A material parameter reverse identification method based on multi-objective optimization

Abstract

The application provides a material parameter reverse identification method based on multi-objective optimization, which comprises the following steps: carrying out a series of tensile-compressive symmetric cyclic plastic deformation tests with different strain levels on a metal material at room temperature; extracting an engineering cyclic stress-strain curve in the tensile-compressive symmetric cyclic plastic deformation data with the maximum strain level and converting the engineering cyclic stress-strain curve into a true cyclic stress-strain curve; extracting the true stress-strain curves of all tensile half cycles, determining the Young's modulus and initial yield strength of each tensile half cycle, and calculating the average Young's modulus and initial yield strength of all cycles; determining a nonlinear follow-up hardening parameter from the true stress-strain curve of the first tensile half cycle; calculating an isotropic stress-cumulative plastic strain curve from the true stress-strain curves of all cycles, and fitting and determining isotropic hardening parameters; and establishing a finite element model, adopting a multi-objective optimization algorithm, and reversely identifying material parameters of a nonlinear mixed hardening constitutive model.

Description

Technical Field

[0001] This invention relates to the field of mechanical property characterization technology for metallic materials, and more specifically, to a method for reverse identification of material parameters based on multi-objective optimization. Background Technology

[0002] Numerical simulation studies of the cyclic plastic deformation behavior of metallic materials have always been a key focus in disciplines such as solid mechanics, materials science and engineering. They have important theoretical value and practical guiding significance for the study of fatigue behavior of metallic components, fatigue life prediction, and the development of anti-fatigue technologies and methods.

[0003] Nonlinear hybrid hardening constitutive models have been widely used in numerical simulations of cyclic plastic deformation behavior of metallic materials because they have a good ability to describe the cyclic hardening / softening behavior and Bauschinger effect of metallic materials during cyclic plastic deformation.

[0004] Rapid and accurate identification of material parameters is crucial for the application of nonlinear hybrid hardening constitutive models. Existing material parameter identification methods mainly fall into two categories: (1) manual identification methods based on theoretical formulas; and (2) automatic reverse identification methods based on optimization. Method (1) starts from the theoretical formulas of the nonlinear hybrid hardening constitutive model, performs simple mathematical operations on the formulas, and combines linear fitting to identify the specific values ​​of material parameters, which is simple and easy to implement. However, method (1) makes very limited use of cyclic plasticity test data, resulting in poor identification accuracy. For example, for parameters including Young's modulus E, initial yield strength σ0, and kinematic hardening parameter C, the identification accuracy is poor. k and γ kThe identification of multiple material parameters, including those from the cyclic plasticity test, typically only uses data from a single cycle of the entire cyclic plasticity test, while data from other cycles are not used. Method (2) usually uses the identification results of Method (1) as initial values, specifies the range of variation for each parameter, and then establishes a numerical simulation model corresponding to the cyclic plasticity test. By setting the difference between the simulation data and the measured data from the cyclic plasticity test as the optimization objective, the optimization method is used to achieve automatic inverse identification of the nonlinear hybrid hardening constitutive model. Compared with Method (1), the optimization objective of Method (2) makes full use of all cyclic plasticity test data, thus making the identification of material parameters of the nonlinear hybrid hardening constitutive model more efficient and accurate. However, Method (2) also has obvious limitations. For example, the data only comes from the cyclic plasticity test data under a single loading condition, so as to achieve inverse identification through single-objective optimization. This leads to Method (2) not being able to fully consider the cyclic plastic deformation behavior of metallic materials under other loading conditions. Alternatively, when using cyclic plasticity tests under multiple loading conditions, the multi-objective optimization problem can be simply and crudely transformed into a single-objective optimization problem by directly summing up the multiple optimization objectives under different loading conditions.

[0005] Therefore, how to establish a method for inverse identification of material parameters in a nonlinear hybrid hardening constitutive model based on multi-objective optimization has become an important problem that urgently needs to be solved. Summary of the Invention

[0006] To address the shortcomings of existing technologies, the present invention aims to provide a low-cost, high-speed, accurate, and simple method for reverse identification of material parameters based on multi-objective optimization.

