JP2009216612A - Simulation method of rubber material - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To accurately simulate a deformation of a filler-filled rubber. <P>SOLUTION: A simulation method for simulating the deformation of a rubber material containing the rubber, the filler and a third component other than the rubber and the filler includes: a step of setting a rubber material model 2 modeled by an element for numerically analyzing the rubber material; a step of setting conditions for the rubber material model 2, and implementing a deformation calculation; and a step of obtaining the required physical quantity from the deformation calculation. The rubber material model includes: a matrix model 3 as a modeled rubber matrix; and a filler model 4 as a modeled filler. A function for representing a relationship between an extension and a stress as its physicality is set to the matrix model 3. The function includes a parameter corresponding to the blending quantity of the third component. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a rubber material simulation method useful for accurately analyzing deformation of a rubber material in which rubber, a filler, and a third component are blended.

  Rubber materials used in tires, sporting goods, and other various industrial products are filled with fillers such as carbon and silica in order to improve their mechanical properties. In particular, silica has been frequently used in recent years for tires. The reason is that the silica compound has a lower rolling resistance than the carbon compound, which contributes to an improvement in the fuel efficiency of the tire. Silica is an oil-free resource and can be said to be an environmentally friendly filler. Therefore, it is very useful for the future development of tires and the like to analyze the deformation of the rubber material filled with the filler containing silica with high accuracy by computer simulation.

  Conventionally, as a rubber material simulation method using a computer, the following Non-Patent Document 1 and Patent Document 1 are known. In particular, in Patent Document 1, a matrix model and a filler model modeled with elements that can be numerically analyzed are set as the rubber material model, and deformation calculation is performed by a finite element method in consideration of the influence of the filler.

Ellen M. Arruda and Marry C. Boyce `` A THREE-DIMENSIONAL CONSTITUTIVE MODEL FOR THE LARGE STRECH BEHAVIOR OF RUBBER ELASTIC MATERIALSS '' Journal of the Mechanics and Physics of Solids Volume 41, Issue 2, Pages 389-412 (February 1993) Japanese Patent No. 3668238

  By the way, when the simulation of the rubber material is performed, the physical properties of the filler model and the matrix model are defined in advance. For example, since the filler model is handled as a hard elastic body, the elastic modulus is set. Further, the physical property (function indicating the relationship between stress and elongation) based on the result of a tensile test of pure rubber not containing a filler was defined as the physical property in the matrix model.

    However, when the filler-containing rubber is simulated by such a method, the stress-elongation result when the tensile deformation is applied may differ from the stress-elongation result obtained in the actual tensile test. . As a result of various experiments, the inventors have found that the influence of the third component other than the rubber and filler added to the filler-filled rubber is large.

  For example, in a rubber material blended with silica, a silane coupling agent (interface binder) is blended in order to chemically bond silica and rubber. Until now, silane coupling agents were thought to act only on the interface between silica and rubber, but nowadays, some silane coupling agents produce cross-linking agents that bind rubbers together during vulcanization. I know that Therefore, the physical properties of the matrix rubber will change depending on the amount of the silane coupling agent. Therefore, it is important to input the influence of such a silane coupling agent into the physical properties of the matrix rubber in order to simulate a silica material with high accuracy.

  In addition, the inventors conducted a tensile test by adding various amounts of silane coupling agents to silica compounded rubber and pure rubber not containing silica, and measured their 100% modulus. The rubber composition is as follows, and the results are shown in FIG.

Formulation (unit: mass%)
--------------------
SBR1502 100
Pearl silica (* 1) 53.2 (or 0)
Silane coupling agent (* 2) Variable Stearic acid 2.0
Zinc oxide 3.0
Anti-aging agent 13 1.0
Sulfur 1.5
Vulcanization accelerator TBBS (* 3) 1.0
Vulcanization accelerator DPG (* 4) 1.0
--------------------
Vulcanization time: 30 minutes at 150 ° C
* 1) “Sea Hoster” manufactured by Nippon Shokubai Co., Ltd., particle size of about 100 nm
* 2) Si266 (bis (3-triethoxysilylpropyl) disulfide) manufactured by Degussa
* 3) Noxeller NS (N-tert-butyl-2-benzothiazolylsulfanamide) manufactured by Ouchi Shinsei Chemical Industry Co., Ltd.
* 4) Noxeller D (diphenylguanidine) manufactured by Ouchi Shinsei Chemical Co., Ltd.

