JP2005351770A - Simulation method for rubber material - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To ensure a simulation near to an actual deformation behavior of a rubber material. <P>SOLUTION: This rubber material simulation method for simulating deformation of the rubber material containing matrix rubber and a filler includes a step for setting a rubber material model including a filler model with at least the one filler divided into a finite number of elements, and a matrix model with the matrix rubber into divided a finite number of elements conformed with an outer circumferential face of the filler model in an initial state, and having an inner circumferential face separable from the outer circumferential face, a step for imparting a condition of tensile deformation to the rubber material model to calculate the deformation, and a step for acquiring a necessary physical quantity from the rubber material model. The step for calculating the deformation includes a step for calculating a separated distance between the inner circumferential face of the matrix model and the outer circumferential face of the filler model, and a step for imparting attractive force along a mutual approaching direction of the inner circumferential face and the outer circumferential face, on the basis of the separated distance. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to a rubber material simulation method useful for analyzing an interaction between fillers of a rubber material.

  Rubber materials used in tires, sporting goods and other various industrial products are greatly deformed when subjected to a load, and are restored to their original state when the load is completely removed. In addition, when the rubber material is deformed, the stress and the strain are not proportional to each other, and the rubber material has nonlinearity that draws a loop in which the path of the stress-strain curve is different between when the load is applied and when the load is unloaded. In order to reduce the labor and cost of trial production, the deformation process of a rubber material is simulated by a computer. Conventionally, as a rubber material simulation method, the following Patent Document 1, Non-Patent Document 1, and the like are known.

JP 2002-365205 A Ellen M. Arruda and Marry C. Boyce `` A THREE-DIMENSIONAL CONSTITUTIVE MODEL FOR THE LARGE STRECH BEHAVIOR OF RUBBER ELASTIC MATERIALSS '' Journal of the Mechanics and Physics of Solids Volume 41, Issue 2, Pages 389-412 (February 1993)

  Patent document 1 pays attention to the point that the rubber material (viscoelastic material) undergoes different deformations according to the strain rate. Specifically, the strain, strain rate, stress, etc. generated in the viscoelastic material are measured in advance under the measurement conditions assuming the actual usage state of the viscoelastic material, and a pseudo longitudinal elastic modulus (correctly rubber is the hook It is not an elastic body satisfying the law.) And a process of deriving a correspondence relationship between strain and strain rate. Then, a predetermined strain rate is given to the product model of the viscoelastic material to be analyzed, and a deformation simulation is performed by appropriately calculating the longitudinal elastic modulus from the above correspondence. Since the method of Patent Document 1 includes an attempt to change the pseudo longitudinal elastic modulus of the viscoelastic material according to the strain rate, it can be evaluated in that hysteresis loss can be incorporated into the simulation.

  In addition, Non-Patent Document 1 proposed by Arruda et al. Describes that a deformation calculation is performed by modeling a rubber material to a polymer level by using a molecular chain network theory.

  A rubber material used as an industrial product usually contains a filler (filler) such as carbon black in a matrix rubber. As the amount of filler added to rubber increases, energy loss increases. FIG. 24 shows an example of experimental results by the inventors. In the figure, it can be seen that as the blending amount CB of carbon black increases, the area of the hysteresis loop corresponding to the energy loss increases.

  FIG. 25 shows a microscopic enlarged photograph of the interface portion between the matrix rubber and the carbon black when the rubber material containing carbon black as a filler is subjected to tensile deformation (“carbon black reinforcement of rubber, part 3. Recent important report on reinforcement "Miho Fukahori Journal of Japan Rubber Association Vol. 77, No. 3 (2004), p. 107). In this photograph, the black sphere is carbon black. The slightly thin black part is matrix rubber. It is known that part of the matrix rubber is peeled off from the interface with the carbon black during tensile deformation. Thereby, voids (voids) indicated by white portions are generated in the rubber material. When such voids are formed, slippage occurs at the interface between the matrix rubber and the filler, and it is considered that energy loss occurs there. Further, the void disappears with unloading, but it is considered that energy loss occurs due to the bonding of molecular chains. The rubber material containing a large amount of filler increases the interface portion between the matrix rubber and the filler. This idea signifies an experimental result that rubber with a large amount of filler increases energy loss.

  However, in the simulation of the conventional rubber material, there is no one that incorporates such peeling at the interface between the filler and the matrix rubber in the deformation calculation. For example, in Japanese Patent Application No. 2003-358167 proposed by the present applicant, a matrix model that models a matrix rubber, a filler model that models a filler, and an interface model between the matrix model and the filler model are set. In order to extend the interface model better than the matrix rubber, one that adjusts the parameters has been proposed. This model can be evaluated in that the interface is taken into account, but does not represent delamination at the interface between the matrix rubber and filler.

  The present invention has been devised in view of the above problems, and includes a filler model obtained by dividing at least one filler particle into a finite number of elements, a matrix rubber, surrounding the filler model and the filler model. A rubber material model including a matrix model divided into a finite number of elements including an interface element having an inner peripheral surface that coincides with the outer peripheral surface and can be separated from the outer peripheral surface. Calculating the separation distance between the inner peripheral surface of the interface element and the outer peripheral surface of the filler model at the time, and the attractive force in the direction of approaching the outer peripheral surface of the filler model to the inner peripheral surface of the interface model based on the separation distance It is based on the inclusion of the step of imparting rust and expresses delamination at the interface between the filler and the matrix rubber to approximate the deformation behavior of the actual rubber. And its object is to provide a simulation method of a rubber material capable of allowing the simulation.

