A method for determining parameters of a tire rubber constitutive model
By using the DMA dynamic testing method, the stress-strain relationship of tire rubber materials at different frequencies and temperatures was obtained, and the parameters of the Neo-Hookean constitutive model were determined. This solved the accuracy problem of finite element analysis of rubber materials in the prior art and achieved more accurate finite element simulation.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SHANDONG JINYU TYRE CO LTD
- Filing Date
- 2026-04-13
- Publication Date
- 2026-06-12
AI Technical Summary
The reliability of finite element analysis of rubber materials in existing technologies relies on quasi-static testing methods, which cannot accurately capture the frequency and temperature correlation of tire rubber, resulting in inaccurate finite element simulation results.
The DMA dynamic testing method was used to obtain the stress-strain relationship of rubber materials at different frequencies and temperatures through simple shear mode, and the relationship between the Neo-Hookean constitutive model parameter C10 and frequency and temperature was determined. The accurate constitutive model parameters were obtained by combining the data fitting.
It improves the accuracy of finite element simulation, avoids complicated experimental testing and data fitting processes, and obtains frequency- and temperature-related hyperelastic constitutive parameters that match the actual test conditions of tires.
Smart Images
Figure CN122192965A_ABST
Abstract
Description
Technical Field
[0001] This application relates to a method for determining the parameters of a tire rubber constitutive model, which belongs to the field of rubber. Background Technology
[0002] Rubber, including natural and synthetic rubber, is an amorphous polymer with added fillers. Its molecules are randomly coiled, and the fillers and long-chain rubber molecules form a network through physicochemical interactions, thus reinforcing the rubber. Rubber possesses many unique physical and chemical properties, such as elasticity, resilience, flexibility, shock absorption, damping, sealing, and insulation. These properties enable rubber to meet a wide range of requirements in industrial and engineering applications, such as tires, hoses, conveyor belts, seals, and damping components. With the rapid development of computing power and nonlinear numerical simulation technology, finite element analysis (FEA) has been successfully applied to structural optimization and reliability analysis of complex rubber components with large deformations. However, the reliability of numerical methods such as finite element analysis in predicting rubber structures and their damage and failure largely depends on the simulation and characterization of the mechanical behavior and heat generation mechanism of rubber materials. Accurately characterizing and simulating the unique mechanical behavior of rubber materials is crucial and a prerequisite for predicting and optimizing the performance of rubber components. Material constitutive relations are axiomatic expressions describing the macroscopic properties of materials, based on extensive experiments and experience, and representing the mechanical and related behaviors (temperature, frequency, time, etc.). Effective and reliable material constitutive equations are crucial for obtaining high-precision finite element simulation results. Therefore, this invention uses the Neo-Hookean model, commonly used in rubber, as an example to study a constitutive testing method for tire rubber and the determination of constitutive parameters.
[0003] To date, dozens of hyperelastic constitutive models for rubber-like materials have been developed. Research on the strain energy function (W) of the hyperelastic constitutive relations of rubber materials can be divided into two categories: one is phenomenological models, which construct suitable strain energy functions; the other is network models based on molecular chain statistics. According to current research, phenomenological theory can yield more accurate mathematical expressions for rubber constitutive properties. This method does not consider the interpretation of the rubber molecular structure or physical meaning, but rather derives mathematical inferences describing the properties of rubber through mathematical reasoning. Rivlin (1948) studied the general expression of the strain energy function from a purely mathematical perspective, assuming that rubber is incompressible. Rivlin believed that the strain energy function can only be an even power function of λi. The simplest even power function satisfying the condition is:
[0004] (1)
[0005] (2)
[0006] (3)
[0007] These three expressions are independent of the choice of coordinate axes and are called strain invariants. Here, λ1, λ2, and λ3 represent the principal tensile ratios in the three directions; I1, I2, and I3 represent the first, second, and third strain invariants, respectively. Based on the assumption that rubber is incompressible, we have I3 = 1.
[0008] For rubber-like materials, a reduced polynomial model is commonly used, with the following formula:
[0009] (4)
[0010] Where W is the strain energy density, C i0 These are material parameters.
[0011] Using the Neo-Hookean model with i=1, then (4) becomes:
[0012] (5)
[0013] Among them, C 10 These are material parameters.
