JP6637845B2 - Simulation method and simulation program - Google Patents

Description

  The present invention relates to a simulation method and a simulation program, and more particularly to a simulation method and a simulation program for simulating viscoelastic properties of a filler-filled rubber in which a filler is dispersed in a matrix polymer.

  A nanoscale composite in which a polymer made of pure rubber is used as a matrix (substrate) and a filler made of carbon black and / or silica is dispersed is called a filler-filled rubber. Filler-filled rubber is used in various applications such as tires, conveyor belts, and seismic isolation rubber for buildings.

  Here, in the filler-filled rubber, it is known that the polymer molecules of the matrix are restricted at the interface between the filler and the matrix polymer. Further, it is also known that the filler distribution has a complicated and random three-dimensional structure. Although it is thought that the reinforcing effect of the filler can be obtained due to such a complicated microstructure, the mechanism is under study.

  For example, in the case of a tire, rolling resistance, braking performance, wear resistance, and the like are important indices indicating tire performance, and it is known that these indices are greatly affected by the microstructure of the rubber filled with filler. Therefore, elucidation of the mechanism of the microstructure of the filler-filled rubber is expected to contribute to the improvement of tire performance, and numerical analysis on the microstructure of the filler-filled rubber has been actively performed.

  For example, in Non-Patent Document 1 by the present inventors, a large-scale FEM (Finite Element Method) analysis using a supercomputer “K” (registered trademark) of RIKEN, and an interphase between a filler and a matrix polymer is obtained. Numerical analysis of the viscoelastic behavior of filler-filled rubber in consideration of the influence of air has been performed. That is, in Non-Patent Document 1, viscoelasticity analysis of a filler-filled rubber, particularly stress relaxation analysis of a filler-filled rubber is performed by combining coarse-grained molecular dynamics (CGMD) and large-scale FEM. .

Hiroshi Kadowaki and 7 others, "Study on Viscoelastic Properties of Filled Rubber Using Large-Scale Numerical Analysis", Japan Society of Mechanical Engineers M & M2015 Material Mechanics Conference, November 21-23, 2015, Proceedings CD- ROM, announcement number 212

  Incidentally, the dynamic elastic modulus of a homogeneous material (single material) made of pure rubber or the like could be obtained by simulation. However, it is almost impossible to determine the dynamic elastic modulus of the filler-filled rubber by the same simulation method as that of the pure rubber in terms of calculation cost due to the influence of the above-described microstructure, and so far it has been obtained by experiments. Was.

  However, if the dynamic elastic modulus of the filler-filled rubber can be determined efficiently by simulation, the dynamic elastic modulus of the actually-filled rubber can be determined by simulation. The dynamic elastic modulus of the filler-filled rubber that has been virtually changed freely can also be obtained by simulation. This leads to elucidation of the mechanism relating to the viscoelastic properties of the filler-filled rubber and furthermore to the viscoelastic behavior, and it can be expected that tire performance such as rolling resistance, braking performance and wear performance will be improved.

  Therefore, an object of the present invention is to provide a simulation method and a simulation program capable of calculating the dynamic elastic modulus of a filler-filled rubber by simulation.

  The inventors of the present invention have made intensive studies to solve the above-mentioned problems. As described above, it is difficult to directly calculate the dynamic elastic modulus of the filler-filled rubber because of its high calculation cost. However, the present inventor has found that by performing a stress relaxation analysis of a filled rubber to calculate a relaxed modulus and calculating a dynamic modulus based on the relaxed modulus, the dynamic modulus of the filled rubber can be calculated. As a result, the present invention has been completed.

The present invention is based on the above findings, and the gist configuration thereof is as follows.
(1) The simulation method according to the present invention is a method for simulating the viscoelastic properties of a filled rubber in which a filler is dispersed in a matrix polymer, wherein the filled rubber is mixed with the matrix polymer and the filler, and the matrix polymer and the (A) modeling into a finite element model including three phases consisting of an interface phase between fillers, and (b) setting viscoelastic properties of the matrix polymer and the interface phase in the finite element model. (C) calculating stress relaxation analysis of the finite element model in which the viscoelastic properties are set, and calculating a relaxation modulus of the filler-filled rubber; And (d) calculating an elastic modulus.
According to the simulation method having such a configuration, the dynamic elastic modulus of the filler-filled rubber can be calculated by simulation.