  The invention according to claim 1 of the present invention is a rubber material simulation method for simulating deformation of a rubber material including a matrix rubber and a filler, and is a filler model in which at least one filler is divided into a finite number of elements. And a matrix material model in which the matrix rubber is divided into a finite number of elements having an inner peripheral surface that coincides with the outer peripheral surface of the filler model in the initial state and can be separated from the outer peripheral surface. Performing a deformation calculation by giving a tensile deformation condition to the rubber material model, and obtaining a necessary physical quantity from the rubber material model, and the step of performing the deformation calculation includes the matrix model Calculating a separation distance between the inner peripheral surface of the filler model and the outer peripheral surface of the filler model; The inner peripheral surface on the basis of between distance and said outer peripheral surface is characterized by comprising the steps of providing an attractive force direction approaching each other.

  The invention according to claim 2 is the rubber material simulation method according to claim 1, wherein the filler is carbon black.

  According to a third aspect of the present invention, when the rubber material model is subjected to load deformation, the attractive force gradually increases with the increase in the separation distance and gradually increases to reach a peak, and further increases in the separation distance. 3. The rubber according to claim 1, wherein the rubber is determined based on a function that draws a parabolic curve that includes a gradually decreasing region that decreases to zero at a predetermined characteristic distance. This is a material simulation method.

  In the invention according to claim 4, in the deformation calculation of the rubber material model, the attractive force linearly decreases from a value at the start of unloading to zero in the unloading deformation from the gradually decreasing region to the initial state. 4. The rubber material simulation method according to claim 3, wherein energy loss is introduced.

  In the invention according to claim 5, the rubber material model includes an interface model defined between the outer peripheral surface of the filler model and the inner peripheral surface of the matrix model. The thickness is zero in an initial state, and the outer peripheral surface and the inner peripheral surface, which overlap each other in the initial state, are separated from each other, and are formed of an element that is deformed to a size surrounded by the node. Item 5. A rubber material simulation method according to any one of Items 1 to 4.

  In the rubber material simulation method of the present invention, in the deformation calculation, a step of calculating a separation distance between the inner peripheral surface of the matrix model and the outer peripheral surface of the filler model, and the filler model on the inner peripheral surface based on the separation distance is calculated. Applying an attractive force in a direction approaching the outer peripheral surface of the. Therefore, the phenomenon that the matrix model peels from the filler model at the time of tensile deformation can be taken into the deformation calculation. This helps to allow simulation closer to actual rubber deformation. Further, an attractive force determined based on the separation distance of the outer peripheral surface of the filler model is applied to the inner peripheral surface of the matrix model. For this reason, the filler model and the matrix rubber can be associated with each other at the interface, and various interactions can be set.

  Further, as in the invention described in claim 4, when the attractive force is linearly decreased in the unloading deformation from the gradually decreasing region to the initial state, energy loss can be introduced into the deformation calculation. Therefore, a simulation closer to the actual deformation behavior of the rubber material is possible.

  Further, the deformation calculation as described above is defined in the rubber material model between the outer peripheral surface of the filler model and the inner peripheral surface of the matrix model and has a thickness in an initial state. It is easily achieved by including an interface model consisting of elements that are zero and that are deformed to a size surrounded by the nodes by displacement of the nodes of the outer peripheral surface and the inner peripheral surface that overlap each other in the initial state. obtain.

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.
FIG. 1 shows a computer apparatus 1 for carrying out the simulation method of the present invention. The computer device 1 includes a main body 1a, a keyboard 1b and a mouse 1c as input means, and a display device 1d as output means. Although not shown, the main body 1a is provided with a processing unit (CPU), a ROM, a working memory, a mass storage device such as a magnetic disk, and CD-ROM and flexible disk drives 1a1 and 1a2. The mass storage device stores a processing procedure (program) for executing a simulation method of the present invention, which will be described later. Therefore, the computer device 1 constitutes a rubber material simulation device.

  FIG. 2 shows an example of the processing procedure of the simulation method. In the present embodiment, first, a rubber material model in which a filler is blended is set (step S1). In FIG. 3, the initial state of the rubber material model 2 as a microscopic structure (unit cell) is visualized and shown in FIG.

  The rubber material model 2 is obtained by dividing a minute region of a rubber material to be analyzed (whether it exists or not) using a finite number of small elements (for example, 2a, 2b, 2c,...). is there. Each element 2a, 2b, 2c... Is defined so as to be capable of numerical analysis. The numerical analysis is possible, for example, by the numerical analysis method such as the finite element method, the finite volume method, the difference method or the boundary element method, the necessary parameters are set according to the rules that allow deformation calculation for each element or the whole system. Means that. Specifically, for each element 2a, 2b, 2c,..., A coordinate value of a node, an element shape, a material characteristic value, and the like are defined. For each of the elements 2a, 2b, 2c, etc., for example, a triangular or quadrilateral element as a two-dimensional plane, and a tetrahedral element, for example, is preferably used as a three-dimensional element. Thus, the rubber material model 2 constitutes numerical data that can be handled by the computer device 1.

