JP5668532B2 - Simulation method and material parameter identification method - Google Patents

Description

  The present invention relates to a simulation method and a material parameter identification method used for analysis of an elastic body.

  Conventionally, various methods have been proposed as simulation methods for analyzing tires. For example, there is a method of analyzing a tire using a finite element method. In order to simulate a tire, it is necessary to identify material parameters (coefficients) of a superelastic model (constitutive equation) that expresses rubber elastic characteristics. There are various methods for identifying material parameters. Proposed.

  For example, in Patent Document 1, a coefficient value included in a Mooney-Rivlin model, which is one of strain energy density functions used in a constitutive equation of a superelastic material, is obtained from a measured value of a material test and a coefficient calculation equation. A technique for calculating a coefficient that satisfies a stability condition necessary for performing a stress simulation more stably in an estimated coefficient calculation apparatus is disclosed.

  Specifically, Patent Document 1 includes a storage device that stores measurement values and calculation process data, and a calculation device connected to the storage device. An approximate solution of a coefficient is obtained by a nonlinear least square method with constraints. A coefficient calculating means for calculating, an error evaluation / solution updating means for evaluating an error of the approximate solution and updating an optimal solution candidate, and a convergence evaluating means for evaluating a convergence state of the approximate solution in the repeated calculation of the approximate solution; When convergence is difficult based on the evaluation of the convergence evaluation means, the initial change means is configured to initialize the candidate of the optimal solution as the initial value of the coefficient, initialize the parameters of the constrained nonlinear least squares method, and redo the constrained nonlinear least squares method. Are disclosed.

JP 2009-211681 A

  However, when the technique disclosed in Patent Document 1 is used, calculation time and processing load may increase because calculation processing by coefficient calculation means or the like is repeatedly executed until the convergence condition is satisfied. .

  The present invention has been made in view of the above, and an object of the present invention is to provide a simulation method and a material parameter identification method capable of identifying a highly accurate material parameter with a small amount of calculation.

  In order to solve the above-described problems and achieve the object, the present invention is a material parameter identification method for identifying a material parameter using a computer, which is based on an actual measurement value of a relationship between stress and strain under a uniaxial stretching condition. And identifying the material parameters of the first elastic model, using the identified material parameters of the first elastic model, the relationship between the stress and strain under the uniaxial extension condition, the stress under the pure shear condition, and Uniaxial stretching based on the relationship of strain and the relationship between stress and strain under uniform biaxial conditions, the relationship between stress and strain under uniaxial stretching conditions and the relationship between stress and strain under pure shear conditions Based on the step of calculating the ratio of the stress between the condition and the pure shear condition, the relationship between the stress and strain under the uniaxial stretching condition, and the relationship between the stress and strain under the uniform biaxial condition Based on the step of calculating the ratio of the stress of the axial extension condition and the uniform biaxial condition, the ratio of the stress of the uniaxial extension condition and the pure shear condition, and the measured value of the stress and strain in the uniaxial extension condition, Based on the step of calculating an estimated value of the relationship between stress and strain under pure shear conditions, the ratio of the stress under the uniaxial stretching condition and the uniform biaxial condition, and the measured value of the relationship between stress and strain under the uniaxial stretching condition Calculating an estimated value of the relationship between stress and strain under a uniform biaxial condition, an actual value of the relationship between stress and strain under the uniaxial stretching condition, and an estimation of the relationship between stress and strain under the pure shear condition Identifying a material parameter of the second elastic model based on the value and an estimated value of the relationship between stress and strain under the uniform biaxial condition.

  Here, it is preferable that the first elastic model is composed of a function including only a first-order invariant of strain. The first elastic model may be, for example, an Arruda-Boyce model or a Yeoh model.

  Further, it is preferable that the second elastic model is constituted by a function including at least a second-order invariant amount of strain or an extension ratio. The second elastic model may be, for example, an Ogden model or a Polynomial model.

  Further, in the step of identifying the material parameter of the second elastic model, a minimum weighted to an error of a strain region frequently used in a simulation performed using the material parameter of the second elastic model to be identified It is preferable to identify the material parameters by the square method.

  Further, in the step of identifying the material parameter of the first elastic model, a minimum weighted to an error of a strain region frequently used in a simulation performed using the material parameter of the second elastic model to be identified It is preferable to identify the material parameters by the square method.