[0014] In the nonlinear finite element analysis of rubber, the study of the Neo-Hookean strain energy function is of great significance and is the most widely used constitutive model.
[0015] Application content
[0016] Temperature field and durability tests on rubber tires are typically conducted on a drum testing machine. Parameters such as load, speed, and running time are then set to study the steady-state rolling of the tire under specified operating conditions. Rubber is a typical temperature- and rate-dependent material. Previous studies have generally employed quasi-static testing methods for hyperelastic constitutive modeling, such as uniaxial tension, planar tension, and isoaxial tension. These methods are quasi-static, yielding rate-independent hyperelastic constitutive parameters.
[0017] To address the problems existing in the prior art, this application proposes a method for determining the parameters of a tire rubber constitutive model. This method employs a testing approach that more closely resembles actual tire testing conditions and fully considers the influence of temperature on the hyperelastic constitutive model parameters during testing. It not only yields hyperelastic constitutive parameters with strain amplitude correlation similar to previous testing methods, but also obtains hyperelastic constitutive parameters with frequency and temperature correlation.
[0018] To solve the above-mentioned technical problems, the technical solution adopted in this application is a method for determining the parameters of a tire rubber constitutive model, comprising the following steps:
[0019] 1) Analyze the material mechanics expression based on the selected DMA dynamic characteristic test mode;
[0020] 2) By using DMA dynamic testing, the stress-strain relationship of the rubber material under different frequency test conditions is obtained, and the relationship between the constitutive model parameters of the rubber material and frequency is obtained;
[0021] 3) The mechanical properties of rubber materials are analyzed by dynamic characteristic testing (DMA) at different temperatures, and the relationship between the constitutive model parameters of rubber materials and frequency and temperature is obtained.
[0022] 4) Determine the parameters of the constitutive model of the rubber material based on the relationship between the parameters of the constitutive model of the rubber material and frequency and temperature.
[0023] In the optimized method for determining the parameters of the above-mentioned tire rubber constitutive model, step 1) selects the simple shear mode as the DMA dynamic characteristic mode.
[0024] The principal draw ratios in the three directions are expressed as
[0025] , , In this diagram, 1, 2, and 3 represent the three principal strain directions, i.e., the eigenvector directions of the strain tensor. Typically, 1 refers to the loading direction, and 2 and 3 are two transverse directions perpendicular to the loading direction. In general 3D deformation, 1, 2, and 3 are not necessarily fixed spatial coordinates (such as x, y, z), but rather local principal axes determined according to the deformation state. Based on the schematic diagram in the attached figure, 1 can be considered x, 2 z, and 3 y. Regarding the direction, it is generally accepted in the industry that 1, 2, and 3 form three mutually perpendicular directions.
[0026] Shear strain is ;
[0027] The strain energy W in the simple shear mode is ;
[0028] The relationship between shear stress and shear strain is as follows: ;
[0029] Where t is the shear stress, γ is the shear strain, W refers to the strain energy function, and C 10 Refers to constitutive parameters.
[0030] In the optimized method for determining the parameters of the above-mentioned tire rubber constitutive model, step 1) selects dynamic testing as the test form and applies dynamic load using simple harmonic waves.
[0031] Strain is expressed as The stress is expressed as Where ε0 is the dynamic strain amplitude, σ0 is the dynamic stress amplitude, δ is the phase angle, and ω is the angular frequency ω=2πf, where f is the frequency;
[0032] Modulus is expressed as , where G' is the energy storage modulus;
[0033] In the simple shear DMA dynamic test, the stress-strain curve is normalized. At this point, the curve passes through the center of the circle, and G is the equivalent modulus. The Neo-Hookean constitutive parameter C is obtained from this. 10 Relationship with energy storage modulus.
[0034] In the optimized method for determining the parameters of the above-mentioned tire rubber constitutive model, step 2) involves conducting DMA dynamic characteristic tests at different frequencies to obtain the stress-strain relationship of the rubber material at different frequencies.
[0035] Using rubber material from the same batch as the target rubber material, a circular sample with a diameter of 10 mm and a thickness of 2 mm was prepared.
