JP2013250673A - Simulation method, simulation device and simulation program for dynamic behavior of viscoelastic body/filler complex - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a simulation method, simulation device and simulation program of the simple dynamic behavior of a viscoelastic body/filler complex in which filler is dispersed in a viscoelastic body such as rubber.SOLUTION: The simulation method of the dynamic behavior of a viscoelastic body/filler complex includes: an input step (a) of inputting computational conditions for simulating a dynamic behavior including the physical property values of a viscoelastic body and filler and the initial arrangement of the filler in the viscoelastic body; an arithmetic step (b) of solving the equation of motion of fluid followed by the viscoelastic body under the input computational conditions; a moving amount calculation step (c) of calculating the moving amount of the filler on the basis of the solution of the equation of motion of the fluid; an arrangement update step (d) of updating the arrangement of the filler by using the calculated moving amount of the filler; and an output step (e) of outputting data relating to the dynamic behavior of the viscoelastic body/filler complex calculated by the repeated processing of the steps (b) to (d). The equation of motion followed by the viscoelastic body is added with the elastic quality of the viscoelastic body.

Description

  The present invention relates to a simulation method, a simulation apparatus, and a simulation program. More specifically, the present invention simulates the dynamic behavior of a material in which a filler is dispersed in a viscoelastic body (hereinafter referred to as “viscoelastic body / filler composite”). The present invention relates to a simulation method, a simulation apparatus, and a simulation program.

  Conventionally, it is known that there is a reinforcing effect when a filler such as carbon black or silica is blended with a viscoelastic body, for example, rubber, and a material in which a filler is blended with rubber (hereinafter referred to as “rubber / filler composite”). Is applied to rubber products such as various tires and conveyor belts. It is known that the filler portion constituting the rubber-filler composite has a complicated network structure in which a plurality of fillers are connected, and the rubber-filler composite exhibits a somewhat uneven filler distribution.

  In order to realize the high performance of the rubber / filler composite having such a complicated filler structure, it is important to grasp the dynamic behavior of the rubber / filler composite when the deformation is given. So far, various methods for simulating the dynamic behavior of such rubber-filler composites have been proposed.

  For example, Patent Document 1 describes a method for predicting the viscoelastic response performance of a rubber product using a finite element method (FEM). Patent Document 2 describes a technique for evaluating a rubber material containing at least rubber and carbon black using a coarse-grained molecular dynamics method.

  However, the method described in Patent Document 1 requires a calculation that satisfies an appropriate boundary condition such as a non-slip condition with respect to the boundary of the solid surface, and a material having many boundaries such as a rubber / filler composite. However, there is a problem that the calculation cost becomes enormous. In addition, in FEM and the like, a mesh for dividing the analysis target into small elements is created, but this mesh needs to be set according to the shape of the filler, and when the position of the filler changes from moment to moment, Another problem is that the calculation cost of re-creating the mesh becomes enormous.

  Moreover, in the method described in Patent Document 2, it is not necessary to consider the boundary conditions required in FEM. However, in order to evaluate macroscopic properties, enormous calculation costs are required, and the rubber-filler composite It is practically impossible to evaluate the macroscopic physical properties of such a large system.

  On the other hand, in Non-Patent Document 1, in a colloidal dispersion system in which colloidal particles are dispersed in a liquid solvent, the dynamic behavior of the colloidal dispersion system is calculated by regarding the colloidal particles not as a rigid body but as a highly viscous liquid domain. A so-called fluid particle dynamics (FPD) method has been proposed. In this FPD method, the high-viscosity domain of colloidal particles and the physical properties of the surrounding viscous liquid are connected smoothly, so the dynamic behavior of the colloidal dispersion system considering the fluid effect can be obtained without revealing the boundary conditions. It is possible to simulate with high efficiency.

  Furthermore, Patent Document 3 describes a method for simulating the dynamic behavior of a charged colloidal dispersion system in which charged particles are dispersed in a liquid solvent by developing the method described in Non-Patent Document 1. A simulation of a colloid electrophoresis phenomenon that occurs in a colloidal dispersion of charged pigment particles between electrodes in an electrophoretic display has been performed.

  As described above, the dynamic behavior of the colloidal dispersion system can be simulated with high efficiency by the FPD method. Therefore, the above-mentioned FPD method is applied to the simulation of the dynamic behavior of the rubber / filler composite, and a high-performance rubber is applied. -Expected to be useful in the design of filler composites.

JP 2005-49333 A JP 2006-64658 A JP 2005-308782 A

H. Tanaka and T.K. Araki, Phys. Rev. Lett. 85 (6), 1338 (2000)

However, in the FPD method described in Non-Patent Document 1, it is assumed that a viscous liquid is used as a solvent. Therefore, when the solvent is a viscoelastic material, such as a rubber-filler composite, in the FPD method. Application to has never been studied. For this reason, it has become unknown under what conditions the simulation can be performed.
Therefore, an object of the present invention is to provide a simulation method, a simulation apparatus, and a simulation program capable of easily simulating the dynamic behavior of a viscoelastic body / filler composite in which a filler is dispersed in a viscoelastic body such as rubber. It is in.

  The inventor diligently studied how to solve the above problems. In the FPD method, the Navier-Stokes equation, which is the equation of motion of the fluid followed by the solvent, includes as parameters the viscosity of the liquid solvent and the solid colloid particles, and the viscosity of the colloid particles is higher than the viscosity of the liquid solvent. By setting, the flow does not enter the particles, and the boundary conditions of the colloidal particles are taken into account. However, in the FPD method, it has been found that even when a filler, which is a rigid particle, is filled in a liquid solvent, the elastic properties of the filler cannot be taken in and the effect of improving rubber elasticity by the filler cannot be evaluated. . Therefore, in order to solve this problem, the inventors have come to the recognition that it is necessary to perform a simulation using a formula that takes into account the elastic characteristics of the viscoelastic body, which is a solvent, and have completed the present invention.

