JP2007265266A - Viscoelastic response performance prediction method, rubber product design method, and viscoelastic response performance prediction device - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method of predicting viscoelastic response performance of a rubber product using a finite element analysis (FEA), a design method and a viscoelastic response performance prediction device. <P>SOLUTION: This viscoelastic response performance prediction method for predicting the viscoelastic response performance of the rubber product using the finite element analysis is characterized by predicting the viscoelastic response performance of the rubber product using a constitutive equation expressing distortion dependency of viscoelastic characteristics of the rubber product using a parameter representing temperature dependency. Mathematical expressions 20, 21 are preferable as the constitutive equation expressing the distortion dependency of the viscoelastic characteristics, and a mathematical expression 33 is preferable as the parameter representing temperature dependency. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

  The present invention relates to a viscoelastic response performance prediction method, a rubber product design method, and a viscoelastic response performance prediction apparatus, and in particular, viscoelasticity that predicts viscoelastic response performance of a rubber product using a finite element analysis method (FEA). The present invention relates to a response performance prediction method, a rubber product design method using the viscoelastic response performance prediction method, and a viscoelastic response performance prediction apparatus.

  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). As a constitutive equation of energy for reflecting the stress-strain relationship of the rubber material used for the FEA calculation in the analysis, the linear elastic equation is shifted to the generalized Mooney-Rivlin equation.

That is, the strain energy W due to the deformation of the rubber can be 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 (FIG. 2). This idea is the basis for describing the deformation of an object with a three-dimensional extent. Furthermore, when the deformations of λ ₁ , λ ₂ , and λ ₃ are expressed as strain energy as shown in the mathematical expressions (1) to (3), the distortion invariants I ₁ , I ₂ , and I ₃ of the Green tensor are complicated Various three-dimensional deformations can be expressed by a simple strain energy function.

  Also, the relationship between stress and strain in each deformation mode is determined by Rivlin et al. Using equations (4) to (6) using strain energy.

Here, σ represents engineering stress and τ represents shear stress. If the equation of the strain energy function W expressed 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 G in each deformation mode can be obtained. . Conventionally, many of the strain energy functions W that have been used for FEA of rubber products use the generalized form of the Mooney-Rivlin equation shown in Equation (7).

Further, assuming that the rubber material is incompressible, it becomes I ₃ −1, and therefore Equation (7) becomes Equation (8) as a function of I ₁ and I ₂ .

  Various strain energy functions have been proposed depending on how many exponent terms in the formula (8) are incorporated to form the formula. For example, when only the primary term is taken, it can be expressed by Equation (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. Furthermore, in the case of approximation up to a quadratic term, Equation (10) is obtained.

This formula (10) is assigned to 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, by introducing a high-order term, an experimentally obtained stress-strain curve can be expressed more accurately.

  However, the above constitutive equations for rubber materials are based on the network deformation theory developed from molecular statistical thermodynamics based on the elongation of rubber molecular chains, and are used as many industrial rubber materials including tires. It does not accurately represent the stress-strain relationship of rubber-based rubber, including viscoelastic properties. It is known as the Payne effect that the addition of a filler gives non-linearity to viscoelastic properties such as storage elastic modulus and tan δ in a strain region of 0 to 100%. The deformation in a tire in a normal rolling wheel state is also mostly strained in this region, and viscoelasticity control in this strained region is particularly important for controlling tire rolling resistance.

Here, a constitutive equation representing the stress-strain relationship of rubber as viscoelastic properties has been proposed (Patent Document 4).
Japanese Patent Laid-Open No. 11-237332 JP 2003-72327 A Japanese Patent Laid-Open No. 2002-1929242 JP 2005-04933 A

  However, the technique described in Patent Document 4 has a problem that parameters included in the constitutive equation must be experimentally obtained for each temperature in order to express the temperature dependence of the viscoelastic characteristics.