[0036] Dynamic loading tests at different frequencies were conducted on the DMA testing equipment, and the stress values under different strains during the tests were recorded; the constitutive model parameters C were obtained through analysis. 10 The relationship between frequency and other frequencies;
[0037] By fitting the data, the relationship between the constitutive model parameter C10 of the rubber material and the frequency was obtained;
[0038] constitutive model parameter C of rubber material 10 The relationship with the change in frequency is expressed as .
[0039] The optimized method for determining the constitutive model parameters of tire rubber described above, in step 3), involves conducting DMA dynamic characteristic tests at different temperatures to obtain the stress-strain relationship of the rubber material at different temperatures; and analyzing the results to obtain the constitutive model parameters C. 10 The relationship between temperature and other factors;
[0040] Through data fitting, the constitutive model C of the rubber material is obtained. 10 Relationship with temperature changes;
[0041] Constitutive Model C of Rubber Materials 10 The relationship with temperature change is expressed as , where T is temperature; e is the base of the exponential function, and e is a natural constant approximately equal to 2.71828.
[0042] In the optimized method for determining the parameters of the above-mentioned tire rubber constitutive model, in step 3), a circular sample with a diameter of 10 mm and a thickness of 2 mm is prepared using rubber material from the same batch as the target rubber material; dynamic loading tests at different temperatures are conducted on a DMA testing device, and the stress values under different strains during the test are recorded.
[0043] The beneficial effects of this application are as follows:
[0044] For tire rubber, this application proposes a dynamic testing method based on DMA dynamic characteristics to obtain hyperelastic constitutive model parameters. The testing method of this application better matches tire drum testing. Previous hyperelastic constitutive parameters were obtained through quasi-static testing methods, which could not accurately capture the rate-dependent properties of rubber materials. This application not only adopts a testing method that is closer to the actual testing conditions of tires, but also fully considers the influence of temperature on the hyperelastic constitutive model parameters during the testing process. This application not only obtains hyperelastic constitutive parameters with the same strain amplitude correlation as previous testing methods, but also obtains hyperelastic constitutive parameters with frequency and temperature correlations.
[0045] The technical solution of this application focuses on the influence of frequency and temperature on the parameters of the rubber constitutive model during tire testing. The parameters obtained through this application can improve the accuracy of finite element simulation and avoid the complicated process of experimental testing and data fitting of hyperelastic constitutive parameters.
[0046] Based on experiments at different temperatures and frequencies, this application's technical solution summarizes the Neo-Hookean constitutive model parameter C of tire rubber under simple shear dynamic deformation. 10 The relationship between temperature and frequency is first determined by the fitting formula to obtain the C values of the rubber material at different frequencies. 10 The relationship between temperature and frequency is then used to determine the constitutive parameters C of the rubber at different temperatures and frequencies. 10 The relationship between frequency and temperature. By fitting data, the specific constitutive model parameters of rubber materials at a certain frequency and temperature can be determined, avoiding a large number of experimental tests and data fitting processes. Attached Figure Description
[0047] Figure 1 This is a schematic diagram of the shear mode deformation during the DMA dynamic characteristic test of this application;
[0048] Figure 2 This is a flowchart illustrating the implementation process of this application. Detailed Implementation
[0049] The technical features of this application are further illustrated below with reference to specific embodiments.
[0050] This application provides a method for determining the parameters of a tire rubber constitutive model, the process of which includes:
[0051] Step 1), based on the selected DMA dynamic characteristic test mode, analyze the material mechanics expression;
[0052] Step 2) Through DMA dynamic testing, the stress-strain relationship of the rubber material under different frequency test conditions is obtained, and the relationship between the constitutive model parameters of the rubber material and frequency is obtained.
[0053] Step 3) Analyze the mechanical properties of rubber materials through dynamic characteristic tests (DMA) at different temperatures, obtain the relationship between the constitutive model parameters of rubber materials and frequency and temperature, and determine the constitutive model parameters of rubber materials.
[0054] The objective of this application can be achieved through the following technical measures:
[0055] In step 1), the rubber used in this application is tire rubber, whose main deformation form is simple shear.
[0056] The DMA dynamic test has three main variations: stretching mode, compression mode, and simple shear mode. This application only illustrates the simple shear mode. The test methods for stretching and shear modes are similar: theoretical derivation is performed first, followed by tests at different temperatures and frequencies. The authors believe that the methods for obtaining the hyperelastic constitutive parameters for these two variations are still within the scope of protection of this application.