  As is apparent from the results of FIG. 14, the modulus of the pure rubber not containing silica does not substantially change even when a silane coupling agent is blended. However, it can be seen that the modulus of silica compounded rubber increases as the compounding amount of the silane coupling agent increases even though the compounding amount of silica is constant. That is, it is presumed that the silane coupling agent has a high probability of forming the above-mentioned crosslinking agent only when combined with silica. Therefore, when simulating silica compounded rubber, the conventional method of simply determining the physical properties of the matrix model portion from pure rubber not compounded with silica is not appropriate.

  The present invention has been devised in view of the above problems, and a rubber material model is set including a matrix model that models a rubber matrix and a filler model that models a filler, and The matrix model is based on the fact that the physical properties are set by a function representing the relationship between stress and strain, and the function includes a parameter that varies depending on the amount of the third component other than rubber and filler. The main object is to provide a simulation method capable of calculating the deformation of a rubber material in which a filler is blended with high accuracy.

  Invention of Claim 1 among this invention is a simulation method which simulates deformation | transformation of the rubber material containing rubber | gum, a filler, and 3rd components other than rubber | gum and a filler, Comprising: Numerical analysis of the said rubber material is carried out. A step of setting a rubber material model modeled by possible elements, a step of performing deformation calculation by setting conditions in the rubber material model, and a step of acquiring a necessary physical quantity from the deformation calculation, The rubber material model includes at least a matrix model in which a rubber matrix is modeled and a filler model in which the filler is modeled. In the matrix model, a function expressing the relationship between elongation and stress is set as the physical property. And the function includes a parameter corresponding to a blending amount of the third component. To.

  The invention according to claim 2 is the rubber material simulation method according to claim 1, wherein the filler is silica and the third component is a silane coupling agent.

  The invention according to claim 3 is the rubber material simulation method according to claim 1 or 2, wherein the function generates a high stress in the matrix rubber model by increasing the parameter.

The invention according to claim 4 is the rubber material simulation method according to any one of claims 1 to 3, wherein the function is defined by the following formula (1).

  In the rubber material simulation method of the present invention, a simulation method for simulating deformation of a rubber material including rubber, a filler, and a third component other than the rubber and the filler is provided. In this case, a rubber material model to be analyzed is modeled including a matrix model obtained by modeling a rubber matrix and a filler model obtained by modeling a filler. In the matrix model, physical properties are set by a function representing the relationship between stress and strain, and a parameter that changes according to the blending amount of the third component is input to the function. In such a simulation method, since the physical properties of the matrix model change based on the blending amount of the third component, it is possible to accurately simulate the deformation of the rubber material blended with the filler. Especially when the filler is silica and the third component is a silane coupling agent as in the invention described in claim 2, the calculation accuracy of the stress generated in the matrix model is more effectively increased. be able to.

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.
FIG. 1 shows a computer apparatus 1 for carrying out the simulation method of the present invention. The computer device 1 includes a main body 1a, a keyboard 1b, a mouse 1c, and a display device 1d. Inside the main body 1a, a large-capacity storage device such as a CPU, a ROM, a working memory, and a magnetic disk is provided. The main body 1a is provided with CD-ROM and flexible disk drive devices 1a1 and 1a2. The mass storage device stores a processing procedure (program) for executing a simulation method of the present invention, which will be described later.

  In the simulation method of the present embodiment, deformation of a rubber material including rubber, a filler, and a third component other than the rubber and the filler is simulated. FIG. 2 shows an example of the processing procedure of the simulation method. In the present embodiment, a model of a rubber material is set in which silica is added as a filler and a silane coupling agent is added as a third component (step S1). Examples of the silane coupling agent include bis (3-triethoxysilylpropyl) polysulfide, bis (2-triethoxysilylethyl) polysulfide, bis (3-trimethoxysilylpropyl) polysulfide, and bis (2-trimethoxysilyl). Ethyl) polysulfide, bis (4-triethoxysilylbutyl) polysulfide, bis (4-trimethoxysilylbutyl) polysulfide and the like. These silane coupling agents may be used alone or in combination of two or more. It may be used. Of these, bis (3-triethoxysilylpropyl) disulfide is preferably used because of the addition effect of the silane coupling agent and the cost.