  The present invention has been made to solve the above problems, and can represent the temperature dependence of the viscoelastic properties of rubber products without relying on experiments, and can easily and accurately measure the viscoelastic response performance of rubber products. It is an object to provide a viscoelastic response performance prediction method, a rubber product design method, and a viscoelastic response performance prediction device that can be well predicted.

  In order to achieve the above object, a viscoelastic response performance prediction method according to the present invention is a viscoelastic response performance prediction method for predicting the viscoelastic response performance of a rubber product using a finite element analysis method, and is temperature dependent. The viscoelastic response performance of the rubber product is predicted using a constitutive equation that represents the strain dependence of the viscoelastic property of the rubber product using a parameter representing the above.

  The viscoelastic response performance prediction apparatus according to the present invention is a viscoelastic response performance prediction apparatus that predicts the viscoelastic response performance of a rubber product using a finite element analysis method, and uses a parameter that represents temperature dependence. The viscoelastic response performance of the rubber product is predicted using a constitutive equation representing the strain dependence of the viscoelastic properties of the rubber product.

  According to the viscoelastic response performance prediction method and the viscoelastic response performance prediction apparatus of the present invention, the constitutive equation representing the strain dependency of the viscoelastic property that can cope with the small strain region of the rubber product is used. Therefore, a constitutive equation representing the strain dependence of the viscoelastic properties that can deal with the small strain region even if the filler is added to give non-linearity to the viscoelastic properties such as storage modulus and tan δ in the small strain region Therefore, the viscoelastic response performance of the rubber product can be accurately predicted. The small strain region is a small deformation region (0.1% to 100%), and the constitutive equation representing the strain dependence of the viscoelastic property capable of coping with the small strain region is a large deformation region (100% to 400%). %) As well as small deformation regions (0.1% to 100%) can be dealt with.

  Moreover, since the parameter showing temperature dependence is used for the constitutive equation showing the strain dependence of a viscoelastic characteristic, the temperature dependence of a viscoelastic characteristic can be represented.

  Therefore, by applying a constitutive equation representing the strain dependence of viscoelastic properties to the finite element analysis method using parameters representing temperature dependence, it is possible to predict the viscoelastic response performance of rubber products without relying on experiments. Both can express the temperature dependence of the viscoelastic properties of the rubber product, and can easily and accurately predict the viscoelastic response performance of the rubber product at the temperature to be predicted.

  The constitutive equation representing the strain dependence of the viscoelastic property according to the present invention may include an equation representing the storage elastic modulus and an equation representing the loss tangent between the storage elastic modulus and the loss elastic modulus.

  Further, the storage elastic modulus G ′ can be expressed by the following mathematical formula.

  Here, γ represents shear strain, A, C, and n represent physical constants depending on the filler network structure, and G′∞ represents each change in viscoelastic response due to the molecular extension effect.

  In addition, physical constants A, n, and G′∞ depending on the filler network structure can be expressed by the following mathematical formula.

Where A ₀ and n ₀ are physical constants depending on the filler network structure, R is a gas constant, E _A is the activation energy of A, E _n is the activation energy of n, and G′∞ ₀ is the activity of G′∞. ΔT represents each temperature, where ΔT is represented by the following equation, where Tg is the glass transition temperature of the rubber polymer and C2 is an arbitrary temperature constant.
ΔT = C2 + T-Tg
Thereby, since the parameters A, n, and G′∞ used in the formula representing the storage elastic modulus G ′ included in the constitutive equation representing the strain dependence of the viscoelastic property can be given temperature dependence, Even if it does not depend on, the temperature dependence of the storage elastic modulus of a rubber product can be expressed, and it becomes possible to easily calculate the viscoelastic property at the temperature to be predicted.

  The loss tangent tan δ can be expressed by the following mathematical formula.

  Where γ is a shear strain, a, c, b, and d are physical constants depending on the filler network structure, and tan δ | γ → 0 represents each of tan δ at strain 0.

  Further, the physical constants a, b, d depending on the filler network structure and tan δ at the strain 0 can be expressed by the following mathematical formula.