That is, the gist of the present invention is as follows.
(1) A method for simulating the dynamic behavior of a viscoelastic body / filler composite in which a filler is dispersed in a viscoelastic body,
a) an input step of inputting calculation conditions for simulating the dynamic behavior, including physical property values of the viscoelastic body and the filler, and an initial arrangement of the filler in the viscoelastic body;
b) a calculation step for solving an equation of motion of the fluid that the viscoelastic body follows under the input calculation conditions;
c) a movement amount calculating step for calculating a movement amount of the filler based on a solution of the equation of motion of the fluid;
d) an arrangement update step of updating the filler arrangement using the calculated amount of movement of the filler;
e) a data output step for outputting data relating to the dynamic behavior of the viscoelastic body / filler composite obtained by the repetitive processing of steps b) to d);
And the equation of motion of the fluid that the viscoelastic body follows is a simulation method that takes into account the elastic properties of the viscoelastic body.

(2) The simulation method according to (1), wherein an equation of motion of the fluid followed by the viscoelastic body is represented by the following equation (A). Where ρ is the mass density, D is the strain rate tensor of the continuum, v is the velocity, p is the pressure, F _p is the external field, Π is the stress tensor of the continuum, and _{ｊ j} ^(b) is the stress tensor of the continuum. component of the viscoelastic body portion, Π _j ^(f) is the component of the filler portion of the stress tensor of the continuum, _{N b} is the number of viscoelastic Maxwell element in the viscoelastic body, _{N f} is the viscoelastic Maxwell element in the filler Τ _j ^(b) is the relaxation time of the Maxwell element in the viscoelastic body, τ _j ^(f) is the relaxation time of the Maxwell element in the filler, and η _j ^(f) is the viscosity of the viscosity element in the Maxwell element in the filler. , η _{j ^(b)} at the viscosity of the viscous element in Maxwell element in the viscoelastic body, phi is a position other than the filler Returns 1 at the position of the filler RETURN function, ∇ is an upper convection derivative (Upper Convective time derivative).

(3) Viscosity η _j ^(f) and η _j ^(b) and relaxation times τ _j ^(f) and τ _j ^(b) in the formula (A) satisfy the following formula (B), (2) The simulation method described in 1.

(4) The data output step further includes a viscoelasticity analysis step of analyzing viscoelasticity of the viscoelastic body / filler composite based on the obtained data. The simulation method according to one item.

(5) An apparatus for simulating the dynamic behavior of a viscoelastic body / filler composite in which a filler is dispersed in a viscoelastic body, the physical properties of the viscoelastic body and the filler, and the viscoelastic body in the viscoelastic body An input unit for inputting a calculation condition for simulating the dynamic behavior, including an initial arrangement of fillers, an arithmetic unit for solving an equation of motion of the fluid that the viscoelastic body follows under the input calculation condition, A movement amount calculator that calculates the movement amount of the filler based on a solution of the equation of motion of the fluid, a placement updater that updates the placement of the filler using the calculated movement amount of the filler, and the arithmetic unit A data output unit for outputting data relating to the dynamic behavior of the viscoelastic body / filler composite obtained by the movement amount calculator and the arrangement updater; Provided, the equation of motion of the fluid in which the rubber is followed, the simulation device, characterized in that is obtained by considering the elastic properties of the rubber.

(6) The simulation apparatus according to (5), wherein an equation of motion of fluid that the viscoelastic body follows is represented by the following expression (A). Where ρ is the mass density, D is the strain rate tensor of the continuum, v is the velocity, p is the pressure, F _p is the external field, Π is the stress tensor of the continuum, and _{ｊ j} ^(b) is the stress tensor of the continuum. component of the viscoelastic body portion, Π _j ^(f) is the component of the filler portion of the stress tensor of the continuum, _{N b} is the number of viscoelastic Maxwell element in the viscoelastic body, _{N f} is the viscoelastic Maxwell element in the filler Τ _j ^(b) is the relaxation time of the Maxwell element in the viscoelastic body, τ _j ^(f) is the relaxation time of the Maxwell element in the filler, and η _j ^(f) is the viscosity of the viscosity element in the Maxwell element in the filler. , η _{j ^(b)} at the viscosity of the viscous element in Maxwell element in the viscoelastic body, phi is a position other than the filler Returns 1 at the position of the filler RETURN function, ∇ is an upper convection derivative (Upper Convective time derivative).

(7) Viscosity η _j ^(f) and η _j ^(b) and relaxation times τ _j ^(f) and τ _j ^(b) in the formula (A) satisfy the following formula (B), (6) The simulation apparatus described in 1.

(8) The analysis unit further includes a viscoelasticity analyzer that analyzes viscoelasticity of the viscoelastic body / filler composite based on the obtained data, and any one of (5) to (7). The simulation apparatus according to item.

(9) a computer configured as a device for determining the dynamic behavior of a viscoelastic body / filler composite in which a filler is dispersed in a viscoelastic body; a) physical properties of the viscoelastic body and the filler, and the viscoelastic body An input step for inputting a calculation condition for simulating the dynamic behavior, including an initial arrangement of the filler in the medium, and b) an equation of motion of the fluid that the viscoelastic body follows under the input calculation condition A calculation step for solving; c) a movement amount calculating step for calculating a movement amount of the filler based on a solution of the equation of motion of the fluid; and d) updating the arrangement of the filler using the calculated movement amount of the filler. And e) outputting data relating to the dynamic behavior of the viscoelastic body / filler composite obtained by the repeated processing of e) and steps b) to d). That is executed and a data output step, the motion equation of the fluid simulation program, characterized in that is obtained by considering the elastic properties of the viscoelastic body.

