JP4299735B2 - Method for predicting and designing elastic response performance of rubber products - Google Patents

Description

  The present invention relates to a method for predicting the elastic response performance of a rubber product using a finite element analysis method (FEA), and a method for designing a rubber product using the prediction method.

  In designing rubber products, there are already several decades of examples of predicting elastic response performance using 3D finite element analysis (FEA) and applying the analysis or simulation results. (For example, see Patent Documents 1 to 3). The energy constitutive equation that reflects the stress-strain relationship of the rubber material used in the FEA calculation is changed from the initial linear elastic equation to the Mooney-Rivlin equation. The introduction of equations is actively performed.

  However, the constitutive equations of these rubber materials are based on the network deformation theory based on entropy elasticity developed from molecular statistical thermodynamics based on the elongation of rubber molecular chains. There is a constitutive equation that can express the elastic response behavior of rubber in the low temperature region or high-speed deformation region (applying the temperature-time conversion rule), which is an important factor in design, using parameters with physical meaning focused on enthalpy elasticity do not do. As a result, there is no constitutive equation that expresses the temperature dependence and strain dependence of the rubber part when considering the application of the three-dimensional model of the filler and rubber to the FEM calculation at the micro level of rubber. It is in a situation where simulation at a micro level considering deformation at low temperature or high speed deformation which is important in the design of rubber materials cannot be performed.

The strain energy (W) due to the deformation of the rubber is expressed as a function of three stretching ratios λ ₁ , λ ₂ , and λ ₃ representing the deformation in the xyz direction on the three-dimensional axis (see FIG. 7). Such an idea is the basis for describing the deformation of a three-dimensional object. Furthermore, when the deformations of λ ₁ , λ ₂ , and λ ₃ are expressed as strain energy as shown in the following formulas (1) to (3), the strain invariants I ₁ , I ₂ , and I ₃ of the Green tensor are used. Thus, complicated and various three-dimensional deformations can be expressed by a simple strain energy function.

  According to Rivlin et al., The stress-strain relationship in each deformation mode is obtained as shown in the following equations (4) to (6) using strain energy (W).

Here, σ represents engineering stress and τ represents shear stress. If the equation of the strain energy function (W) represented by the functions of I ₁ and I ₂ is obtained from the relational expressions of the equations (4) to (6), the stress-strain relationship and the shear modulus in each deformation mode. (G) is obtained. Conventionally, many of the strain energy functions (W) that have been used for the FEA calculation of rubber products use the generalized form of the Mooney-Rivlin equation shown in the following equation (7).

Assuming that the rubber material is incompressible, I ₃ = 1. Therefore, the above equation (7) is expressed by the following equation (8) as a function of I ₁ and I ₂ .

  Various strain energy functions have been proposed depending on how many power exponent terms of the equation (8) are incorporated to construct the equation. For example, when only the primary term is taken, it can be expressed by the following formula (9).

This equation (9) agrees with the strain energy function derived from the network theory based on molecular statistical thermodynamics, and in this case, C _1,0 = (1/2) ρRT / M _C. Here, ρ represents the density of rubber, R represents a gas constant, T represents an absolute temperature, and M _C represents a molecular weight between crosslinking points.

  Further, in the case of approximation up to the second order term, the following equation (10) is obtained.

This formula (10) is substituted into the formula (4) to give a commonly used Mooney-Rivlin formula. In this case, the relationship between the coefficients C ₁ and C ₂ of the Mooney-Rivlin equation is C _1,0 = C ₁ and C _0,1 = C ₂ .

Furthermore, for general purposes, the following equation is used as a constitutive equation capable of calculating a strain-stress curve based on experimental results from a small deformation to a large deformation as a function of only I ₁ .
S = C _1,0 / 2 + 2 * C _2,0 (I ₁ -3) + 3 * C _3,0 (I ₁ -3) ²
(Yeoh, OHRubber Chemi. Tech. 66 754-771 (1993)
Here, C _1,0 , C _2,0 , and C _3,0 are specific coefficients _{depending on the composition} .

  Furthermore, by introducing a high-order term, an experimentally obtained stress-strain curve can be expressed more accurately.

