JP2020085692A - Rubber material simulation method - Google Patents

Abstract

To improve simulation accuracy.SOLUTION: A rubber material simulation method is a method for using a computer to analyze elongation deformation of a rubber material containing a filler. This simulation method includes steps of: inputting a filler model 6 that expresses the filler with multiple first particles 11 on the basis of a dissipative particle dynamics method; inputting a polymer model 7 that expresses a molecular chain of rubber material with multiple second particles 12 on the basis of the dissipative particle dynamics method; applying a definition of Lennard-Jones potential U(r) to a section between the first particles 11 and the second particles 12; creating a structurally-relaxed rubber material model through a molecular dynamics calculation performed by the computer using the filler model 6 and the polymer model 7 which are arranged in a virtual space; and calculating the elongation deformation of the rubber material model.SELECTED DRAWING: Figure 7

Description

The present invention relates to a simulation method for analyzing elongational deformation of a rubber material containing a filler.

Patent Document 1 below proposes a simulation method for analyzing a polymer material containing a filler based on the dissipative particle dynamics method (DPD method). In the simulation method of Patent Document 1 below, a three-dimensional structure filler model composed of a plurality of particles and a linear polymer model composed of a plurality of particles are used. A soft core potential is defined between the polymer model particle and the filler model particle.

JP, 2013-108951, A

The soft core potential is defined only by the repulsive interaction between the particles of the adjacent filler model and the particles of the polymer model. For this reason, the simulation method of Patent Document 1 cannot calculate the phenomenon in which the polymer is adsorbed on the surface of the filler. Therefore, there is room for further improvement in improving the simulation accuracy.

The present invention has been devised in view of the above circumstances, and a main object of the present invention is to provide a rubber material simulation method capable of improving simulation accuracy.

The present invention is a method for analyzing extensional deformation of a rubber material containing a filler by using a computer, wherein the filler is represented by a plurality of first particles based on a dissipative particle dynamics method. Inputting a model into the computer, inputting a polymer model in which the molecular chains of the rubber material are expressed by a plurality of second particles based on the dissipative particle dynamics method into the computer, the first particles A Lenard-Jones potential between the second particle and the second particle, the computer performing a molecular dynamics calculation using the filler model and the polymer model arranged in a virtual space of the computer. And a step of generating a structurally relaxed rubber material model, and the computer calculating a stretching deformation of the rubber material model.

The simulation method of the rubber material according to the present invention may further include a step of defining a bond angle potential between the adjacent first particles.

In the rubber material simulation method according to the present invention, the bond angle potential may be defined by the following equation (1).
here,
U(θ): bond angle potential θ: outside angle of bond angle formed between three adjacent first particles k: spring constant of filler θ ₀ : outside angle of bond angle formed between three adjacent first particles in equilibrium state

A method for simulating a rubber material according to the present invention includes a step of inputting a filler model in which a filler is expressed by a plurality of first particles based on a dissipative particle dynamics method, and A step of inputting a polymer model in which a molecular chain is represented by a plurality of second particles, a step of defining a Lenard-Jones potential between the first particles and the second particles, and a step of defining the Lenard-Jones potential in a virtual space of a computer. The method includes a step of generating a structurally relaxed rubber material model by performing a molecular dynamics calculation using the filler model and the polymer model, and a step of calculating extensional deformation of the rubber material model.

In the simulation method of the present invention, since the filler model and the polymer model are expressed based on the dissipative particle dynamics method, a large space-time is handled as compared with the simulation method based on the all-atom molecular dynamics method. You can

In the simulation method of the present invention, not only repulsive interaction between the first particles and the second particles but also attractive interaction can be defined by the Lenard-Jones potential. Accordingly, in the simulation method of the present invention, the adsorption between the filler model and the polymer model can be expressed, so that the elongation deformation of the rubber material model can be calculated with high accuracy. Can be simulated. Therefore, the simulation method of the present invention can improve the simulation accuracy.