  Further, in order to solve the above-described problems and achieve the object, the present invention is a simulation method for analyzing an elastic body using a computer, in which stress and strain under the uniaxial extension condition of the elastic body are analyzed. The step of identifying the material parameter of the first elastic model based on the actually measured value of the relationship, and using the identified material parameter of the first elastic model, the relationship between stress and strain under the uniaxial stretching condition, Calculating the relationship between stress and strain under shear conditions and the relationship between stress and strain under uniform biaxial conditions, the relationship between stress and strain under uniaxial stretching conditions, and the relationship between stress and strain under pure shear conditions And calculating the ratio of the stress under the uniaxial stretching condition and the pure shear condition based on the relationship between the stress and strain under the uniaxial stretching condition and the relationship between the stress and strain under the uniform biaxial condition. Calculating the ratio of the stress in the uniaxial stretching condition and the uniform biaxial condition based on the following: the ratio of the stress in the uniaxial stretching condition and the pure shear condition, and the actual value of the relationship between the stress and the strain in the uniaxial stretching condition; A step of calculating an estimated value of the relationship between stress and strain under a pure shear condition, the ratio of the stress under the uniaxial stretching condition and the uniform biaxial condition, and the measurement of the relationship between the stress and strain under the uniaxial stretching condition. A step of calculating an estimated value of the relationship between stress and strain under a uniform biaxial condition, a measured value of the relationship between stress and strain under the uniaxial extension condition, and a stress and strain under the pure shear condition Identifying the material parameter of the second elastic model based on the estimated value of the relationship and the estimated value of the stress-strain relationship under the uniform biaxial condition, and the material parameter of the second elastic model Before using And performing a simulation of the elastic body.

  The simulation method and the material parameter identification method according to the present invention have an effect that a highly accurate material parameter can be identified with a small amount of calculation.

FIG. 1 is a configuration diagram of a computer that executes a tire simulation method according to the present embodiment. FIG. 2 is a diagram illustrating an example of a result of identifying material parameters of the Ogden model based on three types of test results. FIG. 3 is a diagram illustrating an example of a result of identifying material parameters of the Ogden model based on one type of test result. FIG. 4 is a diagram illustrating an example of a result of identifying material parameters of the Arruda-Boyce model based on one type of test result. FIG. 5 is a flowchart illustrating the execution procedure of the simulation method according to the present embodiment. FIG. 6 is a diagram illustrating an example of the result of identifying the material parameters of the first elastic model. FIG. 7 is a diagram illustrating an example of an estimated value calculated based on the stress ratio. FIG. 8 is a diagram illustrating an example of the result of identifying the material parameters of the second elastic model. FIG. 9 is a diagram illustrating an example of the results of identifying the material parameters of the Ogden model based on the three types of test results. FIG. 10 is a diagram illustrating an example of a result of identification of material parameters of the Arruda-Boyce model based on one type of test result. FIG. 11 is a diagram illustrating an example of a result of identifying material parameters of the Ogden model based on one type of test result and a known stress ratio. FIG. 12 is a diagram illustrating an example of a result of identifying material parameters using the material parameter identification method according to the present embodiment. FIG. 13 is a diagram for explaining the uniaxial extension test. FIG. 14 is a diagram for explaining the pure shear test. FIG. 15 is a diagram for explaining the equivalent biaxial test.

  Hereinafter, a simulation method and a material parameter identification method according to the present invention will be described with reference to the accompanying drawings. Moreover, although the following description demonstrates the example which applies this invention to the simulation of a tire, this invention is applicable to the simulation of various articles | goods containing an elastic body. The present invention is not limited to the following examples. In addition, constituent elements in the following embodiments include those that can be easily replaced by those skilled in the art or those that are substantially the same.

  The tire simulation method according to the present embodiment is a simulation method using a so-called finite element method (FEM). The tire model divided into a finite number is analyzed using a computer, and the analysis information obtained after the analysis is used. This is a method of calculating an evaluation physical quantity for evaluating the performance of a tire by calculating using a predetermined calculation formula. In the tire simulation method according to this embodiment, material parameters are identified prior to the execution of the simulation.

  First, a computer 50 that executes a tire simulation method according to the present embodiment will be described with reference to FIG. FIG. 1 is a configuration diagram of a computer that executes a tire simulation method according to the present embodiment.

  The computer 50 is, for example, a so-called host computer and is connected to the terminal device 70. The computer 50 is connected to other computers 64a and 64b via the network 63, and can exchange data with each other.

  The computer 50 includes a storage unit 50m that stores various programs, a processing unit 50p that executes various programs, and an input / output unit (I / O) 59 that connects the processing unit 50p and the storage unit 50m.