On the other hand, when considering compressibility, since I ₃ is not equal to 1, there is a case where a change in strain energy due to an elastic response due to hydrostatic pressure is added to the strain energy.
Japanese Patent Laid-Open No. 11-237332 JP 2003-72327 A Japanese Patent Laid-Open No. 2002-1929242

  However, these rubber material constitutive equations are based on the network deformation theory based on entropic elasticity developed from molecular statistical thermodynamics based on the elongation of the molecular chain of rubber, but the design of many industrial rubber materials including tires. There is no constitutive equation that can express the elastic response in consideration of temperature dependency and velocity dependency which are important factors. Therefore, there is no constitutive equation that expresses the temperature dependence and strain dependence of the rubber part when considering the application of the filler and rubber three-dimensional model to the FEM calculation at the micro level of rubber. The simulation at the micro level considering the temperature and strain that are important in the design of rubber materials for automobiles is not possible. As the cause, since the influence of enthalpy elasticity appears in the deformation of rubber at low temperature and high speed deformation, we thought that the above problem can be solved by obtaining a constitutive equation incorporating this effect.

  The present invention has been made in view of the above situation, and uses an finite element analysis method (FEA) to accurately predict the elastic response performance of a rubber product over a wide range of strain and temperature. It is an object of the present invention to provide a performance prediction method and a rubber product design method using the prediction method.

  The elastic response performance prediction method of the present invention according to claim 1 for solving the above-described problem is an elastic response performance prediction method for predicting deformation and fracture behavior of a rubber product using a finite element analysis method (FEA). , The rubber material constituting the rubber product, the micro level at which the filler part and the rubber part can be observed separately, especially the constitutive equation representing the temperature and strain dependence of the elastic modulus of the rubber part and the strain energy The elastic response performance of the rubber product is predicted using a constitutive equation representing the temperature and strain dependence of the rubber product.

  As described above, the elastic response performance prediction method of the rubber product according to the present invention includes a wide deformation region including a large deformation region (usually strain is 100 to 400%) of the rubber portion when the filled rubber is observed at a micro level. In the region, the elastic response performance of rubber products is predicted by the finite element analysis method (FEA) using the constitutive equation representing the temperature and strain dependence of the elastic modulus and the constitutive equation representing the temperature and strain dependence of the strain energy. In particular, with regard to rubber composite products made of rubber materials of the filling system that are important in practical use, there are factors that have physical meaning of temperature and strain dependence, which are important in the design of many industrial rubber products, and The elastic response performance of the rubber part at the micro level can be accurately predicted.

  The constitutive equation in this patent application is an equation that represents the strain energy of the rubber part. By predicting the strain energy distribution and stress concentration distribution during deformation of the rubber part, it is possible to predict heat generation and fracture phenomena at the micro level. It becomes.

  The elastic response performance prediction method of the present invention according to claim 2 is characterized in that the constitutive equation representing the temperature and strain dependence of the elastic modulus and strain energy is a nonlinear equation.

  Generally, in a filled rubber material, in a strain region of about 0.1 to 100%, nonlinearity appears in elastic properties such as storage elastic modulus and tan δ due to the change in the arrangement of the filler network. The elastic response performance of the rubber product used could be predicted at a macro level, but the elastic response should be predicted by separating the filler and rubber parts at the micro level. I couldn't.

  In the present invention, since a nonlinear equation is used as a constitutive equation representing the temperature dependence and strain dependence of elastic modulus and strain energy, even when a filler is added and heat generation or breakage occurs, Since it is considered that fracture will occur, nonlinearity is given to the elastic characteristics, and in a wide range of strain and temperature, the elastic response performance of rubber products is accurately predicted by separating the filler part and the rubber part at a micro level be able to.

  Hereinafter, the nonlinear constitutive equation representing temperature and strain dependence of the elastic modulus and strain energy of the present invention will be described in more detail.

  In general, the elastic modulus (G) of a rubber material can be expressed as follows using statistical thermodynamics.

That is, the elastic modulus (G) is calculated by differentiating the Helmholtz free energy (A) by the strain invariant I ₁ of the Green tensor of the above equation (1).

  Moreover, the average energy (U) of the system in a thermodynamic equilibrium state can be expressed by the following formula (12).

  Here, β is equal to 1 / (kΔT), k is a Boltzmann constant, and ΔT represents a difference from the glass transition temperature (Tg) of the rubber polymer to the measurement temperature. Aα represents the energy level, and Z is a distribution function that normalizes the energy of the system, and can be represented by the following formula (13).

  Here, it can be assumed that Uα is equal to Hamiltonian (H) from the viewpoint of statistical thermodynamics. That is,

  The Hamiltonian (H) can be expressed as a function of temperature conditions and constraints that define microstates in statistical thermodynamics.

In the present invention, two states (r ₁ and r ₂ ) of low temperature and high temperature are set as models representing the temperature dependence of rubber molecules, and the number of states of the rubber molecules in each state is the distortions I _1a and I _1b. By distributing in the energy field, all states are expressed by the function of the following formula (15).