(10) The simulation program according to (9), wherein an equation of motion of fluid that the viscoelastic body follows is represented by the following equation (A). Where ρ is the mass density, D is the strain rate tensor of the continuum, v is the velocity, p is the pressure, F _p is the external field, Π is the stress tensor of the continuum, and _{ｊ j} ^(b) is the stress tensor of the continuum. component of the viscoelastic body portion, Π _j ^(f) is the component of the filler portion of the stress tensor of the continuum, _{N b} is the number of viscoelastic Maxwell element in the viscoelastic body, _{N f} is the viscoelastic Maxwell element in the filler Τ _j ^(b) is the relaxation time of the Maxwell element in the viscoelastic body, τ _j ^(f) is the relaxation time of the Maxwell element in the filler, and η _j ^(f) is the viscosity of the viscosity element in the Maxwell element in the filler. , η _{j ^(b)} at the viscosity of the viscous element in Maxwell element in the viscoelastic body, phi is a position other than the filler Returns 1 at the position of the filler RETURN function, ∇ is an upper convection derivative (Upper Convective time derivative).

(11) The viscosities η _j ^(f) and η _j ^(b) and relaxation times τ _j ^(f) and τ _j ^(b) in the formula (1) satisfy the following formula (B), (10) The simulation program described in 1.

(12) The data output step further includes a viscoelasticity analysis step of analyzing viscoelasticity of the viscoelastic body / filler composite based on the obtained data. The simulation program according to one item.

  ADVANTAGE OF THE INVENTION According to this invention, the simulation method, simulation apparatus, and simulation program which can simulate the dynamic behavior of a viscoelastic body and filler composite body simply can be provided.

It is a figure which shows the physical property of the rubber filler composite body calculated | required by FPD method of a nonpatent literature 1. It is a flowchart which shows one Embodiment of the simulation method of this invention. (A) is a diagram showing the relationship between _{R = (η f / τ f} ) / (η b / τ b) and the filler effect, showing the relationship between (b) is τ _f / τ _b and the filler effect FIG. It is a figure which shows one Embodiment of the simulation apparatus of this invention. (A) is a figure which shows the relationship between the filling rate of a filler, and a Guth-Gold factor, (b) is a figure which shows the relationship between a frequency, an amplitude, and a phase shift. It is a figure which shows the elasticity modulus of the rubber filler composite obtained by the method of this invention and the nonpatent literature 1. FIG.

(Simulation method)
Embodiments of the present invention will be described below with reference to the drawings.
The method for simulating the dynamic behavior of the viscoelastic body / filler composite of the present invention is to obtain the dynamic behavior including a) the physical properties of the viscoelastic body and the filler, and the initial arrangement of the filler in the viscoelastic body. An input step for inputting a calculation condition for the fluid, b) an operation step for solving the equation of motion of the fluid that the viscoelastic body follows under the input calculation condition, and c) the movement of the filler based on the solution of the equation of motion of the fluid A moving amount calculating step for calculating the amount, d) a placement updating step for updating the filler placement using the calculated filler moving amount, and e) a viscoelastic body obtained by the repeated processing of steps b) to d). A data output step for outputting data relating to the dynamic behavior of the filler composite. Here, it is important that the equation of motion of the fluid followed by the viscoelastic body takes into account the elastic properties of the viscoelastic body. In the present invention, the “filler” is a filler that is blended into the viscoelastic body, such as carbon black or silica, for the purpose of reinforcing the viscoelastic body, and means a particulate material. Here, the shape of the filler is not limited to a spherical shape. Further, the “viscoelastic body / filler composite” means a material in which a filler is dispersed in a viscoelastic body. In the following description, the case where the viscoelastic body is rubber will be described as an example, but the present invention is not limited thereto.

  As described above, the FPD method described in Non-Patent Document 1 is a colloidal dispersion system that takes into account the fluid effect of a viscous liquid solvent without the need to explicitly solve the boundary conditions as in FEM described in Patent Document 1. It is possible to simulate the dynamic behavior of the robot with high efficiency. In this FPD method, the motion of a viscous liquid as a solvent is expressed by the following Navier-Stokes equation.

ここで、Ｄ／Ｄｔはラグランジュ微分（＝∂／∂ｔ＋ｖ・∇）、Ｆ_ｐは力場、ρは質量密度、ｖは速度、ｐは圧力、ηは粘度、ζは熱遥動力である。

Here, D / Dt is Lagrangian differential (= ∂ / ∂t + v · ∇), F _p is a force field, ρ is mass density, v is velocity, p is pressure, η is viscosity, and ζ is thermal power.

  When the FPD method is applied to the simulation of the dynamic behavior of the rubber-filler composite, in the above formula (1), by setting the value of the viscosity η to a larger value than when using a viscous liquid as a solvent, It is considered that the reinforcement by the rubber filler can be taken in approximately. Therefore, the dynamic behavior of the rubber / filler composite was simulated using the formula (1), and its viscoelastic properties were examined. Specifically, since it is considered that the thermal energy is very small with respect to the viscosity of the rubber, it is set to zero, and the filling rate (area fraction) of the rigid filler is changed from 0% to 7% in a two-dimensional model, Simulations were performed under periodic shear conditions. Non-dimensionalization was performed under the conditions of density ρ = 1, viscosity η = 0.5, and filler radius R = 3.8. The obtained results are shown in FIG. Here, FIG. 1A shows the relationship between the filler filling rate and the storage elastic modulus G ′, and FIG. 1B shows the relationship between the filler filling rate and the loss elastic modulus G ″. Here, the storage elastic modulus G ′ and the loss elastic modulus G ″ are quantities characterizing dynamic viscoelasticity, as will be described later, and the storage elastic modulus G ′ and the loss elastic modulus G ″ are the values of the viscoelastic body. It is an amount that characterizes elastic properties and viscous properties. As is apparent from these figures, the loss elastic modulus G ″ increases as the filler filling rate increases, while the storage elastic modulus G ′ is 0 regardless of the filler filling rate. It can be seen that the elastic property is not incorporated. That is, when a simulation was performed by the FPD method, it was found that even if a rubber was filled with a filler, the elastic modulus could not be obtained and an appropriate result could not be obtained.