  The storage unit 50m is configured by a volatile storage device such as a RAM serving as a work area for executing various programs or the like, a nonvolatile storage device such as a ROM or a hard disk drive, or a combination thereof. The storage unit 50m stores the simulation program 27.

  The processing unit 50p is configured by a CPU or the like that performs calculations for executing various programs, and includes an information acquisition unit 51, a material parameter identification unit 52, and an analysis unit 53. The information acquisition unit 51, the material parameter identification unit 52, and the analysis unit 53 function when the processing unit 50p executes the simulation program 27 stored in the storage unit 50m.

  The information acquisition unit 51 acquires various types of information necessary for executing the simulation. Based on the information acquired by the information acquisition unit 51, the material parameter identification unit 52 identifies the material parameter of the elastic model used for executing the simulation. The analysis unit 53 executes a simulation using the information acquired by the information acquisition unit 51 and the material parameter identified by the material parameter identification unit 52.

  The input / output unit 59 is connected to the terminal device 70, receives various data output from the terminal device 70, and appropriately inputs the received various data to the processing unit 50p or the storage unit 50m. The input / output unit 59 receives various data output from the processing unit 50p or the storage unit 50m, and appropriately inputs the received various data to the terminal device 70.

  Further, the input / output unit 59 is connected to other computers 64 a and 64 b via the network 63. Therefore, the input / output unit 59 can receive various data output from the processing unit 50p or the storage unit 50m, and can appropriately input the received various data to another computer 64a or 64b via the network 63. It has become. Further, the input / output unit 59 can receive various data output from the other computers 64a or 64b via the network 63, and appropriately input the received various data to the processing unit 50p or the storage unit 50m. ing.

  The terminal device 70 includes a terminal device main body 60, a display device 62 connected to the terminal device main body 60, and an input device 61 connected to the terminal device main body 60. When the terminal device main body 60 receives the data necessary for executing the tire analysis (for example, material characteristics of each member constituting the tire, boundary conditions in the ground contact analysis, etc.) by the input device 61, the input device 61 The data input in is output to the host computer 50. Further, when data is output from the computer 50 to the terminal device 70, the terminal device main body 60 receives the data output from the computer 50, and displays the received data on the display device 62 so as to be visible.

  When the tire simulation method according to the present embodiment is executed, in the computer 50, the processing unit 50p executes the simulation program 27 stored in the storage unit 50m. Subsequently, in the terminal device 70, a process is performed in which the terminal device body 60 inputs data input from the input device 61 to the input / output unit 59 of the host computer 50. Then, the processing unit 50p operates the material parameter identification unit 52 and the analysis unit 53 based on the data input from the terminal device 70 and the other data acquired by the information acquisition unit 51.

  Note that the apparatus configuration for executing the tire simulation method according to the present embodiment is not limited to the above configuration. For example, the other computers 64a and 64b connected via the network 63 are not necessarily required. The computer 50 may be a so-called personal computer or workstation.

  Next, a material parameter identification method included in the tire simulation method according to the present embodiment will be described with reference to FIGS. FIG. 2 is a diagram illustrating an example of a result of identifying material parameters of the Ogden model based on three types of test results. FIG. 3 is a diagram illustrating an example of a result of identifying material parameters of the Ogden model based on only a uniaxial extension test, that is, one type of test result. FIG. 4 is a diagram illustrating an example of the result of identifying the material parameters of the Arruda-Boyce model based on one type of test result as in FIG. 3.

  The graphs shown in FIG. 2, FIG. 3 and FIG. 4 show the relationship between the stress and strain of the elastic body (stress-strain relationship). Here, the vertical axis is the nominal stress (Stress) [MPa], Nominal strain (Strain) [%] is taken on the horizontal axis. Strain is the amount of deformation per unit dimension that occurs when an object is subjected to stress, and the stress is applied to the target elastic body against the length in the tensile direction when no stress is applied to the target elastic body. The ratio of the amount of displacement in the tensile direction generated by this is shown. The strain becomes 100% when the length of the target elastic body is double the original length.

  In the example shown in FIG. 2, the difference between the stress-strain relationship calculated based on the identification result and the actual measurement value is very small. This indicates that a highly accurate identification result can be obtained by identifying the material parameters of the Ogden model based on the three types of test results.

  The Ogden model is one of typical elasticity models, and is a function of the main stretch ratio expressed by an Nth order series. In the third order, the Ogden model is expressed by the following equation (1).

