JP2007101499A - Method for simulating viscoelastic material - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a simulation method for accurately analyzing the state of deformation etc. of viscoelastic materials. <P>SOLUTION: The simulation method includes a step for setting a viscoelastic material model, in which a viscoelastic material is divided by element; a step for computing deformations by setting conditions to the viscoelastic material model; and a step for acquiring required physical quantities, on the basis of the deformation computations. A first energy loss property for generating a first energy loss, based on the amount of strain and a second energy loss property for generating a second energy loss, based on the rate of strain are defined in a first cycle of deformation, comprising loading deformation and unloading deformation in the viscoelastic material model. The deformation computations for a second cycle include a process for generating the first and second energy losses in the first cycle and a process for generating only the second energy loss, when the strain does not exceed the maximum strain of the first cycle in the second cycle. <P>COPYRIGHT: (C)2007,JPO&INPIT

Description

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

  Rubber materials are widely used in tires, sporting goods and other various industrial products. In order to reduce the labor and cost of these prototypes, the deformation state of the rubber material is calculated and predicted in advance using a computer. Conventionally, the following Non-Patent Documents 1 and 2 are known as such methods.

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)

J. S. BERGSTROM and M. C. BOYCE, "CONSTITUTIVE MODELING OF THE LARGE STRAIN TIME-DEPENDENT BEHAVIOR OF ELASTOMERS" Journal of the Mechanics and Physics of Solids Volume 46, No. 5 PP. 931-954 (1998)

By the way, it is known that a rubber material has a Mullins effect. Here, the Malin's effect will be briefly described. As shown in FIG. 15, the rubber is first pulled to a strain ε _{m and} then unloaded (preliminary stretching). At this time, in the stress-strain curve, energy loss occurs through paths 0, R1, R2, and 0. Next, when this pre-stretched rubber is pulled again, the response Kerr curve shows low stress that does not match that during pre-stretch through the paths 0, R3, R2, 0. On the other hand, when the rubber is further pulled to ε _n beyond the strain ε _m , it passes through paths 0, R3, R4, R5, 0. As described above, the Malin's effect is a phenomenon in which rubber is softened after the second cycle when one cycle of deformation of loading and unloading is repeated.

  However, Non-Patent Documents 1 and 2 describe that a rubber material is analyzed by using a molecular chain network theory (which will be described in detail later). However, in the methods described in these documents, The above-mentioned Marin's effect cannot be reproduced.

  In particular, in rubber products such as pneumatic tires, the deformation cycle is periodically repeated. Therefore, in order to accurately simulate viscoelastic materials used for them, it is not appropriate to ignore such a Malins effect.

  The present invention has been devised in view of the above-described problems. In the one cycle of deformation including a load deformation for increasing the strain and an unloading deformation for decreasing the strain in the viscoelastic material model, the strain amount is determined. When defining a first energy loss characteristic for generating an energy loss based on the above and a second energy loss characteristic for generating an energy loss based on a strain rate, and performing a two-cycle deformation calculation, An object of the present invention is to provide a viscoelastic material simulation method capable of reproducing the so-called Malin's effect on the basis of controlling energy loss.

  The invention according to claim 1 of the present invention is a method for simulating deformation of a viscoelastic material, the step of setting a viscoelastic material model obtained by dividing the viscoelastic material by elements capable of numerical analysis; A step of performing a deformation calculation by setting a condition in the viscoelastic material model; and a step of acquiring a necessary physical quantity from the deformation calculation. The viscoelastic material model includes a load deformation for increasing a strain and the strain. In one deformation cycle consisting of unloading deformation to reduce the first energy loss characteristic that generates the first energy loss based on the strain amount, and the second energy loss that generates the second energy loss based on the strain rate When the deformation calculation of at least two cycles is performed, the first and second energy loss characteristics are defined in the first cycle. A process for generating a Rugirosu, in the second cycle, if the strain does not exceed the maximum strain in the first cycle, characterized by comprising a process of generating only the second energy loss.