(2) In the simulation method of the present invention, in the step (d), it is preferable that the dynamic elastic modulus is calculated by Fourier-transforming the relaxation elastic modulus.
With this simulation method, the dynamic elastic modulus of the filler-filled rubber can be calculated more reliably.

(3) The simulation program according to the present invention is a simulation program for causing a computer to execute a simulation of viscoelastic properties of a filler-filled rubber in which a filler is dispersed in a matrix polymer, wherein the filler-filled rubber includes the matrix polymer and the filler, And (a) modeling into a finite element model including three phases consisting of an interface phase between the matrix polymer and the filler, and setting the viscoelastic properties of the matrix polymer and the interface phase in the finite element model (B) performing stress relaxation analysis of the finite element model in which the viscoelastic property is set, and calculating (c) a relaxation modulus of the filler-filled rubber; Calculate the dynamic elastic modulus of filler-filled rubber And step (d) that, characterized in that to the execution.
According to the simulation program having such a configuration, it is possible to cause a computer to execute the simulation of the dynamic elastic modulus of the filler-filled rubber.

(4) In the simulation program according to the present invention, it is preferable that in the step (d), the dynamic elastic modulus is calculated by Fourier-transforming the relaxation elastic modulus.
With this simulation program, the dynamic elastic modulus of the filler-filled rubber can be calculated more reliably.

  According to the present invention, the relaxation elastic modulus is calculated by performing stress relaxation analysis of the filler-filled rubber, and the dynamic elastic modulus is calculated based on the relaxation elastic modulus. A simulation method and a simulation program that can be provided.

5 is a flowchart illustrating a simulation method according to an embodiment of the present invention. 9 is a graph illustrating an example of setting a range of an interface phase in step S10 according to an embodiment of the present invention. FIG. 6 is a diagram illustrating an example in which filler-filled rubber is modeled into a finite element model in step S10 according to an embodiment of the present invention. 9 is a graph illustrating an example of setting viscoelastic properties in step S20 according to an embodiment of the present invention. 5 is a graph showing an example of a stress relaxation curve obtained from a matrix polymer having viscoelastic properties set according to FIG. 4. It is a graph which shows an example of the stress relaxation curve of rubber filled with filler obtained by Step S30 by one embodiment of the present invention. It is a graph which shows an example of the dynamic elastic modulus of the filler filling rubber obtained by Step S40 by one embodiment of the present invention.

  Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the drawings. It should be noted that each drawing merely shows a preferred embodiment or one specific example of the present invention, and that various changes can be made without departing from the gist of the present invention. Naturally, those who have the knowledge will understand.

(Simulation method)
The simulation method according to one embodiment of the present invention is a method for simulating the viscoelastic properties of a filler-filled rubber in which a filler is dispersed in a matrix polymer. As shown in FIG. 1, the following steps S10, S20, S30 and At least step S40 is included. Step S10 models the filler-filled rubber into a finite element model including the matrix polymer and the filler, and a three-phase including an interface phase between the matrix polymer and the filler (step (a)). In step S20, the viscoelastic properties of the matrix polymer and the interface phase are set in the finite element model modeled in step S10 (step (b)). In step S30, the finite element model in which the viscoelastic properties are set in step S20 is subjected to stress relaxation analysis, and a relaxation modulus of the filler-filled rubber is calculated (step (c)). Then, a step S40 calculates a dynamic elastic modulus of the filler-filled rubber based on the relaxation elastic modulus calculated in the step S30 (step (d)). In the simulation method according to one embodiment of the present invention, any step may be included before step S10 or after step S40 as desired. Further, an arbitrary step may be included between steps S10, S20, S30, and S40 as desired.

  Prior to the detailed description of each step, the filler-filled rubber in the present embodiment will be described. Filler-filled rubber is a nanoscale composite in which a filler is dispersed in a matrix polymer using a polymer made of pure rubber (also referred to as "rubber alone") as a matrix (substrate). This is as described above. The filler-filled rubber may contain various additives such as a vulcanizing agent, a vulcanization accelerator, a stabilizer, and a plasticizer. Examples of the filler include carbon black, silica, calcium carbonate, and the like for giving a reinforcing effect to the filler-filled rubber. Particularly, the filler refers to a particulate one, but is not limited to a spherical shape. In the simulation method according to one embodiment, a filler-filled rubber that is actually produced can be modeled and applied, or a material in which a pure rubber material, a filler filling amount and an interface characteristic are virtually freely changed. It is also possible to apply filled rubber. Hereinafter, details of each step will be described in order.