Here, U is the amount of strain energy per unit volume. λ ₁ , λ ₂ , and λ ₃ are main stretch ratios. The stretch ratio is the ratio of the length after deformation to the length of the initial shape. When the strain is ε, the relationship is λ = 1 + ε. Α _i and μ _i are material parameters of the Ogden model.

The three types of tests refer to three tests: a uniaxial extension test (uniaxial tensile test), a pure shear test (uniaxial high-speed biaxial tensile test), and a uniform biaxial test. The uniaxial extension test is the most common test, and is a test in which an elastic body is pulled in a uniaxial direction as shown in FIG. In the case of the uniaxial extension test, the extension ratio in the tensile direction is λ _U , the extension ratio in the tensile direction is λ ₁ , the extension ratios in the cross-sectional direction are λ ₂ and λ _3, and the nominal stress T _U is as follows: It is expressed as Expression (2) to Expression (4).

As shown in FIG. 14, the pure shear test is a test in which an elastic body constrained in one axial direction is pulled in the other axial direction. For pure shear test, the stretch ratio in the pulling direction as lambda _S, and lambda ₁ is a draw ratio of the tensile direction, and lambda ₂ is a stretch ratio in the restraining direction, and lambda ₃ is a stretch ratio in the thickness direction, the nominal the stress _{T S,} is represented by the following formula (5) to (8).

The uniform biaxial test is a test in which an elastic body is pulled in two axial directions with the same magnitude of force, as shown in FIG. For uniform biaxial test, the stretch ratio in each axial direction lambda _B, the pulling direction of the stretching ratio in which lambda ₁ and lambda _2, and lambda ₃ is a stretch ratio in the thickness direction, the nominal stress T _B is These are expressed as the following formulas (9) to (11).

  By identifying the material parameters of the Ogden model based on the actual measurement values measured in these three types of tests, an ideal identification result can be obtained as shown in FIG. However, tests other than the uniaxial extension test may be difficult to implement due to problems such as cost. Although it is possible to identify the material parameters of the Ogden model based on the result of only the uniaxial extension test, in that case, as shown in FIG. 3, the difference between the stress-strain relationship calculated based on the identification result and the actual measurement value Will become bigger.

  When a test other than the uniaxial extension test cannot be performed, it is conceivable to use an elastic model other than the Ogden model in order to improve the accuracy of the identification result. As an elastic model other than the Ogden model, for example, an Arruda-Boyce model derived from statistical consideration of a polymer network can be used. The Arruda-Boyce model is one of elastic models, and is represented by the following formula (12).

Here, I ₁ is a primary invariant of the stretch tensor. Μ and λ _m are material parameters of the Arruda-Boyce model.

  When the material parameters of the Arruda-Boyce model are identified based on the result of only the uniaxial extension test, a relatively good identification result is obtained as shown in FIG. 4, but the calculation is performed based on the identification result in the low strain region. The difference between the measured stress-strain relationship and the measured value is large. The low strain region is, for example, a region where the strain is 0 to 40%, and is frequently used in tire simulation. Therefore, the large deviation in the low strain region becomes a serious problem in tire simulation.

  As described above, when there is only an actual measurement value of only the uniaxial extension test, it is difficult to obtain an accurate identification result by identifying the material parameter using only one elastic model. Therefore, in the material parameter identification method according to the present embodiment, the material parameter of another elastic model is calculated using the material parameter identified using one elastic model.

  Specifically, first, the material parameter of the first elastic model is identified based on the actual measurement value of only the uniaxial extension test. The first elastic model used here is an elastic model in which the ratio of stress in the stress-strain relationship of the three types of tests calculated based on the identification result is almost accurate. The fact that the stress ratio in the stress-strain relationship is substantially accurate means that the stress ratio in each test corresponding to the same strain is substantially the same as the ratio in the actual measurement value.

When the stress ratio in the stress-strain relationship is almost accurate, for example, when three types of tests are actually performed, the stress ratio corresponding to the same strain is
Uniaxial extension test: pure shear test: uniform biaxial test = 1: 1.1: 1.4
The stress corresponding to the same strain in the stress-strain relationship of the three types of tests calculated using the material parameters identified based on the test results of one type for the elastic body. The ratio of
Uniaxial extension test: pure shear test: uniform biaxial test ≒ 1: 1.1: 1.4
It becomes.

  As the first elastic model, for example, an Arruda-Boyce model can be used. Note that the elastic model used as the first elastic model only needs to have a substantially accurate stress ratio in the stress-strain relationship of the three types of tests, at least within the strain range used in the simulation.

