JP6755782B2 - Method for identifying material parameters of rubber-like materials - Google Patents

Description

The present invention relates to a method for identifying material parameters of a rubbery material.

As a simulation method for analyzing rubber products such as tires, for example, there is a numerical analysis method using the finite element method (FEM). In order to simulate such a rubber product, it is necessary to derive, that is, identify the material parameter of the rubber-like material, that is, the material parameter (coefficient) of the superelastic model expressing the rubber elastic property.

Conventionally, in order to identify the material parameters of a superelastic model, the relationship between stress and strain was measured and obtained under three conditions of uniaxial elongation condition, uniaxial restraint uniaxial elongation condition, and biaxial uniform elongation condition, and these three conditions were obtained. The material parameters of the superelastic model have been identified by performing fitting using a superelastic model such as the Ogden model based on the measured values in.

In the identification of the material parameters of the superelastic model using the measured values under these three conditions, the accuracy of the material parameters is not always high, and the distribution of the strain energy density becomes non-uniform without rising. There is a risk.

On the other hand, in Non-Patent Document 1, the ratio of stress to strain (stress ratio) is approximately 1: 1.1: 1 between the uniaxial extension condition, the uniaxial restraint uniaxial extension condition (pure shear condition), and the biaxial uniform extension condition. Focusing on the fact that it is about 4, the stress values of the uniaxially constrained uniaxial extension condition and the biaxial uniform extension condition are calculated from the measured values under the uniaxial extension condition, and by fitting using these calculated values, superelasticity is achieved. It is disclosed to identify the material parameters of the elastic model.

Further, in Patent Document 1, the material parameters of the first elastic model are identified based on the measured values under the uniaxial elongation condition, and the identified material parameters are used to uniaxial elongation condition, pure shear condition and biaxial equality. A method for identifying material parameters of a second elastic model (for example, Ogden model) is disclosed by calculating the relationship between stress and strain under conditions and calculating the ratio of these stresses.

The methods disclosed in Non-Patent Document 1 and Patent Document 1 are based on measured values under uniaxial extension conditions in view of the fact that it is difficult to carry out tests under biaxial uniform extension conditions due to problems such as the spread of testing machines. Uniaxial restraint The relationship between stress and strain under uniaxial extension conditions and biaxial uniform extension conditions is estimated to identify the material parameters of the superelastic model, and the accuracy of the material parameters is not necessarily high.

Japanese Unexamined Patent Publication No. 2012-185042

Takahiro Nagata et al., "Estimation of Biaxial Stretched Rubber Material Model by Uniaxial Test", 14th Computational Mechanics Lecture, 128, 2001, pp. 72-73

The present invention has been made in view of the above points, and an object of the present invention is to provide a method for identifying material parameters of a rubber-like material that can improve the accuracy of material parameters.

The method for identifying the material parameters of the rubber-like material according to the present embodiment is to measure the relationship between stress and strain under uniaxial elongation conditions and the actual measurement of stress and strain under uniaxial restraint uniaxial elongation conditions for the rubber-like material. Obtaining the measured value of the relationship between stress and strain under biaxial uniform elongation conditions, the measured value of the stress and strain relationship under biaxial non-uniform elongation conditions, and the measured value of each obtained condition. Includes calculating the material parameters of the superelastic model by fitting with the superelastic model.

According to this embodiment, the material parameters of the superelastic model are obtained by adding the measured values under the uniaxial extension condition, the uniaxial restraint uniaxial extension condition, and the biaxial uniform elongation condition to the measured values under the biaxial non-uniform elongation condition. By calculating, the accuracy of material parameters can be improved.

The figure which shows the test method of the uniaxial extension condition The figure which shows the test method of the uniaxial restraint uniaxial extension condition The figure which shows the test method of the biaxial uniform extension condition The figure which shows the test method of the biaxial uneven extension condition Graph showing the relationship between stress and strain according to the embodiment Graph showing the relationship between stress and strain according to the comparative example Graph showing strain energy density distribution according to an example Graph showing strain energy density distribution according to a comparative example

Hereinafter, embodiments of the present invention will be described with reference to the drawings.