[0057] Previous methods for testing hyperelastic constitutive model parameters were quasi-static. This application utilizes a DMA dynamic testing device to test the constitutive hyperelastic constitutive parameters of rubber materials. The DMA dynamic characteristic mode selected in this application is the simple shear mode, such as... Figure 1 As shown. The sides of the cube become parallelograms due to shearing, and the amount of shear can be obtained from the tangent of the inclination angle φ of the perpendicular side. There is no strain in the plane perpendicular to the xoy plane (i.e., the shear plane). Therefore, the tensile ratio corresponding to this direction is 1. Since the volume remains constant, the principal tensile ratios in the three directions are...
[0058] (6)
[0059] (7)
[0060] (8)
[0061] The three directions are represented by 1, 2, and 3, which are the eigenvector directions of the strain tensor. Typically, 1 refers to the loading direction, and 2 and 3 are two transverse directions perpendicular to the loading direction. In general 3D deformation, 1, 2, and 3 are not necessarily fixed spatial coordinates (e.g., x, y, z), but rather local principal axes determined by the deformation state. According to the schematic diagram in the attached figure, 1 can be considered x, 2 z, and 3 y. Regarding the direction, it is generally accepted in the industry that 1, 2, and 3 form three mutually perpendicular directions.
[0062] Shear strain is
[0063] (9)
[0064] Shear strain refers to the strain generated under shear stress, as shown in the attached figure.
[0065] Therefore, the strain energy W in the simple shear mode is
[0066] (10)
[0067] The relationship between shear stress and shear strain is as follows:
[0068] (11)
[0069] Where t is the shear stress, γ is the shear strain, W refers to the strain energy function, and C 10 The constitutive parameter refers to the shear strain γ, as shown in the attached figure, which is the strain generated under shear stress.
[0070] In step 1), the test method selected in this application is dynamic test, which uses a simple harmonic wave to apply a dynamic load.
[0071] strain: (12)
[0072] stress: (13)
[0073] in, It is the dynamic strain amplitude. It is the dynamic stress amplitude, δ is the phase angle, ω is the angular frequency ω=2πf, f is the frequency.
[0074] Modulus: (14)
[0075] Where G' is the storage modulus. In the simple shear DMA dynamic test, the stress-strain curve is normalized. At this point, the curve passes through the center of the circle, and G is the equivalent modulus. Therefore, the Neo-Hookean constitutive parameter C is obtained. 10 Relationship with storage modulus. Here, strain, stress, and modulus refer to the strain, stress, and modulus generated when the deformation shown in the attached figure occurs, respectively. Stress: the force per unit area; Strain: the relative deformation of a material under shear stress; Modulus: the ratio of stress to strain, characterizing the material's ability to resist deformation.
[0076] In step 2), dynamic characteristic tests of the rubber material were conducted at different frequencies to obtain the stress-strain relationship of the rubber material at different frequencies. Circular specimens with a diameter of 10 mm and a thickness of 2 mm were prepared using rubber material from the same batch as the target rubber material. Dynamic loading tests at different frequencies were conducted on the DMA testing equipment, and the stress values under different strains were recorded during the tests. The constitutive model parameter C was then analyzed. 10 The relationship between frequency and data fitting yields the constitutive model C for the rubber material. 10 Formula for frequency variation:
[0077] (15)
[0078] In step 3), dynamic characteristic tests of the rubber material were conducted at different temperatures to obtain the stress-strain relationship of the rubber material at different temperatures. Circular specimens with a diameter of 10 mm and a thickness of 2 mm were prepared using rubber material from the same batch as the target rubber material. Dynamic loading tests at different temperatures were conducted on the DMA testing equipment, and the stress values under different strains were recorded during the tests. The constitutive model parameter C was then analyzed. 10 The relationship between temperature and the material's properties was used to obtain the constitutive model C for the rubber material through data fitting. 10 Formula for temperature change:
[0079] (16)
[0080] Where T is temperature (°C).