In the invention according to claim 2, the viscoelastic material model defines a relationship between stress and strain expressed by the following formula (1), and the first energy loss is expressed by the following formula (1). It is generated by increasing the average number of segments N per molecular chain at the time of load deformation based on the amount of strain and maintaining the value when unloading is started at unloading deformation. The viscoelastic material simulation method according to claim 1.

  The viscoelastic material model includes a rubber model in which rubber is modeled, and a filler model in which a filler surrounded by the rubber is modeled so as to be surrounded by the rubber model, and The rubber model includes an interface model having a small thickness surrounding the filler model and a matrix model surrounding the interface model, and at least one of the first and second energy loss characteristics is defined. The viscoelastic material simulation method according to claim 1, wherein the viscoelastic material is a simulation method.

  According to the viscoelastic material simulation method of the present invention, the Malin's effect is reproduced in the second cycle. Therefore, it is possible to obtain an accurate simulation result close to the behavior of actual vulcanized rubber.

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, a mouse 1c, and a display device 1d. The main body 1a is provided with an arithmetic processing unit (CPU), a ROM, a working memory, and a magnetic disk (all not shown), and drive devices 1a1, 1a2 such as a CD-ROM and a flexible disk. The magnetic disk stores a program necessary for executing the simulation method of the present invention.

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

  Although it does not specifically limit as said viscoelastic material, In this embodiment, the case of the vulcanized rubber with which the filler containing rubber | gum and the filler mix | blended in it was filled is shown. However, it does not matter whether the rubber or other viscoelastic material to be analyzed actually exists. The viscoelastic material model is set based on this rubber. In FIG. 3, an example of a unit cell U that is a microscopic structure of the viscoelastic material model 2 is visualized.

  The unit cell U of the viscoelastic material model 2 is set by dividing (substituting) a minute region of rubber to be analyzed into a finite number of small elements 2a, 2b, 2c. The minute region is not particularly limited as long as it can be analyzed, but in this example, it is set as a rectangular region of 300 × 300 nanometers in both vertical and horizontal directions.

  Each element 2a, 2b, 2c... Is defined so as to be capable of numerical analysis. Here, the possibility of numerical analysis means that deformation calculation can be performed on each element or the entire system 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,..., The coordinate value of the node in the coordinate system, the shape of the element, the material property, etc. are defined by numerical data. For each of the elements 2a, 2b, 2c,..., For example, a three or quadrilateral element is used as a two-dimensional plane element, and a tetrahedron element is used as a three-dimensional element. In the present embodiment, the former is used. As a result, the viscoelastic material model 2 forms data that can be deformed by the computer device 1.

  The viscoelastic material model 2 of this embodiment is analyzed in a plane strain state in a deformation simulation described later. Therefore, it is assumed that there is no distortion in the Z-axis direction (the direction perpendicular to the paper surface in FIG. 3).

  In the present embodiment, the viscoelastic material model 2 includes a filler model 3 in which a filler is modeled and a rubber model 4 in which rubber is modeled. Furthermore, the rubber model 4 of this embodiment includes an interface model 5 that is disposed around the filler model 3 and has a small thickness, and a matrix model 6 that surrounds the interface model 5. In FIG. 3, for easy understanding, the filler model 3 is visualized in white, the interface model 5 is slightly light gray, and the matrix model 6 is visualized in the darkest gray.

  The filler model 3 is set based on, for example, carbon black as a filler. However, it is needless to say that the filler is not limited to the above specific examples, and may be other fillers such as silica. The shape of the filler model 3 is preferably determined based on, for example, the shape of carbon black in the actual rubber imaged with an electron microscope.

  FIG. 4 shows a perspective view of carbon black imaged with an electron microscope. As shown in the drawing, the carbon black CB has a structure like a cocoon bundle in which spherical primary particles 7 made of carbon atoms and having a diameter of about several tens of nanometers are irregularly bonded three-dimensionally. The filler model 3 of the present embodiment is set based on a plan view of this structure.