In the method for identifying the material parameters of the rubber-like material according to one embodiment, first, for the target rubber-like material, uniaxial extension conditions, uniaxial restraint uniaxial extension conditions, biaxial uniform elongation conditions, and biaxial non-uniform elongation conditions are satisfied. Obtain the measured value of the relationship between stress and strain under each condition.

A known method can be used as a method for obtaining the measured value of the relationship between stress and strain under the uniaxial extension condition, the uniaxial constraint uniaxial extension condition, and the biaxial uniform extension condition. The present embodiment is characterized in that, in addition to the measured values under these three conditions, the measured values of the relationship between stress and strain under biaxial non-uniform elongation conditions are obtained. Here, the strain is the amount of deformation per unit dimension that occurs when an object is stressed, and the rubber-like shape with respect to the tensile length in a state where no stress is applied to the target rubber-like material. The ratio of the amount of displacement in the tensile direction generated by applying stress to the material is shown.

The measured value under the uniaxial extension condition can be obtained by performing a uniaxial tensile test. The uniaxial tensile test is a test in which a rubber-like material is pulled in the uniaxial direction without being restrained in two axial directions out of three axial directions orthogonal to each other, and a relationship between stress and strain can be obtained. For example, as shown in FIG. 1, it is a test in which a square sheet-shaped rubber-like material 10 is used and pulled in the left-right direction without restraining the vertical direction. In the figure, one end side in the uniaxial direction is gripped by the tool 12. It is shown that the other end side is pulled in the uniaxial direction by fixing with.

The measured value under the uniaxial restraint uniaxial extension condition can be obtained by performing a uniaxial restraint uniaxial tensile test (pure shear test). The uniaxially restrained uniaxial tensile test is a test in which a rubber-like material is pulled in the other uniaxial direction while being restrained in one axial direction, and a relationship between stress and strain can be obtained. For example, as shown in FIG. 2, in a test in which a square sheet-shaped rubber-like material 10 is used and pulled in the left-right direction while being restrained by grippers 14 and 14 so that the strain is 0% in the vertical direction. is there.

The measured value under the biaxial uniform extension condition can be obtained by performing the biaxial uniform tensile test. The biaxial uniform tensile test is a test in which a rubber-like material is pulled in two axial directions at the same speed, and a relationship between stress and strain can be obtained. For example, as shown in FIG. 3, this is a test in which a square sheet-shaped rubber-like material 10 is used and pulled at the same speed in the left-right direction and the up-down direction.

The measured values under the biaxial non-uniform elongation condition can be obtained by performing a biaxial non-uniform tensile test. The biaxial non-uniform tensile test is a test in which a rubber-like material is pulled in two axial directions at different speeds, and a relationship between stress and strain can be obtained. For example, as shown in FIG. 4, this is a test in which a square sheet-shaped rubber-like material 10 is used and pulled at different speeds in the horizontal direction and the vertical direction. The velocity ratio k: 1 in the two axial directions is not particularly limited, and k may be, for example, 1.5 to 4 or 2 to 3 as long as k is larger than 1. Further, as the measured value under the biaxial uneven extension condition, only the measured value at one speed ratio may be obtained, or the measured value at two or more speed ratios may be obtained.

The measured values obtained in this way are as shown in FIGS. 5 and 6 as an example. In FIG. 6, the measured values under three conditions of the uniaxial extension condition, the uniaxial restraint uniaxial extension condition, and the biaxial uniform extension condition are shown. In FIG. 5, the measured values of four conditions are shown by further adding the biaxial uneven extension condition. All measured values are shown by solid lines. The graphs of FIGS. 5 and 6 show the relationship between the stress and strain of the rubber-like material. Here, the vertical axis is the nominal stress (MPa) and the horizontal axis is the elongation ratio λ. ing. The elongation ratio is the ratio of the length after deformation to the length of the initial shape. Therefore, the elongation ratio is 1 when the strain is 0%, and the elongation ratio is 2 when the strain is 100%.