  Moreover, in this embodiment, what performs the deformation | transformation calculation of the rubber material model 2 in a plane strain state is illustrated. Therefore, no distortion occurs in the Z direction (perpendicular to the paper surface). In this example, the length of one side of the rubber material model 2 as a unit cell is a square of 35 nm × 35 nm in both vertical and horizontal directions.

  The rubber material model 2 of this embodiment includes a filler model 3 modeled by dividing at least one filler into a finite number of elements, and a matrix model 4 modeled by dividing matrix rubber into a plurality of elements. Including.

  FIG. 5 shows a schematic view of an electron micrograph of carbon black filled in rubber as an example of the filler. The carbon black 6 is composed of carbon particles having an approximately spherical or egg-shaped primary particle 7 having a diameter of about several tens of nanometers as a minimum structural unit, which is a single unit or a bunch of cocoons that are irregularly coupled three-dimensionally as shown in the figure. It has a structure (this is a “secondary particle”). The filler model 3 of this embodiment is exemplified by a spherical model of carbon black primary particles as the filler.

  The filler model 3 of this embodiment is divided and modeled into quadrilateral planar elements. The quadrilateral includes both elements having the same length of the four sides and elements having the same length of the four sides. Further, in this embodiment, in order to simplify the deformation calculation, the model is axisymmetric with respect to the x axis and the y axis in FIG. Moreover, the filler model 3 has the outer peripheral surface Ea, as FIG. 4 shows. The outer peripheral surface Ea of the filler model 3 is constituted by straight sides connecting the nodes 3A, 3B, 3C,... Appearing outside the elements 3a, 3b, 3c,. The

  Moreover, various fillers represented by carbon black have a hardness several hundred times that of matrix rubber. For this reason, in this embodiment, the filler model 3 is handled as an elastic body. Therefore, the filler model 3 is given a longitudinal elastic modulus as a material characteristic value, and the constitutive equation in the deformation calculation follows Hooke's law. In this embodiment, one of the primary particles of the filler is modeled in the unit cell, but the number can be increased in consideration of the amount of filler added to the rubber. Furthermore, the filler model 3 may be a model of the shape of secondary particles. Further, the filler is not limited to carbon black, and may be silica, for example.

  The matrix model 4 is modeled by dividing the matrix rubber into a finite number of elements. The matrix model 4 of this embodiment is defined in all regions except the filler model 3. Specifically, it surrounds the periphery of the filler model 3 and constitutes the main part of the rubber material model 2. The matrix model 4 in this example is modeled by being divided into quadrilateral planar elements. As described above, the quadrilateral can include both elements having the same length of the four sides and elements having the same length of the four sides. However, it goes without saying that other elements may be used.

  The matrix model 4 has an inner peripheral surface Eb that coincides with the outer peripheral surface Ea of the filler model 3 in the initial state. The initial state is a state in which the body force of the rubber material model 2 is ignored and the external force is zero. The inner peripheral surface Eb of the matrix model 4 is constituted by straight sides that connect between the nodes 4A, 4B, 4C,... That appear inside the elements 4a, 4b, 4c,. Is done. In the initial state, the nodes 4A, 4B, 4C,... Of the matrix model 4 are at the same positions as the nodes 3A, 3B, 3C,. Therefore, the inner peripheral surface Eb of the matrix model 4 coincides with the outer peripheral surface Ea of the filler model 3, and the gap is zero.

  Further, the matrix model 4 is defined such that its inner peripheral surface Eb can be separated from the outer peripheral surface Ea of the filler model 3. “Definable to be separable” is sufficient if the inner peripheral surface Eb of the matrix model 4 has a room that can be separated from the outer peripheral surface Ea of the filler model 3 by the action of an external force. When the inner peripheral surface Eb of the matrix model 4 is separated from the outer peripheral surface Ea of the filler model 3, a calculated void is formed therebetween. In the simulation of a rubber material in which a filler is blended, setting of such boundary conditions can be said to be one of new structures that have never been seen before.

  The constitutive equation of the matrix model 4 employs the affine model molecular chain network theory in this embodiment. In the molecular chain network theory, as shown in FIGS. 6A and 6B, the rubber material a has, as a microscopic structure, a network structure in which randomly oriented molecular chains c are connected at junction points b. Assuming that The junction point b is, for example, a chemical bond between molecules, and includes a crosslinking point.

  As shown in FIG. 6C, one molecular chain c is composed of a plurality of segments d. One segment d is a repeating minimum structural unit in the molecular chain network theory. Further, one segment d is chemically equivalent to one in which a plurality of monomers f in which carbon atoms are connected by a covalent bond are connected, as shown in FIG. 6 (D). Since the individual carbon atoms can freely rotate around the bond axis between the atoms, the segment d can take various forms such as winding as a whole.

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

  In the above non-patent document 1, Aruuda et al. Propose an 8-chain model based on the molecular chain network theory. This proposal is based on the premise that the viscoelastic material forms a cubic network structure h in which minute 8-chain models g are aggregated as viewed macroscopically as shown in the left side view of FIG. As shown in an enlarged view on the right side of FIG. 7, one 8-chain model g includes eight molecular models c each having eight molecular points c positioned at each vertex of a cube from one junction point b1 defined at the center of the cube. It is assumed that each extends to the junction point b2. In the present embodiment, such an 8-chain model is basically adopted as a constitutive equation of the matrix model, and is modified. First, the relationship between the main strain bi of the matrix model and the main stretch λi satisfies the following formula (1).

