JP2007272416A - Method for predicting elastic responsiveness of rubber product, design method and elastic responsiveness prediction device - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method for predicting elastic responsiveness of a rubber product by using finite element analysis (FEA), a design method and an elastic responsiveness prediction device. <P>SOLUTION: In the elastic responsiveness predicting method for predicting elastic responsiveness of a rubber product by using finite element analysis (FEA), the elastic responsiveness of the rubber product is predicted by using a constitutive equation indicating temperature and strain dependency of elastic modulus of a rubber material composing the rubber product, to which an inverse Langevin function g having a statistic bridge point-to-bridge point segment link number n as a variable is applied. As the invert Langevin function, the following mathematical expression 1 is preferred, and as the statistic bridge point-to-bridge point segment rink number, the following mathematical expression 2 is preferred, wherein λ is an elongation ratio or compression ratio, α represents a frequency factor of statistic segment movement, ε represents an activated energy of statistic segment movement, R represents a gas constant, and T represents an absolute temperature. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

  The present invention relates to an elastic response performance prediction method, a design method, and an elastic response performance prediction apparatus for rubber products, and more particularly, a method for predicting an elastic response performance of a rubber product using a finite element analysis method (FEA), The present invention relates to a rubber product design method using a prediction method and an elastic 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 that reflects the stress-strain relationship of the rubber material used in the FEA calculation in the analysis, the linear elastic equation is shifted to the Mooney-Rivlin equation. Recently, the nonlinear constitutive equation in the large deformation region Introduction has been made.

  However, the constitutive equations of these rubber materials are based on the network deformation theory that focuses on entropic elasticity developed from molecular statistical thermodynamics based on the elongation of rubber molecular chains, but for many industrial rubber products such as tires. There is no constitutive equation that can express temperature dependence, which is an important factor in material design, using parameters with physical meaning.

  In view of this, a constitutive equation represented by the following mathematical formula (1) capable of expressing temperature dependence using parameters having physical meaning has been proposed (Patent Document 4).

  Here, when the constitutive equation of the above formula (1) has been proposed, when considering the application of the filler and rubber three-dimensional model to the calculation such as FEM at the micro level of the rubber, particularly the rubber portion. By applying it to the constitutive equation representing temperature dependence and strain dependence, it is possible to perform simulation at the micro level considering temperature and strain which are important in the design of actual industrial rubber materials.

In Patent Document 4 above, the above equation (1) is the right-hand side relative to the second term NT∂S / ∂I ₁ of entropy term S, the generalized form represented by the following equation (2) Application of the Mooney-Rivlin equation is shown.

Here, C _{i, j, k} includes information on ρRT / Mc, ρ represents the density of rubber, R represents a gas constant, T represents an absolute temperature, and Mc represents a molecular weight between crosslinks.
Japanese Patent Laid-Open No. 11-237332 JP 2003-72327 A Japanese Patent Laid-Open No. 2002-1929242 JP 2005-345413 A

However, in the technique described in Patent Document 4, Ci _{, j, k} included in the above equation (2) applied to the entropy term S in the above equation (1) is a parameter proportional to the absolute temperature. Therefore, there is a problem that these coefficients must be obtained experimentally.

  The present invention has been made to solve the above-described problems, and enables elastic temperature calculation of a rubber to be easily calculated without relying on an experiment, and the elastic response performance of a rubber can be easily predicted. It is an object to provide a prediction method, a rubber product design method, and an elastic response performance prediction device.

  In order to achieve the above object, an elastic response performance prediction method according to the present invention is an elastic response performance prediction method for predicting an elastic response performance indicating deformation and fracture behavior of a rubber product using a finite element analysis method (FEA). The rubber product is expressed using a constitutive equation representing the temperature and strain dependence of the elastic modulus of the rubber material constituting the rubber product, and applying an inverse Langevin function with the number of segment links between statistical cross-linking points as a variable. It is characterized by predicting elastic response performance.