  In FIG. 3, an example of the rubber material model 2 as a microscopic structure is visualized. The rubber material model 2 is obtained by replacing a minute region of a rubber material to be analyzed (whether or not actually exist) with a finite number of small elements 2a, 2b, 2c. Each element 2a, 2b, 2c... Is defined so as to be capable of numerical analysis. The possibility of numerical analysis means that deformation calculation for each element or the entire system can be performed by a numerical analysis method such as a finite element method, a finite volume method, a difference method, or a boundary element method. Specifically, for each element 2a, 2b, 2c..., A node coordinate value, an element shape, a material characteristic, etc. in the coordinate system are defined. For each of the elements 2a, 2b, 2c, etc., for example, a triangular or quadrilateral element as a two-dimensional plane, and a tetrahedral element, for example, is preferably used as a three-dimensional element. Thus, the rubber material model 2 constitutes numerical data that can be handled by the computer device 1.

  The rubber material model 2 of this embodiment is analyzed in a plane strain state in a deformation simulation described later. Therefore, there is no distortion in the z direction. In this embodiment, the length of one side of the rubber material model 2 is, for example, 300 nm × 300 nm in both vertical and horizontal directions.

  Further, the rubber material model 2 of the present embodiment includes a matrix model 3 in which a rubber matrix portion is modeled, and dispersed and arranged in the matrix model 3, and in this embodiment, silica as a filler is modeled. A filler model 4 and an interface model 5 interposed between the matrix model 3 and the filler model 4 and forming an interface between them are exemplified.

  The matrix model 3 constitutes a main part of the rubber material model 2 and is expressed using a plurality of elements such as triangles or quadrilaterals. In order to perform the deformation calculation, a function representing the relationship between stress and elongation is defined for each element constituting the matrix model 3 as its physical property. In the rubber material simulation method of this embodiment, in order to express a rubber elastic response, the rubber parts of the matrix model 3 and the interface model 5 are both calculated based on the molecular chain network theory.

  As shown in FIGS. 4A and 4B, the molecular chain network theory is that the rubber material a as a continuum has a microscopic structure in which randomly oriented molecular chains c are connected at junctions b. It is premised on the idea of having a mesh structure. The junction point b is, for example, a chemical bond between molecules, and includes a crosslinking point.

  In the molecular chain network theory, it is assumed that the average position of the junction b does not change for a long time with respect to the atomic fluctuation period, and the perturbation around the junction b is ignored. Furthermore, it is assumed that the end-to-end vector of the molecular chain c having two junctions b and b at both ends co-deforms with the rubber material continuum in which it is embedded. It is burned.

One molecular chain c is assumed to be composed of a plurality of segments d as shown in FIG. Each segment d is chemically equivalent to one in which a plurality of monomers f in which carbon atoms are connected by covalent bonds are connected, as shown in FIG. Since the individual carbon atoms can freely rotate around the bond axis between the atoms, the segment d can take various forms such as winding as a whole. If the number of monomers f is sufficiently large, the macroscopic property of the segment d does not change according to the scaling rule, so that one segment d is treated as a minimum repeating unit in the molecular chain network theory. Furthermore, the shape of the molecular chain defined by the two junctions b and b shown in FIG. 4C is assumed to follow a non-Gaussian statistical distribution. Therefore, the stress generated when the elongation (stretch) λ is applied in the direction connecting the two joining points b and b can be expressed by the following equation (1).