It is a perspective view showing an example of a computer for performing a simulation method of a rubber material. 3 is a structural formula showing an example of a molecular chain of a rubber material. It is a flow chart which shows an example of a processing procedure of a simulation method of a rubber material. It is a conceptual diagram which shows an example of the virtual space where the filler model and the polymer model are arrange|positioned. It is a figure which shows an example of a filler model. It is a figure which shows an example of a polymer model. It is a conceptual diagram explaining an example of the interaction potential of a filler model and a polymer model. It is a flow chart which shows an example of the processing procedure of the simulation method of other embodiments of the present invention. It is a figure which shows an example of the filler model of other embodiment of this invention. 6 is a graph showing an example of a relationship between a bond angle potential U(θ) and an external angle θ of a bond angle. It is a graph which shows the relationship between stress and strain of an example and comparative examples A and B. (A) is a perspective view showing a rubber material model before extensional deformation, (b) is a perspective view showing a rubber material model of an example after extensional deformation, and (c) is a comparative example B after extensional deformation. It is a perspective view showing a rubber material model.

An embodiment of the present invention will be described below with reference to the drawings.
In the rubber material simulation method of the present embodiment (hereinafter, may be simply referred to as “simulation method”), the elongation deformation of the rubber material mixed with the filler is analyzed using a computer.

FIG. 1 is a perspective view showing an example of a computer for executing a rubber material simulation method. The computer 1 includes a main body 1a, a keyboard 1b, a mouse 1c and a display device 1d. The main body 1a is provided with, for example, an arithmetic processing unit (CPU), a ROM, a working memory, a storage device such as a magnetic disk, and disk drive devices 1a1 and 1a2. In addition, software or the like for executing the simulation method of the present embodiment is stored in the storage device in advance.

As the filler, for example, silica, carbon black, alumina or the like is adopted. As the molecular chain of the rubber material, for example, natural rubber, synthetic rubber, resin or the like is adopted.

FIG. 2 is a structural formula showing an example of the molecular chain 2 of the rubber material. As the molecular chain 2 of this embodiment, cis-1,4 polyisoprene shown in FIG. 2 (hereinafter, may be simply referred to as “polyisoprene”) is exemplified. Molecular chain 2 constituting the polyisoprene methine group (e.g., -CH =,> C =) , methylene group _(-CH 2 -), and a monomer isoprene constituted by a methyl group (-CH ₃₎ (Isoprene molecule) 3 is connected at a polymerization degree n. In addition, as the molecular chain, one other than polyisoprene may be used.

FIG. 3 is a flowchart showing an example of the processing procedure of the rubber material simulation method. In the simulation method of this embodiment, first, a virtual space of a computer (hereinafter, simply referred to as “virtual space”) is defined in the computer 1 (step S1). FIG. 4 is a conceptual diagram showing an example of the virtual space 4 in which the filler model 6 and the polymer model 7 are arranged.

The virtual space (cell) 4 corresponds to a part of the rubber material. The virtual space 4 has at least a pair of surfaces 5 and 5 facing each other, and in the present embodiment, three pairs of surfaces 5 and 5 facing each other, and is defined as a rectangular parallelepiped or a cube (a cube in this example). ..

Periodic boundary conditions are defined for each surface 5, 5. By using the virtual space 4 as described above, in the coarse-grained molecular dynamics calculation described later, for example, the filler model 6 and the polymer model 7 described later, which are obtained from the surface 5a on one side, It can be calculated that a part of the polymer model 7 comes in from the surface 5b on the other side. Therefore, the virtual space 4 can be treated as if the surface 5a on one side and the surface 5b on the other side are continuous (connected).

Each length L1 of one side of the virtual space 4 can be set appropriately. The length L1 of the present embodiment is preferably three times or more of a radius of gyration (not shown) that is an amount indicating the expansion of the polymer model 7 described later. Accordingly, in the coarse-grained molecular dynamics calculation described later, it is possible to prevent the occurrence of collision with the own image due to the periodic boundary condition and appropriately calculate the spatial expansion of the polymer model 7. Further, the size of the virtual space 4 is set to a stable volume at 1 atmospheric pressure, for example. Thereby, the virtual space 4 can define the volume of at least a part of the rubber material to be analyzed. The virtual space 4 is stored in the computer 1.

Next, in the simulation method of this embodiment, the filler model 6 representing the filler is input to the computer 1 based on the dissipative particle dynamics method (step S2). FIG. 5 is a diagram showing an example of the filler model 6.

The filler model 6 represents a filler (not shown) by a plurality of first particles 11. The filler model 6 of the present embodiment has a three-dimensional structure by defining the first bond potential P1 that constrains between the first particles 11 and 11.