Where E _a is the activation energy of a, E _b is the activation energy of b, E _d is the activation energy of d, E _D is the activation energy of D, a ₀ , b ₀ , d ₀ , and D ₀ Represents each of the physical constants depending on the filler network structure.

  As a result, the parameters a, b, d, and tan δ | γ → 0 used in the expression representing the loss tangent tan δ included in the constitutive equation representing the strain dependence of the viscoelastic property can be given temperature dependence. The temperature dependence of the loss tangent of the rubber product can be expressed without depending on the experiment, and the viscoelastic property of the rubber product at the predicted temperature can be easily calculated.

Further, the above-described shear strain γ is expressed by the following mathematical expression, where I ₁ is the invariant of strain.
γ ² = I ₁ -3
Accordingly, the storage elastic modulus G ′ and the loss tangent tan δ are expressed by the following mathematical expressions.

  The rubber product design method according to the present invention is characterized in that a rubber product is designed using the viscoelastic response performance prediction method of the present invention described above.

  By predicting the viscoelastic response performance of the rubber product using the constitutive equation representing the strain dependence of the viscoelastic properties of the rubber product of the present invention described in detail above, a desired simulation of the rubber product is performed through a required simulation. It is possible to design an optimal rubber material for achieving the performance, and it is possible to provide an efficient and accurate method for designing a useful rubber product.

  As described above, according to the viscoelastic response performance prediction method, the rubber product design method, and the viscoelastic response performance prediction apparatus of the present invention, the strain dependency of the viscoelastic property is expressed using the parameter indicating the temperature dependency. By applying the constitutive equation to the finite element analysis method and predicting the viscoelastic response performance of rubber products, the temperature dependence of the viscoelastic properties of rubber can be expressed without relying on experiments. An effect is obtained that the viscoelastic response performance of a rubber product can be easily and accurately predicted.

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

  As shown in FIG. 1, a viscoelastic response performance prediction device 50 for performing viscoelastic response performance prediction according to an embodiment of the present invention is a viscoelastic response performance prediction program for executing viscoelastic response performance prediction. Thus, the computer processing system is configured to execute processing described 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 viscoelastic response performance prediction program is stored in advance on a hard disk or the like.

  In the viscoelastic response performance prediction method according to the embodiment of the present invention, a constitutive equation representing the strain dependence of the viscoelastic property in the small strain region of the rubber product is used. This constitutive equation will be described below.

  A nonlinear constitutive equation of the filling rubber is proposed as the following formula (22) by C.K.L.Davies, D.De, A.G.Thomas (DDT) et al. The relationship between the parameters in the equation and the strain dependence of the elastic modulus has been experimentally verified.

  Here, n is the strain dependence of the elastic modulus and is a function of the particle size and filling amount of the filler, A is a function of the polymer network and the volume fraction of the filler, and C is the strain at which the elastic modulus starts to decrease. , K represents the full size of the polymer molecule.

On the other hand, Kraus is based on the idea of the Payne effect that is manifested by the formation of a filler network. The difference between the shear property G ′ ₀ in the low deformation region and the shear property G′∞ in the large deformation region is shown for the filled rubber. According to Kraus's model, which is the elastic modulus due to the network generation of the filler, the magnitude of the elastic modulus due to the network generation is proportional to the product of the number of contacts of each filler particle and the elastic modulus at the contact point (23) Has proposed.

Here, N _con represents the number of contact points with which the filler particles are in elastic contact, K ₀ is an elastic constant at the contact points, and R represents the distance between the center points of the filler particles.

The storage of filled rubber is performed by combining the following equation (22), which is a nonlinear constitutive equation by CKLDavies, D. De, AGThomas (DDT), and equation (23), which is Kraus's filler network equation. A constitutive equation representing the strain dependence of viscoelastic properties such as elastic modulus G ′, loss elastic modulus G ″, and tan δ is derived. That is, the number of contacts N _con in which the filler particles are in elastic contact with each other can be expressed as Equation (24) using Equation (22) from the comparison between Equation (22) and Equation (23). .