  The overall response characteristic of the network structure can be obtained in consideration of the contribution of individual molecular chains. However, it is mathematically difficult to calculate in consideration of all of them. For this reason, in the molecular chain network theory, an averaging method is introduced. In this embodiment, it conforms to the molecular chain network theory adopting the 8-chain model. That is, in this method, as shown in FIG. 5A, the rubber material which is a superelastic body is macroscopically a cubic network structure h in which minute 8-chain models g. Is assumed. In addition, as shown in an enlarged view in FIG. 5B, one 8-chain model g is composed of eight joint points b2 provided at each vertex from one joint point b1 defined at the center of the cube. The calculation is performed assuming that the molecular chain c extends.

  In the present embodiment, the disappearance of the junction point b corresponding to the strain of the material is considered in the molecular chain network theory. In an actual rubber material, it is known that intertwined portions of molecular chains (that is, the junction point b) disappear due to a large strain during a deformation process under load. That is, the number of junction points b decreases. For example, as shown in FIG. 6A, when tensile stress in the direction of the arrow acts on the molecular chains c1 to c4 joined at one joining point b, the joining points b are stretched by the molecular chains c1 to c4. Disappears under great strain. As a result, as shown in FIG. 6B, the molecular chains c1 and c2 that have been two in the past become one long molecular chain c5. The same applies to the molecular chains c3 and c4. Such a phenomenon occurs sequentially as the load deformation of the rubber material proceeds. In addition, the decrease in the number of junction points b increases the number of segments included in one molecular chain c.

Next, the above phenomenon is specifically applied to the mesh structure h shown in FIG. The network structure h is assumed to have k 8-chain models g coupled in the width direction, the height direction, and the depth direction. When the total number of junction points b included in the mesh structure h is represented by the symbol m as “entanglement number”, it is represented by the formula (2). Similarly, the number n of molecular chains c included in the network structure h (that is, the number of molecular chains included in the unit volume of the matrix model 3) n is expressed by Expression (3).
m = (k + 1) ³ + k ³ (2)
n = 8k ³ Formula (3)

Here, if k is a sufficiently large number, the following expression (4) is obtained by omitting the third order term of k from the above expression (2). Furthermore, from the relationship between the expressions (3) and (4), the entanglement number m is expressed by the expression (5) using n.
m = 2k ³ Formula (4)
m = n / 4 Formula (5)

Furthermore, since the mesh structure h has no material in and out before and after deformation, the total number N _A of the segments can be assumed to be always constant. Therefore, Expressions (6) and (7) are established.
N _A = n · N Equation (6)
_{N = N A / n = N} A / 4m ... formula (7)

Here, the disappearance of the junction point b corresponds to a decrease in the number m of entanglements in the matrix model 3 and an increase in the number of segments included in one molecular chain c. Therefore, in order to introduce the decrease in the junction points accompanying the above-described load deformation (elongation) into the molecular chain network theory using the 8-chain model, the average number of segments per molecular chain in the above formula (1) N is increased according to the parameter λ _c for strain. Examples of the strain-related parameter λ _c include strain, elongation, strain rate, or first-order invariant of strain. Further, the load deformation is a deformation in which the strain of the model increases in a very short time, and conversely, the deformation in the unloading deformation is a deformation in which the strain decreases.

Further, in the present embodiment, only for the molecular chain network theory applied to the matrix model 3, an increase in the junction points according to the blending amount of the silane coupling agent is considered. In FIG. 7, the volume content of silica is fixed to 20%, the volume content μ of the silane coupling agent μ (= v _c / v _o [%]; v _c : the volume of the silane coupling agent, v _o : The relationship between the nominal stress and the nominal strain when a tensile test is performed until the strain reaches 50% is shown for a plurality of types of silica-blended rubber in which only the total volume of rubber) is changed. As is apparent from FIG. 7, it is understood that the silica-containing rubber is cured (the elastic modulus is increased) as the volume content μ of the silane coupling agent is increased. The reason is that, as described above, the silane coupling agent that exists outside the interface between the rubber and silica particles causes a condensation reaction during vulcanization and is generated as a crosslinking agent that chemically bonds the rubber molecular chains together. It is guessed that it worked. Bonding of rubber molecular chains by such a silane coupling agent corresponds to generating a new bonding point in the matrix rubber and reducing the number of segments contained in one molecular chain. Therefore, in order to introduce the action of the silane coupling agent into the molecular chain network theory, in this embodiment, based on the increase in the amount of the silane coupling agent, the amount per molecular chain in the formula (1) is determined. Decrease the average number of segments N.