The first particles 11 are arranged so as to aggregate inside the virtual space 4 (shown in FIG. 4 ). The placement of the first particles 11 can be performed, for example, based on the position of the primary particles of the filler in the electron beam transmission image obtained by capturing the actual rubber material. As a result, the filler model 6 can accurately represent the actual shape of the filler.

The first particle 11 is treated as a mass point of the equation of motion in the molecular dynamics calculation described later. That is, parameters such as mass, diameter, electric charge or initial coordinates are defined for the first particles 11.

The first bond potential P1 is defined as the potential U _Harmonic of the following formula (2).

here,
k: spring constant between particles r _ij : distance between particles r ₀ : equilibrium length
The distance r _ij and the equilibrium length r ₀ are defined as the distance between the centers of particles.

The first bond potential P1 defined by the above formula (2) is a Harmonic type. The Harmonic type is a potential in which a so-called linear spring is defined, and a restoring force that is proportional to the magnitude of deviation from the equilibrium length r ₀ is exerted. In the first bond potential P1 as described above, the upper limit value of the distance r _ij is not limited to the equilibrium length r ₀ (that is, the extension length is not set). Each variable is represented by paper 1 (Robert D. Groot and Patrick B. Warren, "Dissipaticle dynamics: Bridging the gap between atomistic and mesoscopic simulation", J. Chem Phys. vol.107, No.11, 15 September 1997, p4423. -4435), it can be set appropriately. The filler model 6 is stored in the computer 1.

Next, in the simulation method of this embodiment, the polymer model 7 (shown in FIG. 4) representing the molecular chain 2 is input to the computer 1 based on the dissipative particle dynamics method (step S3). FIG. 6 is a diagram showing an example of the polymer model 7.

In the polymer model 7, the molecular chain 2 (shown in FIG. 2) is represented by a plurality of second particles 12. The polymer model 7 of this embodiment has a three-dimensional structure including a plurality of second particles 12 and a second bonding chain 14 connecting the second particles 12 and 12.

The second particles 12 are obtained by substituting a structural unit forming a monomer of the polymer or a part of the monomer. The second particles 12 of the present embodiment are replaced with one second particle 12 with an arbitrary number (for example, several tens) of the monomers 3 as a structural unit. As a result, a plurality of (for example, 10 to 500) second particles 12 are set in the polymer model 7.

The second particles 12 are treated as mass points of the equation of motion in the molecular dynamics calculation described later. That is, parameters such as mass, diameter, electric charge or initial coordinates are defined in the second particles 12.

The second binding chain 14 binds between the second particles 12, 12. The second bond chain 14 is defined by the second bond potential P2. The second bond potential P2 is defined by the potential F ^S _ij of the following formula (3).

here,
C: Spring constant r _ij : Distance between particles r _s : Equilibrium bond length n _ij : r _ij /|r _ij | (unit vector)
The distance r _ij and the equilibrium bond length r _s are defined as the distance between the centers of particles.

The second bond potential P2 defined by the above equation (3) is a DPD type bond potential. The DPD type bond potential is a stretching potential whose bond length follows a Gaussian distribution and is generally used in DPD simulations. Each variable should be set appropriately based on, for example, thesis 2 (Satoru Yamamoto, "Mesoscopic Simulation for the Drying Process of Polymer Films", Journal of the Rheological Society of Japan, vol.32, No.5, p295-301). You can

The number of polymer models 7 arranged in the virtual space 4 can be appropriately set based on the size of the virtual space 4 and the like. An example of the number of polymer models 7 is about 50 to 1000. The polymer model 7 is stored in the computer 1.

Next, in the simulation method of the present embodiment, interaction potentials P3 and P4 are defined between the first particles 11 and 11 of the adjacent filler model 6 and between the second particles 12 and 12 of the adjacent polymer model 7. (Step S4). FIG. 7 is a conceptual diagram illustrating an example of the interaction potentials of the filler model 6 and the polymer model 7.

The interaction potential P3 between the first particles 11 and 11 of the adjacent filler model 6 and the interaction potential P4 between the second particles 12 and 12 of the adjacent polymer model 7 are based on the dissipative particle dynamics method. It is defined by the potential F ^c _ij of the following formula (4).

here,
a _ij : Potential strength r _ij : Distance between particles r _c : Cut-off distance n _ij : r _ij /|r _ij | (unit vector)
The distance r _ij and the cutoff distance r _c are defined as the distance between the centers of the particles.