Here, FIGS. 5 and 6 use a vulcanized rubber sample having a length of 70 mm, a width of 70 mm, and a thickness of 1.0 mm as a rubber-like material (the sample shape excluding the gripping allowance is 50 mm square), and the strain rate is 100% /. This is the result of measuring the strain from 0% to 100% in 30 seconds. The velocity ratio (λ ₁ : λ ₂ ) in the two axial directions under the biaxial non-uniform extension condition was 2: 1.

As shown in FIG. 6, the solid line A1 showing the measured value of the uniaxial extension condition is on the lowest stress side, and the solid line B1 showing the measured value of the uniaxial restraint uniaxial extension condition is on the high stress side, and is on the highest stress side. There is a solid line C1 showing the measured value of the biaxial uniform extension condition. On the other hand, in one embodiment shown in FIG. 5, a biaxial uneven extension condition is provided between the solid line B1 indicating the measured value of the uniaxially constrained uniaxial extension condition and the solid line C1 indicating the actual measurement value of the biaxial uniform extension condition. There is a solid line D1 showing the measured value.

In the present embodiment, the material parameters of the superelastic model are then calculated by fitting using the superelastic model to the actually measured values of the above-mentioned conditions. That is, the material parameters of the superelastic model are identified based on the measured values. Identification of material parameters can be done using a computer. As one embodiment, the material parameter identification device includes an actual measurement value acquisition unit that acquires the actual measurement values of the above four conditions, and a material parameter identification unit that identifies the material parameters of the superelastic model based on the actual measurement values. However, these can be realized by having a processor installed in the computer execute the program.

As the superelastic model, various known superelastic models can be used, and examples thereof include an Ogden model, a Mooney model, a Mooney-Rivlin model, and a Neo-Hookean model. The case where the Ogden model is used will be described below.

The Ogden model (3rd order) is a strain energy density function represented by the following equation (1).

In the formula, W is the amount of strain energy per unit volume. λ ₁ , λ ₂ , and λ ₃ are elongation ratios in each axial direction, and λ ₁ × λ ₂ × λ ₃ = 1. α _n and μ _n are constants and are material parameters of the Odgen model.

The fitting method is not particularly limited, and a known method, for example, the least squares method can be used. That is, it can be calculated by approximating the material parameters α _n and μ _n of the Ogden model to the above measured values by the method of least squares.

Specifically, in the Ogden model, the relationship between the nominal stress S and the elongation ratio λ under the above four conditions is expressed by the following equations (2) to (5).

Therefore, if the measured values shown in FIG. 5 and the values of α _n and μ _n such that the values calculated by these equations (2) to (5) are equal to each other can be obtained by fitting calculation using the least squares method. Good.

FIG. 5 shows the identification result (fitting result) according to the embodiment identified based on the measured values of the four conditions. Further, FIG. 6 shows the identification results of the comparative example identified based on the measured values under the three conditions. The fitting results are all shown by dotted lines. The fitting result under the uniaxial extension condition is indicated by the dotted line A2, the fitting result under the uniaxial constraint uniaxial extension condition is indicated by the dotted line B2, and the fitting result under the biaxial uniform extension condition is indicated by the dotted line C2. The fitting results under the uneven stretching condition are shown by the dotted lines D2.

As a result of fitting, there is a particularly large difference in the fitting accuracy under the biaxial uniform extension condition, and in the example identified based on the measured values of the four conditions shown in FIG. 5, the identification was performed based on the measured values of the three conditions shown in FIG. The fitting accuracy was improved by 6.8% in the sum of the squares of the differences as compared with the comparative example. As shown in FIG. 6, there is a relatively large gap between the solid line B1 showing the measured value of the uniaxially constrained uniaxial extension condition and the solid line C1 showing the actually measured value of the biaxial uniform extension condition. It is considered that this affects the fitting accuracy under the biaxial uniform extension condition. That is, by adding the actual measurement value of the biaxial uneven extension condition, as shown in FIG. 5, the actual measurement value of the biaxial uneven extension condition can be obtained between the two solid lines B1 and C1 having a large gap. Since the solid line D1 shown can be added and the gap between the data to be fitted can be reduced, it is considered that the fitting accuracy is improved.