  In the above formula (15), κ represents the repulsive energy of the rubber molecule.

Furthermore, since two states having different energy states of rubber molecules are represented by r ₁ = −1 and r ₂ = 1, when expressed as a statistical thermodynamic average <r ₁ · r ₂ >, this product is +1. In this case, two extreme energy states are shown, in which the rubber molecules are in the same state, and in the case of -1, the rubber molecules are in different states. Here, it is considered that the strain energy field contributes to all molecules by the same amount. Therefore, by setting I ₁ = I _1a = I _1b , the above equation (15) can be simply expressed as the following equation (16). Can do.

  From the equation (11) and the equation (16), the nonlinear constitutive equation representing the temperature and strain dependence of the elastic modulus of the present invention can be expressed as follows.

Here, N ₁ and N ₂ represent coefficients related to the elastic modulus, and correspond to P and Q in the above formula (I).

  By the way, the free energy of Helmholtz is expressed as follows.

  Substituting Z into the Helmholtz free energy equation, the strain energy representing the elastic form of rubber can be expressed as follows. P is a correction coefficient representing the number of micro states.

  Thus, the constitutive equation representing the temperature and strain dependence of strain energy can be expressed as Formula (II).

  The formula (III) representing the entropy change S is called a generalized MOONEY-RIVLIN formula, and is a general formula (see “The Physics of Rubber Elasticity” by L.R.G. Treloar, Oxford Press (1958)).

  When the entropy change S is simplified as j = k = O, i = m in equation (III), equation (IV) is obtained.

Further, the entropy change S may be expressed by the following equation.
S = C _1,0 / 2 + 2 * C _2,0 (I ₁ -3) + 3 * C _3,0 (I ₁ -3) ²
Here, C _1,0 , C _2,0 , and C _3,0 are specific coefficients _{depending on the composition} . It can be obtained by curve fitting of the strain energy change of the experimental result.

  As described above, the constitutive equation representing the temperature and strain dependence of the elastic modulus of the present invention and the constitutive equation representing the temperature and strain dependence of the strain energy are represented by the above-described equations (V) and (VI). The elastic response performance of a rubber product can be predicted by using a constitutive equation obtained by applying the relational expression between the pressure and the strain.

  In the constitutive equation representing the temperature and strain dependence of the strain energy, when the strain energy (A) exceeds the fracture limit energy value of the rubber material, the strain energy or stress is set to zero, thereby breaking the strain energy. It is possible to predict the elastic response performance of rubber products including phenomena.

In the equations of σ (tensile or compressive stress) and λ (elongation ratio or compression ratio), τ (shear stress) and γ (shear strain) represented by the formulas (V) and (VI), σ × γ Or, when τ × γ exceeds the limit value σ * × γ ^* or τ * × γ ^* of the rubber material, by setting the σ or τ to zero, the elasticity of the rubber product including the fracture phenomenon Response performance can be predicted.

Then, Tc calculated by the following equation may be applied to the fracture limit value σ * × γ ^* or τ * × γ ^{* of the} rubber material.
Tc = T0 * r * h
T0 is the net accumulated strain energy, h is the hysteresis ratio, and r is the correction coefficient.

  The elastic response of the rubber product using the constitutive equation representing the temperature and strain dependence of the elastic modulus of the rubber material constituting the rubber product of the present invention as detailed above and the constitutive equation representing the temperature and strain dependence of the strain energy. By predicting the performance, it is possible to design the optimum rubber material at the micro level to achieve the desired performance of the rubber product through the required simulation, in particular through the simulation at the micro level of rubber. Therefore, it is possible to provide an efficient and accurate method for designing a useful rubber product.

  As described above, the present invention predicts the temperature and strain dependence of the elastic modulus of the rubber part when observing at the micro level of the rubber material constituting the rubber product when predicting the elastic response performance of the rubber product. Since the constitutive equation and the constitutive equation representing the temperature and strain dependence of strain energy are used, the elastic response performance of rubber products over a wide range of strain and temperature, the deformation behavior of the rubber part at the micro level is accurately measured. Can be predicted.

  In addition, by using the elastic response performance prediction method of the present invention, it is possible to provide a useful rubber product design method capable of efficiently and accurately designing a rubber product and a rubber raw material.

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

  As shown in FIG. 1, an elastic response performance prediction apparatus (50) for executing prediction of elastic response performance according to an embodiment of the present invention uses an elastic response performance prediction program for executing elastic response performance prediction. It is comprised by the computer arithmetic processing system which performs the process mentioned later. Such a computer system includes, for example, a CPU, a ROM, a RAM, a hard disk, an input / output terminal, and other necessary units. The elastic response performance prediction program is stored in advance on a hard disk or the like.

