JP2009259043A - Simulation method for rubber material - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a simulation method capable of accurately simulating the large deformation of a rubber material. <P>SOLUTION: The simulation method for calculating the deformation of the rubber material by a very small time increment includes: processing (a step S1) for setting a rubber analysis model obtained by modelling the rubber material with a plurality of elements to be numerically analyzed; processing (a step S2) for setting the value of a first material parameter for regulating the volume change of the rubber analysis model, so as to allow the rubber analysis model to substantially indicate a non-compression characteristic; processing (a step S5) for performing the first deformation calculation of the rubber analysis model through the use of at least the first material parameter and a predetermined analysis condition; processing (a step S6) for calculating the distortion or stress of each element after the first deformation calculation; and parameter change processing (a step S8) for changing the value of the first material parameter so as to allow the element to have the substantial volume change when the distortion or stress exceeds a predetermined value. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to a simulation method useful for accurately reproducing a large deformation of a rubber material.

  In recent years, in order to efficiently evaluate the performance of various rubber materials, various computer simulation methods using a finite element method have been performed (for example, see Patent Documents 1 and 2 below). In these simulations, for example, material characteristics are defined in a rubber analysis model obtained by dividing a rubber material by a finite number of elements, and the deformation behavior is calculated for each minute time increment by applying predetermined boundary conditions. .

  In addition, the rubber analysis model normally defines incompressible characteristics that do not substantially change in volume even when deformed. Specifically, as a material characteristic of the rubber analysis model, a Poisson's ratio is set to a value close to 0.5, or a bulk elastic modulus is set to be large.

In addition, as a material constant of the elastic body, the bulk modulus k, the modulus E, and the Poisson's ratio ν satisfy the relationship of the following formula (1).
k = E / {3 (1-2ν)} (1)
In the above formula (1), if the Poisson's ratio ν is set to 0.5, the bulk modulus k becomes infinite and an error occurs during computer calculation. In conventional deformation simulations of rubber materials, a value very close to 0.5, for example, about 0.490 to 0.495, is set as the Poisson's ratio in order to prevent such problems.

JP 2003-320824 A Japanese Patent No. 3314082

  By the way, if the rubber material is deformed by applying a tensile load, in an emergency just before the breakage, partial breakage or voids are generated inside the rubber material, and the apparent volume of the rubber material is increased. . However, it is difficult to reproduce such a volume change in a conventional rubber analysis model in which the Poisson's ratio is fixed to a constant value so as to exhibit incompressibility characteristics. For this reason, the deformation behavior of the rubber analysis model at the time of large deformation often deviates greatly from that of actual rubber, and there is a problem that accurate reproduction is difficult.

  The present invention has been devised in view of the above circumstances, and when the deformation or stress of an element of a rubber analysis model after deformation exceeds a predetermined value, a substantial volume change occurs in the element. A rubber material that is useful for accurately reproducing the behavior at the time of large deformation just before fracture based on the inclusion of the process of changing the value of the first material parameter that defines the volume change so as to cause The main purpose is to provide a simulation method.

The invention according to claim 1 of the present invention is a simulation method for performing deformation calculation of a rubber material in minute time increments, and is a rubber analysis model in which a rubber material is modeled by a plurality of elements capable of numerical analysis. A process of setting, a process of setting a value of a first material parameter that defines a volume change of the rubber analysis model, so that the rubber analysis model exhibits substantially non-compression characteristics;
A process of performing a first deformation calculation of the rubber analysis model using at least the first material parameter and a predetermined analysis condition, and a process of calculating the strain or stress of each element after the first deformation calculation And a parameter changing process for changing the value of the first material parameter so that a substantial volume change occurs in the element when the strain or stress exceeds a predetermined value. And

  According to a second aspect of the present invention, in the rubber according to the first aspect, the first material parameter is a Poisson ratio, and the parameter changing process decreases the value of the Poisson ratio based on strain or stress of the element. This is a material simulation method.

  The invention according to claim 3 is the rubber material simulation method according to claim 2, wherein an initial value of the Poisson's ratio is 0.49 or more and less than 0.5.