From the above, in the matrix model 3 of this embodiment, the average number of segments N per molecular chain is equal to the initial number of segments N ₀ per molecular chain as shown in the following formula (8). It is defined as a function in which the term of the parameter f (λ _c ) corresponding to the strain is added and the parameter g (μ) corresponding to the blending amount of the silane coupling agent is subtracted.
N (λ _c , μ) = N ₀ + f (λ _c ) −g (μ) (8)
Here, the symbols are as follows.
N ₀ : initial number of segments per molecular chain μ: v _c / _vo
v _c : volume of silane coupling agent v _o : total volume of rubber

  Therefore, the equation (1) using the average segment number N obtained by the above equation (8) can change the stress depending on the strain of the matrix model 3 and the blending amount of the silane coupling agent. In particular, by including the term of the parameter {g (μ)}, the deformation calculation of the silica-containing rubber material can be performed in consideration of the influence of the silane coupling agent on the matrix rubber, so that the calculation accuracy is improved. . The coefficients of the functions f and g can be determined as appropriate (details will be described in the embodiments).

  Next, the filler model 4 shows a model of silica as a filler. In this embodiment, it is modeled by a plurality of triangular or quadrilateral elements. The silica is composed of single particles that are very hard compared to rubber having a diameter of about 60 to 300 nm. The filler model 4 has physical properties that are substantially equal to the physical properties of the silica to be analyzed. The filler model 4 is handled as an elastic body, not a viscoelastic body.

  In the present embodiment, five filler models 4 are included in the rubber material model 2 so that the volume content μ of the filler model 4 is 20%. The particle sizes of the filler models 4 are all set equal. However, the number of filler models 4 included in the rubber material model 2 may be appropriately determined according to the amount of silica compounded in the rubber to be analyzed. In addition to silica, carbon black or the like may be used as the filler.

  The interface model 5 corresponds to a thin interface portion including a silane coupling agent that chemically bonds silica and matrix rubber, and is modeled. The interface model 5 continuously surrounds the filler model 4 with a small thickness t, and is composed of two layers of elements in this embodiment. Further, the inner peripheral surface of the interface model 5 is in contact with the filler model 4 and the outer peripheral surface thereof is in contact with the matrix model 3. For example, conditions that do not separate each other can be set for each of these contact surfaces. The thickness t of the interface model 5 is not particularly limited, but is set to about 10 to 30%, more preferably about 15 to 25% of the diameter of the filler model 4 in view of various experimental results. It is desirable to be done.

The interface model 5 also defines the relationship between stress and elongation according to the above equation (1). However, the average number of segments N in the equation (1) used in the interface model 5 is set to a fixed value as in the following equation (9). In addition, it is considered that the interface portion is harder than the matrix rubber due to the influence of the silane coupling agent. For this reason, in this embodiment, the initial value N ₀ of the average number of segments N in Expression (9) is made smaller than the initial value N ₀ of the average number of segments N in Expression (8) applied to the matrix model 3. It is difficult to stretch. Thereby, the interface model 5 of the present embodiment defines viscoelastic properties that are harder than the matrix model 4 (however, the interface model 5 is set to be softer than the filler model 4).
N (λ _c ) = N ₀ (9)

  In the deformation simulation of the rubber material model 2 to be described later, the parameter λc is constantly calculated for each element of the matrix model 3 and the interface model 5 at the time of load deformation, and these are substituted into the equations (8) and (9). The Thereby, the number of segments N of the element is constantly updated and taken into the simulation.

  Next, deformation conditions for deforming the rubber material model 2 are set. In the present embodiment, conditions for applying tensile deformation to the rubber material model 2 by adding an arbitrary average strain rate in the y direction of FIG. 3 are defined. On the contrary, after reaching the predetermined strain amount, the strain is gradually reduced to zero at the same strain rate as described above. However, it goes without saying that various deformation conditions can be determined.

