue as an example. The maximum and minimum values ​​of the train load, the frequency characteristic mean, variance, and the amplitude range of the three-fold mean square error confidence interval are obtained through the statistical characteristic analysis of the train load, providing the load premise for the actual track structure loading condition. The characteristic value is used as the key parameter of the loading function to calculate the maximum compressive stress of the track structure components. Based on this as the loading condition, the nominal stress of the loading is According to the sine function expression, as shown in formula (4): ; In the formula are the load average nominal stress and nominal stress amplitude, respectively. is the load frequency, and three load characteristic parameters are used to define the load variation. These characteristics are obtained through train load statistics and stress calculation based on track structure; effective stress is the nominal stress. times, so the effective stress The nominal stress can be divided by To express.

[0026] The effective stress of load versus time Derivative, where time The relevant variables are and , by partial derivation, we can get formula (5): ; In the formula, due to the existence , which affects the existence of the solution when deriving the analytical solution of loading cycle and damage later. At this time, the damage is frozen and assumed to be constant over time. According to the form of the neglected term, when the damage is small, Relatively close to 1, and then Multiplying together, the calculated value of the entire term will be much smaller than the first term Calculated value. But when the damage accumulates, will approach 0, and the calculated value of the entire term will increase rapidly, affecting the overall calculation results, resulting in a large deviation between the analytical expression of the second and third stages of fatigue loading and the direct finite element solution. However, through trial calculations, it was found that this term has little effect on the discussion of parameter laws, so a certain degree of accuracy is sacrificed to find the analytical solution of the result in order to improve the efficiency of parameter calibration; therefore, formula (5) will be Ignore , and only the first term remains on the right side of equation (5).

[0027] To make the expression concise, let , substituting the simplified formula (5) into the formula (2) after combining similar terms, we can obtain the damage evolution function related to the load average nominal stress and nominal stress amplitude as well as the load frequency.

[0028] Due to the current It is the differential with respect to time t, and the load is an excitation of equal amplitude and frequency, so the load is discussed in each complete loading cycle. The advantages of this are: ① Through periodic integration, these complex time-varying characteristics can be averaged or simplified to obtain an equation about the cycle period that is easier to handle and analyze; ② For problems that require long-term simulation or iterative solutions, directly solving the original differential equation is very time-consuming. Through periodic integration, the problem can be simplified to an equation about the cycle period, thereby greatly reducing the amount of calculation; ③ The equation after periodic integration has a simpler form, and it is easier to apply various mathematical tools for parameter analysis, further revealing the sensitivity of material fatigue parameters and quickly calibrating material parameter values.

[0029] The loading period is the inverse of the frequency , so in one cycle, the derivative of the left side of the equation with respect to time t is will be transformed into the derivative with respect to the number of cycles , integrate the right side of the equation within one cycle, and get the integral of the damage with respect to the number of cycles as shown in equation (6): ; Considering that the damage does not increase during unloading, When unloading, it automatically becomes 0. The integral term on the right side of equation (6) is calculated according to a cycle. Segmented means that only in the interval and is not equal to 0, and the integral term on the right side of the final equation becomes .

[0030] Substituting the integral result into equation (6), we get the differential expression of the number of cycles and damage as equation (7): ; Since the right side of equation (7) cannot be directly integrated, the numerical integration method is used to construct multiple sets of equations with equal spacing to solve them, and the discrete equation system format can be obtained. The discrete equation can adjust the equal spacing value by itself. Take the spacing equal to 1 as an example, as shown in equation (8): ; Belongs to the natural number set, represented by 0, 1, 2…; Similarly, according to the uniaxial tension case, the semi-analytical expression of tensile damage and fatigue loading times is as follows: ; in .

[0031] Step 4: Solve the semi-analytical expression through numerical integration method and draw a graph of the calculation results.