FIG. 7 is a graph showing the strain energy density distribution according to the embodiment, and is an Ogden model represented by the above equation (1) using the material parameters α _n and μ _n obtained from the fitting results shown in FIG. The result of calculating the distribution of strain energy density from the function of. Since the product of the elongation ratios λ ₁ , λ ₂ and λ ₃ in each axial direction is an invariant of λ ₁ × λ ₂ × λ ₃ = 1, the strain energy density W is calculated from the equation (1) based on this. .. Here, λ ₁ is the left-right direction in FIGS. 1 to 4 (that is, the tensile direction under the uniaxial extension condition and the uniaxial restraint uniaxial extension condition, and one axial direction under the biaxial uniform condition and the biaxial non-uniform extension condition. Λ ₂ is the elongation ratio in the vertical direction in FIGS. 1 to 4 (that is, in the constraint direction under the uniaxial constraint uniaxial extension condition, and in the biaxial uniform extension condition and the biaxial non-uniform extension condition). It is the elongation ratio in the tensile direction in the other axial direction), and λ ₃ is the elongation ratio in the thickness direction.

FIG. 8 is a graph showing the strain energy density distribution according to the comparative example, and is calculated by using the material parameters α _n and μ _n obtained from the fitting results shown in FIG.

As is clear from the comparison with FIG. 8, in the embodiment shown in FIG. 7, the biaxial axis is between the fitting result under the biaxial uniform extension condition and the fitting result under the uniaxial extension condition and the uniaxial constraint uniaxial extension condition. There were fitting results under uneven stretch conditions, and as a result, the strain energy density distribution was more uniform. In addition, in the identification of the material parameters of the superelastic model using the measured values under the conventional three conditions, the strain energy density distribution sometimes became non-uniform without rising, but the biaxial non-uniform extension condition By adding and fitting the measured values in, it became possible to improve the accuracy of the strain energy density distribution.

The material parameters of the rubber-like material obtained as described above can be used in a simulation for analyzing a rubber product containing the rubber-like material. That is, the material parameters are used to simulate a rubber product modeled with numerically analyzable elements. According to the present embodiment, since the accuracy of the material parameters is improved as described above, the analysis accuracy can also be improved in the simulation of the rubber component performed using the material parameters.

The simulation can be performed by using a computer as basic hardware, and is not particularly limited. As an embodiment, it can be used for tire simulation. That is, using the above material parameters, a tire model created by dividing a tire into a finite number of elements is simulated. The tire analysis method by simulation is not particularly limited, and can be used for various analyzes such as ground contact analysis, rigidity analysis, and rolling analysis.

Although one embodiment of the present invention has been described above, this embodiment is presented as an example and is not intended to limit the scope of the invention. These novel embodiments can be implemented in various other embodiments, and various omissions, replacements, and changes can be made without departing from the gist of the invention. These embodiments and modifications thereof are included in the scope and gist of the invention, and are also included in the scope of the invention described in the claims and the equivalent scope thereof.

10 ... Rubber-like material, 12, 14 ... Grab

Claims (2)

For rubber-like materials, the measured value of the relationship between stress and strain under uniaxial elongation conditions, the measured value of the relationship between stress and strain under uniaxial restraint uniaxial elongation conditions, and the relationship between stress and strain under biaxial uniform elongation conditions. Obtaining the measured value and the measured value of the relationship between stress and strain under biaxial non-uniform elongation conditions, and
To calculate the material parameters of the superelastic model by fitting using the superelastic model to the measured values of the obtained conditions.
Methods for identifying material parameters of rubbery materials, including. The method for identifying material parameters according to claim 1, wherein the superelastic model is an Ogden model.