  Carbon black is hundreds of times harder than viscoelastic materials such as rubber. For this reason, in the present embodiment, the filler model 3 is treated as an elastic body and no energy loss occurs. The area or volume of the filler model 3 occupied in the cell unit U is preferably determined based on the filler filling amount of the rubber to be analyzed.

  In this embodiment, the interface model 5 continuously surrounds the filler model 3 and is set with a small thickness. The rubber and the filler are physically or chemically bonded at the interface, and it is assumed that various phenomena that are not seen in other parts, such as a large slip and a friction associated therewith, are generated. Therefore, setting such a model 5 and giving appropriate parameters thereto is useful for performing an accurate simulation. The interface model 5 does not necessarily need to continuously surround the filler model 3. Further, the thickness t of the interface model 5 is not particularly limited. However, as a result of various experiments, it is preferably 1 to 20 nanometers, more preferably 5 to 10 nanometers.

  The matrix model 6 is set so as to surround the interface model 5 outside the interface model 5. In the unit cell U, the matrix model 6 is a part excluding the filler model 3 and the interface model 5 and constitutes a main part of the rubber model 4.

  In the present embodiment, the first energy loss characteristic and the second energy loss characteristic are defined in the rubber model 4.

  FIG. 5 is a conceptual diagram for explaining the deformation principle of the rubber model 4 having such energy loss characteristics. The individual elements of the rubber model 4 are considered to be equivalent to a combination of the first deformation network A and the second deformation network B in parallel. The stress S generated in the entire system is the sum of the stress generated in the first deformation network A and the stress generated in the second deformation network B. Therefore, by obtaining the stress of each of the networks A and B, the stress of the entire system can be calculated. Since both networks A and B are connected in parallel, the strain (elongation) generated in each network A and B is the same. Hereinafter, the stress calculation procedure of the rubber model 4 based on the deformation networks A and B will be described.

<First modified network A>
The first deformation network A is considered to be equivalent to a spring, and the stress is a value corresponding to the elongation (strain) of the spring. This spring elongation is not affected by the strain rate. Accordingly, the first deformation network A exhibits a stress that does not depend on the strain rate.

  For the calculation of the stress of the first deformation network A, an 8-chain model based on the molecular chain network theory is used. As the stress, for example, the following formula (1) can be applied in the velocity format.

  As shown in FIGS. 6A and 6B, the molecular chain network theory is that the rubber material a as a continuum is a microscopic structure da, and randomly oriented molecular chains c are connected at a junction b. It is a theory that thinks that it has a mesh structure. The junction point b is, for example, a chemical bond between molecules, and includes a crosslinking point.

  As shown in FIG. 6C, it is assumed that 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. 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.

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

  Further, as shown in FIG. 7, in the molecular chain network theory, the viscoelastic material is considered as a cubic network structure h in which microscopic 8-chain models g are aggregated macroscopically. As shown in an enlarged view on the right side, one 8-chain model g represents molecular chains c extending from one junction point b1 defined at the center of the cube to eight junction points b2 provided at each vertex. The stress is calculated based on these. This 8-chain model is described in Non-Patent Document 1.

  However, in the present embodiment, in the first modified network A, a general 8-chain model proposed by Non-Patent Document 1 or the like is used to generate the first energy loss based on the strain amount as follows. Used with some modifications.

  The rubber material is allowed to have a large strain that can reach several hundred percent in the deformation process under load. This mechanism is considered to be obtained by unraveling (disappearing) the entangled portion (junction point b) of the molecular chain c intricately entangled. Specifically, as shown in FIG. 8 (A), when tensile stress in the direction of the arrow acts on the molecular chains c1 to c4 entangled at one junction point b, each molecular chain c1 to c4 expands. The joint point b is considered to be damaged (disappeared) under a large strain. Then, as shown in FIG. 4B, the 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. And it is thought that energy loss arises by the failure | damage of this junction point b. In the simulation of the present embodiment, the influence due to the breakage (disappearance) of the junction point b of the molecular chain c is taken into the equation (1).