The interaction potentials P3 and P4 defined by the above equation (4) exert a repulsive force between particles when the distance r _ij between particles becomes equal to or _smaller than a predetermined cutoff distance r _c . Further, when the distance r _ij between particles is equal to the predetermined cutoff distance r _c , the interaction potentials P3 and P4 are zero and no repulsive force is generated between particles. Therefore, the interaction potentials P3 and P4 are defined only by the repulsive interaction between the first particles 11 and 11 of the adjacent filler model 6 and between the second particles 12 and 12 of the adjacent polymer model 7. Each variable can be appropriately set based on, for example, thesis 2. The interaction potentials P3 and P4 are stored in the computer 1.

Next, in the simulation method of this embodiment, the Lenard-Jones potential U _LJ (r _ij ) is defined between the first particles 11 and the second particles 12 (step S5). The Lenard-Jones potential is defined by the following equation (5).

here,
r _ij : Distance between particles r _c : Cutoff distance ε: Strength of LJ potential defined between particles σ: Corresponding to particle diameter Note that the distance r _ij and the cutoff distance r _c are between the centers of the particles. Is defined as the distance of.

The Lenard-Jones potential U _LJ (r _ij ) becomes infinitely larger as the distance r _ij between the first particle 11 and the second particle 12 approaches 0 from 2 ^1/6 σ. At this time, the Lenard-Jones potential U _LJ (r _ij ) can cause a repulsive interaction between the first particles 11 and the second particles 12.

The Lenard-Jones potential U _LJ (r _ij ) becomes minimum when the distance r _ij between the first particle 11 and the second particle 12 is 2 ^1/6 σ, and its value becomes −ε. That is, the Lenard-Jones potential U _LJ (r _ij ) is minimized when the first particles 11 and the second particles 12 are in contact with each other.

The Lenard-Jones potential U _LJ (r _ij ) becomes larger and approaches 0 as the distance r _ij between the first particle 11 and the second particle 12 becomes larger than 2 ^1/6 σ. At this time, the Lenard-Jones potential U _LJ (r _ij ) can cause attractive interaction between the first particles 11 and the second particles 12.

Thus, the Lenard-Jones potential U _LJ (r _ij ) can define not only the repulsive interaction between the first particle 11 and the second particle 12 but also the attractive interaction. Each variable of the Lenard-Jones potential U _LJ (r _ij ) is described in, for example, Paper 3 (Kurt Kremer & Gary S. Grest, “Dynamics of entangled linear polymer melts: A molecular-dynamics simulation”, J. Chem Phys. vol. .92, No.8, 15 April 1990, p5057-5086). The Lenard-Jones potential U _LJ (r _ij ) is stored in the computer 1.

Next, in the simulation method of this embodiment, the computer 1 generates a rubber material model whose structure is relaxed (step S6). In step S6, the filler model 6 and the polymer model 7 arranged in the virtual space 4 shown in FIG. 4 are used to perform a molecular dynamics calculation based on the dissipative particle dynamics method, and thus the structure-relaxed rubber material is obtained. A model 20 is generated.

In the molecular dynamics calculation of the present embodiment, for example, Newton's equation of motion is applied assuming that the filler model 6 and the polymer model 7 follow classical mechanics for a predetermined time in the virtual space 4. Then, the movements of the filler model 6 and the polymer model 7 at each time are tracked for each unit time of the simulation. The calculation of structural relaxation can be performed using, for example, COGNAC or VSOP included in the soft material general simulator (J-OCTA) manufactured by JSOL Co., Ltd. Thereby, in step S6, the rubber material model 20 whose structure is relaxed can be generated. A rubber material model 20 based on the dissipative particle dynamics method is stored in the computer 1.

In the simulation method of this embodiment, the repulsive interaction and attractive mutual interaction between the first particle 11 and the second particle 12 are calculated by the Lenard-Jones potential U _LJ (r _ij ) defined by the above equation (5). The action can be defined. Accordingly, the simulation method of the present embodiment can represent the adsorption of the filler model 6 and the polymer model 7 in the simulation based on the dissipative particle dynamics method.