Here, “N” is the average number of segments per molecular chain c, “C ^R ” is the shear modulus of matrix rubber, among which “n” is the number of molecular chains c per unit volume, “ “k _B ” is the Boltzmann constant, “T” is the absolute temperature, and “L” is the Langevin function. The velocity type constitutive equation can be expressed by the following equation (2) using Kirchhoff's stress rate and strain rate.

“Σ _jj ” is the Cauchy stress tensor, “δ _ij ” is the Kronecker delta, A _ij is the left stretch tensor, and “Κ” is the penalty parameter for the penalty method introduced to constrain the volume of the matrix model to be constant. In this embodiment, the value is 100.

  Furthermore, the matrix model 4 of this embodiment is modified from the 8-chain model of Arruda et al. In order to reproduce the energy loss of the rubber itself. First, the rubber material is allowed to have a large strain that can reach several hundred percent due to the extension of the molecular chain c intertwined in a complicated manner. In other words, in the deformation process under load, the entangled portions of the molecular chain c are unwound and the number of the junction points b is reduced.

  As shown in FIG. 8A, when the tensile stress in the direction of the arrow acts on the molecular chains c1 to c4 joined at one joining point b, each molecular chain c1 to c4 expands and the joining point b is large. It is considered to be damaged (disappeared) under strain. As shown in FIG. 8B, the two molecular chains c1 and c2 that have been two so far behave as if they were one long molecular chain c5. The same applies to the molecular chains c3 and c4. Such a phenomenon is considered to occur sequentially as the load deformation of the rubber material proceeds. In this case, a lot of energy is lost. In the present embodiment, this state is taken into the equation (1). For this reason, the average number N of segments per molecular chain c is defined as a variable parameter indicating a different value during load deformation and unload deformation. Here, the load deformation is a deformation in which the distortion of the matrix model 4 increases during a very short time, and conversely, the unloading deformation is a deformation in which the distortion decreases.

Regarding the above points, the macroscopic three-dimensional network structure h in FIG. 7 will be referred to again. The network structure h is assumed to have k 8-chain models g coupled in the width direction, the height direction, and the depth direction. However, k is a sufficiently large number. When the total number of junction points b included in the network structure h is represented by the sign “m” as “entanglement number”, the entanglement number m is expressed by the equation (3) and the molecular chain c included in the network structure h. The number, that is, the number “n” of molecular chains contained in the unit volume of the matrix model 4 can be expressed by the equation (4).
m = (k + 1) ³ + k ³ (3)
n = 8k ³ Formula (4)

Here, since k is a sufficiently large number, when the terms other than the third-order term of k are omitted, the above equations can be expressed by the following equations (5) and (6).
m = 2k ³ Formula (5)
n = 8k ³ (6)

Further, from the relationship between the expressions (5) and (6), the entanglement number m can be expressed by the expression (7) using n.
m = n / 4 Formula (7)

Further, even if the matrix model 4 is deformed, the total number N _A of the molecular chain segments contained therein does not change, so that the equations (8) and (9) hold.
N _A = n · N (8)
_{N = N A / n = N} A / 4m ... formula (9)

  As is clear from the equation (9), in order to make the average segment number N a variable parameter in the load application process and the subsequent strain recovery process, the entanglement number m of the molecular chain c of the rubber is changed. You can do it. Thus, the energy loss of the matrix model 4 itself can be taken into the deformation calculation by setting the average segment number N as a variable parameter having different values between the load deformation and the unload deformation. .

The average segment number N can be determined by various methods. As an example, it is desirable to increase the load based on a parameter related to the strain at the time of load deformation. The parameter related to the strain is not particularly limited, and examples thereof include strain, strain rate, or first-order invariant I ₁ of strain. In the present embodiment, the average segment number N is defined by the following formula (10). This indicates that the average segment number N is a function of the first-order invariant I ₁ of distortion (more specifically, the parameter λ _c that is the square root) of each element of the matrix model 4.

Expression (10) is an example determined by various experiments, and A to E are all constants. This can be easily determined from actual measurement results such as a simple uniaxial tensile test of a rubber test piece. For example, first, a stress-strain curve of a rubber material to be analyzed is obtained. And said n and N are defined so that the curve at the time of the load unloading may be followed. Thus, the total number N _A (= n · N) of the molecular chain segments is determined. Then, since when the load, the total number of segments N _A molecule chain c during both unloading the same, to match the curve of the load at the load, determine the average number of segments N in each strain (where N is varied .) Then, parameters A to E in Expression (10) are determined so as to match the determined average segment N at the time of loading. In this embodiment, N = 16 is used, and N at the end of load is set to be equal to that at unloading. In this embodiment, the above constants are set as follows.
A = + 2.9493
B = −5.8029
C = + 5.5220
D = -1.3582
E = + 0.1325

FIG. 9 shows the relationship between the average number of segments N and the parameter λ _C. As λ _C, which is a parameter related to distortion, increases, the average segment number N increases smoothly. In this example, the upper limit of the parameter λ _C is 2.5. In the deformation calculation described later, the parameter λ _C is always calculated for each element of the matrix model 4 at the time of load deformation. The calculated λ _C is substituted into Equation (10), and the average number of segments N of the element in the distortion state is calculated. The average segment number N is calculated at an appropriate timing and is taken into the constitutive equation of the matrix model 4. In the present embodiment, the average number of segments N is a constant value when the matrix model 4 is unloaded and deformed.