Here, K ₀ is the elastic constant at the contact point, R is the distance between the center of gravity of the filler particles, and α is a constant. Furthermore, Kraus has proposed Formula (25) as a formula representing the loss elastic modulus G ″ based on the same concept.

Here, C ₁ is a constant, k _b is a decay rate constant of the filler network, N _con is the number of contact points with which the filler particles are in elastic contact, and f _b is a strain contribution. On the other hand, in Kraus theory, G ″ is defined to be proportional to only the collapse and recovery rate of the filler network, and therefore G ″ approaches zero as long as the strain is zero. This is different from the experimentally observed G ″ behavior and suggests that the correction of Equation (25) is necessary.

In this regard, D.C. Ulmer has added a term proportional to the number of contact points N _con of the filler as another contribution of G ″ to Equation (25), and proposes Equation (26) regarding the strain dependence of G ″.

Here, C ₂ represents a constant representing the contribution ratio of the second term, and f _b is represented by the mth power of the shear strain γ. Then, from the mathematical formula (24) and the mathematical formula (26), a nonlinear elastic equation and a nonlinear viscoelastic equation are derived as constitutive equations representing the strain dependence of the viscoelastic property in the small strain region of the rubber product. That is, the nonlinear elastic equation is an equation representing the storage elastic modulus, and specifically, the storage elastic modulus G ′ is obtained from Equation (27). The nonlinear viscoelastic equation is an equation representing the loss elastic modulus, and specifically, the loss elastic modulus G ″ is obtained from Equation (28).

  Here, D, b, B, C, and P represent physical constants that depend on the filler network structure. Furthermore, the terms of G′∞ and G ″ ∞ represent the molecular extension effect from the concept of C.K.L.Davies, D.De, A.G.Thomas (DDT). If this term is ignored in the low strain region before stretching, the strain dependence of the loss tangent tan δ between the storage elastic modulus and the loss elastic modulus, which is a nonlinear viscoelastic equation, is tan δ = G ″ / G ′. From the relationship, it can be expressed by Equation (29).

  Here, a, b, c, and d are physical constants depending on the filler network structure, and tan δ | γ → 0 represents tan δ at zero strain.

By the way, since there is a relationship of γ ² = I ₁ −3, when the equations (27) to (29) are rewritten as a function of the shear strain γ, they can be expressed by the following equations (30) to (32).

  Here, the temperature dependence of each parameter included in the above equations (30) and (32) is best expressed by the improved Arrhenius plot shown below. Therefore, each parameter is expressed using the following equation (33). Represents the temperature dependence of.

Where R is a gas constant, E _k is an activation energy of k, T is a temperature, Tg is a glass transition temperature, and C2 is an arbitrary temperature constant.

  When the above equation (33) is applied to the physical constants A, n, a, b, d depending on the filler network structure and tan δ at the strain 0, as shown in the following equations (34) to (39): Since each parameter can have temperature dependency, it is possible to express temperature dependency continuously.

Where R is the gas constant, E _A is the activation energy of A, E _n is the activation energy of n, a is the activation energy, E _b is the activation energy of b, E _d is the activation energy of d, E _D is the activation energy of tan δ | γ → 0, a ₀ , b ₀ , d ₀ , and D ₀ are physical constants depending on the filler network structure, ΔT represents each temperature, and Tg is a glass of rubber polymer When the transition temperature, C2, is an arbitrary temperature constant, ΔT = C2 + T−Tg.

  Since G′∞ is a parameter affected by the mesh, Equation (40) is applied as an equation for entropy elasticity.

However, G′∞ ₀ represents the activation energy of G′∞.

  The parameters c and C are treated as constants because they hardly show temperature dependence.

  FIG. 3 shows a comparison result between a straight line indicating the temperature dependence of each parameter obtained by the mathematical expressions (34) to (40) described above and an actual measurement value (plots of FIGS. 3 to 5) of the temperature dependence of each parameter. As shown in FIG.