Such adsorption (physical adsorption) is caused by the van der Waals interaction and electrostatic interaction between the filler (not shown) and the molecular chain 2 (shown in FIG. 2). It is a phenomenon of adsorption, and it is considered that the strength of adsorption influences the strength of the rubber material. Therefore, in the simulation of the present embodiment, since the adsorption of the filler model 6 and the polymer model 7 can be expressed, the simulation accuracy can be improved.

Next, in the simulation method of this embodiment, the computer 1 calculates the extension deformation of the rubber material model 20 (step S7). In step S7, a uniaxial tensile test for pulling the rubber material model 20 shown in FIG. 4 in a predetermined direction is calculated. The direction in which the rubber material model 20 is pulled can be selected as appropriate, and in this example, it is pulled in the z-axis direction.

The deformation calculation of the rubber material model 20 is performed in the z-axis direction by one end of the rubber material model 20 (the surface 5a on one side of the virtual space 4 shown in FIG. 4) and the other end of the rubber material model 20 (see FIG. 5). The expansion of the rubber material model 20 is calculated so that the surfaces 5b) on the other side of the illustrated virtual space 4 are separated from each other. The elongation calculation of the rubber material model 20 is calculated based on the molecular dynamics calculation.

In step S7, the thermal motions of the filler model 6 and the polymer model 7 are calculated by the elongation calculation of the rubber material model 20. Such thermal motion is caused by strain applied to the rubber material model 20, bond potentials P1 and P2 (shown in FIGS. 5 and 6), interaction potentials P3 and P4 (shown in FIG. 7), and Lenard-Jones potential U. It is calculated based on _LJ (r _ij ) and the equation of motion.

The deformation condition can be set as appropriate. As an example of the deformation conditions, the strain applied to the rubber material model 20 is about 0.1 to 0.3, and the deformation speed V1 per strain (shown in FIG. 4) is about 1000 to 10000τ.

In step S7, a physical quantity associated with the deformation of the rubber material model 20 is calculated for each unit step (MD step) of the molecular dynamics calculation from the start of the deformation of the rubber material model 20 to the end of the deformation. The physical quantity can be appropriately adopted. The physical quantity of this embodiment includes the stress of the rubber material model 20. In step S7, the stress component in the extension direction of the rubber material model 20 is calculated for each strain applied to the rubber material model 20. Thereby, in the simulation method of this embodiment, a stress-strain curve showing the relationship between the stress and the strain of the rubber material model 20 can be acquired. The strength of the rubber material can be evaluated based on this stress-strain curve.

In the simulation method of this embodiment, since the filler model 6 and the polymer model 7 are expressed based on the dissipative particle dynamics method, a large space-time is handled as compared with the simulation method based on the all atom molecular dynamics method. You can In the simulation method of the present embodiment, the filler model 6 and the polymer model 7 are combined in the simulation based on the dissipative particle dynamics method by the Lenard-Jones potential U _LJ (r _ij ) defined by the above equation (5). Adsorption can be expressed. Therefore, in the simulation method of the present embodiment, the strength of the rubber material model 20 and the physical quantity can be calculated with high accuracy, and the simulation accuracy can be improved.

Next, in the simulation method of this embodiment, the computer 1 determines whether or not the physical quantity of the stretched and deformed rubber material model 20 is good (step S8). The standard of the physical quantity for judging whether or not it is good can be appropriately set according to the strength required for the rubber material and the like.

When it is determined in step S8 that the physical quantity of the rubber material model 20 is good (“Y” in step S8), the rubber material is manufactured based on the conditions set in the rubber material model 20 (step S9). ). On the other hand, if it is determined in step S8 that the rubber material model 20 having a good physical quantity does not exist (“N” in step S8), the various conditions set in the rubber material model 20 are changed (step S10), Steps S1 to S8 are performed again. Thereby, the simulation method of the present embodiment can develop a rubber material having excellent strength based on the interaction between the filler and the polymer.

As shown in FIG. 5, since the first bond potential P1 of the filler model 6 of the present embodiment is defined by the potential U _Harmonic of the above formula (2), the binding force between the first particles 11 and 11 is weak. .. Therefore, in the step S7 of calculating the elongation deformation of the rubber material model 20 shown in FIG. 4, the filler model 6 may be largely deformed (expanded) by being stretched between the first particles 11 and 11. The expanded filler model 6 may be largely deformed due to collision with its own image due to the periodic boundary condition, and the shape of the filler model 6 may not be maintained. In order to prevent such expansion and deformation of the filler model 6, it is desirable that the filler model 6 has a high binding force between the adjacent first particles 11 and 11. FIG. 8 is a flowchart showing an example of a processing procedure of a simulation method according to another embodiment of the present invention. FIG. 9: is a figure which shows an example of the filler model 6 of other embodiment of this invention. In addition, in this embodiment, the same configurations as those of the previous embodiments are denoted by the same reference numerals, and the description thereof may be omitted.