  In addition, the rubber material model 2 of the present embodiment is further exemplified by including an interface model 5. The interface model 5 is defined between the outer peripheral surface Ea of the filler model 3 and the inner peripheral surface Eb of the matrix model 4. The interface model 3 represents the void shown in FIG. 25 in the deformation calculation.

  FIG. 10A shows the rubber material model 2 partially enlarged. In the initial state where the nodes 3B, 3C, 3D,... On the outer peripheral surface Ea of the filler model 3 and the nodes 4B, 4C, 4D,. The thickness is zero (not visible in the figure). On the other hand, as shown in FIG. 10 (B), when the rubber material model 2 is subjected to tensile deformation and the nodes 3B, 3C, 3D... And the nodes 4B, 4C, 4D. , And changes to a size surrounded by the nodes. Specifically, the side connecting the two nodes 3B and 4B that overlap in the initial state, the side connecting 3C and 4C, the side forming the outer peripheral surface Ea of the filler model 3, and the inner peripheral surface of the matrix model 4 It is transformed into a quadrilateral element 5b... Surrounded by sides constituting Eb.

In the interface model 5, the inner peripheral surface Eb of the matrix model 4 and the outer peripheral surface of the filler model 3 approach each other based on the separation distance between the outer peripheral surface Ea of the filler model 3 and the inner peripheral surface Eb of the matrix model 4. A direction attractive force (Traction) is applied to each of the surfaces Ea and Eb (or nodes thereof). The interface model 5 expresses the adhesive force (adhesive friction force) between the outer peripheral surface Ea of the filler model 3 defined to be separable and the inner peripheral surface Eb of the matrix model 4, and the interaction between these interfaces ( (Stickiness) can be simulated. When the second phase particles are peeled from the metal matrix and voids are formed at the interface, for example, the following experiment by needleman et al. This document was developed based on a metal composition, but can be similarly applied to the separation of the rubber matrix and the filler and the generation of voids associated therewith by appropriately changing the order.
A Continuum Model for Void Nucleation by Inclusion Debonding / A. needleman (Journal of Applied Mechanics SEP. 1987 Vol.54 P.525-531)

FIG. 11 is a graph showing the relationship between the attractive force and the separation distance employed in this embodiment. This graph was determined with reference to the above literature. The vertical axis represents the attractive force t _AB . The abscissa indicates a dimensionless amount (u / δ) obtained by dividing the separation distance u _AB by the feature distance δ. Here, the subscripts A and B of the attractive force t or the separation distance u are, as shown in FIG. 12, the midpoint B of the two nodes 1 and 2 on the outer peripheral surface Ea of the interface model 5 and the initial state. This indicates that the distance is from the midpoint A of the nodes 3 and 4 on the inner peripheral surface Eb side of the matrix model 4 of the interface model 5 that overlaps the nodes 1 and 2. The “characteristic distance δ” is a limit length when the attractive force t _AB becomes zero when the separation distance u _AB between the midpoints A and B in FIG. 12 is increased.

In this embodiment, the attractive force t _AB gradually increases with the increase of the separation distance u _AB and gradually increases as it reaches the peak (σ max), and further decreases with the increase of the separation distance u _AB , and A parabolic curve including a gradually decreasing region that reaches zero when the separation distance u _AB reaches the characteristic distance δ is drawn. That is, the attractive force t _AB is a function of the separation distance u _AB . Only when the attractive force t _AB reaches the peak value σmax, it is considered that the molecular rubber slips only in the matrix rubber. This peak value σmax represents the strength of the interface, and in this example, it is generated at approximately 1/3 of the characteristic distance δ. Further, the attractive force t _AB disappears when the separation distance u _AB becomes the characteristic distance δ. Thereby, complete slip occurs between the midpoints A and B.

By changing the two parameters of the interface distance σmax, which is the peak value of the characteristic distance δ and the attractive force t _AB , the strength against peeling of the interface between the filler model 3 and the matrix model 4 can be changed variously. . This means that various behaviors against interfacial delamination can be adjusted according to the filler and / or matrix rubber constituting the rubber material.

Next, formulation of the function of the attractive force t _AB will be described. In this example, a two-dimensional shape is shown, but a three-dimensional shape can also be used. The potential energy φ (u _AB ) between the midpoints A and B in the state of the separation distance u _AB can be determined by the formula (11) by adopting the needleman basic formula.

The attractive force acting between the midpoints A and B can be obtained by partial differentiation of the potential energy φ (u _AB ) with respect to the separation distance u _AB . This is shown in equation (12).

As described above, slipping occurs at the interface between the matrix rubber and the filler when the gap becomes large. In addition, even when the voids disappear, energy loss occurs due to molecular chain bonding. Therefore, in the present embodiment, the attractive force t _AB is linearly decreased from the value at the start of unloading to zero in the unloading deformation from the gradually decreasing region to the initial state. For example, when unloading is started from an arbitrary point Px on the gradual decrease area of FIG. 11, as shown by the chain line arrow in the figure, the attractive force t _AB is the same as the point Px and the origin as the separation distance u _AB decreases. Decrease while taking the value on the straight line connecting. For this reason, the attractive force t _AB draws a loop representing energy loss in one cycle of loading and unloading, as indicated by solid line arrows and chain line arrows. Therefore, in the deformation calculation of the rubber material model 2 of the present embodiment, an energy loss due to separation at the interface between the matrix model 4 and the filler model 3 is introduced, which is closer to the actual deformation behavior. Further, by setting the parameters more appropriately, it is possible to calculate energy loss that approximates the actual rubber material.