Here, consider the relationship between the decrease in the entanglement of the molecular chain c and the segment d in the 8-chain model. In the mesh structure h shown in FIG. 7, k 8-chain models g are combined in the width direction, the height direction, and the depth direction, respectively, where k is a sufficiently large number. 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 is expressed by the following equation (2). Similarly, the number of molecular chains c contained in the network structure h, that is, the number n of molecular chains contained in a unit volume of the rubber model 4 can be expressed by the formula (3).
m = (k + 1) ³ + k ³ (2)
n = 8k ³ Formula (3)

Here, since k is a sufficiently large number, when the terms other than the third-order term are omitted, the above equation (2) can be expressed by the following equation (4).
m = 2k ³ Formula (4)
Furthermore, from the relationship between the above formulas (3) and (4), the entanglement number m can be expressed by formula (5) using n.
n = 4m Formula (5)

Here, even if the rubber model 4 is deformed, there is no change in the total number N _A of the molecular chain segments contained in the unit volume. Therefore, if the average number of segments contained in one molecular chain is N, Equation (6) holds.
N _A = n · N Equation (6)

Solving equation (6) for N yields equation (7).
N = N _A / n ... (7)

Moreover, Formula (8) is obtained from Formula (7) and Formula (5).
N = N _A / 4m (8)

  As is apparent from the equation (8), when the number of entanglements m of the molecular chain c of rubber decreases due to load deformation, the average number of segments N included in one molecular chain c increases. Accordingly, by increasing the average number of segments N in the equation (1) according to the load, in other words, the amount of strain, the decrease in the entanglement of the molecular chain c can be incorporated into the equation (1). This can simulate the deformation mechanism of rubber with higher accuracy and improve calculation accuracy. Here, the load deformation is a deformation in which the strain of the viscoelastic material model 2 increases during a very short time, and a deformation in which the strain decreases is referred to as an unloading deformation.

  The average number of segments N can be increased by various methods based on the amount of strain. For example, it is desirable that the average segment number N increases based on the strain amount during load deformation itself or a parameter related thereto. Specific parameters include the first-order invariant I1 of strain. In the present embodiment, the average segment number N is defined by the following formula (9). Equation (9) indicates that the average number of segments N is a function of the first-order invariant I1 of distortion (more specifically, the parameter λc which is the square root) of each element.

The above A to E in the formula (9) 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. Next, the average number of segments N in each strain is obtained so as to match the curve at the time of loading. Then, the parameters A to E in the equation (9) are determined so as to coincide with the determined average segment N at the time of loading.

FIG. 9 shows the relationship between the average number of segments N and the parameter λc. 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
As λ _c, which is a parameter related to the amount of strain, increases, the average segment number N increases smoothly. In this example, the upper limit of λ _c is about 2.5. In the deformation simulation described later, λ _c is always calculated for each element of the rubber model 4 during load deformation. The calculated λ _c is substituted into Equation (9), and the average number of segments N of the element in the distortion state is calculated. The average segment number N is calculated at a predetermined timing, and is taken into equation (1).

  When the rubber model 4 is unloaded and deformed, the average number of segments N included in one molecular chain c is maintained until the average number of segments N when the unloading is started becomes zero. . Therefore, at the time of unloading, the molecular chain is not entangled (or disappears), so that the rubber model 4 can be deformed with a lower stress than that at the time of loading. Thereby, the softening phenomenon at the time of unloading is reproduced in the simulation. Moreover, in such a 1st deformation network A, stress differs in the same strain state at the time of loading and unloading. Thereby, in the stress-strain curve, the first energy loss is generated, which draws a hysteresis loop.

  Thus, the value when the average number of segments N per molecular chain in the formula (1) is increased based on the amount of strain at the time of load deformation and unloading is started at the time of unloading deformation. By maintaining the above, the first energy loss characteristic in which the first energy loss based on the strain amount is generated is defined in the first deformation network A.