[0081] The constitutive testing method and constitutive model parameter determination method for tire rubber in this application, based on experiments at different temperatures and frequencies, summarizes the Neo-Hookean constitutive model parameter C of tire rubber under simple shear dynamic deformation through research and analysis. 10 The relationship between temperature and frequency is first determined by the fitting formula to obtain the C values of the rubber material at different frequencies. 10 The relationship between temperature and frequency is then used to determine the constitutive parameters C of the rubber at different temperatures and frequencies. 10 The relationship between frequency and temperature. By fitting data, the specific constitutive model parameters of rubber materials at a certain frequency and temperature can be determined, avoiding a large number of experimental tests and data fitting processes.
[0082] Of course, the above description is not intended to limit this application, nor is this application limited to the examples given above. Any changes, modifications, additions, or substitutions made by those skilled in the art within the scope of this application should fall within the protection scope of this application.
Claims
1. A method for determining the parameters of a tire rubber constitutive model, characterized in that: Includes the following steps: 1) Analyze the material mechanics expression based on the selected DMA dynamic characteristic test mode; 2) By using DMA dynamic testing, the stress-strain relationship of the rubber material under different frequency test conditions is obtained, and the relationship between the constitutive model parameters of the rubber material and frequency is obtained; 3) The mechanical properties of rubber materials are analyzed by dynamic characteristic testing (DMA) at different temperatures, and the relationship between the constitutive model parameters of rubber materials and frequency and temperature is obtained. 4) Determine the parameters of the constitutive model of the rubber material based on the relationship between the parameters of the constitutive model of the rubber material and frequency and temperature.
2. The method for determining the parameters of the tire rubber constitutive model according to claim 1, characterized in that: In step 1), the selected DMA dynamic characteristic mode is simple shear mode; The principal stretch ratios in the three directions are expressed as follows: 、 、 ； Shear strain is expressed as ; The strain energy W in the simple shear mode is ; The relationship between shear stress and shear strain is as follows: ; Where t is the shear stress, γ is the shear strain, W refers to the strain energy function, and C 10 Refers to constitutive parameters.
3. The method for determining the parameters of the tire rubber constitutive model according to claim 1, characterized in that: In step 1), the selected test method is dynamic test, which uses a simple harmonic wave to apply a dynamic load; Strain is expressed as The stress is expressed as ,in, It is the dynamic strain amplitude. It is the dynamic stress amplitude, δ is the phase angle, ω is the angular frequency ω=2πf, f is the frequency; Modulus is expressed as , where G' is the energy storage modulus; In the simple shear DMA dynamic test, the stress-strain curve is normalized. At this point, the curve passes through the center of the circle, and G is the equivalent modulus. The Neo-Hookean constitutive parameter C is obtained from this. 10 Relationship with energy storage modulus.
4. The method for determining the parameters of the tire rubber constitutive model according to claim 1, characterized in that: In step 2), DMA dynamic characteristic tests are performed at different frequencies to obtain the stress-strain relationship of the rubber material at different frequencies; Using rubber material from the same batch as the target rubber material, a circular sample with a diameter of 10 mm and a thickness of 2 mm was prepared. Dynamic loading tests at different frequencies were conducted on the DMA testing equipment, and the stress values under different strains during the tests were recorded; the constitutive model parameters C were obtained through analysis. 10 The relationship between frequency and other frequencies; The constitutive model parameters C of the rubber material were obtained through data fitting. 10 Relationship with frequency variation; constitutive model parameter C of rubber material 10 The relationship with the change in frequency is expressed as .
5. The method for determining the parameters of the tire rubber constitutive model according to claim 1, characterized in that: In step 3), DMA dynamic characteristic tests are conducted at different temperatures to obtain the stress-strain relationship of the rubber material at different temperatures; the constitutive model parameter C is then analyzed. 10 The relationship between temperature and other factors; Through data fitting, the constitutive model C of the rubber material is obtained. 10 Relationship with temperature changes; Constitutive Model C of Rubber Materials 10 The relationship with temperature change is expressed as , where T is temperature.
6. The method for determining the parameters of the tire rubber constitutive model according to claim 5, characterized in that: In step 3), a circular sample with a diameter of 10 mm and a thickness of 2 mm is prepared using rubber material from the same batch as the target rubber material; dynamic loading tests at different temperatures are conducted on the DMA testing equipment, and the stress values under different strains are recorded during the test.