The attraction force t _AB that decreases linearly at the time of unloading is a linear expression in which the value of the point Px is substituted for (u _AB / δ) in the term [1- (u _AB / δ)] ² of equation (12). Is calculated using When the attractive force t _AB does not exceed the peak value σmax, the attractive force t _AB is defined so as to return to the initial state through the same route as when loaded. In this case, reversible deformation occurs and no energy loss occurs. As described above, when the attractive force t _AB does not exceed the peak value σ max, slipping at the interface is not started, so that no energy loss occurs. Needless to say, the attractive force t calculated using the interface model 5 acts on each node of the inner peripheral surface Eb of the matrix model 4 and the outer peripheral surface Ea of the filler model 3.

  Next, in the simulation method of the present embodiment, deformation calculation is performed by giving a tensile deformation condition to the rubber material model 2 (steps S2 and S3).

  The tensile deformation conditions include a strain rate for applying a tensile strain to the rubber material model 2 and a maximum strain amount. Moreover, the simulation of this embodiment is performed based on the homogenization method (asymptotic expansion homogenization method). If the analysis target region is constituted by repetition of an arbitrary microscopic structure and the degree of repetition is very dense, the region may not be directly discretized by the finite element method. In the homogenization method, as shown in FIG. 13, the entire rubber material M having the microscopic structure of the rubber material model 2 (unit cell of the rubber material model 2 shown in FIG. 3) periodically and vertically. Two independent variables are used: a macroscopic scale xI representing the microscopic structure and a microscopic scale yI representing the microscopic structure. In the homogenization method, asymptotic expansion of independent variables in different scale fields of microscopic scale yI and macroscopic scale xI is performed to a certain extent that reflects the model structure of the microscopic structure shown in FIG. The average mechanical response of the entire rubber material M having the following size can be obtained.

The asymptotic expansion homogenization method itself has already been established in the numerical calculation method (see, for example, the following document), and is not essential for the present invention, so it will not be described in detail here. Higa, Y. And Tomita, Y ,, Computational Prediction of Mechanical Properties of Nickel-based superalloy with gamma Prime Phase Precipitates, Proceedings of ICM8 (Victoria, BC, Canada), Advance Materials and Modeling of Mechanical Behavior, (Edited by Ellyin, F, and Proven, JW), III (1999), 1061-1066, Fleming Printing Ltd.
Yoshikazu Higa, Yoshihiro Hamada, Modeling and Simulation of Deformation Behavior by Homogenization Method of Particle Reinforced Composites, Transactions of the Japan Society of Mechanical Engineers, A66 (2000), 1441-1446.

Further, in the present embodiment, the rubber material model M shown in FIG. 13 is a rectangular shape of 100 nm × 100 nm, and a uniform uniaxial tensile strain Γ ₂ (its strain rate is 1.0 × 10 ⁻⁵ / s), a constant strain rate is given to the microscopic structure (unit cell) shown in FIG. In addition, an unloading condition for gradually reducing the strain to zero at the same strain rate when the strain load Γ _{2 in the} x2 direction reaches 0.40 is added to the tensile strain condition of the average strain rate in the x2 direction of FIG. .

  FIG. 14 shows an example of a more specific processing procedure of deformation calculation (deformation simulation). In the deformation calculation, first, data is input to the computer apparatus 1 (step S31). The input data includes numerical data constituting the rubber material model 2, the preset deformation or boundary conditions, and the like.

  Next, a stiffness matrix of each element is created (step S32), and then the stiffness matrix of the entire structure is assembled (step S33). Displacement of known nodes and nodal forces are introduced into the stiffness matrix of the entire structure (step S34), and the stiffness equation is analyzed. Then, unknown node displacement is determined (step S35), and physical quantities such as strain, stress and principal stress of each element are calculated and output (steps S36 to 37). In step S38, it is determined whether or not to end the calculation. If the result is negative, step S32 and subsequent steps are repeated.

  The above-described deformation calculation can be performed using, for example, an engineering analysis application software using a finite element method (for example, LS-DYNA developed by Livermore Software Technology, USA).

  In the deformation calculation of the present embodiment, constants and the like are set as follows.

<Matrix model>
C ^R = 0.4 (MPa)
N = 16
n = 1.0 × 10 ²⁶
T = 296K
k _B = 1.38066 × 10 ⁻²⁹
N _A = n · N = 1.6 × 10 ²⁷
Average number of segments N during load deformation
N = 16
Average number of segments N during unloading deformation (constant)
N = 16

In this example, the average number of segments N at the time of load deformation is constant, and attention is paid only to the energy loss at the interface. For example, the average number of segments N is defined by the following equation (the constant can be changed as appropriate). By doing so, the energy loss of the matrix model can be taken into account.
N = -3.2368 + 20.6175 λc-21.8168 λc ² +10.8227 λc ³ -1.9003 λc ⁴
The three-dimensional 8-chain model is used as the rubber material model so as not to change in the thickness direction (Z-axis direction in FIG. 3).