[0007] To solve the above problems, the technical solution of the present invention is as follows:

[0008] A method for inverse identification of material parameters based on multi-objective optimization includes the following steps:

[0009] A series of tension-compression symmetrical cyclic plastic deformation tests were conducted on metallic materials at different strain levels at room temperature to obtain cyclic plastic deformation data of metallic materials at room temperature;

[0010] Extract the engineering cyclic stress-strain curves from the tension-compression symmetric cyclic plastic deformation data at the maximum strain level, and convert the engineering cyclic stress-strain curves into true cyclic stress-strain curves.

[0011] Extract the true stress-strain curves of all tensile half-cycles, determine the Young's modulus and initial yield strength of each tensile half-cycle, and calculate the average Young's modulus and initial yield strength of all cycles.

[0012] The nonlinear kinematic hardening parameters were determined from the actual stress-strain curves of the first stretching half-cycle.

[0013] Calculate the isotropic stress-cumulative plastic strain curve from the true stress-strain curves of all cycles, and fit to determine the isotropic hardening parameters;

[0014] A finite element model was established, and a multi-objective optimization algorithm was used to reverse identify the material parameters of the nonlinear hybrid hardening constitutive model.

[0015] Preferably, the formula for extracting the engineering cyclic stress-strain curve from the tension-compression symmetric cyclic plastic deformation data at the maximum strain level and converting it into the true cyclic stress-strain curve is as follows:

[0016]

[0017]

[0018] σ T =σ E (1+ε E )

[0019] ε T =Ln(1+ε E )

[0020] In the formula, P is the load, A0 is the original cross-sectional area of ​​the metal specimen, L0 is the original length of the gauge length of the metal specimen, ΔL is the change in length of the gauge length of the metal specimen, and σ T and ε T These represent the true stress and the true strain, respectively, σ E and ε E These are engineering stress and engineering strain, respectively.

[0021] Preferably, the steps of extracting the true stress-strain curves of all stretching half-cycles, determining the Young's modulus and initial yield strength of each stretching half-cycle, and calculating the average Young's modulus and initial yield strength of all cycles specifically include: extracting the true stress-strain curves of all stretching half-cycles, performing linear fitting on the elastic segment of each stretching half-cycle, thereby determining the Young's modulus E of each stretching half-cycle. i The second derivative of the true stress-strain curve for each stretching half-cycle is calculated, and the true stress value corresponding to the true strain at the minimum point of the second derivative is determined as the initial yield strength σ for each stretching half-cycle. 0i Calculate the Young's modulus E and the initial yield strength σ0 of all cycles.

[0022] Preferably, the formula for calculating the average Young's modulus E and the initial yield strength σ0 over all cycles is as follows:

[0023]

[0024]

[0025] Where i∈[1,n], and n is the total number of cycles.

[0026] Preferably, the step of determining the nonlinear kinematic hardening parameters from the true stress-strain curve of the first tensile half-cycle specifically includes: according to the formula: The true stress-strain curve of the first stretching half-cycle is transformed into a true stress-plastic strain curve, where ε and ε e and ε p Given the total strain, elastic strain, and plastic strain, and N back stress components, the true stress-plastic strain curve of the first tensile half-cycle is divided into M segments, M = N + 1. The nonlinear kinematic hardening parameter C on each segment is calculated. k and γ k .

[0027] Preferably, the calculation of the nonlinear kinematic hardening parameter C on each curve segment... k and γ k The formula is:

[0028]

[0029]

[0030] Where k∈[1,N].

[0031] Preferably, the step of calculating the isotropic stress-cumulative plastic strain curve from the true stress-strain curves of all cycles and fitting to determine the isotropic hardening parameters specifically includes: extracting the true stress-strain curves of all tensile and compressive half-cycles, and determining the maximum stress σ of each tensile half-cycle. tmax_i and the yield strength σ for each compression half-cycle c_i According to the formula: Calculate the isotropic stress σ for each cycle. 0 i According to the isotropic stress σ 0 i Calculate the cumulative plastic strain for each cycle Then, according to the formula: By fitting the isotropic stress-cumulative plastic strain curve, the values ​​of the isotropic hardening parameters Q and b are obtained.