  In this way, the inventor has found that the conventional FPD method cannot correctly simulate the dynamic behavior of the viscoelastic body, so that the equation of motion of fluid that can incorporate the elastic properties of rubber as a solvent can be taken into account. We have come to realize that it is necessary to perform a simulation using Therefore, the inventor came to derive the equation of motion of the fluid used in the following simulation method of the dynamic behavior of the rubber / filler composite of the present invention after trial and error.

ここで、Πは連続体の応力テンソルであり、

と表され、また、ｖは、連続の条件から以下の式（４）の関係を満たす。

That is, first, the momentum conservation law of an incompressible viscous fluid is given by the following equation (2).

Where Π is the stress tensor of the continuum,

In addition, v satisfies the relationship of the following formula (4) from the continuous condition.

In the present invention, as a result of intensive studies to consider the elastic properties of the viscoelastic body, the constitutive equation of the stress tensor Π in the above formula (2) is considered separately for the filler portion and the rubber portion, and the Maxwell model Based on the above, it was found that the elastic properties of the viscoelastic body can be taken into account by rewriting with the following formulas (5) and (6).

である。よって、

である。また、Ｄは、

である。上記式（５）および（６）は、Ｍａｘｗｅｌｌモデルに基づいたものであるが、原理的には、非線形性を含むＭａｘｗｅｌｌモデル以外のモデルに基づいた粘弾性構成則を導入することも可能である。また、φは、フィラーの位置で１を返しフィラー以外の位置で０を返す連続的な関数である。 Here, the stress tensor of the continuum [pi, D is strain rate tensor of the continuum, _{N b} is the number of viscoelastic Maxwell element of rubber parts, _{N f} viscoelastic Maxwell element number of the filler portion, eta is the viscosity of Maxwell element The viscosity of the element, τ is the relaxation time of the Maxwell element, and η / τ represents the elastic modulus of the elastic element in the Maxwell element. Also, ∇ is the upper convective time derivative (Upper Convective Time Derivative),

It is. Therefore,

It is. D is

It is. The above formulas (5) and (6) are based on the Maxwell model, but in principle it is also possible to introduce a viscoelastic constitutive law based on a model other than the Maxwell model including nonlinearity. . Φ is a continuous function that returns 1 at the filler position and 0 at positions other than the filler.

In describing the constitutive equation of the stress tensor に お け る in the above formula (2) separately for the filler part and the rubber part, several methods are conceivable. The inventors examined various equations before deriving the above equations (5) and (6). For example, the constitutive equation of the stress tensor Π can be expressed as the following formulas (11) and (12).

  However, if the above formulas (11) and (12) are used, viscoelastic relaxation characteristics are uneven in the vicinity of the interface of the moving filler, and the results are worse than in the formulas (5) and (6). Furthermore, it was found that the convergence of the calculation deteriorates and it is difficult to perform the simulation. Therefore, as a result of intensive studies on an expression that does not cause such a problem, an expression in which φ is taken into consideration only for D, such as the above expressions (5) and (6), has been derived. The validity of these equations is shown by verifying the filler effect described later.

あるいは、

を用いることができる。ここで、ｒはフィラー中心からの距離、Ｒはフィラー半径、ａはフィラーとゴム部分を接続する界面の厚さ、δｘは離散格子の空間刻みである。 Here, when φ is one and spherical, for example,

Or

Can be used. Here, r is the distance from the filler center, R is the filler radius, a is the thickness of the interface connecting the filler and the rubber part, and δx is the spatial increment of the discrete lattice.

  When the filler is not spherical, instead of using the above formulas (13) and (14), a function that appropriately represents the shape of the filler is used, or 1 is returned at the position of the filler and a position other than the filler. Can also be configured to return 0. In this way, a case where the filler is not spherical can be dealt with.

  In this way, it was possible to derive the fluid equation of motion (6) in which the filler is expressed as a domain having viscoelastic characteristics greatly different in the rubber as the solvent. By solving the equation (6) and the equations (2), (4) and (5) (corresponding to the equation (A) in the claims) simultaneously, the dynamic behavior of the viscoelastic body / filler is obtained. Can be simulated. In Equation (6), the ratio of the viscous component to the elastic component of the constitutive equation of the stress Π changes according to the shape of the filler at the position where the filler is present, so the boundary condition between the filler and rubber is explicitly treated. Therefore, the calculation cost can be reduced, and the dynamic behavior of the filler-dispersed material can be simulated. In addition, the position of the center of gravity of the filler changes due to the stress from the rubber, which is a viscoelastic body, and the interaction force between the fillers. It becomes possible to simulate about. Hereinafter, each step of the simulation method of the present invention will be described with reference to the flowchart of FIG. Here, the case where the viscoelastic body is rubber is described as an example, but the present invention is not limited thereto.