  Next, in the simulation method of the present embodiment, a deformation simulation is performed using the set rubber material model 2 (step S3). A specific processing procedure of the deformation simulation is shown in FIG. In the deformation simulation, first, various data of the rubber material model 2 are input to the computer device 1 (step S31). The input data includes information such as the positions of nodes and material properties defined for each element.

  The computer apparatus 1 creates a stiffness matrix for each element based on the input data (step S32), and then assembles a stiffness matrix for the entire structure (step S33). Displacement of known nodes and nodal forces are introduced into the stiffness matrix of the entire structure (step S34), and the stiffness equation is analyzed. Then, unknown node displacement is determined (step S35), and physical quantities such as strain, stress and principal stress of each element are calculated and output (steps S36 to 37). In step S38, it is determined whether or not to end the calculation. If negative, step S32 and subsequent steps are repeated. Such simulation (deformation calculation) can be performed using, for example, engineering analysis application software using the finite element method (for example, LS-DYNA developed and improved by Livermore Software Technology, Inc., USA). it can.

This simulation is performed based on a homogenization method (asymptotic expansion homogenization method). As shown in FIG. 9, the homogenization method expresses x _I representing the whole rubber material M having the microscopic structure (unit cell) shown in FIG. 3 and the microscopic structure. Two independent variables with y _I are used. By asymptotically expanding each independent variable in different scale fields, the microscopic scale and the macroscopic scale, the average mechanical response of the entire rubber material reflecting the model structure of the microscopic structure shown in FIG. Can be sought. In other words, if the analysis target area is composed of repetitions of an arbitrary microscopic structure, and the repetition degree is so dense that the area cannot be discretized directly by the finite element method, the analysis target is replaced with a homogeneous equivalent model. By analyzing the whole and returning the analysis result to the microscopic structure at an arbitrary point, the deformation of the microscopic structure itself can be obtained approximately.

  When the deformation calculation is performed, a necessary physical quantity can be acquired from the result (step S4). As the physical quantity, a stress-strain curve is particularly effective for examining the deformation behavior of the silica-containing rubber. Further, the time-series deformation state of each element of the rubber material model 2 can be visualized and displayed. At this time, it is desirable to color each element according to the stress.

  In the present invention, the contribution of the third component (silane coupling agent) other than rubber and filler to the rubber physical properties is carved out. For this reason, if the matrix portion of the filler-filled rubber in which the amount of the third component is changed by several levels is determined in advance, some characteristics of the filler-filled rubber when the blending amount of the third component is further varied are tested. Predictable without data. Accordingly, the development efficiency of the rubber material is improved. In the above embodiment, silica is used as the filler, and an example of the silane coupling agent is shown as the third component. However, in the present invention, carbon is used as the filler, and the third component is used instead of these combinations. It can also be applied to the case where a dialkyltin oxide + thiuram compound, a dithiocarbamate compound, a thiazole compound, or a sulfenamide compound is used. Moreover, about the parameter regarding the compounding quantity of a 3rd component, although the volume content rate was mentioned as an example in the said embodiment, if it changes according to a compounding quantity, various things, such as mass, mass ratio, and surface area ratio, will be mentioned, for example. It goes without saying that can be adopted.

1) Analytical model An overall macroscopic model of a silica-blended rubber material having the microscopic structure shown in FIG. 3 was periodically set. The size of the macroscopic model was a rectangular shape of 2 mm × 2 mm.

2) Specific constitutive equation of 8-chain model In this example, in order to describe the viscoelastic behavior of rubber more accurately, as shown in FIG. 10, a viscoelastic 8-chain model and a model composed of a damper Was used. The actual molecular chain of rubber has viscosity due to friction with surrounding molecular chains. In order to express such friction, a standard model of a spring / damper having viscous resistance was introduced into each molecular chain of the 8-chain model A. Therefore, the stress σ _c generated when the elongation λ _c is applied in the direction connecting the two junctions of the monomolecular chain can be expressed by the following equation from the above equation (1).

Further, assuming that the stretch of each element of the single molecular chain shown in FIG. 10 is λ _α , λ _β , and λ _γ , λ _α = λ _c . The other subscripts α, β, and γ are also associated with the elements shown in FIG. However, λ _c = λ _β · λ _γ .