  Subsequently, in the material parameter identification method according to the present example, a pure shear test and a uniform biaxial test are performed using the stress ratio in the stress-strain relationship (first estimated value) calculated based on the identification result. In this case, an estimated value (second estimated value) of the measured value obtained in this case is calculated. The estimated values obtained when performing the pure shear test and the uniform biaxial test are calculated based on the stress ratio in the stress-strain relationship of the three types of tests calculated based on the identification results. Calculated by multiplying.

  Since the stress ratio in the stress-strain relationship of the three types of tests calculated based on the identification results is almost accurate, the estimated value calculated here is almost equal to the actually measured value obtained by actually performing the test. Conceivable.

  Subsequently, in the material parameter identification method according to the present embodiment, the material parameter of the second elastic model is based on the measured value of the uniaxial extension test, the estimated value of the pure shear test, and the estimated value of the uniform biaxial test. Is identified. The second elastic model used here is an elastic model that can obtain a highly accurate identification result by identifying material parameters based on the results of three types of tests. As the second elastic model, for example, the Ogden model can be used. Note that the elastic model used as the second elastic model only needs to obtain a highly accurate identification result at least within the strain range used in the simulation.

  Since the estimated values used here are considered to be approximately equal to the actual measured values obtained by actually performing the test, the estimated values are regarded as actual measured values, and the material parameters of the second elastic model are identified. It can be expected that an accurate identification result will be obtained.

  Next, the execution procedure of the simulation method according to the present embodiment will be described with reference to FIGS. FIG. 5 is a flowchart illustrating the execution procedure of the simulation method according to the present embodiment. FIG. 6 is a diagram illustrating an example of the result of identifying the material parameters of the first elastic model. FIG. 7 is a diagram illustrating an example of an estimated value calculated based on the stress ratio. FIG. 8 is a diagram illustrating an example of the result of identifying the material parameters of the second elastic model. In FIG. 5, it is assumed that the simulation program 27 is executed in advance and the information acquisition unit 51 and the like are already functioning.

  When the computer 50 detects an input of a simulation start instruction by the user, the computer 50 sets a simulation condition in step S12. Specifically, various conditions input by the user are detected, and the detected conditions are set as simulation conditions. Here, the simulation conditions are the type of analysis, the model to be analyzed, the conditions, and the like.

  After setting the simulation conditions in step S12, the computer 50 sets a strain range to be set as the material parameter identification range based on the input conditions in step S14. In the following description, it is assumed that 0 to 100% is set as the strain range.

  Subsequently, in step S16, the computer 50 acquires an actual measurement value of a stress-strain relationship under a uniaxial stretching condition. The actual measurement value of the stress-strain relationship acquired here is, for example, data in which the stress corresponding to each strain increased in increments of 10% is recorded. The actual measurement value may be acquired from the terminal device 70 or may be acquired from another computer 64a or 64b.

  Subsequently, in step S18, the computer 50 identifies the material parameter (first material parameter) of the first elastic model using the actually measured value of the stress-strain relationship under the uniaxial stretching condition. Here, as an identification method, for example, a least square method can be used.

  Subsequently, in Step S20, the computer 50 calculates a first estimated value of the stress-strain relationship under the uniaxial stretching condition using the identified material parameter. In step S22, the computer 50 calculates a first estimated value of the stress-strain relationship under the pure shear condition using the identified material parameter. In step S24, the computer 50 uses the identified material parameter to equalize. A first estimate of the stress-strain relationship under biaxial conditions is calculated.

  By the calculations in step S20, step S22, and step S24, as shown in FIG. 6, the first estimated value of the stress-strain relationship of the three tests is calculated. Although these first estimated values may deviate from the actually measured values, the stress ratio in the stress-strain relationship is almost accurate.

  Subsequently, in step S26, the computer 50 performs uniaxial stretching based on the first estimated value of the stress-strain relationship under the uniaxial stretching condition and the first estimated value of the stress-strain relationship under the pure shear condition. The ratio of the stress under the condition and the stress under the pure shear condition is calculated. In step S28, the computer 50 performs uniaxial stretching based on the first estimated value of the stress-strain relationship under the uniaxial stretching condition and the first estimated value of the stress-strain relationship under the uniform biaxial condition. The ratio of the stress under the condition and the stress under the uniform biaxial condition is calculated.