First, in step S1, calculation conditions for obtaining dynamic behavior including physical property values of rubber and filler and initial arrangement of filler in rubber are input. Here, the physical property values of rubber and filler are specifically the viscosity and elastic modulus of rubber, its frequency characteristics, the size, number of fillers, viscosity, elastic modulus and its frequency characteristics.
The calculation conditions for simulating the dynamic behavior of rubber-filler composites are specifically the number of filler placement updates, time and space mesh, and the magnitude and frequency of periodic shear strain applied to the system. Etc. The filler is arranged by randomly specifying the coordinates of the center of gravity.

ここで、εおよびａはフィラー間の相互作用の大きさを決定するパラメータである。 Next, in step S2, the equation of motion (6) of the fluid followed by the rubber as the solvent is solved under the calculation conditions input in step S1. Specifically, first, the force F _i acting on each filler is calculated using the model potential U (r). As the model potential U (r), for example, a Leonard-Jones (LJ) type potential represented by the following formula (15) can be used.

Here, ε and a are parameters that determine the magnitude of the interaction between the fillers.

ここで、Ｎはフィラーの数であり、「力Ｆ_ｉ」は、ベクトル量である。こうして求められた各フィラーに働く力Ｆ_ｉに、i番目のフィラーのφ_i（ｒ）を掛け合わせて全てのフィラーについて足し合わせることにより、流体の運動方程式（６）における力場Ｆ_pが得られる。

ここで、Ｖ_ｉは、要素ｉの体積である。この力場Ｆ（ｒ）を式（６）に代入して式（６）を解くことにより、各フィラーの速度ｖ_ｉが求められる。また、「速度ｖ_ｉ」は、ベクトル量である。 By using such a model potential U (r), the force F _i acting on each particle can be obtained by calculating the gradient of U (r) (by partial differentiation of U (r) with the filler position vector r _i. ) You can ask.

Here, N is the number of fillers, and “force F _i ” is a vector quantity. The force field F _p in the fluid equation of motion (6) is obtained by multiplying the force F _i acting on each filler thus obtained by φ _i (r) of the i-th filler and adding up all the fillers. It is done.

Here, V _i is the volume of the element i. By substituting this force field F (r) into the equation (6) and solving the equation (6), the velocity v _{i of} each filler can be obtained. Further, “velocity v _i ” is a vector quantity.

Subsequently, in step S3, the amount of movement of the filler is calculated based on the solution of the fluid equation of motion. More specifically, each velocity after Δt (for example, after 0.01 second) is obtained by substituting the velocity v _{i of} each filler calculated by solving the equation of motion (6) in step S2 into the following equation (18). calculating the movement amount [Delta] r _i of filler. Here, the “movement amount Δr _i ” is a vector amount.

Thereafter, in step S4, the filler arrangement is updated using the obtained amount of filler movement. That is, obtained in step S3, the movement amount [Delta] r _i of each filler after Delta] t, by adding together the coordinates of each filler, and updates the arrangement of the filler. By repeating the above steps S2 to S4, the dynamic behavior of the rubber / filler composite can be simulated.

  When evaluating viscoelasticity, a specified periodic shear boundary condition is set in the system to solve the simultaneous equations (2) and (4) to (6). As the boundary condition, for example, a periodic boundary condition by the Lees-Edwards method is imposed on an amount obtained by subtracting the shear rate from the actual velocity v. Even when the filler crosses the boundary, the calculation is performed by imposing periodic boundary conditions according to the position and speed of the filler according to the shear rate.

Finally, in step S5, it is determined whether or not the processing in steps S2 to S4 has been performed a predetermined number of times (for example, 10000 times when Δt = 0.01 seconds and 100 seconds are simulated). If it is, the data regarding the dynamic behavior of the obtained rubber / filler composite is output and the process ends. If not, the process returns to step S2. Here, the data relating to the dynamic behavior of the rubber / filler composite includes data such as the arrangement and speed of the filler in each repetition.
In this way, the dynamic behavior of the rubber / filler composite can be simulated by repeatedly performing the processes of steps S2 to S4.

Here, the relationship between the viscosity η and elastic modulus τ of the rubber in formula (6) and the filler effect is examined. FIG. 3 shows the relationship between the viscosity and elastic modulus of the rubber / filler composite and the filler effect. FIG. 3A shows R ₁ = (η _f / τ _f ) / (η _b / τ _b ). FIG. 3B shows the relationship between R ₂ = τ _f / τ _b and the filler effect. In the present invention, the “filler effect” means the actual increase in storage modulus and loss modulus relative to the increase in storage modulus and loss modulus expected by the filler (increment obtained by simulation). (Theoretical value = 1 + 2.5 g, g is a filling rate), R ₁ is the ratio of the elastic modulus of the elastic element in each Maxwell element in the filler part and the rubber part, η _f / τ _f and η _b / tau _b, the elastic modulus of the elastic element in the Maxwell element in each filler portion, means the elastic modulus of the elastic element in the Maxwell element in the rubber portion. The above results were obtained under conditions where the rubber / filler composite was given 5% strain and vibration having a frequency of 0.01. The numerical values of the simulation were made dimensionless under the conditions of density ρ = 1, viscosity η = 0.5, and filler radius R = 3.8.

As is clear from these figures, the filler effect increases monotonically with increasing values of R ₁ or R ₂ . That is, the larger the value of R ₁ or R ₂ , the greater the filler effect. For example, by setting R ₁ to 50 or more and R ₂ to 100 or more, a filler effect of 90% or more can be obtained, and it can be considered that the filler as a rigid body can be expressed in the equation of motion of fluid.