<Step of modeling into a finite element model: S10>
First, step S10 for modeling into a finite element model will be described. Here, as described above, the filler-filled rubber contains a matrix polymer and a filler. And, as described above, in the vicinity of the interface between the filler and the matrix polymer, since the polymer molecules of the matrix are constrained, in modeling the filler-filled rubber, providing an interface phase between the matrix polymer and the filler. I do. Therefore, in step S10, the filler-filled rubber is modeled as a finite element model including three phases including a matrix polymer, a filler, and an interface phase. Here, the “finite element model including three phases” may include a finite element model including only the three phases of the matrix polymer, the filler, and the interface phase, or include a phase other than the three phases. This means that a finite element model may be configured by four or more phases. However, in the following description, for simplification of the description, the present embodiment will be described using a finite element model constituted by the three phases.

  Here, the width of the interfacial phase of the filler-filled rubber is generally considered to be about several nm. However, in order to accurately apply the simulation method according to the present embodiment, in step S10, prior to setting the region of the interface phase, the material interface of the filler-filled rubber is previously determined by coarse-grained molecular dynamics (CGMD). It is preferable to analyze it. For example, a Kremer-Grest model is used to construct a matrix polymer molecule of a filler-filled rubber, and the behavior of the molecule near the interface with the filler is analyzed using a molecular dynamics simulator `` LAMMPS '' (Large-scale Atomic / Molecular Massively). What is necessary is just to analyze using Parallel Simulator). Fig. 2 shows the mean-square displacement (MSD) of a given filler-filled rubber, which indicates the distance from the interface and the mobility of the polymer molecules of the matrix polymer, using a molecular dynamics simulator "LAMMPS". It is an example of the obtained graph. The distance is normalized by the reference distance σ. In the example shown in FIG. 2, the interaction between the matrix polymer and the filler is limited to about 6σ from the interface (about 4nm in this model) in both the strong and weak cases. Is confirmed. In such a case, a region from the interface to about 6σ on the polymer side can be used as the interface phase of the filler-filled rubber. In the following description, in order to explain more specifically, a region sufficiently distant from the interface is converted to a temperature of 323 K, and the corresponding temperature when the interaction of polymer molecules is strong is set such that the smaller the value of the MSD, the higher the temperature. It is assumed that the state is the same as when the temperature is low, and the temperature is set to 267K, and the corresponding temperature when the interaction is weak is set to 295K. Note that the numerical values exemplified here are merely examples, and the present invention is not limited to these numerical values at all. In setting the region width of the interface phase, a known molecular dynamics method can be applied in addition to the above-described CGMD.

  The following method can be used to model the filler-filled rubber into a finite element model including the three phases. For example, the actually prepared filler-filled rubber is measured by a three-dimensional transmission electron microscope measurement to obtain a 3D-TEM image, and from the 3D-TEM image, a matrix polymer pixel, a filler pixel, and a filler are obtained by image processing. Pixels in the interface phase near the surface can be classified into three types. In setting the range of the interface phase, it is preferable to apply the range in which the above-described constraint occurs. Then, a lattice-like three-dimensional unit finite element can be obtained by the Voxel method or the like such that one pixel of each obtained pixel corresponds to one element. FIG. 3 is an example of a finite element model modeled by rectangular finite elements by the Voxel method. The code M corresponds to the matrix polymer element, the code F corresponds to the filler element, and the code I corresponds to the interface phase element. In the case of FIG. 3, each element is a rectangular parallelepiped with a side of about 1 nm, and the number of elements is about 66 million elements (850 × 850 × 91), which is about 200 million degrees of freedom. If the finite element model is several million to tens of million elements, the stress relaxation analysis in step S30 described below can be sufficiently performed. More elements may be set, but if the number of elements is too small, it is considered that the accuracy of the stress relaxation analysis described later will be affected.

  In the above description, one pixel corresponds to one element in order to make the description concrete, but a plurality of pixels may correspond to one element in order to reduce calculation cost. In terms of accuracy, it is preferable that one pixel corresponds to one element. Although the example of FIG. 3 is a finite element generated from the actually measured 3D-TEM image, the filler filling amount and the filler distribution in the 3D-TEM image are corrected to create a corrected model of the filled rubber. It is also possible. In addition, it goes without saying that a virtual finite element model of the rubber filled with filler may be created.