  In the method for predicting elastic response performance in the embodiment of the present invention, a constitutive equation representing the temperature and strain dependence of the elastic modulus of the rubber material constituting the rubber product, and a constitutive equation representing the temperature and strain dependence of the strain energy, particularly Preferably, the elastic response performance of the rubber product is predicted using the previous formula (I) and the previous formula (II).

  Using the constitutive equation representing the temperature and strain dependence of the elastic modulus of the rubber material represented by the previous formula (I) of the present invention, the temperature dependence of the stress-strain curve for the actual unfilled compounded rubber material The calculation result is shown in FIG. It can be well understood that a curve showing a non-linear relationship related to temperature and strain dependence of stress is shown.

  Further, in the constitutive equation representing the temperature and strain dependence of the strain energy, when the strain energy (A) exceeds the fracture limit energy value of the rubber material, a fracture phenomenon in which the strain energy or stress is zero is included. The result of calculating the temperature dependence of the stress-strain curve is shown in FIG. It can be understood well that a curve showing a nonlinear relationship related to the temperature and strain dependence of stress including the fracture phenomenon is shown.

Furthermore, in order to increase accuracy, it is necessary to consider the temperature dependence of the fracture limit value σ * or τ * × γ * of the rubber material. For this, Tc calculated by the following equation can be applied to the fracture limit value of the rubber material.
Tc = To ・ r ・ h
Here, To represents the energy consumed for actual molecular chain breakage with net accumulated strain energy. Other than that, it is considered that destruction energy is dissipated as heat. h represents the hysteresis ratio and can be replaced with tan δ. r is a correction coefficient. The temperature dependence of Tc is obtained from the temperature dependence of Tanδ, and this makes it possible to calculate the temperature dependence of the elastic response with high accuracy.

  The constitutive equation representing the temperature and strain dependence of the elastic modulus of the rubber material of the present invention and the constitutive equation representing the temperature and strain dependence of the strain energy can be applied to the experimental results measured at various temperatures. It can also be applied to the results of elastic strain and temperature dependence at the service temperature of rubber products. Furthermore, since the strain dependency of the rubber that has received the strain history and the temperature history changes, it can be applied to the result of the elastic strain and the temperature dependency after receiving the strain history and the temperature history.

  The temperature dependence of strain-stress behavior was measured using the following unfilled crosslinked SBR, and the experimental values were compared with the predicted values. The rubber formulation used is SBR 1500: 100 parts, 1 part stearic acid, 3 parts ZnO, 1 part accelerator, 1.75 parts sulfur.

  The temperature dependence of the fracture limit value of the rubber material applies the temperature-time conversion rule to the result of the temperature dependence of tanδ shown in Fig. 4 (0.3% strain, viscoelasticity measuring device manufactured by Toyo Seiki at 52Hz) Although the rubber deformation speed at the fracture surface was shifted by a low temperature of about 20 ° C considering that the measured deformation speed of tan δ was faster than 3.12 mm / s), the fracture limit value at the temperature corresponding to the experiment (● plot in Fig. 5) Values were used. Also, To = 100.

The value used for the constitutive equation is S = C _1,0 / 2 + 2 * C _2,0 (I ₁ -3) + 3 * C _3,0 (I ₁ -3) ²
In this case, C1.0 = 2.5, C2.0 = 0.25, and C3.0 = 0.15.

  In Equation 1, k = 40, P = 1, Mc = 2000, R = 8.2, and N1 = 90000000 were used.

  FIG. 6 shows the predicted value of the elastic response of the rubber predicted using the above parameters, and the strain-stress plot results measured by changing the temperature with a tensile tester manufactured by Toyo Seiki (tensile speed of 300 mm / s). . Both agree well, and it can be said that the temperature dependency and strain dependency of the elastic modulus of the rubber are accurately expressed.

  The above results show the deformation behavior and temperature dependence up to the destruction of rubber material very well, and apply this constitutive equation to FEM for deformation and fracture phenomenon at low temperature and high speed at micro level. Can be expressed with high accuracy.