In the above-described simulation method of the dynamic behavior of the viscoelastic body / filler composite of the present invention, in the data output step, the viscoelastic analysis is performed to analyze the viscoelasticity of the viscoelastic body / filler composite based on the obtained data. By further performing the step, physical properties such as viscosity and elastic modulus of the viscoelastic body / filler composite can be obtained. Explaining rubber as an example of the viscoelastic body, specifically, the physical property value of the rubber / filler composite is obtained as a storage elastic modulus G ′ and a loss elastic modulus G ″ of dynamic viscoelasticity. The G ′ and G ″ are obtained as follows. That is, a periodic shear strain γ ₀ sin (ωt) having a magnitude (amplitude) γ ₀ and a frequency ω is inclined in a direction (x direction) perpendicular to the direction of the shear flow in the system direction (x direction). The average stress σ at that time can be calculated from the following equation (19).

となった時の、Ａ／γ_０、Ｂ／γ_０がそれぞれ、Ｇ’、Ｇ’’である。これらＧ’およびＧ’’は、完全弾性体ではＧ’≠０、Ｇ’’＝０であり、完全粘性体ではＧ’＝０、Ｇ’’≠０である。また、ｔａｎδ＝Ｇ’’／Ｇ’が小さいほど、弾性体に近い粘弾性材料を意味する。
こうして、従来のＦＰＤ法では実質的に不可能だった、粘弾性体・フィラー複合体のマクロな物性を求めて評価することができ、ゴム等の粘弾性材料の設計に有効利用することができる。 This stress

A / γ ₀ and B / γ ₀ are G ′ and G ″, respectively. These G ′ and G ″ are G ′ ≠ 0 and G ″ = 0 in the complete elastic body, and G ′ = 0 and G ″ ≠ 0 in the complete viscous body. Moreover, the smaller tan δ = G ″ / G ′, the viscoelastic material closer to the elastic body.
Thus, it is possible to evaluate and evaluate the macroscopic physical properties of the viscoelastic body / filler composite, which is practically impossible with the conventional FPD method, and can be effectively used for designing viscoelastic materials such as rubber. .

(Simulation device)
Next, a simulation apparatus for the dynamic behavior of the viscoelastic body / filler composite of the present invention will be described. FIG. 4 is a diagram showing an embodiment of a simulation apparatus for the dynamic behavior of the viscoelastic body / filler composite of the present invention. The device 1 includes an input unit 11, an analysis unit 12, and a display unit 13. The analysis unit 12 includes a calculator 21, a movement amount calculator 22, an arrangement updater 23, and a data output unit 24. Hereinafter, although the case where a viscoelastic body is rubber | gum is demonstrated to an example, it is not limited to this.

  The input unit 11 inputs calculation conditions for simulating dynamic behavior including physical properties of rubber and filler, and initial arrangement of the filler in the rubber. For example, a keyboard or a mouse can be used as the input unit 11.

  The analysis unit 12 solves the physical property values of the rubber and the filler inputted by the input unit 11 and the equation (6) that is the equation of motion of the fluid that the rubber follows under the calculation conditions for obtaining the dynamic behavior. The dynamic behavior of the rubber / filler composite is simulated by calculating the amount of movement. Hereinafter, each structure of the analysis part 12 is demonstrated.

  The computing unit 21 solves the equation of motion of the fluid that the rubber follows under the arrangement of the filler input by the input unit 11. Here, equation (6), which is a fluid equation of motion that takes into account the elastic properties of rubber, is stored in advance in a storage device (not shown), and is read from the storage device when solving the equation of motion of the fluid. It is configured to be.

  The movement amount calculator 22 calculates the movement amount of the filler based on the solution of the fluid equation of motion obtained by the calculator 21.

  The arrangement updating unit 23 updates the filler arrangement using the filler movement amount calculated by the movement amount calculator 22.

  The data output unit 24 outputs data related to the dynamic behavior of the rubber / filler composite obtained by the calculator 21, the movement amount calculator 22, and the arrangement updater 23.

  The analysis unit 12 including the computing unit 21, the movement amount calculator 22, the arrangement updater 23, and the data output unit 24 can be configured by a CPU, a memory, a storage device, and the like in a personal computer.

  The display unit 13 displays data related to the rubber / filler composite output by the data output unit 24. For example, a display can be used as the display unit 13.

Further, in the equation of motion (6) of the fluid followed by the rubber, the relationship between R = (η _f / τ _f ) / (η _b / τ _b ) and the filler effect, and τ _f / τ _{b shown in FIG.} And the filler effect, by setting R to 50 or more and τ _f / τ _b to 100 or more, a filler effect of 90% or more can be obtained, and the filler as a rigid body can be expressed in the fluid equation of motion.
Thus, the dynamic behavior of the viscoelastic body / filler composite can be simply simulated by the simulation apparatus of the present invention.

  In the simulation apparatus of the present invention described above, the analysis unit 12 is based on data relating to the dynamic behavior of the viscoelastic body / filler composite obtained by the calculator 21, the movement amount calculator 22, and the arrangement updater 23. A viscoelasticity analyzer for analyzing viscoelastic characteristics of the viscoelastic body / filler composite may be further included. Thus, it is possible to evaluate and evaluate the macroscopic physical properties of the viscoelastic body / filler composite, which is practically impossible with conventional simulation apparatuses, and can be effectively used for designing viscoelastic materials such as rubber. .

(Simulation program)
A computer can be suitably used as the above-described simulation apparatus 1 of the present invention, and such a computer stores a program describing processing contents for realizing each function of the means shown in FIG. This program can be realized by reading out and executing this program by a central processing unit (CPU) of the computer. Thus, by causing a computer to execute the simulation program of the present invention, the dynamic behavior of the viscoelastic body / filler composite can be easily simulated.