For example, in the example shown in FIG. 6, when the strain is 50%, the ratio of the stress under the uniaxial stretching condition and the stress under the pure shearing condition is Y ₅₀ / X ₅₀ , which is equal to the stress under the uniaxial stretching condition. The ratio of stress under the axial condition is Z ₅₀ / X ₅₀ . When the strain is 100%, the ratio of the stress under the uniaxial stretching condition to the stress under the pure shear condition is Y ₁₀₀ / X ₁₀₀ , and the ratio of the stress under the uniaxial stretching condition and the stress under the uniform biaxial condition. _is a Z _{100 / X} 100.

  Subsequently, in step S30, the computer 50 multiplies the measured value of the stress-strain relationship under the uniaxial extension condition by the ratio of the stress under the uniaxial extension condition to the stress under the pure shear condition to obtain the stress under the pure shear condition. A strain-related estimated value (second estimated value) is calculated. Then, in step S32, the computer 50 multiplies the measured value of the stress-strain relationship under the uniaxial extension condition by the ratio of the stress under the uniaxial extension condition to the stress under the uniform biaxial condition to obtain the stress under the equivalent biaxial condition. -An estimated value (second estimated value) of the strain relationship is calculated.

Here, as shown in FIG. 7, the estimated value of the pure shear condition and the estimated value of the uniform biaxial condition are calculated for each measured value of the uniaxial extension condition. For example, the estimated value of the stress when the strain is 50% under the pure shear condition is calculated by multiplying the actual measured value of the stress when the strain is 50% under the uniaxial stretching condition by Y ₅₀ / X ₅₀ . Further, the estimated value of the stress when the strain is 50% under the uniform biaxial condition is calculated by multiplying the measured value of the stress when the strain is 50% under the uniaxial stretching condition by Z ₅₀ / X ₅₀ . Similarly, the estimated value of the stress when the strain is 100% under the pure shear condition is calculated by multiplying the measured value of the stress when the strain is 100% under the uniaxial stretching condition by Y ₁₀₀ / X ₁₀₀ . Further, the estimated value of the stress when the strain is 100% under the uniform biaxial condition is calculated by multiplying the actually measured value of the stress when the strain is 100% under the uniaxial extension condition by Z ₁₀₀ / X ₁₀₀ .

  Note that the ratio multiplied by the actual measurement value of the uniaxial stretching condition may be an average value or a median value obtained by statistically processing the stress ratio instead of the stress ratio when the strain is the same. For example, the pure shear condition is obtained by multiplying the measured value of the stress when the strain is 50% under the uniaxial stretching condition by the average value within the strain range of the ratio of the stress under the uniaxial stretching condition and the stress under the pure shearing condition. The estimated value of the stress when the strain is 50% may be calculated.

  Subsequently, in step S34, the computer 50 performs the actual measurement value of the stress-strain relationship under the uniaxial stretching condition, the estimated value (second estimated value) of the stress-strain relationship under the pure shear condition, and the equal biaxial condition. The material parameter of the second elastic model is identified using the estimated value (second estimated value) of the stress-strain relationship at. Here, as an identification method, for example, a least square method can be used.

  When the stress-strain relationship of the three types of tests is calculated using the material parameters thus identified, as shown in FIG. 8, the difference between the stress-strain relationship based on the measured value of the uniaxial extension condition and the identification result is small, and the identification result It can be seen that the accuracy of is high. The identified material parameters also correspond to multiaxial fields, and if the measured values are calculated for pure shear conditions and uniform biaxial conditions, the difference between the measured values and the stress-strain relationship based on the identification results is small. Be expected.

  Subsequently, the computer 50 executes a simulation using the identified material parameters as step S36. When a plurality of members having different compounds to be used are included in the tire, it is preferable to identify the material parameters by executing the procedure of steps S12 to S34 for each member. For example, when different compounds are used for the cap tread, the bead filler, and the inner liner, it is preferable to identify the material parameters separately. The simulation executed in step S36 is a tire simulation modeled with elements that can be numerically analyzed using the material parameters of the second elastic model, and includes, for example, ground contact analysis, stiffness analysis, rolling Includes analysis.

  Next, effects of the simulation method and the material parameter identification method according to the present embodiment will be described with reference to FIGS. FIG. 9 is a diagram illustrating an example of the results of identifying the material parameters of the Ogden model based on the three types of test results. FIG. 10 is a diagram illustrating an example of a result of identification of material parameters of the Arruda-Boyce model based on one type of test result. FIG. 11 is a diagram illustrating an example of a result of identifying material parameters of the Ogden model based on one type of test result and a known stress ratio. FIG. 12 is a diagram illustrating an example of a result of identifying material parameters using the material parameter identification method according to the present embodiment. The elastic bodies to be analyzed in the examples shown in FIGS. 9 to 12 are the same.