  Note that R is a gas constant, C = c = 0.002, and C2 = 52.

  It can be said that the straight lines obtained by the mathematical formulas (34) to (40) are in good agreement with the plot of the actual measurement values, and the mathematical formulas (34) to (40) well represent the temperature dependence of each parameter. You can see that

  Further, by applying the above formulas (34) to (40), a curve obtained by calculating the storage elastic modulus G ′, the loss elastic modulus G ″, and the loss tangent tan δ of the rubber product, and actual measurement values thereof (FIG. 6). FIG. 6 shows the result of comparison with the plot of FIG. The loss elastic modulus G ″ curve was calculated by multiplying the storage elastic modulus G ′ curve by the loss tangent tan δ curve.

  Any calculated curve is in good agreement with the measured value plot, and it is possible to express the temperature dependence of the storage elastic modulus G ′, loss elastic modulus G ″, and loss tangent tan δ of the filled rubber with high accuracy. It can be confirmed that.

  In the conventional viscoelastic constitutive equation, for each parameter included in the equation representing the storage elastic modulus G ′, loss elastic modulus G ″, and loss tangent tan δ of rubber, for the experimental results measured according to each temperature, Since the parameters were determined by performing the parameter fitting, the temperature dependence could be expressed only discretely. However, by applying the above equations (34) to (40), the storage elastic modulus G ′ of the rubber , The loss elastic modulus G ″, and the loss tangent tan δ by making the parameter included in the equation temperature dependent, it is possible to express the temperature dependency continuously, and the viscoelastic characteristics at a desired temperature can be expressed. It can be expressed by a method such as interpolation or extrapolation. Thereby, in the above mathematical formulas (34) to (40), it is only necessary to calculate the temperature dependence of the viscoelastic property by substituting the temperature to be predicted, so that the viscoelastic property at each temperature can be simply expressed. The viscoelastic characteristic response performance at each temperature can be easily predicted by applying to a predictive design technique such as FEM.

  In addition, it is possible to express the temperature dependence of storage elastic modulus G ′, loss elastic modulus G ″, and loss tangent tan δ of rubber with high accuracy. It is possible to predict the performance of rubber products and tires with higher accuracy.

  Moreover, since these equations can be applied to the experimental results measured at various temperatures, the equations can also be applied to the results of viscoelastic strain dependence at the use temperature of the rubber product. 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 viscoelastic strain dependency after receiving the strain history and the temperature history.

  In addition, by applying the viscoelastic constitutive equation using the temperature-dependent parameters described above to inverse engineering, predict viscoelastic properties to maximize the performance of the product from the product usage conditions, It is also possible to optimize materials by feeding back to material development.

  It is also possible to design an optimal rubber material for achieving the desired performance of the rubber product through rubber simulation, and to provide an efficient and accurate method for designing a useful rubber product. Can do.

  As described above, according to the viscoelastic response performance prediction method according to the present embodiment, the constitutive equation representing the strain dependency of the viscoelastic property using the parameter indicating the temperature dependency is applied to the finite element analysis method. Thus, by predicting the viscoelastic response performance of the rubber product, it is not necessary to experimentally determine the parameters indicating the storage modulus and loss tangent at the temperature to be predicted, and the viscoelastic properties of the rubber product can be obtained without relying on the experiment. The viscoelastic response performance of the rubber product at the prediction target temperature can be easily and accurately predicted.

  In addition, the viscoelastic response performance of rubber products can be accurately predicted by applying a constitutive equation representing the strain dependence of viscoelastic properties that can cope with the small strain region of rubber products to the finite element analysis method. .

  Further, by expressing the temperature dependence of the parameters of the equations representing the storage elastic modulus G ′ and the loss tangent tan δ with an improved Arrhenius plot, the temperature dependence of each parameter can be accurately represented, and the temperature of each parameter By expressing the dependency with high accuracy, the temperature dependency of the storage elastic modulus G ′ and the loss tangent tan δ can be expressed with high accuracy.