となることが期待される。ここで、ωは周波数、Ｇ’およびＧ’’はゴム・フィラー複合体の粘弾性体の粘弾性係数であり、Ｇ’は貯蔵弾性率、Ｇ’’は損失弾性率である。また、Ｇ’_ｂｕｌｋおよびＧ’’_ｂｕｌｋはフィラーがない場合のゴムの粘弾性係数である。 <Verification of validity of simulation method>
First, the validity of the simulation method of the present invention was verified. Here, rubber was assumed as the viscoelastic body. It is known as Einstein's law or Guth-Gold's law that when the rigid filler is diluted thinly in the liquid or elastic body, the fluid effect is that the overall viscosity and elastic modulus change depending on the number of fillers. It has been. Theoretically, when small fillers are dispersed, the Reynolds number is usually sufficiently small, and the behavior can be analyzed by the Stokes approximation ignoring the influence of inertia. One filler is considered as a sparse limit. It agrees with the result when
It is considered that the same is true for the fluid effect of the filler in the viscoelastic medium. Therefore, it was verified that the Guth-Gold rule, which is one of the effects of increasing the elastic modulus, was established when a filler was blended with rubber. That is, when the above law is established, the area fraction g of the filler is

It is expected to be Here, ω is a frequency, G ′ and G ″ are viscoelastic coefficients of the viscoelastic body of the rubber / filler composite, G ′ is a storage elastic modulus, and G ″ is a loss elastic modulus. G ′ _bulk and G ″ _bulk are viscoelastic coefficients of rubber in the absence of filler.

Fig.5 (a) is a figure which shows the relationship between the filling rate of a filler, and Guth-Gold factor A by the simulation method of this invention. Here, this relationship is obtained by applying 10% strain and vibration having a frequency of 0.01 to the rubber / filler composite. The numerical values were made dimensionless under the conditions of density ρ = 1, viscosity η = 0.5, and filler radius R = 3.8. As is clear from this figure, it can be seen that the Guth-Gold factor increases as the filling rate of the filler increases, and the stress of the rubber / filler composite increases.
FIG. 5B shows the relationship between the frequency given to the rubber / filler composite, the Guth-Gold factor, and the phase shift according to the simulation method of the present invention. The amplitude A and phase shift Δ are obtained by fitting the stress σ (t) with the filler with σ ₀ (t) = Aσ ₀ (t + Δ / ω) with respect to the stress σ ₀ (t) with no filler. This is the amount obtained. Here, the strain applied to the rubber / filler composite is 10%, the area fraction g of the filler is about 8%, and τ _b is 100. First, looking at the relationship between the Guth-Gold factor and the frequency, it can be seen that the dependence on the frequency is hardly seen up to about 0.02, but increases rapidly when it exceeds 0.03. It can also be seen that the phase shift increases monotonically with increasing frequency.
This frequency dependence is considered to be influenced by the inertial term of the filler part. In the case of rubber or the like, the influence of inertia is small, and in order to simulate an experimentally obtained frequency, it is necessary to perform simulation for a long time with a considerably small frequency. Assuming that the Reynolds number is approximately ρR ² ω / η≈30Ω, under this condition, the frequency dependence is sufficiently small when 30ω << 1, and the viscoelasticity of the rubber-filler composite can be obtained. I understand.

<Evaluation of viscoelastic properties of rubber / filler composites>
The viscoelastic properties of the rubber / filler composite were examined by the simulation method of the present invention (invention example) and the FPD method (comparative example) of Non-Patent Document 1. Specifically, the filler was moderately aggregated with respect to the two-dimensional model, arranged at an area fraction of 20%, and the LJ interaction between the fillers was calculated as ε = 0.5. The viscosity and elastic modulus at a strain of 3%, 10%, and 30% with respect to a frequency of 0.01 were obtained. The numerical values were made dimensionless under the conditions of density ρ = 1, viscosity η = 0.5, filler radius R = 3.8. R ₁ = 50 and R ₂ = 100. The obtained result is shown in FIG. As is apparent from this figure, the storage elastic modulus G ′ includes a contribution from the viscoelastic body as a contribution other than the filler, and G ″ also changes accordingly. In the invention example, the rubber filler It can be seen that the viscoelastic properties of the composite are correctly determined.

DESCRIPTION OF SYMBOLS 1 Simulation apparatus 11 Input part 12 Analysis part 13 Display part 21 Calculator 22 Movement amount calculator 23 Arrangement updater 24 Data output device S1 Input step S2 Calculation step S3 Movement amount calculation step S4 Arrangement update step S5 Data output step

Claims (12)

A simulation method of dynamic behavior of a viscoelastic body / filler composite in which a filler is dispersed in a viscoelastic body,
a) an input step of inputting calculation conditions for simulating the dynamic behavior, including physical property values of the viscoelastic body and the filler, and an initial arrangement of the filler in the viscoelastic body;
b) a calculation step for solving an equation of motion of the fluid that the viscoelastic body follows under the input calculation conditions;
c) a movement amount calculating step for calculating a movement amount of the filler based on a solution of the equation of motion of the fluid;
d) an arrangement update step of updating the filler arrangement using the calculated amount of movement of the filler;
e) a data output step for outputting data relating to the dynamic behavior of the viscoelastic body / filler composite obtained by the repetitive processing of steps b) to d);
And the equation of motion of the fluid that the viscoelastic body follows is a simulation method that takes into account the elastic properties of the viscoelastic body. The simulation method according to claim 1, wherein an equation of motion of fluid that the viscoelastic body follows is expressed by the following equation (A). Where ρ is the mass density, D is the strain rate tensor of the continuum, v is the velocity, p is the pressure, F _p is the external field, Π is the stress tensor of the continuum, and _{ｊ j} ^(b) is the stress tensor of the continuum. component of the viscoelastic body portion, Π _j ^(f) is the component of the filler portion of the stress tensor of the continuum, _{N b} is the number of viscoelastic Maxwell element in the viscoelastic body, _{N f} is the viscoelastic Maxwell element in the filler Τ _j ^(b) is the relaxation time of the Maxwell element in the viscoelastic body, τ _j ^(f) is the relaxation time of the Maxwell element in the filler, and η _j ^(f) is the viscosity of the viscosity element in the Maxwell element in the filler. , η _{j ^(b)} at the viscosity of the viscous element in Maxwell element in the viscoelastic body, phi is a position other than the filler Returns 1 at the position of the filler RETURN function, ∇ is an upper convection derivative (Upper Convective time derivative).