<Filler model>
Longitudinal elastic modulus E: 100 MPa
Poisson's ratio: 0.3

<Interface model>
Feature distance δ = 25nm
Interface strength (peak value) σmax = 0.1 MPa
Strength ratio γ = 1.0 between normal direction n and shear direction s
Note that the characteristic distance δ, the interface strength σmax, and the like may be determined with reference to experimental results of actual materials, or may be set as appropriate to match the results.

  Moreover, the computer apparatus 1 can acquire a required physical quantity from the result of the deformation calculation (step S4). FIGS. 15A to 15D visualize the deformation behavior from the initial state to the separation between the matrix model 4 and the filler model 4. The physical quantity here includes strain and stress of each element. The deformation calculation is performed in minute time increments, and such visualization can be easily performed by acquiring the strain and stress of each element at appropriate time intervals. In particular, elements with large stress are given a dark gray color.

  As shown in FIG. 15, the separation between the filler model 3 and the matrix model 4 starts at the upper and lower poles of the filler model 3 and gradually spreads to the equator. It can be seen that the stress concentration of the matrix model 4 is released by the generation of the voids. By further tensile deformation, the void (interface model) grows greatly in the direction of the tensile axis, causing high elongation and stress on the inner peripheral surface Eb of the matrix model 4.

In FIG. 16, as other physical quantities, the vertical axis represents the average stress Σ ₂ (calculated by the following equation (13)) in the macroscopic structure of the rubber material model 2, and the horizontal axis represents the tensile strain Γ _2. Each is shown. Here, the upper limit of the tensile strain Γ2 is 0.4. In the graph in the figure, the solid line shows the boundary conditions set so that the interface does not peel off. In other graphs with chain lines, the interface strength σmax is changed to 0.1, 0.2 and 0.3 MPa, respectively. In the case where the matrix model 4 and the filler model 3 do not peel at the interface, the stress and strain are approximately proportional and no energy loss occurs. However, it can be seen that energy loss occurs in any of the examples in which peeling at the interface is taken in and energy loss is caused there.

“A” is the area of the unit cell, a ⁱ is the area of the element i, and σ ⁱ ₂₂ is the tensile stress of the element i.

FIG. 17 also shows an example of physical quantities obtained by deformation calculation, where the vertical axis shows the logarithmic volume strain Ev in the macroscopic structure of the rubber material model 2 and the horizontal axis shows the tensile strain Γ _2. ing. Here, the upper limit of the tensile strain Γ ₂ is 0.4. In the graph in the figure, the legend is the same as that in FIG. Moreover, the test result of an actual rubber material is shown by a thick solid line. In this test, it can be confirmed that the interface strength σmax = 0.3 MPa is very close to the test result of the actual rubber material. In the example in which the interface strength is increased, it is simulated that the volume strain increases very slowly because a high restraint force acts on the interface. However, it can be seen that a sudden volume distortion occurs at the stage where the attractive force becomes zero and complete slipping occurs at the interface.

FIG. 18 shows the result of deformation calculation for four types of interface model 5 with different feature distances δ. The left vertical axis shows the mean stress Σ ₂ in the macroscopic structure of the rubber material model 2, the right vertical axis shows the logarithmic volume strain Ev of the rubber material model, and the horizontal axis shows the tensile strain Γ ₂ of the rubber material model. Has been. Here, the upper limit of the tensile strain Γ ₂ is 0.4. In the example in which the characteristic distance δ is increased, the distance until complete slippage at the interface is started is large. Thus, it can be seen that, similar to the interface strength results above, it has been simulated that the volumetric strain increases very slowly. In the figure, l ₀ is the vertical length of the rubber material model M 100 nm.

FIG. 19 shows another implementation result.
In this example, the filler model 3 is modeled in a substantially elliptic shape. The major axis G of the elliptical filler model 3 is inclined at an angle θ with respect to the x-axis shown in the figure. The filler model 3 in this example is used as a model of carbon black secondary particles, for example.

  20 and 21 show the results of deformation calculation under the same conditions as described above by changing the aspect ratio represented by the ratio (La / Lb) between the minor axis a and the major axis b of the filler model 3. Is shown. In this example, the angle θ formed by the long axis is 0 °. In FIG. 20, the vertical axis represents the mean stress Σ2 of the macroscopic structure, and the horizontal axis represents the tensile strain Γ2. In FIG. 21, the vertical axis represents the logarithmic volume strain Ev. In either case, the upper limit of the tensile strain Γ2 is 0.4. As is apparent from the results of FIGS. 20 to 21, in the rubber material model including the filler model having a high aspect ratio, low resistance to tensile deformation and a rapid increase in volume strain are observed in all strain sections.

22 and 23, the aspect ratio represented by the ratio (La / Lb) between the minor axis La and the major axis Lb of the filler model 3 is constant at 2.0, and the angle θ of the major axis is changed. (Θ = 0 to 90 °), and the result of the deformation calculation under the same conditions as described above is shown. In FIG. 22, the vertical axis represents the average stress Σ ₂ of the macroscopic structure, and the horizontal axis represents the tensile strain Γ ₂ . In FIG. 23, the vertical axis represents the logarithmic volume strain Ev. As is clear from these results, when the angle θ of the major axis of the filler model 3 is increased, high resistance to tensile deformation is observed in all strain sections, and energy loss is reduced. Further, when the angle θ is large, the volumetric strain is slowly increased.