[0032] In step 4, specifically: (1) Define function variables; The variables that need to be defined in equations (14) and (15) are: , ; (2) Set the initial values ​​of the variables and adjust each initial value as needed; Get the constant, under pressure: ; Tension: ; (3) Establish an iterative format; (3.1) Set the maximum number of cycles, and define the value yourself; (3.2) Set the initial value of the iteration count variable; (3.3) Set a two-dimensional empty set, the row dimension of which is the maximum number of loops N , the column dimension is 2, which are used to store the cycle ratio n / N and damage variables d , use for loop to construct iterative calculation format about damage; (4) Draw a graph of the calculation results; The normalized cycle ratio of the loading times is n / N As the horizontal axis, the damage variable is used as the vertical axis to draw a graph, including setting the graph title, setting the legend, and setting the horizontal and vertical axis names of damage and number of cycles.

[0033] In this embodiment, the normalized damage-fatigue times obtained by finite element and semi-analytical solutions are plotted on Figure 2 and Figure 3 It can be seen that the curves of the two are relatively consistent in the first and second stages of fatigue, and there is a certain error in the third stage. In the parameter calibration application scenario, both the semi-analytical solution and the finite element solution can characterize the variation law of the curve under different parameter conditions. The calculation time used by the finite element is about 105s, while the calculation time of the analytical solution is only about 2.67s, and the calculation efficiency is significantly improved.

[0034] In parameter calibration, there is no need to build a high-order tensor stress update algorithm that participates in finite element calculations. The idea is to ensure the specificity of the calculation as much as possible, without considering how to derive the consistent tangent stiffness tensor, how to participate in finite element calculations, and how to update it, and focus more on the evolution of damage. At the same time, the relationship between fatigue damage and time is transformed into a relationship with the number of cyclic loadings, which more intuitively describes the cumulative effect of damage behavior.

[0035] In this method, the driving factors of damage are optimized, from elasticity and plasticity to elastic fatigue damage, avoiding the problem that plasticity is not applicable at low stress levels. The tensile fatigue damage function needs to be re-formulated through fatigue tests with train load characteristics, which solves the irrationality of using a consistent type of damage equation to describe the damage of tensile and compressive anisotropic concrete, making the fatigue damage equation more targeted for the damage behavior analysis of ballastless track under train load.

[0036] The present invention and its embodiments are described schematically above, and the description is not restrictive. The drawings show only one embodiment of the present invention, and the actual structure is not limited thereto. Therefore, if a person skilled in the art is inspired by it and designs a structural method and an embodiment similar to the technical solution without creativity without departing from the purpose of the invention, they shall all fall within the protection scope of the present invention.

Claims

1. A semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy, characterized by: The following steps are involved: Step 1: Use equivalent free energy to express the material damage state of complex stress state and simple stress state, and use it for unified measurement of complex stress and simple stress state, so that it is equivalent to the uniaxial tensile and compressive stress values ​​under uniaxial loading conditions; Step 2: Based on the free energy defined by the effective stress of tension and compression, combined with the three-stage damage characteristics of material fatigue, a function reflecting the three-stage changes is constructed using the natural constant as the base and the damage variable as the exponent; Step 3, obtain a semi-analytical expression by analytically expressing the tensile and compressive damage and the number of fatigue loadings; Step 4: Solve the semi-analytical expression through numerical integration method and draw a graph of the calculation results.

2. The semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy according to claim 1 is characterized by: In step 1, the complex stress state includes Simple stress states include uniaxial tension and uniaxial compression.

3. The semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy according to claim 2 is characterized by: In step 1, the effective stress after tension and compression decomposition is used to define the tensile and compressive damage free energy. The tensile and compressive stress decomposition is realized by the fourth-order projection tensor. The evolution of tensile and compressive fatigue damage is described by the stress state respectively, as shown in formula (1): ； In the formula, pull is +, press is -, is the effective stress tensor in tension and compression The component at the position, the multiplied variable in the expression is repeated twice as a repeated index, For repeated indices, all repeated indices need to be traversed and summed within their indicator definition domain, that is, they obey the Einstein summation convention.