The simulation according to claim 2, wherein the viscosity η _j ^(f) and η _j ^(b) and the relaxation times τ _j ^(f) and τ _j ^(b) in the formula (A) satisfy the following formula (B). Method.

  The simulation method according to any one of claims 1 to 3, wherein the data output step further includes a viscoelasticity analysis step of analyzing viscoelasticity of the rubber / filler composite based on the obtained data. An apparatus for simulating the dynamic behavior of a rubber / filler composite in which a filler is dispersed in a viscoelastic body,
An input unit for inputting calculation conditions for simulating the dynamic behavior, including physical property values of the viscoelastic body and the filler, and an initial arrangement of the filler in the viscoelastic body;
A calculator that solves the equation of motion of the fluid that the viscoelastic body follows under the input calculation conditions; a calculator of amount of movement that calculates the amount of movement of the filler based on the solution of the equation of motion of the fluid; A placement updater that updates the placement of the filler using the amount of movement of the filler, and the motion of the viscoelastic body / filler composite obtained by the computing unit, the movement amount calculator, and the placement updater. An analysis unit having a data output device for outputting data relating to the mechanical behavior;
And the equation of motion of the fluid followed by the viscoelastic body takes into account the elastic properties of the viscoelastic body. The simulation apparatus according to claim 5, wherein an equation of motion of fluid that the viscoelastic body follows is expressed by the following equation (A). Where ρ is the mass density, D is the strain rate tensor of the continuum, v is the velocity, p is the pressure, F _p is the external field, Π is the stress tensor of the continuum, and _{ｊ j} ^(b) is the stress tensor of the continuum. component of the viscoelastic body portion, Π _j ^(f) is the component of the filler portion of the stress tensor of the continuum, _{N b} is the number of viscoelastic Maxwell element in the viscoelastic body, _{N f} is the viscoelastic Maxwell element in the filler Τ _j ^(b) is the relaxation time of the Maxwell element in the viscoelastic body, τ _j ^(f) is the relaxation time of the Maxwell element in the filler, and η _j ^(f) is the viscosity of the viscosity element in the Maxwell element in the filler. , η _{j ^(b)} at the viscosity of the viscous element in Maxwell element in the viscoelastic body, phi is a position other than the filler Returns 1 at the position of the filler RETURN function, ∇ is an upper convection derivative (Upper Convective time derivative).

The simulation according to claim 6, wherein the viscosity η _j ^(f) and η _j ^(b) and the relaxation times τ _j ^(f) and τ _j ^(b) in the formula (A) satisfy the following formula (B). apparatus.

  The simulation apparatus according to claim 5, wherein the analysis unit further includes a viscoelasticity analyzer that analyzes viscoelasticity of the viscoelastic body / filler composite based on the obtained data. . In a computer configured as a device for determining the dynamic behavior of a viscoelastic body / filler composite with a filler dispersed in a viscoelastic body,
a) an input step of inputting calculation conditions for simulating the dynamic behavior, including physical property values of the viscoelastic body and the filler, and an initial arrangement of the filler in the viscoelastic body;
b) a calculation step for solving an equation of motion of the fluid that the viscoelastic body follows under the input calculation conditions;
c) a movement amount calculating step for calculating a movement amount of the filler based on a solution of the equation of motion of the fluid;
d) an arrangement update step of updating the filler arrangement using the calculated amount of movement of the filler;
e) a data output step for outputting data relating to the dynamic behavior of the viscoelastic body / filler composite obtained by the repetitive processing of steps b) to d);
And the equation of motion of the fluid takes into account the elastic properties of the viscoelastic body. The simulation program according to claim 9, wherein an equation of motion of fluid that the viscoelastic body follows is expressed by the following equation (A). Where ρ is the mass density, D is the strain rate tensor of the continuum, v is the velocity, p is the pressure, F _p is the external field, Π is the stress tensor of the continuum, and _{ｊ j} ^(b) is the stress tensor of the continuum. component of the viscoelastic body portion, Π _j ^(f) is the component of the filler portion of the stress tensor of the continuum, _{N b} is the number of viscoelastic Maxwell element in the viscoelastic body, _{N f} is the viscoelastic Maxwell element in the filler Τ _j ^(b) is the relaxation time of the Maxwell element in the viscoelastic body, τ _j ^(f) is the relaxation time of the Maxwell element in the filler, and η _j ^(f) is the viscosity of the viscosity element in the Maxwell element in the filler. , η _{j ^(b)} at the viscosity of the viscous element in Maxwell element in the viscoelastic body, phi is a position other than the filler Returns 1 at the position of the filler RETURN function, ∇ is an upper convection derivative (Upper Convective time derivative).

The viscosity η _j ^(f) and η _j ^(b)) and relaxation times τ _j ^(f) and τ _j ^(b) in the formula (A) satisfy the following formula (B). Simulation program.

  The simulation according to any one of claims 9 to 11, wherein the data output step further includes a viscoelasticity analysis step of analyzing viscoelasticity of the viscoelastic body / filler composite based on the obtained data. program.

Images