  The elastic response performance prediction apparatus according to the present invention is an elastic response performance prediction apparatus that predicts elastic response performance showing deformation and fracture behavior of a rubber product using a finite element analysis method. The elastic response performance of the rubber product is predicted using a constitutive equation that expresses the temperature and strain dependence of the elastic modulus of the rubber and applies an inverse Langevin function with the number of segment links between statistical crosslinks as a variable.

  As described above, the elastic response performance prediction method and the elastic response performance prediction device of the rubber product according to the present invention have a large deformation region (usually a strain of 100 to 400%) when the filled rubber is observed at a micro level. The elastic response performance of rubber products is predicted by finite element analysis using a constitutive equation representing the temperature and strain dependence of the elastic modulus in a wide range of deformation including For rubber composite products made of rubber materials, the elastic response performance of the rubber part at the micro level with parameters that have physical meaning of temperature and strain dependence, which is important in the design of many industrial rubber products Can be accurately predicted.

  In addition, the inverse Langevin function with the number of segment links between crosslinks as a function of temperature as a variable is applied to the constitutive equation representing the temperature and strain dependence of the elastic modulus of rubber materials to predict the elastic response performance of rubber products. Therefore, the temperature dependence can be calculated without experimentally determining the coefficient of the constitutive equation, and the elastic response performance of the rubber product can be easily predicted.

  The inverse Langevin function is a function that represents a phenomenon in which the stress rapidly increases as the degree of molecular freedom with respect to rubber extension decreases. The number of segment links between cross-linking points is a statistical number of segments that represents the mobility due to the temperature of the cross-linking rubber molecular chain as a parameter having a physical meaning.

  Moreover, the structural equation representing the temperature and strain dependence of the elastic modulus of the rubber material according to the present invention can be a nonlinear equation.

  As a result, a nonlinear equation is used as a constitutive equation representing the temperature and strain dependence of the elastic modulus, so that even when a filler is added and heat generation or destruction occurs, heat generation or destruction occurs in the rubber part. Therefore, nonlinearity is given to the elastic characteristics, and the elastic response performance of the rubber product can be accurately predicted by separating the filler portion and the rubber portion at a micro level in a wide range of strain and temperature.

  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.

  The constitutive equation representing the temperature and strain dependence of the elastic modulus (G) of the present invention can be expressed as follows.

  Where R is a gas constant, N is a mesh number coefficient, к is an intermolecular cohesive energy that contributes to energy elasticity, T is an absolute temperature, ΔT is a difference from the glass transition temperature Tg of the rubber polymer, and S is a rubber deformation time Represents each entropy change.

Further, ∂S / ∂I _{1 in} the second term on the right side of the above mathematical formula (4) can be expressed as follows.

Where λ represents an expansion ratio or compression ratio, C ₁ is a constant related to ρRT / Mc, and represents a parameter corresponding to the C ₁ coefficient of the Mooney-Rivlin equation, ρ is density, and Mc is rubber. The average molecular weight between mesh points, g, represents each of the inverse Langevin functions.

  By applying an inverse Langevin function using the number of segment links between crosslink points n and the extension ratio λ as variables as a function of temperature to the above formula (4) as in the above formula (5), the coefficient of the constitutive equation The temperature dependence can be calculated without experimentally obtaining.

  Moreover, the number n of segment links between statistical cross-linking points can be expressed as follows.

  However, (alpha) represents the frequency factor of statistical segment motion, and (epsilon) represents each of the activation energy of statistical segment motion. Note that ε can be predicted from the microstructure of a molecule as in a known document (Y. Abe, PJ Flory, Journal of Chemical Physics volume 52, p. 2814 (1976)).

  By expressing the statistical link number n between the cross-linking points as in the above formula (6), the statistical link number n between the statistical cross-linking points is a function of the absolute temperature T. By using n as a variable of the inverse Langevin function of the above equation (4), the temperature dependence can be calculated without experimentally obtaining the coefficient of the constitutive equation.

According to the present invention, the generalized form of the Mooney-Rivlin equation (2) above is not applied to the entropy term S of the above equation (1), and the bridge has a physical meaning. It is necessary to experimentally find C _{i, j, k} included in the above equation (2) by applying the inverse Langevin function with the statistical segment number n representing the mobility of the point rubber molecular chain as a variable. Therefore, it is possible to easily calculate the temperature dependence of rubber without relying on experiments.