It is the schematic which showed the prediction apparatus of the elastic response performance. The result of having calculated the temperature dependence of the stress-strain curve of a rubber material using the constitutive equation showing the temperature and strain dependence of the elastic modulus of this invention is illustrated. In the constitutive equation representing the temperature and strain dependence of the elastic modulus of the present invention, when the strain energy (A) exceeds the fracture limit energy value of the rubber material, the result of calculation assuming that the strain energy or stress is zero, The result of having calculated the temperature dependence of the stress-strain curve of material is shown in figure. It is a graph which shows the temperature dependence of tan-delta. It is a graph which shows the fracture limit energy value for every temperature. It is a graph which shows the predicted value and experimental value for each temperature. It is the figure which showed the relationship between the expansion ratio (lambda) 1, (lambda) 2, (lambda) 3 and the stress t1, t2, t3 showing the deformation | transformation of a xyz direction on a three-dimensional axis | shaft.

Explanation of symbols

  50 Elastic response performance prediction device

Claims (10)

In an elastic response performance prediction method for predicting deformation and fracture behavior of a rubber product using a finite element analysis method (FEA),
Temperature and strain dependence equations temperature and strain dependence of the constitutive equations and strain energy following table vital given by the following equation table vital (I) of the modulus of elasticity of the rubber material constituting the rubber product (II An elastic response performance prediction method, wherein the elastic response performance of the rubber product is predicted using a constitutive equation represented by

[In Formula (I), G represents elastic modulus, S represents entropy change at the time of rubber deformation, P and Q represent coefficients related to elastic modulus, I ₁ represents invariant of strain, and T represents Represents absolute temperature. β is equal to 1 / (kΔT), k is a Boltzmann constant, and ΔT is a difference from the glass transition temperature Tg of the rubber polymer. ]
[In Formula (II), A represents strain energy, S represents entropy change at the time of rubber deformation, I ₁ represents invariant of strain, and T represents absolute temperature. β is equal to 1 / (kΔT), k is a Boltzmann constant, and ΔT is a difference from the glass transition temperature Tg of the rubber polymer. ]   The elastic response performance prediction method according to claim 1, wherein the constitutive equation representing temperature and strain dependence of the elastic modulus and strain energy is a nonlinear equation. The elastic response performance prediction method according to claim 1 or 2 , wherein the entropy change S is expressed by the following formula (III).

[In the formula (III), C _{i, j, k} represents a constant related to ρRT / Mc, ρ represents density, R represents a gas constant, T represents absolute temperature, and Mc represents an average molecular weight between rubber mesh points. I _{1 to} I ₃ represent distortion invariants. ] The entropy change S is elastic response performance prediction method according to claim 1 or claim 2, characterized by being represented by formula (IV) shown below.

[In Formula (IV), C _1,0 represents a constant related to ρRT / Mc, ρ represents density, R represents a gas constant, T represents absolute temperature, and Mc represents an average molecular weight between network points of rubber. I ₁ represents an invariant of strain. ] The entropy change S is elastic response performance prediction method according to claim 1 or claim 2, characterized by being represented by the following formula.
S = C _1,0 / 2 + 2 * C _2,0 (I ₁ -3) + 3 * C _3,0 (I ₁ -3) ²
[C _1,0 , C _2,0 , C _3,0 are specific coefficients _depending on the blending. ] The constitutive equation representing the temperature and strain dependence of the elastic modulus and the constitutive equation representing the temperature and strain dependence of the strain energy are changed into the relational expressions of stress and strain represented by the following formulas (V) and (VI). elastic response performance prediction method according to any one of claims 1 to 5, with the constitutive equations obtained by applying, characterized in that predicting the elastic response performance of rubber products.

[In the formulas (V) and (VI), σ represents tensile or compressive stress, τ represents shear stress, λ represents elongation or compression ratio, and γ represents shear strain. ] The constitutive equation representing the temperature and strain dependence of the strain energy is characterized in that the strain energy or stress is set to zero when the strain energy (A) exceeds the fracture limit energy value of the rubber material. The elastic response performance prediction method according to any one of 1 to 6 . In the equations of σ (tensile or compressive stress) and λ (elongation ratio or compression ratio), τ (shear stress) and γ (shear strain) represented by the formulas (V) and (VI), σ × γ or τ 7. The elastic response performance prediction according to claim 6 , wherein when .gamma. Exceeds a fracture time limit value .sigma. *. Times..gamma. * Or .tau. *. Times..gamma. * Of the rubber material, the .sigma. Or .tau. Is set to zero. Method. The elasticity according to any one of claims 1 to 8 , wherein Tc calculated by the following equation is applied to the fracture time limit value σ * × γ * or τ * × γ * of the rubber material. Response performance prediction method.
Tc = T0 * r * h
[T0 is the net accumulated strain energy, h is the hysteresis ratio, and r is the correction coefficient. ] A rubber product design method, wherein a rubber product is designed using the elastic response performance prediction method according to any one of claims 1 to 9 .

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