  The invention according to claim 4 is the rubber material simulation method according to claim 1, wherein the first material parameter is a bulk modulus.

  According to a fifth aspect of the present invention, the initial value of the bulk modulus of elasticity is 20 times or more of the modulus of elasticity of an element of the rubber analysis model, and the parameter changing process is performed on the strain or stress of the rubber analysis model. The rubber material simulation method according to claim 4, wherein the bulk modulus is reduced based on the rubber material.

  In the invention according to claim 6, after the parameter changing process, a process of performing a second deformation calculation using the changed first material parameter, a result of the second deformation calculation, and the first The rubber according to any one of claims 1 to 5, further comprising: a process of calculating a difference from a result of deformation calculation; and a process of reducing the time increment when the difference exceeds a predetermined value. This is a material simulation method.

  In the rubber material simulation method of the present invention, the value of the first material parameter is initially set in the rubber analysis model so as to show incompressibility. On the other hand, when the strain or stress of the element of the rubber analysis model after deformation exceeds a predetermined value, the first material parameter for defining the volume change so that the substantial volume change occurs in the element. Processing to change the value is performed. Therefore, in the deformation calculation, the rubber analysis model can exhibit incompressibility characteristics when the deformation is small, and can exhibit compression characteristics that cause a volume change when the deformation is large. Therefore, according to the simulation method of the present invention, the rubber analysis model can reproduce the shape change accompanied with the volume change at the time of large deformation indicated by the actual rubber material. Therefore, even the computer simulation can accurately reproduce the deformation behavior of the actual rubber material.

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.
The present invention provides a rubber material simulation method for examining the deformation behavior of a rubber material. Such a simulation method is performed using a computer apparatus 1 as shown in FIG. 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 appropriately provided with a processing unit (CPU), a ROM, a working memory, a storage device such as a magnetic disk, and disk drive devices 1a1, 1a2. Note that a processing procedure (program) for executing a method to be described later is stored in the storage device in advance.

FIG. 2 shows a flowchart of the simulation method of the present embodiment.
In the present embodiment, first, a rubber analysis model is set (step S1). The rubber analysis model is set by modeling a rubber material to be analyzed (whether or not it actually exists) with an element capable of numerical analysis. The rubber material may have any shape as long as it contains rubber in the main part. For example, the rubber material may be a block or a sheet made of rubber, or a composite of rubber and cord material. Furthermore, the rubber material may have a shape of a more specific rubber product (both not shown) such as a golf ball or a pneumatic tire.

  FIG. 3A shows a perspective view of the rubber material 2 to be analyzed in the present embodiment. The rubber material 2 is a thin hexahedron as a whole, and a slit 2a made of a cut having a small width is formed at the center thereof. In the simulation method of the present embodiment, the deformation behavior when a tensile load P that opens the slit 2a is applied to such a rubber material 2 is examined. However, it is needless to say that the shape and the like of the rubber material 2 are not limited to the illustrated modes and can be freely determined.

  In FIG.3 (b), the side view of the rubber analysis model 3 which modeled the said rubber material 2 is visualized and shown. The rubber analysis model 3 is set by dividing the rubber material 2 using a finite number of small elements e (mesh division). This process can be arbitrarily performed by the user on the computer apparatus 1 using mesh creation software or the like.

  The rubber analysis model 3 is specified as numerical data that can be handled by the computer device 1. Specifically, the node coordinate value, node number, element number, etc. of each element e are defined in association with each other as numerical data. These numerical data are used for calculation in the computer apparatus 1 and are easily visualized as shown in FIG. 3B using software.

  As the element e, for example, a three-dimensional solid element such as a tetrahedral element having four nodes or a hexahedral element having six nodes is preferably used. The rubber analysis model 3 of this embodiment is all modeled by regular hexahedron solid elements. In order to investigate in more detail the deformation around the slit 3a in which the notch effect appears, the peripheral portion of the slit 3a is composed of relatively small elements (the nodes are dense) compared to other portions. Is desirable. When the rubber material 2 includes a composite material such as a cord layer, a two-dimensional planar element such as a shell element in which orthogonal anisotropy is defined may be used as the element e.