  Further, the inverse Langevin function g (λ, n) can be expressed as follows.

Further, a configuration obtained by applying the above-described constitutive equation representing the temperature and strain dependence of the elastic modulus of the present invention to the relational expression between stress and strain expressed by the following mathematical formulas (8) and (9). The equation can be used to predict the elastic response performance of a rubber product.

  Where σ represents tensile or compressive stress, τ represents shear stress, and γ represents shear strain.

  As described above, by predicting the elastic response performance 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 described in detail, through the required simulation, In particular, it is possible to design an optimal rubber material at the micro level to achieve the desired performance of the rubber product through simulation at the micro level of the rubber, and the useful rubber product is efficient and accurate. The design method can be provided.

  As described above, according to the elastic response performance prediction method, the rubber product design method, and the elastic response performance prediction device of the present invention, the inverse Langevin function using the number of segment links between crosslinks as a function of temperature as a variable, Applying to the constitutive equation that expresses the temperature and strain dependence of the elastic modulus of rubber material to predict the elastic response performance of rubber products, calculate the temperature dependence without experimentally calculating the coefficient of the constitutive equation The elastic response performance of the rubber product can be easily predicted.

  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 that performs elastic response performance prediction according to an embodiment of the present invention will be described later with an elastic response performance prediction program for executing elastic response performance prediction. It is comprised by the computer arithmetic processing system which performs a process. 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 of 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, particularly preferably the above-described mathematical formulas (4) to (7). Is used to predict the elastic response performance of the rubber product.

  In the above formula (6), obtained by applying 4.2 kcal / mol to the activation energy ε of the statistical segment motion and substituting 54557 for the frequency factor α of the statistical segment motion. FIG. 2 shows a comparison result between the straight line and the actually measured value of temperature dependency of the number of segment links between the statistical cross-linking points of the rubber (plot in FIG. 2). It can be said that the straight line obtained by Equation (6) shows good agreement with the plot of the actual measurement values. In addition, the numerical value of 4.2 kcal / mol of the activation energy ε corresponds to the activation energy for exceeding the steric hindrance in the rotational direction generated between the rubber molecular chain segments between the crosslinking points. This activation energy can also be predicted using an Ising chain model, which is a one-dimensional molecular mirror model, or molecular dynamics (MD).

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

  Further, the temperature dependence of the strain stress curve of the rubber is predicted by using the above-described constitutive equations (4) to (9), and the strain stress curve is calculated using the following unfilled crosslinked SBR pure rubber. The temperature dependence was measured, and the predicted value by the above prediction was compared with the experimental value. The rubber formulation used is SBR 1500: 100 parts, 1 part stearic acid, 3 parts ZnO, 1 part accelerator, 1.75 parts sulfur.

  In addition, the value of the parameter used in said numerical formula (4)-(9) is a value shown in the following tables. К was used as a value having temperature dependence as shown in FIG.

  Fig. 3 shows the predicted value of the elastic response of rubber predicted using the values of the above parameters, and the strain-stress plot measured by changing the temperature with a tensile tester manufactured by Toyo Seiki (tensile speed of 300 mm / s). Shown in

  The temperature dependence of the stress-strain curve expressed using the equations obtained from statistical thermodynamic calculations is in good agreement with the actual measurement results, and the temperature dependence of the strain-stress curve of rubber is expressed accurately. I can say that. In addition, the above results show the temperature dependence of the rubber material very well, and the elastic response performance showing deformation and fracture behavior at the micro level can be expressed accurately by applying this constitutive equation to FEM. Can do.

  Since the constitutive equation representing the temperature and strain dependence of the elastic modulus of the rubber material of the present invention can be applied to the experimental results measured at various temperatures, the elastic strain and temperature at the use temperature of the rubber product. It can also be applied to dependency results. 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.