<Second modified network B>
The second deformation network B is considered to be equivalent to a series connection body of the spring e and the dashpot p. The dashpot p is a damping device in which a piston provided with an orifice or the like moves in a cylinder cylinder sealed with a fluid (oil etc.) that complies with Newton's law of viscosity, and has a resistance proportional to the moving speed of the piston. Arise. Moreover, the resistance becomes an energy loss.

  The total elongation (strain) of the second deformation network B is the sum of the elongation of the spring e and the elongation (strain) of the dashpot. The stress generated in the second deformation network B also has a value corresponding to the elongation (strain) of the spring e. However, the elongation of the spring e is not simply determined by the load and is affected by the resistance of the dashpot p depending on the strain rate. That is, the second deformation network B shows a stress depending on the strain rate.

  Here, the 8-chain model based on the molecular chain network theory is also applied to the spring e of the second deformation network B. Therefore, the stress generated in the spring e is also calculated using the equation (1). In this calculation, the average number of segments N included in one molecular chain c may change as in the case of the first deformation network A, or may be a fixed value.

In the second modified network B, a second energy loss characteristic is defined. The second energy loss characteristic generates a second energy loss based on the strain rate. That is, in the second deformation network B, resistance depending on the strain rate is generated in the dashpot p, which affects the elongation ratio λ _c of the spring e. The elongation ratio λ _c can be calculated from the main stretch in each direction of the molecular chain c, as in the proviso of equation (1).

Next, in the second deformation network B, each main stretch λ _i ^(Be) is calculated by the following procedure. Note that the subscript i of λ represents each direction (i = 1, 2,...), And in two dimensions means x and y. The subscript (Be) on the right shoulder is a code for distinguishing from the main stretch of the first deformation network A.

First, the equivalent shear stress τ ^(B) in the second deformation network B is calculated by the equation (10). Σ ^{(B) ′ in} Expression (10) is the equivalent stress of the second deformation network B obtained by the calculation in the previous step. Further, the strain rate of the dashpot p is calculated from the equation (11) using the values of the equivalent shear stresses τ ^(B) and λ _c ^(Bp) . Note that λ _c ^{(Bp) in} equation (11) is the dashpot expansion ratio, and this value is calculated from the previous calculated value λ _i ^(Bp) (calculated by equation (14) described later ⁾ . Calculated according to the proviso method in 1).

From the strain rate, the deformation rate D ^{(Bp) of the} dashpot p is calculated from the equation (12), and the elongation of the dashpot p is calculated from the deformation rate D ^(Bp) by the equations (13) and (14). Is done. Here, the subscript t indicates the time step of calculation, and Δt is the increment.

Then, each stretch λ _i ^(Be) of the spring e of the second deformation network B is obtained from the following equation (15). In equation (15), λ _i is the overall expansion ratio of the second deformation network B.

The λ _c ^(Be) is substituted into the equation (1), whereby the stress of the second deformation network B is calculated. This stress depends on the strain rate by a mechanism equivalent to the action of the dashpot. In the stress-strain diagram of the deformation network B, a hysteresis loop is drawn so as to have an energy loss depending on the strain rate. About the dependence of a strain rate, it can adjust by determining each material constant of said Formula (11) suitably.

  Then, the total stress of the rubber model 4 is calculated by the sum of the stress of the first deformation network A and the stress of the second deformation network B.

  Such first and second energy loss characteristics may be given to at least one of the interface model 5 or the matrix model 6 in the rubber model 4, but it is preferable that they be defined in both.

  It is desirable that the interface model 5 and the matrix model 6 have different viscoelastic characteristics. The viscoelastic characteristics referred to herein include characteristics represented by a stress-strain curve at an arbitrary deformation rate, as shown in FIG. For example, as shown in FIG. 10, it is desirable to set the interface model 5 to be softer than the matrix model 6. This relatively increases the strain near the interface and helps to simulate the large slip between the rubber and filler at the interface. Moreover, it can also comprise so that the area of a hysteresis loop may differ. In this aspect, the viscoelastic characteristics are determined so that the strain of the interface model 5 is larger than that of the matrix model 6 in the case of the same stress. However, it is not limited to such an aspect.

  Next, in the simulation method of the present embodiment, a deformation simulation is performed using the material model 2 (step S3). A specific processing procedure of the deformation simulation is shown in FIG. In the deformation simulation, first, various data are 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.

  In the computer apparatus 1, a stiffness matrix of each element is created based on the input data (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, the unknown node displacement is determined (step S35), and physical quantities such as stress, principal stress and / or strain of each element are calculated and output using the above formulas (steps S36 to 37). This physical quantity is stored in the computer apparatus 1 as appropriate. In step S38, it is determined whether or not to end the calculation. If negative, step S32 and subsequent steps are repeated.

  The simulation can be performed using, for example, engineering analysis application software using a finite element method (for example, LS-DYNA developed and improved by Livermore Software Technology, Inc., USA).

  In addition, this simulation illustrates a mode performed based on a homogenization method (asymptotic expansion homogenization method). In the homogenization method, if the analysis target area is composed of repetitions of an arbitrary microscopic structure, and the repetition degree is very dense, the analysis target area cannot be discretized directly by the finite element method. By substituting for the analysis of 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.

More specifically, in the homogenization method, as shown in FIG. 12, x _I representing the whole material M periodically having the unit cell U of the material model 2 and the microscopic structure are represented. Two independent variables with y _I are 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 material M reflecting the model structure of the unit cell U can be obtained. 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.

In the present embodiment, the deformation simulation is performed by setting the constants and the like of the expressions (1) and (10) as follows.
C ^R = 0.268
N = 6.6
T = 296
k _B = 1.38066 × 10 ⁻²⁹
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

Further, in the deformation simulation, uniaxial tensile deformation was generated in the analysis region of the entire material M. Tensile condition, the x ₂ direction strain rate = 0.01 in FIG. 12 (/ s), the deformation of the four cycles were performed in order. The amount of tension in each cycle was as follows.
1st cycle: 100% (elongation ratio 2.0)
Second cycle: 100% (elongation ratio 2.0)
3rd cycle: 200% (elongation ratio 3.0)
4th cycle: 200% (elongation ratio 3.0)

  In the deformation simulation, the average number of segments N of the rubber model 4 was calculated by the equation (9) in each strain state, and the value was sequentially substituted into the equation (1). The material model was constrained so as not to change in the thickness direction (Z-axis direction in FIG. 3). Furthermore, the average segment number N of the interface model 5 and the matrix model 6 was set as follows.

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

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

  When the deformation calculation is performed, a necessary physical quantity is obtained for each calculation step timing from the result (step S4). FIGS. 13A and 13B are graphs showing the relationship between the stress-elongation ratios of the first and second cycles of the viscoelastic material model, and FIGS. 14A and 14B are viscoelasticity. A graph showing the stress-elongation ratio relationship at the third to fourth cycles of the material model is shown.

  As is clear from the figure, it can be confirmed that the viscoelastic material model 2 reproduces the Malin's effect after the second cycle.

  That is, the loop drawn by the first cycle stress-elongation ratio graph includes both the first energy loss E1 and the second energy loss E2. That is, in the load deformation, conceptually, the bond point b of the molecular chain c is broken based on the amount of strain, so the average number of segments per molecular chain is increased and the first energy loss is generated. Is done. Further, in the deformation network B, a dashpot resistance (energy loss) based on the strain rate is generated, so that a second energy loss based on this is generated.

  Moreover, in FIG. 13, it turns out that the 1st energy loss is not produced | generated in the 2nd cycle of the same distortion amount as the 1st cycle, and only the 2nd energy loss is produced | generated. This is based on the fact that the bond point b of the molecular chain c is not newly damaged because the strain amount in the second cycle does not exceed the maximum strain in the first cycle. In this case, the load deformation takes the same path as the unloading deformation in the first cycle. On the other hand, since the second energy loss based on the strain rate also occurs in the second cycle, the stress-elongation ratio curve follows the same path as the unloading deformation in the first cycle as shown in FIG. Will pass.