[0032] Preferably, the steps of establishing a finite element model, employing a multi-objective optimization algorithm, and inversely identifying the material parameters of the nonlinear hybrid hardening constitutive model specifically include: establishing a finite element model, performing numerical simulations of a series of room temperature tensile-compression symmetric cyclic plastic deformation processes at different strain levels, integrating the simulation process into a multi-objective optimization framework, and simultaneously using a series of different simulated real stress-time curves obtained from numerical simulations and a series of different measured real stress-time curves obtained from a series of room temperature tensile-compression symmetric cyclic plastic deformation tests of the metallic material as optimization objectives. The formula is as follows:

[0033]

[0034] Where j represents the j-th objective, j∈[1,S], S is the total number of objectives, i.e., the total number of tensile-compression symmetrical cyclic plastic deformation tests and corresponding numerical simulations at different strain levels, and m j This represents the total number of data points on the simulated and measured stress-time curves for tension-compression symmetric cyclic plastic deformation at the j-th strain level. and These represent the simulated stress value and the measured stress value of the i-th data point, respectively.

[0035] Preferably, the step of establishing a finite element model and using a multi-objective optimization algorithm to inversely identify the material parameters of the nonlinear hybrid hardening constitutive model further includes: assigning Young's modulus E, initial yield strength σ0, and nonlinear kinematic hardening parameter C to the model. k and γ k The isotropic hardening parameters Q and b are set as input variables. The results calibrated in the above steps are used as the initial values ​​of each input variable. A multi-objective optimization algorithm is used to perform multi-objective optimization operations that simultaneously minimize all optimization objectives. A unique optimal solution is selected from the optimal solution set through multi-criteria decision-making techniques. The Young's modulus E, initial yield strength σ0, and nonlinear kinematic hardening parameter C corresponding to the unique optimal solution are then used as input variables. k and γ k The values ​​of isotropic hardening parameters Q and b are determined as the material parameters of the nonlinear hybrid hardening constitutive model obtained by reverse identification.

[0036] Compared with existing technologies, this invention can truly achieve accurate identification of material parameters in nonlinear hybrid hardening constitutive models by simultaneously considering the cyclic plastic response of metallic materials under multiple strain levels. By using multi-criteria decision-making techniques, a unique optimal solution can be obtained from the optimal solution set, effectively solving the problem that the unique optimal solution cannot be accurately obtained in the parameter inversion process of traditional single-objective optimization. The constitutive parameter value corresponding to the unique optimal solution will be determined as the material parameters of the nonlinear hybrid hardening constitutive model obtained by the final inverse identification. The method of this invention has the advantages of low cost, high-speed and accurate computation, and simplicity. It can be widely applied to the inverse identification of material parameters in nonlinear hybrid hardening constitutive models of various metallic materials, and has high practical value and important practical significance in computational mechanics, experimental mechanics, and engineering applications. Attached Figure Description

[0037] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings:

[0038] Figure 1 This is a flowchart of a material parameter reverse identification method based on multi-objective optimization provided in an embodiment of the present invention;

[0039] Figure 2 A comparison diagram of a series of tension-compression symmetric cyclic stress-strain curves at different strain levels and experimentally measured curves, provided in Embodiment 1 of the present invention, using a nonlinear hybrid hardening constitutive model based on multi-objective optimization inverse identification.

[0040] Figure 3 This is a comparison chart of a series of tension-compression symmetric cyclic stress-strain curves at different strain levels and experimentally measured curves, obtained by using a nonlinear hybrid hardening constitutive model based on multi-objective optimization and inverse identification for Embodiment 2 of the present invention. Detailed Implementation

[0041] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention and will make the technical solution and beneficial effects of the present invention readily apparent.

[0042] Specifically, Figure 1 The flowchart of the material parameter reverse identification method based on multi-objective optimization provided in the embodiments of the present invention is as follows: Figure 1 As shown, the method includes the following steps:

[0043] S1: Conduct a series of tension-compression symmetrical cyclic plastic deformation tests on metallic materials at different strain levels at room temperature to obtain cyclic plastic deformation data of metallic materials at room temperature;

[0044] S2: Extract the engineering cyclic stress-strain curve from the tension-compression symmetric cyclic plastic deformation data at the maximum strain level, and convert the engineering cyclic stress-strain curve into the real cyclic stress-strain curve;

[0045] Specifically, the engineering cyclic stress-strain curves from the tension-compression symmetric cyclic plastic deformation data at the maximum strain level are extracted according to the following formula and transformed into true cyclic stress-strain curves:

[0046]

[0047]

[0048] σ T =σ E (1+ε E )

[0049] ε T =Ln(1+ε E )

[0050] Where P is the load, A0 is the original cross-sectional area of ​​the metal specimen, L0 is the original length of the gauge length of the metal specimen, ΔL is the change in length of the gauge length of the metal specimen, and σ T and ε T These represent the true stress and the true strain, respectively, σ E and ε E These are engineering stress and engineering strain, respectively.