<Step of setting viscoelastic properties: S20>
Subsequent to step S10 described above, in step S20, the viscoelastic properties of the matrix polymer and the interface phase are set in the finite element model modeled in step S10. Since the filler may be handled as a linear elastic body, it is not necessary to set viscoelastic properties. As the viscoelastic properties of the matrix polymer and the interface phase, a dynamic elastic modulus can be used, and it is preferable to use a storage elastic modulus G ′ and a loss elastic modulus G ″. Here, a viscoelastic body of a Prony series model can be used for the matrix polymer and the interface phase. As for the viscoelastic properties of the interfacial phase, the frequency properties of the matrix polymer are shifted to a lower temperature side by the WLF (Williams, Landel, Ferry) equation based on the corresponding temperature obtained by the coarse-grained molecular dynamics (CGMD) described above. The storage elastic modulus G ′ and the loss elastic modulus G ″ thus obtained can be used. In this step, as the viscoelastic properties, other than the Prony series model, a Bergstrom-Boyce model, a Parallel Network model, a Parallel Rheological Framework model, and the like may be used as the viscoelastic properties of other known viscoelastic body models, a matrix. The temperature-frequency characteristics of the polymer may be determined in advance by experiments.

  FIG. 4 shows, as an example, the storage modulus G ′ and the loss modulus G ″ of the matrix polymer and the interfacial phase. In the figure, 323K means the viscoelastic property of the matrix polymer because it corresponds to a region sufficiently distant from the above-mentioned interface, and 267K and 295K mean the viscoelastic properties of the interface phase in which the interaction is strong and weak respectively. . The storage elastic modulus G 'and the loss elastic modulus G' 'at 267K and 295K are obtained by performing temperature-frequency conversion according to the WLF equation. In addition, in order to show a difference from FIG. 6 described later, FIG. 5 shows an FEM analysis in which a momentary strain at each of the above-mentioned temperatures is introduced into the matrix polymer alone (that is, pure rubber) assuming no filler. A graph showing a stress relaxation curve according to is shown for reference. 5 and 6, the elapsed time and the stress are normalized (AU: arbitrary unit).

<Step of calculating relaxation modulus of filler-filled rubber: S30>
Following step S10 and step S20, in step S30, the finite element model for which the viscoelastic properties are set in step S20 is subjected to stress relaxation analysis by FEM to calculate the relaxation modulus of the filler-filled rubber.

  In this step S30, for example, FEM analysis in which instantaneous strain is introduced into the filler-filled rubber is performed, and a relaxation modulus G (t) can be obtained based on a stress relaxation curve that attenuates with the lapse of time after the introduction of the strain. The distortion to be introduced is not limited to instantaneous distortion, and may be distortion for a certain period of time. An example of the stress relaxation curve obtained in step S30 is shown in the graph of FIG. In the graph of FIG. 6, the interaction in the graph of FIG. 4 was calculated using the viscoelastic property of weak (295 K), and the supercomputer “K” (registered trademark) of RIKEN was used. This is calculated by the FEM program “FrontISTR” (3072 parallel) corresponding to large-scale parallel calculation. The graph of FIG. 6 is a stress relaxation curve based on a large-scale FEM model of the three phases of the matrix polymer, the filler, and the interface layer. The graph partially differs in tendency from the stress relaxation analysis of the matrix polymer alone shown in FIG. Is also confirmed.

<Step of calculating dynamic elastic modulus of filler-filled rubber: S40>
In step S40, the dynamic elastic modulus of the filler-filled rubber is calculated based on the relaxation elastic modulus G (t) calculated in step S30. This step is a particularly characteristic step in the present embodiment. Heretofore, it has been almost impossible to determine the dynamic elastic modulus of the filler-filled rubber by a simulation method similar to that of pure rubber in terms of calculation cost due to the influence of the microstructure of the filler-filled rubber. In contrast, in the present embodiment, once the relaxation modulus G (t) is obtained up to step S30, the dynamic modulus of the filler-filled rubber is calculated using the obtained relaxation modulus G (t). I do. By doing so, the present inventor has found that the dynamic elastic modulus of the filler-filled rubber can be calculated by simulation.