Further, when a strain energy density function W corresponding to work per unit volume based on the volume before deformation is used, the stress σ _c is expressed as the following equation.

  From the above two formulas, the following formula holds identically.

In the case of the 8-chain model, if the main stretches are λ ₁ , λ ₂ , and λ ₃ , the molecular chain stretch λ _c can be expressed as √ {(λ ₁ ² + λ ₂ ² + λ ₃ ² ) / 3}. Because it can, the following equation holds.

From the above three equations, the stress σ _i ^{A in} the main stretch direction of the 8-chain model ^A and the stretch λ _i are given by the following relationship.

The main and stretch direction of the stress sigma _i ^B of the 8 chain model B, the stretch lambda _i 'is given by the following relation.

In this embodiment, an incompressible rubber material is handled. Therefore, the hydrostatic pressure p is used to satisfy the incompressibility. At this time, using the above two equations, the constitutive equation of the incompressible rubber material is as follows.
σ _i = σ _i ^A + σ _i ^B −p

The speed format of the above constitutive equation can be expressed as follows, and this equation is used in this embodiment.

2) Average number of segments N of one molecular chain in the matrix model
The following formula was adopted.
N (λ _c , μ) = N ₀ + f (λ _c ) −g (μ)
= N ₀ + a ₀ + a ₁ λ _c + a ₂ λ _c ² −√ (b ₀ + b ₁ μ)
Here, an example of how to determine specific functions of f (λ _c ) and g (μ) will be described. First, the change of the entanglement point accompanying the material deformation of the matrix model depends only on λ _c (elongation). Therefore, first, a simulation is performed with a pure rubber not filled with silica, and the coefficient of f (λ _c ) is determined so that the result matches the actual pure rubber test result. First, in the first load deformation cycle, the number of entanglement points at the time of loading changes (decreases), and at the time of unloading, it does not change from the value at the start of unloading. Then, the coefficient of the function f (λ _c ) is determined so that the simulation result matches the actual test result. It should be noted that the change in the number of entanglement points is irreversible at the time of reloading after being unloaded once. That is, at the time of re-loading, it is assumed that the average segment number N does not change in the deformation region below the maximum elongation reached in the previous load deformation cycle. That is, the value of N in the unloading of the first deformation cycle and the reloading of the second deformation cycle are the same. When the deformation further proceeds and exceeds the maximum elongation experienced in the first deformation cycle, the value of N is determined to change (decrease) again. Next, the result of this simulation is compared with the tensile test result of the silica compounded rubber in which the compounding amount of the silane coupling agent is changed, and the relationship between the difference ΔN and μ (FIG. 11) is examined. From this result, the coefficient of g (μ) is determined. Specifically, the following parameters were set.
N _αA = N _{αA_0} + a ₀ + a ₁ λ _c + a ₂ λ _c ² -√ (b ₀ + b ₁ μ)
(However, N is the change in this form, N _.alpha.A, N _{alpha B} in FIG. 10, only N _.alpha.A of N _{[beta] A)}
N _{αA_0} = 8.3
a ₀ = 1.20
a ₁ = −2.19
a ₂ = 0.99
b ₀ = −7.17
b ₁ = 9.52
N _αB = 9.8, N _βA = 10.0
N _A = n ・ N = 4.75 × 10 ²⁶
C ^R _αB = 0.08, C ^R _βA = 0.185
C ^R _αA is a value determined from the value of N _αA from C ^R = n · k _B · T

3) Average number of segments N of one molecular chain in the interface model
N (λ _c ) = N ₀ = 2.0 (fixed value)
a ₀ = 1.20
a ₁ = -2.19
Others are the same as the above values.
N _αA = 2.0
N _αB = 9.8
N _βA = 10.0
C ^R _αA = 0.97, C ^R _αB = 0.08, C ^R _βA = 0.185