[0051] S3: Extract the true stress-strain curves of all tensile half-cycles, determine the Young's modulus and initial yield strength of each tensile half-cycle, and calculate the average Young's modulus and initial yield strength of all cycles.

[0052] Specifically, the true stress-strain curves of all stretching half-cycles are extracted, and linear fitting is performed on the elastic segment of each stretching half-cycle to determine the Young's modulus E of each stretching half-cycle. i The second derivative of the true stress-strain curve for each stretching half-cycle is calculated, and the true stress value corresponding to the true strain at the minimum point of the second derivative is determined as the initial yield strength σ for each stretching half-cycle. 0i Then, calculate the average Young's modulus E and the initial yield strength σ0 for all cycles according to the following formula:

[0053]

[0054]

[0055] Where i∈[1,n], and n is the total number of cycles.

[0056] S4: Determine the nonlinear kinematic hardening parameters from the actual stress-strain curve of the first stretching half-cycle;

[0057] Specifically, the true stress-strain curve of the first stretching half-cycle is transformed into the true stress-plastic strain curve according to the following formula:

[0058]

[0059] Among them, ε, ε e and ε p Let the total strain, elastic strain, and plastic strain be represented, respectively. Then, take N back stress components and divide the true stress-plastic strain curve of the first tensile half-cycle into M segments, M = N + 1. Calculate the nonlinear kinematic hardening parameter C on each segment of the curve according to the following formula. k and γ k :

[0060]

[0061]

[0062] Where k∈[1,N].

[0063] S5: Calculate the isotropic stress-cumulative plastic strain curve from the true stress-strain curves of all cycles, and fit to determine the isotropic hardening parameters;

[0064] Specifically, the true stress-strain curves of all tensile and compression half-cycles are extracted, and the maximum stress σ of each tensile half-cycle is determined. tmax_i and the yield strength σ for each compression half-cycle c_i Then, the isotropic stress σ for each cycle is calculated according to the following formula. 0 i :

[0065]

[0066] The cumulative plastic strain for each cycle is calculated according to Table 1 below.

[0067]

[0068] Table 1

[0069] Where Δεi is the strain range of the i-th cycle.

[0070] Then, according to the formula: By fitting isotropic stress-cumulative plastic strain curves, the values ​​of isotropic hardening parameters Q and b are obtained.

[0071] S6: Establish a finite element model, use a multi-objective optimization algorithm, and reverse identify the material parameters of the nonlinear hybrid hardening constitutive model.

[0072] Specifically, a finite element model is established to numerically simulate a series of tensile-compression symmetrical cyclic plastic deformation tests of the metallic material at different strain levels at room temperature, as conducted in step S1. The simulation process is integrated into a multi-objective optimization framework. The optimization objective uses a series of different simulated stress-time curves obtained from the numerical simulation and a series of different measured stress-time curves obtained from the tensile-compression symmetrical cyclic plastic deformation tests of the metallic material at different strain levels at room temperature, along with a series of different squared error loss functions, as shown in the following equation:

[0073]

[0074] Where j represents the j-th objective, j∈[1,S], S is the total number of objectives, i.e., the total number of tensile-compression symmetrical cyclic plastic deformation tests and corresponding numerical simulations at different strain levels, and m j This represents the total number of data points on the simulated and measured stress-time curves for tension-compression symmetric cyclic plastic deformation at the j-th strain level. and Let E and σi represent the simulated stress value and the measured stress value at the i-th data point, respectively. Then, let Young's modulus E, initial yield strength σ0, and nonlinear kinematic hardening parameter C be used. k and γ k The isotropic hardening parameters Q and b are set as input variables. The values ​​calibrated in steps S3 to S6 are used as the initial values ​​of each input variable. Constraints are set within a reasonable range for each parameter. A multi-objective optimization algorithm is used to perform multi-objective optimization operations that simultaneously minimize all optimization objectives. Finally, a unique optimal solution is selected from the optimal solution set using a multi-criteria decision-making technique. The Young's modulus E, initial yield strength σ0, and nonlinear kinematic hardening parameter C corresponding to the unique optimal solution are set as follows: k and γ k The values ​​of the isotropic hardening parameters Q and b are determined as the material parameters of the nonlinear hybrid hardening constitutive model obtained by the final inverse identification.