In the simulation method of this embodiment, the bond angle potential U(θ) is defined between the adjacent first particles 11 (11) (step S11). The bond angle potential U(θ) is for binding between the adjacent first particles 11 and 11. The bond angle potential U(θ) of the present embodiment is defined by the following equation (1), for example.

here,
U(θ): bond angle potential θ: outside angle of bond angle formed between three adjacent first particles k: spring constant of filler θ ₀ : outside angle of bond angle formed between three adjacent first particles in equilibrium state

The external angle θ ₀ of the bond angle in the equilibrium state can be appropriately set according to the structure of the filler model 6. FIG. 10 is a graph showing an example of the relationship between the bond angle potential U(θ) and the external angle θ of the bond angle. The bond angle potential U(θ) in FIG. 10 is set such that the external angle θ ₀ of the bond angle in the equilibrium state is 60°.

The bond angle potential U(θ) becomes 0 when the external angle θ of the bond angle is 60° or −60°. On the other hand, the bond angle potential U(θ) increases as the outside angle θ of the bond angle deviates from 60° or −60°. In addition, the bond angle potential U(θ) can be more tightly bound between the adjacent first particles 11 as the spring constant k is larger.

As described above, the bond angle potential U(θ) is obtained when the external angle θ of the bond angle formed between the three adjacent first particles 11 of the filler model 6 is different from the external angle θ ₀ of the bond angle in the equilibrium state. The action can be defined to return the bond angle external angle θ to the equilibrium bond angle external angle θ ₀ . Thereby, the external angle θ of the bond angle formed between the three adjacent first particles 11 of the filler model 6 can be maintained at the external angle θ ₀ of the bond angle in the equilibrium state.

With such a bond angle potential U(0), the adjacent first particles 11 and 11 can be firmly constrained so as to maintain an equilibrium state, and therefore the elongation deformation of the rubber material model 20 shown in FIG. 4 is calculated. In step S7, expansion and deformation of the filler model 6 can be prevented. Therefore, the simulation method of this embodiment can effectively represent the adsorption of the filler model 6 and the polymer model 7, and can improve the simulation accuracy.

In step S11, a plurality of types of bond angle potentials U(θ) having different equilibrium bond angles outside angle θ ₀ may be defined according to the structure of the filler model 6. In this embodiment, the equilibrium bond angle external angle θ ₀ is set to 0°, and the equilibrium bond angle external angle θ ₀ is set to 60°. (Θ) is defined. Thereby, the deformation of the filler model 6 can be effectively prevented. The bond angle potential U(θ) is stored in the computer 1. Note that the bond angle potential U(θ) is not limited to the above formula (1) as long as it can constrain the adjacent first particles 11, 11. For example, by defining the potential to fix the filler shape, it is possible to define a filler model in which filler particles are linked to form a tufted aggregate, or a filler model having a non-crystalline amorphous structure on the surface. it can.

Although the particularly preferred embodiments of the present invention have been described above in detail, the present invention is not limited to the illustrated embodiments and can be modified into various modes.

A structurally relaxed rubber material model was generated and the elongation deformation of the rubber material model was calculated (Example, Comparative Example A, Comparative Example B and Comparative Example C). In Examples, Comparative Examples A, and B, a filler model in which a filler is expressed by a plurality of first particles and a polymer in which a molecular chain is expressed by a plurality of second particles are based on the dissipative particle dynamics method. The model has been entered. On the other hand, in Comparative Example C, the filler model and the polymer model were input based on the Kremer-Grest coarse-grained model.

The Lenard-Jones potential was defined between the first particle and the second particle of Example and Comparative Example C. Furthermore, in Example and Comparative Example C, the bond angle potential was defined between the adjacent first particles.