4) Parameters such as filler model Longitudinal elastic modulus of filler model: 100 MPa
Poisson's ratio of filler model: 0.3
Material temperature: T = 296K
k _B = 1.38066 × 10 ⁻²⁹
C _β ^R = 0.185 MPa
C _α ^R = 0.08 MPa
N _A = n · N = 4.75 × 10 ²⁶
N _A0 = 8.38
N _αB = 9.8
Volume content μ of filler model: 20%
Longitudinal elastic modulus E of filler model: 100 MPa
Poisson's ratio ν of filler model: 0.3

5) for generating a uniform uniaxial tensile deformation in deformation conditions the macroscopic model, added deformation speed 100 mm / min constant in x ₂ direction in FIG. 9, the predetermined maximum strain (2.5 or 4.0) Gradually increased. Further, after reaching the maximum strain, the strain was gradually reduced to zero at the same deformation rate as described above. The macroscopic model was calculated based on a molecular chain network theory using an 8-chain model so as not to change in the thickness direction (Z-axis direction in FIG. 3).

6) Calculation Results First, FIG. 12 shows a stress-strain curve of a matrix model alone used for the rubber material model of the silica-containing rubber of this example. Moreover, the experimental value and simulation result by the tensile test of pure rubber are also shown. The pure rubber in the simulation is obtained by omitting the term of g (μ) relating to the blending amount of the silane coupling agent from the function of the number of segments N contained in one molecular chain in the formula (8). From FIG. 12, it is reproduced that the matrix model in the silica-containing rubber has a higher modulus than that of the pure rubber.

  Next, in FIG. 13A, simulations and experimental values are shown together for the stress-strain curve of the rubber material model according to this example. As is clear from FIG. 13A, it can be seen that the simulation result shows a very good correlation with the experimental value. FIG. 13B shows the silica-compound rubber using the matrix model in which the physical properties are set by removing the term of the parameter g (μ) of the third component from the equation (8) used in FIG. A simulation result is shown (comparative example). In this example, it can be confirmed that the accuracy is worse than that in FIG.

It is a perspective view which shows an example of the computer apparatus used by this embodiment. It is a flowchart which shows the process sequence of this embodiment. It is a diagram showing one embodiment of a rubber material model (microscopic structure). (A) is a viscoelastic material, (B) is a diagram for explaining the structure of one molecular chain, (C) is an enlarged view of one molecular chain, and (D) is an enlarged view of a segment. (A) is a perspective view which shows the network structure of a viscoelastic material, (B) is a perspective view which shows an example of an 8-chain model. (A), (B) is a diagram explaining the fracture | rupture of the junction of a molecular chain. It is a stress-strain curve when changing the volume content (micro | micron | mu) of the silane coupling agent of a silica compounding rubber. It is a flowchart which shows the procedure of a deformation | transformation simulation. The relationship between the microscopic structure explaining the homogenization method and the entire structure is shown. It is a block diagram of the 8-chain model of an Example. It is a graph which shows the relationship between the volume content rate (micro | micron | mu) of a silane coupling agent, and (DELTA) N. The stress-strain curve of a matrix model is shown. (A) is an Example method, (b) shows the stress-strain curve of the silica compounded rubber by the comparative example method. It is a graph explaining the influence of the volume content rate of a silane coupling agent.

Explanation of symbols

1 Computer device 2 Rubber material model 3 Matrix model 4 Filler model 5 Interface model

Claims (4)

A simulation method for simulating deformation of a rubber material including rubber, a filler, and a third component other than rubber and the filler,
Setting a rubber material model obtained by modeling the rubber material with an element capable of numerical analysis;
Performing deformation calculation by setting conditions in the rubber material model;
Obtaining a necessary physical quantity from the deformation calculation, and
The rubber material model includes at least a matrix model that models a rubber matrix and a filler model that models the filler, and the matrix model has a function that represents a relationship between elongation and stress as its physical properties. As well as
The rubber function simulation method, wherein the function includes a parameter corresponding to a blending amount of the third component.   The rubber material simulation method according to claim 1, wherein the filler is silica and the third component is a silane coupling agent.   3. The rubber material simulation method according to claim 1, wherein the function generates a high stress in the matrix rubber model by increasing the parameter. The rubber function simulation method according to claim 1, wherein the function is defined by the following formula (1).

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