[0075] Example 1

[0076] 2024 aluminum alloy, with its low density and good strength, plasticity, toughness, and fatigue resistance, has wide applications in many industrial fields, especially in the aerospace industry, where it is an important high-strength aluminum alloy for aviation. This invention employs the method disclosed herein to inversely identify the material parameters of a nonlinear hybrid hardening constitutive model of 2024 aluminum alloy. This involves conducting four sets of tension-compression symmetric cyclic plastic deformation tests (ε) at different strain levels. max_1=0.005, ε max_2 =0.0075, ε max_3 =0.01, ε max_4 =0.0125,), finite element simulation was performed using ANSYS software, the multi-objective optimization framework was implemented using Matlab, the multi-objective optimization algorithm was the NSGA II algorithm, and the multi-criteria decision-making technique was the Linear-MCDM technique. The identified parameters were substituted into the finite element model for simulation. The comparison between the simulated cyclic stress-strain curves and the corresponding experimentally measured cyclic stress-strain curves was shown in the figure. Figure 2 As shown in the figure. The comparison of results shows that the material parameters of the nonlinear hybrid hardening constitutive model of 2024 aluminum alloy identified by the inverse identification method based on multi-objective optimization disclosed in this invention can simultaneously and accurately describe the cyclic plastic deformation behavior of 2024 aluminum alloy under a series of different strain levels, thus proving that the method is reasonable, effective, and accurate, and that the entire multi-objective optimization process is correct.

[0077] Example 2

[0078] TiB2 / 2024 aluminum-based composites possess low density and excellent strength, plasticity, toughness, and fatigue resistance, making them widely used in numerous industrial fields, especially in aerospace, where they are crucial high-strength aluminum alloys. This invention employs the method disclosed herein to inversely identify the material parameters of a nonlinear hybrid hardening constitutive model of 2024 aluminum alloy. This involves conducting four sets of tension-compression symmetric cyclic plastic deformation tests (ε) at different strain levels. max_1 =0.005, ε max_2 =0.0075, ε max_3 =0.01, ε max_4 =0.0125,), finite element simulation was performed using ABAQUS software, the multi-objective optimization framework used was modeFRONTIIER, the multi-objective optimization algorithm used was the MOGA II algorithm, and the multi-criteria decision-making technique used was GA-MCDM technology. The identified parameters were substituted into the finite element model for simulation. The comparison between the simulated cyclic stress-strain curves and the corresponding experimentally measured cyclic stress-strain curves was shown in the figure. Figure 3 As shown in the figure. The comparison of results shows that the material parameters of the nonlinear hybrid hardening constitutive model of 2024 aluminum alloy identified by the inverse identification method based on multi-objective optimization disclosed in this invention can simultaneously and accurately describe the cyclic plastic deformation behavior of 2024 aluminum alloy under a series of different strain levels, thus proving that the method is reasonable, effective, and accurate, and that the entire multi-objective optimization process is correct.

[0079] In summary, this invention discloses a method for inverse identification of material parameters based on multi-objective optimization. It is simple, low-cost, accurate, and reliable, and can be widely applied to the inverse identification of material parameters in nonlinear hybrid hardening constitutive models of various metallic materials. It has high practical value and important practical significance in computational mechanics, experimental mechanics, and engineering applications.

[0080] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A method for material parameter inverse identification based on multi-objective optimization, characterized in that, The method comprises the following steps: A series of tension-compression symmetric cyclic plastic deformation tests of different strain levels at room temperature are carried out on the metal material to obtain the cyclic plastic deformation data of the metal material at room temperature; An engineering cyclic stress-strain curve in the maximum strain level tension-compression symmetric cyclic plastic deformation data is extracted, and the engineering cyclic stress-strain curve is converted into a true cyclic stress-strain curve; extracting the true stress-strain curve of all tensile half cycles, determining the Young's modulus and initial yield strength of each tensile half cycle, calculating the average Young's modulus and initial yield strength of all cycles, specifically including: extracting the true stress-strain curve of all tensile half cycles, respectively linear fitting the elastic segment of each tensile half cycle, thereby determining the Young's modulus E of each tensile half cycle i , and the true stress value corresponding to the true strain of the minimum point of the second derivative is determined as the initial yield strength σ 0i of each tensile half cycle, calculating the average Young's modulus E and initial yield strength σ0 of all cycles; The nonlinear follow-up hardening parameters are determined from the true stress-strain curve of the first stretching half cycle, specifically including: converting the true stress-strain curve of the first stretching half cycle into a true stress-plastic strain curve, wherein ε, ε e and ε p are total strain, elastic strain and plastic strain respectively, taking N back stress components, dividing the true stress-plastic strain curve of the first stretching half cycle into M segments, M=N+1, and calculating the nonlinear follow-up hardening parameters C k and γ k on each segment, and the calculation formula is: Wherein, k ∈ [1, N]; An isotropic stress-accumulative plastic strain curve is calculated from the true stress-strain curves of all cycles, and isotropic hardening parameters are determined by fitting; A finite element model is established, and a multi-objective optimization algorithm is used to reversely identify the material parameters of the nonlinear mixed hardening constitutive model.