  The simulation of the rubber material as described above can evaluate viscoelastic characteristics including energy loss in more detail for rubber materials in which the blending ratio of the filler to the rubber material and the type of filler are variously changed. As a specific application, for example, the particle size and aspect ratio of the filler model are variously simulated to determine the particle shape that gives the optimum energy loss, and an actual filler is prototyped based on this shape. Can be evaluated. It is also possible to conduct a simulation by combining various fillers and matrix rubbers having different physical properties, and to investigate a composition that can obtain an optimum energy loss. In the simulation of the present invention, it is possible to simulate the actual behavior rather than the separation between the filler model and the matrix model, and the analysis can be performed with high accuracy prior to trial production of the actual rubber material using the result. Make it possible.

It is a perspective view which shows an example of the computer apparatus used by this embodiment. It is a flowchart which shows the process sequence of this embodiment. It is a diagram showing one embodiment of a rubber material model (microscopic structure). It is the A section enlarged view. It is a diagram which shows the shape of carbon black. (A) is a viscoelastic material, (B) is a diagram for explaining the structure of one molecular chain, (C) is an enlarged view of one molecular chain, and (D) is an enlarged view of a segment. The left view is a perspective view showing a network structure of a viscoelastic material, and the right view is a perspective view showing an example of an 8-chain model. (A), (B) is a diagram explaining the fracture | rupture of the junction of a molecular chain. It is a graph which shows the relationship between parameter (lambda) c and the average number of segments N per molecular chain. (A) is an initial state, (B) is a schematic diagram showing the interface state of the rubber material model after deformation. It is a graph which shows the relationship between attractive force and separation distance. It is an expansion explanatory view of an interface. It is a conceptual diagram which shows the relationship between the microscopic structure explaining the homogenization method, and the whole structure. It is a flowchart which shows the procedure of deformation | transformation calculation. (A)-(D) are the deformation | transformation figures of the rubber material model which show visually the result of a deformation | transformation calculation. It is a graph which shows the relationship between an average stress and a tensile strain as a deformation | transformation calculation result of a rubber material model. It is a graph which shows the relationship between volume strain and tensile strain as a deformation | transformation calculation result of a rubber material model. It is a graph which shows the relationship with an average stress, a volume strain, and a tensile strain as a deformation | transformation calculation result of the rubber material model at the time of changing the characteristic distance (delta). It is a diagram which shows other embodiment of a rubber material model. It is a graph which shows the relationship between an average stress and a tensile strain as a deformation | transformation calculation result of the rubber material model when the aspect ratio of a filler model is varied. It is a graph which shows the relationship between volume strain and tensile strain as a deformation | transformation calculation result of the rubber material model when the aspect ratio of a filler model is varied. It is a graph which shows the relationship between an average stress and a tensile strain as a deformation | transformation calculation result of the rubber material model at the time of changing the angle of the major axis of a filler model. It is a graph which shows the relationship between a volume strain and a tensile strain as a deformation | transformation calculation result of the rubber material model at the time of changing the angle of the major axis of a filler model. It is a graph which shows the tension test result of the rubber material which changed the compounding quantity of carbon black. It is a microscope picture of carbon black compounding rubber.

Explanation of symbols

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

Claims (5)

A rubber material simulation method for simulating deformation of a rubber material including a matrix rubber and a filler,
A filler model in which at least one filler is divided into a finite number of elements;
A step of setting a rubber material model including a matrix model in which the matrix rubber is divided into a finite number of elements having an inner peripheral surface that matches the outer peripheral surface of the filler model in the initial state and can be separated from the outer peripheral surface. ,
Including a step of performing a deformation calculation by giving a tensile deformation condition to the rubber material model, and a step of obtaining a necessary physical quantity from the rubber material model,
And the step of performing the deformation calculation, calculating a separation distance between the inner peripheral surface of the matrix model and the outer peripheral surface of the filler model,
A method for simulating a rubber material, comprising: applying an attractive force in a direction in which the inner peripheral surface and the outer peripheral surface approach each other based on the separation distance.   The rubber material simulation method according to claim 1, wherein the filler is carbon black. The attractive force gradually increases with the increase in the separation distance and reaches a peak at the time of load deformation of the rubber material model, and
4. The method according to claim 1, further comprising: a function that draws a parabolic curve that decreases smoothly as the distance increases further, and that includes a gradually decreasing region that reaches zero at a predetermined characteristic distance. Item 3. The rubber material simulation method according to Item 1 or 2.   In the unloading deformation from the gradually decreasing region to the initial state, the attractive force linearly decreases from a value at the start of unloading to zero, thereby introducing an energy loss in the deformation calculation of the rubber material model. The method for simulating a rubber material according to claim 3. The rubber material model includes an interface model defined between the outer peripheral surface of the filler model and the inner peripheral surface of the matrix model,
The interface model is composed of an element whose thickness is zero in an initial state, and an element that deforms to a size surrounded by the nodes by separating the nodes of the outer peripheral surface and the inner peripheral surface that overlap each other in the initial state. The method for simulating a rubber material according to any one of claims 1 to 4, wherein: