JP2005121536A - Method for simulating viscoelastic material - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To analyze the deformation state of a viscoelastic material, such as rubber, precisely. <P>SOLUTION: A method for simulating the viscoelastic material that simulates the deformation of the viscoelastic material includes a step for setting a viscoelastic material model in which a viscoelastic material is modelled by an element capable of numeric analysis; a step for calculating deformation by setting conditions to the viscoelastic material model; and a step for acquiring a required physical quantity from the deformation calculation. In the viscoelastic material model, the relation between stress and distortion is defined, and the number of average segments N per molecular chain differs between loaded deformation and unloaded deformation for expressing energy loss. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to a viscoelastic material simulation method useful for accurately analyzing a deformation state of a viscoelastic material such as rubber.

  Viscoelastic materials such as rubber are used in tires, sports equipment, and other various industrial products. The viscoelastic material is greatly deformed when subjected to a load, and can be restored to its original state when the load is completely removed. In addition, the viscoelastic material has a non-linearity in which the stress and strain are not proportional to each other during deformation, and the stress-strain curve path is different between when the load is applied and when the load is unloaded. In order to reduce the labor and cost of the trial production, the deformation process of the viscoelastic material is simulated in advance using a computer. As conventional viscoelastic material simulation methods, 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 a viscoelastic material shows a different longitudinal elastic modulus according to a 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 the correspondence relationship between the longitudinal elastic modulus and the strain or strain rate is determined. A derivation step is performed. 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 the trial that the longitudinal elastic modulus of a viscoelastic material changes according to a strain rate, it can evaluate in the point which can take in hysteresis loss in simulation. On the other hand, in order to obtain the correspondence between the strain rate and the longitudinal elastic modulus, many experiments are required. This point has room for further improvement.

  In addition, Non-Patent Document 1 proposed by Arruda et al. Describes that a viscoelastic material is modeled and calculated up to a polymer level by using a molecular chain network theory. Here, the molecular chain network theory will be briefly described.

  In the molecular chain network theory, as shown in FIGS. 19A and 19B, the polymer material a as a continuum has a microscopic structure in which randomly oriented molecular chains c are connected at junctions b. The premise is that it has a mesh structure. The junction point b is, for example, a chemical bond between molecules, and includes a crosslinking point.

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

  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. It is further assumed that the end-to-end vector of the molecular chain c having two junctions b, b at both ends co-deforms with the continuum of the material in which it is embedded.

  Aruuda et al. Have proposed a further eight-chain model based on molecular chain network theory. As shown in FIG. 4A, the viscoelastic material can be considered macroscopically as a cubic network structure h in which minute 8-chain models g are assembled. As shown in an enlarged view in FIG. 4B, one 8-chain model g is composed of eight junction points provided at each vertex from one junction point b1 in which the molecular chain c is defined at the center of the cube. It is assumed that it extends to b2.

The viscoelastic material is handled as a superelastic body (a material that hardly changes in volume and returns to its original shape after unloading) in the simulation. A superelastic body is defined as a substance having a strain energy function W that produces a conjugate kirchhoff stress S _ij by being differentiated by a Green strain component E _ij as shown in the following formula (2). The In other words, the strain energy function hypothetically indicates the presence of potential energy stored when the viscoelastic material is deformed. Therefore, the stress and strain relationship can be obtained from the differential gradient of the strain energy function W.

Aruuda et al. Believe that the entropy change suddenly increases when the deformation of the viscoelastic material increases (the molecular chain stretches and orients) according to the non-Gaussian chain theory. ) Shows the strain energy function W. Then, by substituting this strain energy function W into the above equation, the relationship between the stress and strain of the viscoelastic material can be extracted.

  By performing a deformation simulation of a viscoelastic material using the relationship between stress and strain of Aruuda et al., For example, as shown in FIG. 20, a non-linear relationship between stress and strain is obtained in a uniaxial tensile deformation simulation. Can do. This result shows a good correlation with the actually measured value when the load is deformed.

  However, Aruuda et al., Non-Patent Document 1 does not consider energy loss, which is one of the characteristics of viscoelastic materials. For this reason, when a simulation for unloading the load from the state Q in FIG. 20 is performed, strain recovery is performed through substantially the same path as the curve. With this, a hysteresis loop is not formed and energy loss cannot be calculated.

  The inventors tried various experiments based on the model of Aruuda et al. First, it is considered that the viscoelastic material is allowed to have a large strain that can reach several hundreds of percent by extending the long molecular chain c intertwined in a complicated manner as described above. Therefore, the inventors unraveled the intertwined portions of the molecular chains c of the viscoelastic material in the deformation process under load (reduction of the number of joint points b), or further entanglement (unbonding) by unloading the load. An assumption was made that an increase in the number of points b) may occur. In other words, this assumption means that the average number of segments N per molecular chain in the equation (1) is a variable parameter during loading or unloading deformation. The inventors verified this assumption and found that the hysteresis of the viscoelastic material can be expressed.

  As described above, the present invention provides a viscoelastic material simulation method capable of more accurately performing deformation simulation of a viscoelastic material on the basis of incorporating a characteristic capable of expressing energy loss into the viscoelastic material model itself. It is intended to provide.

The invention described in claim 1 of the present invention is a viscoelastic material simulation method for simulating deformation of a viscoelastic material, and sets a viscoelastic material model in which the viscoelastic material is modeled by an element capable of numerical analysis. Performing a deformation calculation by setting a condition on the viscoelastic material model, and obtaining a necessary physical quantity from the deformation calculation. The viscoelastic material model is expressed by the following formula (1): Is defined, and the average number of segments N per molecular chain in the formula (1) is different between load deformation and unload deformation, thereby reducing energy loss. This is a viscoelastic material simulation method characterized by

  The invention according to claim 2 is characterized in that the average segment number N increases according to a parameter relating to strain at the time of load deformation and takes a constant value at the time of unloading deformation. This is a material simulation method.

The invention according to claim 3 is the viscoelastic material simulation method according to claim 2, wherein the parameter relating to the strain is a first-order invariant I ₁ of the strain.

  The invention according to claim 4 is the viscoelastic material simulation method according to claim 2, wherein the parameter relating to the strain is a strain amount or a strain rate.

  In the simulation method of the viscoelastic material of the present invention, the energy loss of the viscoelastic material can be expressed. Therefore, it is possible to perform a highly accurate simulation according to the characteristics of the actual viscoelastic material. Further, the energy loss can be expressed by determining the average number of segments N per molecular chain in the formula (1) as a different parameter between the load deformation and the unload deformation. A large number of experimental data collection steps are not required, and analysis costs and labor can be further reduced.

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 as an input means, a mouse 1c, and a display device 1d as an 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. The computer device 1 is preferably EWS or the like.

  FIG. 2 shows an example of the processing procedure of the simulation method. In this embodiment, a viscoelastic material model is first set (step S1). In FIG. 3, an example of the viscoelastic material model 2 is visualized.

  The viscoelastic material model 2 is obtained by replacing a viscoelastic material to be analyzed (whether or not actually exists) with a finite number of small elements 2a, 2b, 2c. Each element 2a, 2b, 2c... Is defined so as to be capable of numerical analysis. The possibility of numerical analysis means that deformation calculation for each element or the entire system can be performed by a numerical analysis method such as a finite element method, a finite volume method, a difference method, or a boundary element method. Specifically, for each element 2a, 2b, 2c..., A node coordinate value, an element shape, a material characteristic, etc. in the coordinate system are defined. For each element 2a, 2b, 2c..., For example, a triangular or quadrangular element as a two-dimensional plane, and a three-dimensional element, for example, a tetrahedron element is preferably used. Thereby, the viscoelastic material model 2 can constitute numerical data that can be handled by the computer apparatus 1.

  The viscoelastic material model 2 of this embodiment is analyzed in a plane strain state in a deformation simulation described later. Therefore, there is no distortion in the Z direction. The length of one side of the viscoelastic material model 2 is, for example, 2 mm in each length and width. The viscoelastic material model 2 is defined by nine planar elements 2a to 2i forming a quadrilateral.

For each of the elements 2a to 2i, the relationship between the stress and strain of the formula (1) is defined as material characteristics. Therefore, the relationship between stress and strain can be obtained for each element 2a to 2i. The derivation process of Equation (1) will be briefly described. The viscoelastic material has a small volume change due to deformation, and can be ignored. Therefore, the non-compressible viscoelastic material, a non-compression condition can be the density is constant, Kirchhoff stress S _ij can be expressed by Equation (4). E _ij is a component of Green strain, p is hydrostatic pressure (determined by boundary conditions), X _i is a position of an arbitrary object point P in a state C ₀ where neither stress nor strain is 0, and x _j is a deformed state. This is the position of the object point P in C.

Since the Cauchy stress component σ _ij and the kirchhoff stress component S _mn have the relationship of the following equation (5), the equation (4) can be obtained from the equation (4).

Here, when the volume is constant, J in Equation (5) indicating the volume change rate of the object can be regarded as 1. Further, since the strain energy function of Equation (3) is a function of only the first-order invariant I ₁ of the strain of the left Cauchy-Green deformation tensor A _ij , Equations (6) to (7) are obtained.

Further, by using the following relational expressions (8 to 10), the speed format display of Expression (7) becomes Expression (11).

  Then, under the constant volume condition, there is no essential difference even if the Jaumann velocity of the Cauchy stress shown in Equation (11) is replaced with the Jaumann velocity of the kirchhoff stress, and the deformation rate tensor D is replaced with the strain rate tensor. Thus, the speed format display of the constitutive equation of the incompressible viscoelastic material can be expressed by the above equation (1).

  In the viscoelastic material model 2 of the present embodiment, in the above formula (1), the average number of segments N per molecular chain is defined as a parameter that differs between load deformation and unload deformation. Thereby, each element 2a thru | or 2i of the viscoelastic material model 2 can express an energy loss. This is fundamentally different from Arruda's proposal. Here, the load deformation is a deformation in which the strain of the viscoelastic material model 2 increases during a minute time, and conversely, the unloading deformation is a deformation in which the strain decreases.

  FIG. 4A shows a macroscopic three-dimensional network structure h of a viscoelastic material, and FIG. 4B shows an enlarged view of one of the 8-chain models g constituting the network structure h. Show. The network structure h in this example is assumed to have k 8-chain models g coupled in the width direction, the height direction, and the depth direction.

Now, when the total number of junction points b included in the mesh structure h is represented by the symbol m as “entanglement number”, the entanglement number m can be represented by Expression (12). Similarly, when the number of molecular chains c included in the network structure h (that is, the number of molecular chains included in the unit volume of the viscoelastic material model 2) is n, this n is expressed by the formula (13). be able to.
m = (k + 1) ³ + k ³ Formula (12)
n = 8k ³ (13)

Here, if k is a sufficiently large number, the above equations can be expressed by the following equations (14) and (15), respectively, except for the third-order term of k. The number m can be expressed by equation (16) using n.
m = 2k ³ Formula (14)
n = 8k ³ (15)
m = n / 4 Formula (16)

Furthermore, assuming that the total number N _A of molecular chain segments contained in the viscoelastic material model does not change even when deformed, equations (17) to (18) hold.
N _A = n · N (17)
_{N = N A / n = N} A / 4m ... formula (18)

  In order to express the energy loss of the viscoelastic material model 2 in the simulation, the inventors say that the entanglement number m of the molecular chain c of the viscoelastic material changes in the loading process and the subsequent strain recovery process. I tried the idea. That is, as shown in FIG. 5A, for example, when 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. Was considered to be damaged by large strain.

  As a result, as shown in FIG. 5B, the two molecular chains c1 and c2 that have been two so far are considered to 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 viscoelastic material model 2 proceeds.

The breakage of the joint point b as described above is nothing but a decrease in the number of entanglements m in the viscoelastic material model 2. Since the viscoelastic material model itself has no material in and out, assuming that the total number N _A of the segments of the molecular chain c contained therein does not change, as the load deformation of the viscoelastic material model 2 proceeds, one molecule It is clear from the equation (18) that the average number of segments N included in the chain c increases. That is, the average segment number N is not constant, and it is considered appropriate to use a variable parameter that is different between load deformation and unload deformation. Thereby, the energy loss at the time of deformation of the viscoelastic material model can be reproduced. This will be described in more detail later.

In order to more preferably reproduce such a situation on a computer, it is particularly preferable that the average number of segments N is increased according to a parameter relating to the strain at the time of load deformation. The parameter relating to the strain is not particularly limited, and examples thereof include a strain amount, a strain rate, or a first-order invariant I ₁ of strain. In the present embodiment, the average number of segments N is the square root of the invariant I ₁ of the first-order strain in each element 2a to 2i of the viscoelastic material model 2 as shown in the following formula (19). This is a function of a certain parameter λc.

Equation (19) is an example determined by various experiments, and A to E are constants and can be determined from actual measurement results such as a simple uniaxial tensile test of a rubber specimen. it can. In this example, the above constants are set as follows.
A = + 2.9493
B = −5.8029
C = + 5.5220
D = -1.3582
E = + 0.1325

  FIG. 6 shows the relationship between the average number of segments N and the parameter λc when the elements 2a to 2i of the viscoelastic material model 2 are subjected to load deformation. As λc, which is a parameter related to strain, increases, the average segment number N increases smoothly. In this example, the upper limit of the parameter λc is 2.5. In the deformation simulation of the viscoelastic material model 2 to be described later, the parameter λc is always calculated for each element 2a to 2i of the viscoelastic material model 2 at the time of load deformation. The calculated λc is substituted into equation (19). Thereby, the average number of segments N in the strain state of the element can be calculated.

On the other hand, at the time of unloading deformation of the viscoelastic material model 2, the average number of segments N is a constant value. As a method for determining the average segment number N, for example, there is the following method. In the viscoelastic material to be analyzed, when there is one stress-strain curve, first, n and N are determined so as to match the unloading curve (that is, N _A = n · N is determined). Then, when the load, because the total number of segments N _A unloading during both molecular chain is identical, to match the load of the curve to determine the average number of segments N in each strain (where N is changed.) . Then, parameters A to E in Expression (19) are determined so as to match the determined average segment N at the time of loading. In this embodiment, N = 6.6 is used, and N at the end of load is set to be N at unloading.

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

  The computer apparatus 1 creates a stiffness matrix for each element based on the input data (step S32), and then assembles a stiffness matrix for the entire structure (step S33). Displacement of known nodes and nodal forces are introduced into the stiffness matrix of the entire structure (step S34), and the stiffness equation is analyzed. Then, the unknown nodal displacement is determined (step S35). Then, physical quantities such as strain, stress and principal stress of each element are calculated and output (steps S36 to S37). In step S38, it is determined whether or not to end the calculation. If negative, step S32 and subsequent steps are repeated.

  In this embodiment, the viscoelastic material model 2 is subjected to a uniaxial tensile test, and the tensile strain is gradually increased from 0 to 2.0 in increments of 0.01, and acts on the viscoelastic material model 2 for each strain. Calculate the stress to be On the contrary, after the strain reaches 2.0, the strain is gradually reduced to 0 in steps of 0.01. In each strain state, the average number of segments N is calculated, and this value is substituted into the strain energy function of equation (1) for sequential calculation. As the viscoelastic material model, the three-dimensional 8-chain model of Arruda et al. Is used so as not to change in the thickness direction (Z-axis direction in FIG. 3).

  The simulation is performed using the computer apparatus 1. If the material properties of the viscoelastic material model 2 and the deformation conditions can be determined as described above, the calculation itself in steps S32 to S37 can be performed by, for example, engineering analysis application software using the finite element method (for example, Livermore Software, USA). -LS-DYNA etc. developed and improved by Technology Co., Ltd.)

Moreover, in this embodiment, the constant of Formula (1) etc. were set as follows.
C ^R = 0.268 N = 6.6 T = 296
k _B = 1.38066 × 10 ⁻²⁹ n = C ^R / K _B /T=6.558×10 ²⁵ N _A = n · N = 4.328 × 10 ²⁶

  When the deformation calculation is performed, a necessary physical quantity can be acquired from the result (step S4). FIG. 8 shows the relationship between true stress and strain as a physical quantity of the viscoelastic material model. During the load deformation of the viscoelastic material model 2, a nonlinear first curve L1 was obtained. Further, at the time of unloading deformation, a second curve L2 different from the first curve L1 was obtained. The second curve L2 showed lower elasticity than the first curve L1, and the hysteresis loop could be reproduced. By calculating the closed area of this loop, the energy loss in one deformation cycle can be calculated.

  The shape of each of the curves L1 and L2 can be variously set by changing the function representing the average number of segments N per molecular chain by appropriately setting the coefficients A to E in the equation (19). Therefore, for example, energy loss can be examined in detail for each material by setting various coefficients A to E according to the rubber material to be analyzed. This is very useful for improving the performance of tires and golf balls using viscoelastic materials as the main part. Further, when the stress-strain curve of the material to be simulated is known, by adjusting the coefficients A to E and the like accordingly, the viscoelastic material representing the same stress-strain relationship as the actual viscoelastic material can be obtained. Simulation can be performed easily.

As a comparative example of the present invention, a deformation simulation similar to that of Arruda et al. Was performed with the average number of segments N per molecular chain being constant. In this example, the average segment number N was set as follows.
N = 6.6

  FIG. 9 shows the relationship between the true stress and strain of the viscoelastic material model 2 as a result of the comparative example. In this case, the viscoelastic material model 2 passes through the same curve L3 during load deformation and unload deformation. Therefore, the energy loss that is a characteristic of the viscoelastic material cannot be expressed. This is not related to the value of N, and is based on the fact that N is the same during load deformation and unload deformation.

In the above embodiment, the average segment number N is a function of λc which is the square root of the primary strain invariant I _1. However, strain rate and strain amount can be used instead of λc. In this case, similarly to the equation (19), the energy loss can be expressed by changing the coefficient between the load deformation and the unload deformation.

FIG. 10 shows another embodiment of the present invention.
In this embodiment, the viscoelastic material model 2 is exemplified by a rubber modeled with a rubber (filler) blended in a rubber matrix. That is, the viscoelastic material model 2 includes a matrix model 3 (the darkest part) in which a rubber matrix portion is modeled, and a filler (filler) that is distributed and arranged in the matrix model 3. Examples include a filler model 4 (white portion) and an interface model 5 (slightly black portion) that is interposed between the matrix model 3 and the filler model 4 and forms an interface therebetween. The vertical and horizontal lengths of the viscoelastic material model 2 in FIG. 10 are 300 nm × 300 nm.

  The matrix model 3 constitutes the main part of the viscoelastic material model 2 and is expressed using, for example, triangular or quadrilateral elements. For each element constituting the matrix model 3, the relationship between the stress and strain of the above-described equation (1) is defined as material characteristics. It goes without saying that constants and the like in the formula (1) can be different from those in the above embodiments according to conditions.

  In the present embodiment, the filler model 4 is a model of carbon black as a filler. However, the filler is not limited to carbon black, and may be silica, for example. In this example, the physical shape of the filler model 4 is determined based on a shape obtained by imaging the shape of carbon black filled in actual rubber with an electron microscope. Specifically, the shape (minimum unit) of the carbon black 6 is, as schematically shown in FIG. 11, spherical primary particles 7 made of carbon atoms and having a diameter of about several tens of nm are irregularly bonded three-dimensionally. Has a tufted structure.

  Further, since carbon black has a hardness (longitudinal elastic modulus) several hundred times that of viscoelastic materials such as rubber, the filler model 4 is handled as an elastic body instead of a viscoelastic body in this embodiment. Accordingly, the filler model 4 is given a longitudinal elastic modulus as a material characteristic, and stress and strain are proportional in deformation calculation. The amount of the filler model 4 is determined in consideration of the amount of rubber blended.

  An interface model 5 is set around the filler model 4. As a physical structure, it has not been confirmed that such a special layer is formed at the interface between the filler and the rubber matrix. However, as described above, various phenomena occur at the interface between the filler and the matrix rubber. In particular, a slip or friction phenomenon between the rubber matrix and the filler at the interface causes a relatively large energy loss.

  The interface model 5 of the present embodiment continuously surrounds the filler model 4, is provided with the interface model 5 with a small thickness, and has different viscoelastic properties than the matrix model 3. This makes it possible to reproduce a large energy loss at the interface. The interface model 5 does not necessarily need to continuously surround the filler model 4, but it is preferable to surround most of the interface model 5. The thickness t of the interface model 5 is not particularly limited, but if it is too large, the energy loss tends to be excessively expressed. Conversely, if it is too small, it is difficult to reproduce the energy loss at the interface. As a result of various experiments, the thickness t is desirably 1 to 20 nm, and more preferably 5 to 10 nm, for example.

  Further, since the interface model 5 is modeled as a viscoelastic material, the material characteristics are defined based on the formula (1) as in the matrix model 3. The interface model 5 of the present embodiment defines viscoelastic properties that are softer than the matrix model 3. Specifically, the inclination of the elastic component in the stress-strain (or elongation) curve (corresponding to the longitudinal elastic modulus of the elastic body) is reduced. In addition, each element of the interface model 5 of this example is set so that the energy loss generated per strain cycle is larger than that of the matrix model 3.

This simulation illustrates an embodiment performed based on a homogenization method (asymptotic expansion homogenization method). As shown in FIG. 12, the homogenization method includes x _I representing the entire rubber material M periodically having the microscopic structure (unit cell) shown in FIG. 10, and y representing the microscopic structure. two independent variables and _I is used. By asymptotically expanding each independent variable in different scale fields, a microscopic scale and a macroscopic scale, the average mechanical response of the entire rubber material M reflecting the model structure of the microscopic structure shown in FIG. Can be requested. In other words, if the analysis target area is composed of repetitions of an arbitrary microscopic structure, and the repetition degree is so dense that the area cannot be discretized directly by the finite element method, the analysis target is replaced with a homogeneous equivalent model. By analyzing the whole and returning the analysis result to the microscopic structure at an arbitrary point, the deformation of the microscopic structure itself can be obtained approximately. The asymptotic expansion homogenization method is described in detail, for example, in the following document.
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.
Specifically, in addition to the equation (1), a macroscopic equilibrium equation of the following equation (20) and a characteristic displacement function (Y-periodic) of the equation (21) are used.

Moreover, in this embodiment, the constant of Formula (1) etc. were set as follows.
C ^R = 0.268
N = 6.6
T = 296
k _B = 1.38066 × 10 ⁻²⁹
n = C ^R / K _B /T=6.558×10 ²⁵
N _A = n · N = 4.328 × 10 ²⁶
Volume content of filler model 30%
Longitudinal elastic modulus E of filler model: 100 MPa
Poisson's ratio ν of filler model: 0.3

In the deformation simulation, a rectangular analysis region of 2 mm × 2 mm is set as the macroscopic model M, and an average strain rate 1 in the x2 direction of FIG. 12 is generated in order to generate uniform uniaxial tensile deformation in the macroscopic model M. 0.0 × 10 ⁻⁵ / s is added and the strain is gradually increased from zero to 2.5 in steps of 0.01, and the stress acting on the rubber material model 2 is calculated for each strain. On the contrary, after the strain reached 2.5, the strain was gradually reduced to zero at the same strain rate as described above in increments of 0.01. In each strain state, the average number of segments N is calculated, and this value is substituted into Equation (1) for sequential calculation. The rubber material model is a three-dimensional 8-chain model of Arruda et al. So that it does not change in the thickness direction. The average number of segments N of the matrix model 3 and the interface model 5 was set as follows.

<Matrix model>
・ Average number of segments N during load deformation
N = -3.2368 + 20.6175 λc-21.8168 λc ² + 10.8227λc ³ -1.9003 λc ⁴
・ Average number of segments N during unloading deformation (constant)
N = 6.6
・ Total number of molecular chain segments N _A (constant)
N _A = 4.3281 × 10 ²⁶

<Interface model>
・ Average number of segments N during load deformation
N = -5.9286 + 20.6175 λc-21.8168 λc ² + 10.8227λc ³ -1.9003 λc ⁴
・ Average number of segments N during unloading deformation (constant)
N = 3.91
・ Total number of molecular chain segments N _A (constant)
N _A = 3.203 × 10 ²⁵

  When the deformation calculation is performed, a necessary physical quantity can be acquired from the result (step S4). FIG. 13 shows the results of the deformation simulation performed independently for each of the matrix model 3, the filler model 4, and the interface model 5. The filler model 4 shows the highest elasticity and no energy loss occurs. The interface model 5 shows viscoelastic characteristics that cause a larger strain than the matrix model 3 even at the same stress, and the energy loss (the loop area of the curve) is also large.

  A curve La in FIG. 14 shows the relationship between the true stress and strain in the entire rubber material model M in which the above models are incorporated as the microscopic structure (cell unit) in FIG. At the time of load deformation, a non-linear first curve La1 was obtained. Further, at the time of unloading deformation, a second curve La2 different from the first curve La1 was obtained. The second curve La2 was softer than the first curve La1 (low elasticity), and a hysteresis loop could be reproduced. By calculating the closed area of this loop, the energy loss in one deformation cycle can be calculated.

  By the way, when the same deformation simulation as Arruda et al. Was performed with the average number of segments N per molecular chain being constant, the result of not being able to express energy loss through the same curve during load deformation and unload deformation. It becomes. In the above embodiment, the average segment number N is a function of the parameter λc. However, a strain rate or a strain amount can be used instead of λc. In this case, similarly to the equation (19), the energy loss can be expressed by changing the coefficient between the load deformation and the unload deformation.

  The shape of each of the curves La1 and La2 can be set in various ways by appropriately setting the coefficients A to E and changing the function for determining the average number of segments N per molecular chain in the equation (19). . Therefore, by setting various coefficients A to E according to the rubber material to be analyzed, energy loss and the like can be examined in detail for each material.

  The curve Lb is the same as the viscoelastic material model 2 shown in FIG. 10 except that the interface model 5 is given the same material characteristics as the matrix model 3 (that is, a model that does not consider the interface). Results are shown. Since there is no soft region at the interface, the stress-strain curve rises more than La. Moreover, since there is no large energy loss of the interface model 5, it turns out that the energy loss of the whole model is also small.

  Further, the curve Lc shows the result of simulation by giving the interface model 5 material characteristics opposite to those described above. That is, the viscoelastic material model 2 that obtains the curve Lc defines viscoelastic characteristics so that the interface model 5 exhibits higher elasticity than the matrix model 3 as shown in FIG. The average number of segments N for each model is set as follows.

<Matrix model>
・ Average number of segments N during load deformation
N = -3.2368 + 20.6175 λc-21.8168 λc ² + 10.8227λc ³ -1.9003 λc ⁴
・ Average number of segments N during unloading deformation (constant)
N = 6.6
・ Total number of molecular chain segments N _A (constant)
N _A = 4.3281 × 10 ²⁶

<Interface model>
・ Average number of segments N during load deformation
N = -5.4800 + 20.6175 λc-21.8168 λc ² + 10.8227λc ³ -1.9003 λc ⁴
・ Average number of segments N during unloading deformation (constant)
N = 4.36
・ Total number of molecular chain segments N _A (constant)
N _A = 8.569 × 10 ²⁶

  As a result, as shown in FIG. 14, it can be confirmed that the viscoelastic characteristics of the entire system of the viscoelastic material model 2 have a higher gradient and higher elasticity than the curve Lb.