  In addition, by accurately expressing the storage modulus G ′ and loss tangent tan δ included in the constitutive equation representing the strain dependence of the viscoelastic properties, the constitutive equation is applied to the finite element analysis method, and the viscoelastic response of the rubber product Performance can be predicted more accurately.

It is the schematic which showed the prediction apparatus of the viscoelastic response performance which concerns on embodiment of this invention. FIG. 6 is a diagram showing a relationship between stretch ratios λ1, λ2, λ3 and stresses t1, t2, t3 representing deformation in the xyz direction on a dimensional axis. (A)-(C) is a figure which shows the straight line and actual value which calculated parameters n, A, and G'infinity using Numerical formula (34), (35), (40), respectively. (A), (B) is a figure which shows the straight line which calculated parameters a and b using Numerical formula (36), (37), and an actual value, respectively. (A) and (B) are diagrams respectively showing a straight line obtained by calculating the parameters d and tan δ | γ → 0 using Equations (38) and (39), and an actual measurement value. Each of (A), (B), and (C) shows the temperature dependence of the storage elastic modulus G ′, loss elastic modulus G ″, and loss tangent tan δ of the filled rubber in the formulas (30) to (32), (34 It is a figure which shows the curve and actual value which were calculated using ()-(39).

Explanation of symbols

50 Elastic response performance prediction device

Claims (9)

A viscoelastic response performance prediction method for predicting viscoelastic response performance of rubber products using a finite element analysis method,
Viscoelastic response performance prediction characterized by predicting viscoelastic response performance of a rubber product using a constitutive equation representing strain dependency of the viscoelastic property of the rubber product using a parameter representing temperature dependency Method.   The viscoelastic response performance prediction method according to claim 1, wherein the constitutive equation includes an equation representing a storage elastic modulus and an equation representing a loss tangent between the storage elastic modulus and a loss elastic modulus. The viscoelastic response performance prediction method according to claim 2, wherein the storage elastic modulus G ′ is represented by the following mathematical formula.

Here, γ represents shear strain, A, C, and n represent physical constants depending on the filler network structure, and G′∞ represents each change in viscoelastic response due to the molecular extension effect. The viscoelastic response performance prediction method according to claim 3, wherein physical constants A, n, and G′∞ depending on the filler network structure are represented by the following mathematical formulas.

However, A ₀ and n ₀ are physical constants which depend on the filler network structure, R represents gas constant, E _A is the activation energy of the A, E _n is n activation energy, G'∞ _0's G'∞ Activation energy, ΔT represents each of the temperatures, where Tg is the glass transition temperature of the rubber polymer and C2 is an arbitrary temperature constant, ΔT = C2 + T−Tg. The viscoelastic response performance prediction method according to claim 2, wherein the loss tangent tan δ is expressed by the following mathematical formula.

Where γ is a shear strain, a, c, b, and d are physical constants depending on the filler network structure, and tan δ | γ → 0 represents each of tan δ at strain 0. 6. The viscoelastic response performance prediction method according to claim 5, wherein tan δ at physical constants a, b, d and strain 0 depending on the filler network structure is represented by the following mathematical formula.

Where E _a is the activation energy of a, E _b is the activation energy of b, E _d is the activation energy of d, E _D is the activation energy of tan δ | γ → 0, a ₀ , b ₀ , d ₀ , And D ₀ represent each of the physical constants depending on the filler network structure. The viscoelastic response performance prediction method according to any one of claims 3 to 6, wherein the shear strain γ is represented by the following mathematical formula, where I ₁ is an invariant of strain.
γ ² = I ₁ -3   A rubber product design method, wherein a rubber product is designed using the viscoelastic response performance prediction method according to any one of claims 1 to 7. A viscoelastic response performance prediction device for predicting the viscoelastic response performance of a rubber product using a finite element analysis method,
A viscoelastic response performance prediction device that predicts viscoelastic response performance of a rubber product using a constitutive equation that represents strain dependency of the viscoelastic property of the rubber product using a parameter that represents temperature dependency.

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