  The curve of the third cycle passes the same path as the second cycle until the elongation ratio is 2.0, which is the same as the first to second cycles, when the load is deformed, but the elongation ratio exceeds 2.0. Then, since the bond point b of the new molecular chain c is broken, the stress increases, and the first energy loss based on the strain amount is generated again. Similarly to the first or second cycle, a second energy loss may occur depending on the strain rate.

  In the 4th cycle, as in the 2nd cycle, the strain does not exceed the maximum strain in the 3rd cycle, so that the bond point b of the molecular chain c is not newly damaged and disappears. For this reason, the load deformation in the fourth cycle follows the same path as the unload deformation in the third cycle. For this reason, the first energy loss based on the strain amount is not generated, and only the second energy loss corresponding to the strain rate occurs.

  As described above, according to the simulation method of the present embodiment, it is possible to accurately reproduce the Marin's effect found in viscoelastic materials, particularly vulcanized rubber, on the simulation. Therefore, it is possible to perform the performance evaluation of the rubber material of the pneumatic tire that deforms periodically under a repeated load with higher accuracy.

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 the microscopic structure of a viscoelastic material model. It is a diagram which shows the shape of carbon black. It is a conceptual diagram explaining the deformation | transformation principle of a rubber model. (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. It is a perspective view which shows an example of the network structure of a viscoelastic material, and the 8-chain model which comprises it. (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 stress-strain curve of an interface model and a matrix model at an arbitrary deformation rate. It is a flowchart which shows the procedure of a deformation | transformation simulation. The relationship between the whole structure explaining the homogenization method and its unit cell is shown. (A) is the 1st cycle of deformation | transformation simulation, (B) is the 2nd cycle, and is a graph which shows the relationship of stress-elongation ratio, respectively. (A) is the 1st cycle of deformation | transformation simulation, (B) is the 2nd cycle, and is a graph which shows the relationship of stress-elongation ratio, respectively. It is a stress-strain diagram explaining the Marin's effect of a rubber material.

Explanation of symbols

1 Computer device 2 Material model 3 Filler model 4 Rubber model 5 Interface model 6 Matrix model

Claims (3)

A method of simulating deformation of a viscoelastic material,
Setting a viscoelastic material model obtained by dividing the viscoelastic material by elements capable of numerical analysis;
Performing deformation calculation by setting conditions in the viscoelastic material model;
Obtaining a necessary physical quantity from the deformation calculation,
The viscoelastic material model includes a first energy loss characteristic that generates a first energy loss based on the strain amount in one deformation cycle including a load deformation that increases strain and an unload deformation that decreases the strain. And a second energy loss characteristic that generates a second energy loss based on the strain rate is defined,
When performing deformation calculation of at least two cycles, a process of generating the first and second energy losses in the first cycle;
And a process of generating only the second energy loss when the strain does not exceed the maximum strain of the first cycle in the second cycle. In the viscoelastic material model, a relationship between stress and strain represented by the following formula (1) is defined,
The first energy loss increases the average number of segments N per molecular chain in the following formula (1) based on the amount of strain during load deformation, and starts unloading during unloading deformation. The viscoelastic material simulation method according to claim 1, wherein the viscoelastic material is generated by maintaining a value obtained when the viscoelastic material is maintained.

The viscoelastic material model includes a rubber model that models rubber, and a filler model that is modeled so that a filler surrounded by the rubber is surrounded by the rubber model,
The rubber model includes an interface model having a small thickness surrounding the filler model and a matrix model surrounding the interface model, and at least one of the first and second energy loss characteristics is defined. The viscoelastic material simulation method according to claim 1, wherein the viscoelastic material is simulated.

Images