  FIG. 16 intermittently visualizes and shows the progress of the deformation to recovery state in the uniaxial tensile deformation simulation of the viscoelastic material model 2 (curve La). FIGS. 16A to 16E show load deformation, and FIGS. 16F to 16J show unload deformation. Moreover, the place where the color is changing to white indicates a region with a large distortion. This result accurately reproduces that a large strain is concentrated between the interface of the filler model 4 and between the filler models 4 and 4.

  FIG. 17 shows the strain distribution in the tensile direction of each element at the maximum strain, (A) shows the curve Lb in FIG. 13, and (B) shows the curve La. The thinner the color, the greater the distortion. In the case of (A) in which the interface is not taken into consideration, the generation of strain is broadly and gently generated as a whole, whereas in the case of (B) in which the interface is taken into consideration, around the filler model 4 It is well reproduced that the strain is concentrated.

  In FIG. 18, the energy loss of one cycle of strain is calculated for each element, and the magnitude is shown by color. The lighter the color, the greater the energy loss. The viscoelastic material model 2 is represented by the maximum deformation shape of FIG. 17, (A) is the curve Lb of FIG. 14, and (B) is the curve La. In the case of (B) considering the interface, it can be confirmed that a large energy loss occurs at the interface of the filler model 4.

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 which shows one Embodiment of a viscoelastic material model. (A) is a perspective view which shows the network structure of a viscoelastic material, (B) is a perspective view which shows an example of an 8-chain model. (A), (B) is a diagram explaining the fracture | rupture of the junction of a molecular chain. It is a graph which shows the relationship between parameter (lambda) c and the average number of segments N per molecular chain. It is a flowchart which shows the procedure of a deformation | transformation simulation. It is a graph which shows the relationship between true stress-strain as a simulation result of a viscoelastic material model. It is a graph which shows the relationship with a true stress-strain as a simulation result of the viscoelastic material model of a comparative example. It is a diagram which shows other embodiment of a viscoelastic material model. It is a diagram which shows the shape of carbon black. The relationship between the microscopic structure explaining the homogenization method and the entire structure is shown. The stress-strain curve of a matrix model, a filler model, and an interface model is shown. As another embodiment, a true stress-strain curve obtained from a deformation simulation result of a viscoelastic material model. The stress-strain curve of a matrix model and an interface model is shown. (A)-(J) are the diagrams which visualize and show the deformation process of a viscoelastic material model. (A) is a model in which the interface layer is not considered, and (B) is a tensile strain distribution diagram of the model in which the interface layer is considered. (A) is a model in which the interface layer is not considered, and (B) is an energy loss distribution diagram of the model in which the interface layer is considered. (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. As a simulation result by Arruda et al., A stress-strain curve is shown.

Explanation of symbols

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

Claims (4)

A viscoelastic material simulation method for simulating deformation of a viscoelastic material,
Setting a viscoelastic material model in which the viscoelastic material is modeled by an element capable of numerical analysis;
Performing deformation calculation by setting conditions in the viscoelastic material model;
Obtaining a necessary physical quantity from the deformation calculation, and
In the viscoelastic material model, the relationship between stress and strain represented by the following formula (1) is defined, and the average number of segments N per molecular chain in the formula (1) A simulation method of a viscoelastic material, characterized in that energy loss can be expressed by being different depending on unloading deformation.
  2. The viscoelastic material simulation method according to claim 1, wherein the average number of segments N increases according to a parameter relating to strain at the time of load deformation, and takes a constant value at the time of unloading deformation. The viscoelastic material simulation method according to claim 2, wherein the parameter relating to the strain is a first-order invariant I ₁ of the strain.   The viscoelastic material simulation method according to claim 2, wherein the parameter relating to the strain is a strain amount or a strain rate.