  As shown in FIG. 9, when the material parameters of the Ogden model are identified based on the three types of test results, the difference between the stress-strain relationship calculated based on the identification results and the actual measurement value is small, and ideal identification results Can be obtained. However, it may be difficult to prepare three kinds of test results due to problems such as cost.

  As shown in FIG. 10, when the material parameters of the Arruda-Boyce model are identified based on one type of test result, the difference between the stress-strain relationship calculated based on the identification result and the actual measurement value is large in the low strain region. Therefore, an identification result suitable for the simulation of a tire that is frequently used in the low strain region cannot be obtained.

The example shown in FIG. 11 is known for an actual measurement value of a uniaxial extension condition and an elastic body having the same composition as the elastic body to be analyzed. Uniaxial extension test: pure shear test: equal biaxial test = 1: 1. 1: 1.4
The results of identifying the material parameters of the Ogden model based on the pure shear condition and the estimated value of the uniform biaxial condition obtained by multiplying the actual stress value of the uniaxial stretching condition by the stress ratio. Even in this example, the difference between the stress-strain relationship calculated based on the identification result and the actually measured value is large in the low strain region, and the identification result suitable for the tire simulation cannot be obtained. Further, in this example, the difference between the stress-strain relationship calculated based on the identification result and the actually measured value is relatively small as a whole, but it is used for an elastic body such as a tire whose characteristics greatly change depending on the composition. In the field of compounds and the like, even if the ratio of the stress of an elastic body having a similar composition is used, an identification result that is greatly different from the actual measurement value may be obtained.

  As shown in FIG. 12, when the material parameter is identified using the material parameter identification method according to the present embodiment, the stress − calculated based on the identification result is used even though the actually measured value is one. The deviation between the strain relationship and the measured value is small even in the low strain region, and an accuracy close to that obtained when three types of measured values are used is obtained.

  The divergence between the stress-strain relationship calculated based on the identification results obtained by the above four methods and the actual measurement values is as shown in the following table, and the material parameter identification method according to this example has three types of actual measurement values. It can be seen that the material parameters can be identified with an accuracy close to the case of using.

  As described above, according to the simulation method and the material parameter identification method according to the present embodiment, the material parameter can be accurately identified from one kind of actually measured value. Further, since it is not necessary to repeatedly perform the calculation until the convergence condition is satisfied, the material parameter can be identified with a small amount of calculation.

  The simulation method and material parameter identification method according to the present embodiment can be variously modified without departing from the gist of the present invention. For example, in the identification process of step S18 and step S34 of the execution procedure shown in FIG. 5, the material parameter may be identified by the least square method by weighting the error in the strain region frequently used in the simulation. In the case of tire simulation, the strain region frequently used in the simulation is a low strain region (for example, 0 to 40%). Moreover, a suitable weight is 10 to 100 times, for example. Thus, the accuracy of the simulation can be improved by identifying the material parameter by weighting the error in the strain region frequently used in the simulation.

  In the above embodiment, an example is shown in which the Arruda-Boyce model is used as the first elastic model. However, the first elastic model is a stress-strain of three types of tests calculated based on the identification result. Any elastic model in which the ratio of stress in the relationship is almost accurate may be used. The elastic model having such characteristics is, for example, an elastic model composed of a function composed only of a first-order invariant of strain. In addition to the Arruda-Boyce model, for example, there is a Yeoh model as an elastic model composed of a function consisting of only primary invariants.

  The Yeoh model is one of the superelastic models, and is represented by the following formula (13).

Here, C ₁₀ , C ₂₀ , and C ₃₀ are material parameters in the Yeoh model.

  In the above-described embodiment, an example in which the Ogden model is used as the second elastic model has been described. However, the second elastic model can be obtained by identifying material parameters based on the results of three types of tests, thereby improving accuracy. Any elastic model that can obtain a high identification result. The elastic model having such characteristics is, for example, an elastic model composed of a function including at least a second-order invariant of strain or an extension ratio. In addition to the Ogden model, for example, there is a Polynomial model as an elastic model composed of a function including at least a second-order invariant or an expansion ratio.

  The Polynomial model is one of the superelastic models and is represented by the following formula (14). Here, a quadratic polynomial is given as an example.