4. The semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy according to claim 3 is characterized by: In step 2, the slope of the function is large in the first stage, and the damage accumulation is relatively fast. The second stage is relatively flat, the accumulation speed is slow, and it accounts for the largest proportion of the three stages. It can control the damage speed in the first stage. The larger the damage speed, the faster it decays, and the smaller the damage speed, the slower it decays. Controls the damage speed in the third stage. The larger the damage, the faster the damage accumulates. The smaller the damage, the slower the damage accumulates. Optimize the cumulative effect of the first and second stages of the damage curve under different loading stress levels, and use The distance between the damage evolution relationships under different stress levels is reduced, and the parameters related to the damage function are calibrated through test data. The tensile and compressive fatigue damage functions of ballastless track concrete are reorganized through fatigue tests, as shown in formulas (2) and (3), where represents the tensile and compressive damage variables, , is the characteristic parameter of fatigue damage function; ； 。 5. The semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy according to claim 4 is characterized by: In step 3, under the action of 1D uniaxial tension and compression fatigue load, the tension and compression damage free energy is Simplified to effective tensile and compressive stress ; Therefore, the left sides of equations (2) and (3) remain unchanged, and the right sides of the equations are replaced by the effective tensile and compressive stresses, respectively. Then, the terms on the right side of the equations related to the damage variable are replaced by By moving them to the left side of the equation and merging similar terms, we can obtain the damage expression after separating the variables, and derive the analytical expression based on this. The maximum and minimum values ​​of train loads, the mean value of frequency characteristics, the variance and the amplitude range of the confidence interval of three times the mean square error are obtained through the statistical characteristics analysis of train loads, which provides the load premise for the actual track structure loading conditions. The characteristic value is used as the key parameter of the loading function to calculate the maximum compressive stress of the track structure components. Based on this as the loading condition, the nominal stress of the loading is According to the sine function expression, as shown in formula (4): ； In the formula are the load average nominal stress and nominal stress amplitude, respectively. is the load frequency, and three load characteristic parameters are used to define the load variation. These characteristics are obtained through train load statistics and stress calculation based on track structure; effective stress is the nominal stress. times, so the effective stress Divide the nominal stress by To express; The effective stress of load versus time Derivative, where time The relevant variables are and , by partial derivation, we get formula (5): ； Sacrificing accuracy to find the analytical solution of the result to improve the efficiency of parameter calibration; Therefore, formula (5) Ignore, only the first term remains on the right side of equation (5); To make the expression concise, let , substitute the simplified formula (5) into the formula (2) after combining similar terms, and obtain the damage evolution function related to the load average nominal stress and nominal stress amplitude and load frequency; where the loading period is the inverse of the frequency , so in one cycle, the left side of the equation is about time The derivative of will be transformed into the derivative with respect to the number of cycles , integrate the right side of the equation within one cycle, and get the integral of the damage with respect to the number of cycles as shown in equation (6): ； Considering that the damage does not increase during unloading, When unloading, it automatically becomes 0. The integral term on the right side of equation (6) is calculated according to a cycle. Segmented means that only in the interval and is not equal to 0, and the integral term on the right side of the final equation becomes ; Substituting the integral result into equation (6), we get the differential expression of the number of cycles and damage as equation (7): ； Since the right side of equation (7) cannot be directly integrated, the numerical integration method is used to construct multiple sets of equations with equal spacing to obtain a discrete equation format. The discrete equation can adjust the equal spacing value by itself. Take the spacing equal to 1 as an example, as shown in equation (8): ； Belongs to the natural number set, represented by 0, 1, 2…; Similarly, according to the uniaxial tension case, the semi-analytical expression of tensile damage and fatigue loading times is as follows: ； in .

6. The semi-analytical method for fatigue damage of ballastless track based on tension and compression anisotropy according to claim 5 is characterized by: In step 4, specifically: (1) Define function variables; The variables that need to be defined in equations (14) and (15) are: ， ； (2) Set the initial values ​​of the variables and adjust each initial value as needed; Get the constant, under pressure: ; Tension: ; (3) Establish an iterative format; (3.1) Set the maximum number of cycles, and define the value yourself; (3.2) Set the initial value of the iteration count variable; (3.3) Set a two-dimensional empty set, the row dimension of which is the maximum number of loops N , the column dimension is 2, which are used to store the cycle ratio n / N and damage variables d , use for loop to construct iterative calculation format about damage; (4) Draw a graph of the calculation results; The normalized cycle ratio of the loading times is n / N As the horizontal axis, the damage variable is used as the vertical axis to draw a graph, including setting the graph title, setting the legend, and setting the horizontal and vertical axis names of damage and number of cycles.

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