2. The material parameter inverse identification method based on multi-objective optimization according to claim 1, characterized in that, The formula for extracting the engineering cyclic stress-strain curve in the maximum strain level tension-compression symmetric cyclic plastic deformation data and converting it into a true cyclic stress-strain curve is: wherein P is the load, A0 is the original cross-sectional area of the metal test piece, L0 is the original length of the gauge section of the metal test piece, ΔL is the change in length of the gauge section of the metal test piece, σ T and ε T are the true stress and true strain, respectively, and σ E and ε E are the engineering stress and engineering strain, respectively.

3. The material parameter inverse identification method based on multi-objective optimization according to claim 1, characterized in that, The formula for calculating the average Young's modulus E and the initial yield strength σ0 of all cycles is: Wherein, i ∈ [1, n], n is the total number of cycles.

4. The material parameter inverse identification method based on multi-objective optimization according to claim 1, characterized in that, The step of calculating the isotropic stress-accumulated plastic strain curve from all the real stress-strain curves of the cycles and fitting the isotropic hardening parameters specifically comprises: extracting the real stress-strain curves of all the tensile half cycles and compression half cycles, determining the maximum stress σ tmax_i and the yield strength σ c_i of each tensile half cycle and each compression half cycle, calculating the isotropic stress σ of each cycle according to the formula: 0 i calculating the accumulated plastic strain of each cycle according to the isotropic stress σ 0 i i and fitting the isotropic stress-accumulated plastic strain curve to obtain the values of the isotropic hardening parameters Q and b according to the formula: .​ 5. The material parameter inverse identification method based on multi-objective optimization according to claim 1, characterized in that, The step of establishing a finite element model, using a multi-objective optimization algorithm, and reversely identifying the material parameters of the nonlinear mixed hardening constitutive model comprises: establishing a finite element model, numerically simulating the room temperature tension-compression symmetric cyclic plastic deformation process of a series of different strain levels, and integrating the simulation process into a multi-objective optimization framework; a series of different simulation true stress-time curves obtained by numerical simulation and a series of different measured true stress-time curves obtained by a series of different strain levels of the metal material at room temperature are used as a series of different square error loss functions as optimization objectives, and the formula is: where j represents the jth target, j ∈ [1, S], S is the total number of targets, i.e., the total number of series of tension-compression symmetric cyclic plastic deformation tests and corresponding numerical simulations at different strain levels, m j represents the total number of data points on the stress-time curve of the jth strain level of tension-compression symmetric cyclic plastic deformation simulation and measurement, and respectively represent the simulation stress value and the measured stress value of the ith data point.

6. The material parameter inverse identification method based on multi-objective optimization according to claim 5, characterized in that, The step of establishing the finite element model and inversely identifying the material parameters of the nonlinear mixed hardening constitutive model by using a multi-objective optimization algorithm further comprises: setting the values of Young's modulus E, initial yield strength σ0, nonlinear follow-up hardening parameters C k and γ k , and isotropic hardening parameters Q and b as input variables, setting the results of the calibration of Young's modulus E, initial yield strength σ0, nonlinear follow-up hardening parameters C k and γ k , and isotropic hardening parameters Q and b as initial values of the respective input variables, performing a multi-objective optimization operation for minimizing all optimization objectives by using a multi-objective optimization algorithm, selecting a unique optimal solution from the optimal solution set by using a multi-criteria decision-making technique, and determining the values of Young's modulus E, initial yield strength σ0, nonlinear follow-up hardening parameters C k and γ k , and isotropic hardening parameters Q and b corresponding to the unique optimal solution as the material parameters of the nonlinear mixed hardening constitutive model inversely identified.

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