Here, when the storage elastic modulus G ′ (ω) and the loss elastic modulus G ″ (ω) of the filler-filled rubber corresponding to the predetermined angular frequency ω are obtained as the dynamic elastic modulus of the filler-filled rubber, this step S40 is performed. In the above, it is preferable that the relaxation elastic modulus G (t) is calculated (numerically integrated) by performing a Fourier transform using the following equations (1) and (2).

  FIG. 7 is a graph showing the storage elastic modulus G ′ and the loss elastic modulus G ″ of the filler-filled rubber obtained according to the present embodiment, subjected to the Fourier transform. In the graph of FIG. 7, each plot is a value calculated by numerical integration, and a solid line and a broken line are curves obtained by fitting. The frequency characteristics of the storage elastic modulus G ′ and the loss elastic modulus G ″ obtained here are similar in tendency to those obtained by experiments.

  As described above, by the simulation method according to the present embodiment, the dynamic elastic modulus of the filler-filled rubber can be calculated by simulation.

(Simulation program)
The above-described simulation method according to the present invention can be executed by using a computer. Such a computer stores, for example, a program describing processing contents for realizing each step shown in FIG. 1 in a storage unit of the computer. And the program is read and executed by a central processing unit (CPU) of the computer. That is, the simulation program according to the present invention is a simulation program that causes a computer to execute a simulation of the viscoelastic properties of a filler-filled rubber in which a filler is dispersed in a matrix polymer, wherein the filler-filled rubber is the matrix polymer and the filler, and (A) modeling into a finite element model including three phases consisting of an interface phase between the matrix polymer and the filler, and setting viscoelastic properties of the matrix polymer and the interface phase in the finite element model A step (b), a step (c) of performing stress relaxation analysis on the finite element model in which the viscoelastic properties are set, and calculating a relaxation modulus of the filler-filled rubber. Dynamic elastic modulus of filled rubber And calculating steps (d), it is executed. In the simulation program according to the present invention, it is preferable that in the step (d), the dynamic elastic modulus is calculated by Fourier-transforming the relaxation elastic modulus. The description of the same steps as those in the embodiment of the simulation method described above is omitted. Thus, by causing the computer to execute the simulation program of the present invention, the dynamic elastic modulus of the filler-filled rubber can be calculated by simulation.

According to the present invention, the dynamic elastic modulus of the filler-filled rubber can be calculated by simulation. By using this simulation method and / or simulation program, it is possible to clarify the mechanism relating to the viscoelastic properties of the filler-filled rubber, and it is also possible to expect improvement in tire performance.

Claims (4)

Using a computer having a central processing unit and a storage unit, a simulation method for simulating the viscoelastic properties of a filled rubber in which a filler is dispersed in a matrix polymer,
The computer stores the finite element model in which the filler-filled rubber is modeled into a finite element model including three phases including the matrix polymer and the filler, and an interface phase between the matrix polymer and the filler. (A) storing in the unit ;
(B) the computer setting viscoelastic properties of the matrix polymer and the interface phase in the finite element model, and storing the finite element model in which the viscoelastic properties are set in the storage unit ;
The computer, using the central processing unit, performs stress relaxation analysis on the finite element model in which the viscoelastic properties are set, calculates a relaxation modulus of the filler-filled rubber, and stores the relaxation modulus. (C) storing in the
And (d) calculating the dynamic elastic modulus of the filler-filled rubber based on the relaxed elastic modulus using the central processing unit .   The simulation method according to claim 1, wherein in the step (d), the dynamic elastic modulus is calculated by performing a Fourier transform on the relaxation elastic modulus. A simulation program that causes a computer to execute a simulation of viscoelastic properties of a filler-filled rubber in which a filler is dispersed in a matrix polymer,
Modeling the filler-filled rubber into a finite element model comprising three phases consisting of the matrix polymer and the filler, and an interphase between the matrix polymer and the filler; (a)
Setting (b) the viscoelastic properties of the matrix polymer and the interfacial phase in the finite element model;
(C) calculating stress relaxation analysis of the finite element model in which the viscoelastic properties are set, and calculating a relaxation modulus of the filler-filled rubber;
(D) calculating a dynamic elastic modulus of the filler-filled rubber based on the relaxation elastic modulus.   The simulation program according to claim 3, wherein in the step (d), the dynamic elastic modulus is calculated by performing a Fourier transform on the relaxation elastic modulus.