Here, I ₂ is a secondary invariant of the stretch tensor. C ₁₀ , C ₀₁ , C ₁₁ , C ₂₀ , and C ₀₂ are material parameters in the second-order Polynomial model.

27 simulation program 50 host computer 50m storage unit 50p processing unit 51 information acquisition unit 52 material parameter identification unit 53 analysis unit 59 input / output unit 60 terminal device main body 61 input device 62 display device 63 network 64a, 64b computer

Claims (8)

A material parameter identification method for identifying a material parameter using a computer,
Identifying the material parameters of the first elastic model based on the measured values of the relationship between stress and strain under uniaxial stretching conditions;
Using the identified material parameters of the first elastic model, the relationship between stress and strain under uniaxial stretching conditions, the relationship between stress and strain under pure shear conditions, and the relationship between stress and strain under equal biaxial conditions A calculating step;
Calculating the ratio of stress under uniaxial stretching condition and pure shear condition based on the relationship between stress and strain under uniaxial stretching condition and the relation between stress and strain under pure shear condition;
Calculating the ratio of the stress in the uniaxial stretching condition and the uniform biaxial condition based on the relationship between the stress and strain in the uniaxial stretching condition and the relationship between the stress and strain in the uniform biaxial condition;
Calculating an estimated value of the relationship between the stress and strain under the pure shear condition based on the ratio of the stress under the uniaxial extension condition and the actual value of the relationship between the stress and strain under the uniaxial extension condition; ,
Based on the ratio of stress between the uniaxial stretching condition and the uniform biaxial condition and the measured value of the relationship between stress and strain under the uniaxial stretching condition, an estimated value of the relationship between stress and strain under the uniform biaxial condition is calculated. Steps,
Based on the measured value of the relationship between stress and strain under the uniaxial stretching condition, the estimated value of the relationship between stress and strain under the pure shear condition, and the estimated value of the relationship between stress and strain under the uniform biaxial condition, Identifying a material parameter of the second elastic model. A method of identifying a material parameter, comprising:   The material parameter identification method according to claim 1, wherein the first elastic model is configured by a function including only a first-order invariant of strain.   3. The material parameter identification method according to claim 1, wherein the second elastic model is configured by a function including at least a second-order invariant of strain or an extension ratio.   In the step of identifying the material parameter of the second elastic model, a least square method weighting an error of a strain region frequently used in a simulation performed using the material parameter of the second elastic model to be identified The material parameter identification method according to any one of claims 1 to 3, wherein the material parameter is identified by:   In the step of identifying the material parameter of the first elastic model, a least square method weighting an error of a strain region frequently used in a simulation performed using the material parameter of the second elastic model to be identified The material parameter identification method according to claim 1, wherein the material parameter is identified by:   The material parameter identification method according to any one of claims 1 to 5, wherein the first elastic model is an Arruda-Boyce model or a Yeoh model.   The material parameter identification method according to claim 1, wherein the second elastic model is an Ogden model or a Polynomial model. A simulation method for analyzing an elastic body using a computer,
Identifying a material parameter of the first elastic model based on an actual measurement value of a relationship between stress and strain under a uniaxial extension condition of the elastic body;
Using the identified material parameters of the first elastic model, the relationship between stress and strain under uniaxial stretching conditions, the relationship between stress and strain under pure shear conditions, and the relationship between stress and strain under equal biaxial conditions A calculating step;
Calculating the ratio of stress under uniaxial stretching condition and pure shear condition based on the relationship between stress and strain under uniaxial stretching condition and the relation between stress and strain under pure shear condition;
Calculating the ratio of the stress in the uniaxial stretching condition and the uniform biaxial condition based on the relationship between the stress and strain in the uniaxial stretching condition and the relationship between the stress and strain in the uniform biaxial condition;
Calculating an estimated value of the relationship between the stress and strain under the pure shear condition based on the ratio of the stress under the uniaxial extension condition and the actual value of the relationship between the stress and strain under the uniaxial extension condition; ,
Based on the ratio of stress between the uniaxial stretching condition and the uniform biaxial condition and the measured value of the relationship between stress and strain under the uniaxial stretching condition, an estimated value of the relationship between stress and strain under the uniform biaxial condition is calculated. Steps,
Based on the measured value of the relationship between stress and strain under the uniaxial stretching condition, the estimated value of the relationship between stress and strain under the pure shear condition, and the estimated value of the relationship between stress and strain under the uniform biaxial condition, Identifying the material parameters of the second elastic model;
And a step of simulating the elastic body using material parameters of the second elastic model.

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