  In addition, in Formulas (4) and (6), it is only necessary to calculate the predicted value of the elastic response of the rubber using a constitutive equation in which the temperature to be predicted is substituted, so there is no need to experimentally obtain the coefficient of the constitutive equation. The temperature dependence of the elastic modulus of rubber can be calculated without relying on experiments, and the elastic response performance of rubber can be easily predicted.

  As described above, according to the prediction method of the elastic response performance according to the present embodiment, the inverse Langevin function using the number of segment links between crosslinks as a function of temperature as a variable, the temperature of the elastic modulus of the rubber material, and In order to predict the elastic response performance of rubber products by applying it to the constitutive equation representing the strain dependence, the coefficient of the constitutive equation is experimentally considered by considering the temperature dependence of the number of segment links between cross-linking points, which is a physical parameter. Therefore, the temperature dependence of rubber can be calculated without relying on experiments, and the elastic response performance of rubber products can be easily predicted.

  Further, by applying a constitutive equation representing the temperature and strain dependence of the elastic modulus of the rubber material to the FEM, it is possible to accurately represent the elastic response performance showing the deformation and fracture behavior of the rubber product at the micro level.

  In addition, through the simulation at the micro level of rubber, it becomes possible to design the optimal rubber material for achieving the desired performance of the rubber product at the micro level, and the useful rubber product with high efficiency and high accuracy. The design method can be provided.

It is the schematic which showed the prediction apparatus of the elastic response performance which concerns on embodiment of this invention. It is a graph which shows the temperature dependence of the number of segment links between statistical bridge points. It is a graph which shows the predicted value and measured value regarding the temperature dependence of the stress strain curve of rubber. It is a graph which shows the temperature dependence of к.

Explanation of symbols

50 Elastic response performance prediction device

Claims (8)

In the elastic response performance prediction method for predicting the elastic response performance showing the deformation and fracture behavior of rubber products using the finite element analysis method,
The elasticity of the rubber product is expressed using a constitutive equation that expresses the temperature and strain dependence of the elastic modulus of the rubber material constituting the rubber product and applies the inverse Langevin function with the number of segment links between statistical crosslinks as a variable. An elastic response performance prediction method characterized by predicting response performance.   The elastic response performance prediction method according to claim 1, wherein the constitutive equation is a nonlinear equation. The elastic response performance prediction method according to claim 1, wherein the constitutive equation is a mathematical expression shown below.

Where G is the elastic modulus, R is the gas constant, N is the network coefficient, к is the intermolecular cohesive energy that contributes to energy elasticity, T is the absolute temperature, ΔT is the difference from the glass transition temperature Tg of the rubber polymer, S Represents entropy change at the time of rubber deformation, I ₁ represents each invariant of strain, and ∂S / ∂I ₁ is a mathematical expression shown below.

Where λ is an expansion ratio or compression ratio, C ₁ is a constant related to ρRT / Mc, ρ is a density, Mc is an average molecular weight between rubber mesh points, g is an inverse Langevin function, and n is a segment between the statistical cross-linking points. Represents each number of links. The elastic response performance prediction method according to claim 3, wherein the statistical link point n between cross-linking points is a mathematical expression shown below.

However, (alpha) represents the frequency factor of statistical segment motion, and (epsilon) represents each of the activation energy of statistical segment motion. 5. The elastic response performance prediction method according to claim 3, wherein the inverse Langevin function g (λ, n) is a mathematical expression shown below.

The elastic response performance of a rubber product is predicted using a constitutive equation obtained by applying the constitutive equation to a relational expression between stress and strain represented by the following mathematical formulas: 6. The elastic response performance prediction method according to any one of items 5.

Where σ represents tensile or compressive stress, τ represents shear stress, and γ represents shear strain.   A rubber product design method, wherein a rubber product is designed using the elastic response performance prediction method according to claim 1. In the elastic response performance prediction device that predicts the elastic response performance that shows the deformation and fracture behavior of rubber products using the finite element analysis method,
The elasticity of the rubber product is expressed using a constitutive equation that expresses the temperature and strain dependence of the elastic modulus of the rubber material constituting the rubber product and applies the inverse Langevin function with the number of segment links between statistical crosslinks as a variable. Elastic response performance prediction device that predicts response performance.

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