A soft core potential was defined between the first particle and the second particle of Comparative Examples A and B based on the conventional dissipative particle dynamics method. Further, in Comparative Example A, the bond angle potential was defined between the adjacent first particles. On the other hand, in Comparative Example B, in B, the bond angle potential was not defined between the adjacent first particles.
The common specifications and parameters of each potential are as follows.
Virtual space:
One side length L1: 17.24σ
Volume: 5121 σ ³
Filler model:
Number of filler models: 4
Number of first particles constituting one filler model: 791
Density of the first particles in the filler model: 3 (1/σ ³ )
Polymer model:
Number of polymer models: 120
Second particles constituting one polymer model: 100 particles Crosslink density: 0.04 (1/σ ³ )
Structural relaxation calculation:
Time step: 0.01
Number of steps: 1,000,000
Deformation calculation:
Distortion: 0.01τ
Total steps: 100,000
Total time: 1,000τ
Strain rate: 0.001 (1/τ)
Elongation rate: 0.0172 (σ/τ)
Poisson's ratio: 0.5
Bond potential between the first particles of the filler model (U _Harmonic ):
Equilibrium length r ₀ : 0.78σ
Spring constant k: 100 (ε/σ ² )
Bond potential (F ^s _ij ) between the second particles of the polymer model:
Equilibrium bond length r _s : 0.86σ
Spring constant C: 4
Interaction potential (soft core potential) P3, P4:
Cutoff distance r _c : 1.0σ
Potential intensity a _ij : 25
Lenard-Jones potential (Example, Comparative Example C):
Cutoff distance: 2.5σ
Potential intensity ε: 1.0σ
Value σ corresponding to the diameter of the first particle and the second particle: 1.0σ
Bond angle potential between first particles (Example, Comparative Example B):
Spring constant of filler k: 200ε
Outer angle θ ₀ of equilibrium bond angle: 0°, 60°

FIG. 11 is a graph showing the relationship between stress and strain in Examples and Comparative Examples A and B. In the example, it was possible to confirm the behavior in which stress rises from around strain 0.7. Such behavior is similar to the behavior of the rubber material during extensional deformation. On the other hand, in Comparative Examples A and B, it was not possible to confirm the behavior of stress rising from around strain 0.7. Therefore, in the examples, the adsorption between the filler model and the polymer model could be expressed by the Lenard-Jones potential. Furthermore, in the example, the hardness of the filler can be expressed by the bond angle potential, and the simulation accuracy can be improved. In addition, in the example, the calculation time could be shortened to 1/50 of the calculation time of the comparative example C. Therefore, the embodiment can significantly reduce the calculation cost.

FIG. 12A is a perspective view showing a rubber material model before extensional deformation. FIG. 12B is a perspective view showing a rubber material model after extension deformation according to the embodiment. FIG. 12C is a perspective view showing a rubber material model of Comparative Example B after extensional deformation. 12A to 12C, the polymer model is omitted.

The filler model after extensional deformation of the example shown in FIG. 12B was able to maintain the shape of the filler before extensional deformation shown in FIG. 12A due to the bond angle potential between the first particles. On the other hand, in the filler model after elongation deformation of Comparative Example B shown in FIG. 12C, the shape of the filler model is greatly deformed, and the shape of the filler before elongation deformation shown in FIG. 12A cannot be maintained. It was Therefore, the rubber material model of the example could improve the simulation accuracy.

6 Filler model 7 Polymer model 11 First particle 12 Second particle

Claims (3)

A method for analyzing extensional deformation of a rubber material blended with a filler using a computer,
Inputting a filler model in which the filler is represented by a plurality of first particles to the computer based on a dissipative particle dynamics method;
Inputting into the computer a polymer model in which the molecular chains of the rubber material are represented by a plurality of second particles based on the dissipative particle dynamics method;
Defining a Lenard-Jones potential between the first particle and the second particle,
The computer, by performing a molecular dynamics calculation by using the filler model and the polymer model arranged in a virtual space of the computer, to generate a structurally relaxed rubber material model, and the computer, A method for simulating a rubber material, including the step of calculating extensional deformation of a rubber material model. The method for simulating a rubber material according to claim 1, further comprising the step of defining a bond angle potential between the adjacent first particles. The method for simulating a rubber material according to claim 2, wherein the bond angle potential is defined by the following formula (1).
here,
U(θ): bond angle potential θ: outside angle of bond angle formed between three adjacent first particles k: spring constant of filler θ ₀ : outside angle of bond angle formed between three adjacent first particles in equilibrium state