  Next, in this embodiment, material characteristics and an overall stiffness matrix are set in the rubber analysis model 3 (step S2). These are input and set by the user using the computer apparatus 1.

  As the material characteristics, for example, the elastic modulus E, density ρ, and Poisson's ratio as the first material parameter relating to volume change of each element e of the rubber analysis model 3 are defined. Each value of the material property is determined according to the rubber material 2 to be analyzed, and the value is assigned to each element e. Since the rubber analysis model 3 of the present embodiment is made of a homogeneous material, the same material characteristics are defined for all the elements e.

  The Poisson's ratio is set to a value at which the rubber analysis model 3 substantially exhibits non-compression characteristics as an initial value. That is, as described above, it is not 0.5 so as not to cause a calculation error, but it is set to a value close to it, preferably 0.49 or more and less than 0.50 (0.49 in this embodiment). Is desirable. Therefore, “the rubber analysis model 3 exhibits substantially non-compression characteristics” does not mean perfect non-compression characteristics with a Poisson's ratio of 0.5, but approximates 0.5 as described above. The characteristics indicated by the object having the Poisson's ratio are sufficient.

The overall stiffness matrix “K” is obtained by adding the element stiffness matrix [D] set for each element for the entire model, and can be obtained by the following equation (2) using the element stiffness matrix [K] e. it can.
[K] = Σ [K] e (2)
The element stiffness matrix [K] e is calculated using the stiffness matrix [D] and the B matrix as shown in the following equation (3). The stiffness matrix [D] is difficult to deform with respect to the load of the element, that is, the relationship between the stress σ and the strain ε as shown in the following formula (4), and can be determined according to the material characteristics. Further, the B matrix shows the relationship between the displacement δ and the strain ε as shown in the following formula (5).
[K] e = ∫ [B] ⁻¹ [D] [B] dv (3)
[Σ] = [D] {ε} (4)
{Ε} = [B] {δ} (5)
In the present embodiment, the stiffness matrix [D] includes the Poisson's ratio as the first material parameter as in the following formula (6).

  Next, it is determined by the computer apparatus 1 whether or not the analysis is completed (step S3). Conditions for terminating the analysis can be variously determined as necessary, and examples thereof include an elapsed time from when a load is applied and whether or not the rubber analysis model 3 is broken. In the present embodiment, the latter is adopted. If the analysis end condition is satisfied, the process ends (Y in step S3).

If it is determined that the analysis has not ended (N in step S3), the boundary condition is updated in the simulation method of the present embodiment (step S4). The updated boundary conditions are:
Examples include the magnitude, orientation, load speed, and / or time increment for deformation calculation of the tensile load P.

Next, the first deformation calculation of the rubber analysis model 3 is performed by the finite element method (step S5). In the first deformation calculation, the nodal displacement of each element e, that is, the deformation shape of the rubber analysis model 3 after the deformation is calculated. The overall stiffness matrix [K], the nodal displacement vector {δ} indicating the displacement of each nodal point, and the external force vector {P} have the relationship of the following equation (7), so the inverse matrix [K] ^{−1 of the} overall stiffness matrix: To obtain the nodal displacement vector {δ}.
{P} = [K] {δ} (7)

  Next, the maximum principal strain is calculated for each element of the rubber analysis model 3 (step S6). The maximum principal strain more representatively indicates the magnitude of strain generated in various directions of each element e. Therefore, by checking the value, the degree of deformation of each element can be checked accurately. However, physical quantities such as maximum stress and average stress may be used instead of such parameters.

  Next, based on the change in the maximum principal strain, the presence or absence of an element that needs to change the value of the first material parameter (Poisson's ratio) so as to cause a substantial volume change in the rubber analysis model 3 is examined. (Step S7). Table 1 below shows an example of the relationship between the maximum principal strain of each element e and the Poisson's ratio value to be set based on the value. FIG. 4 shows the graph. In this embodiment, when the maximum principal strain of an element is 0 to 3.0 and the deformation is relatively small, the initial value (that is, 0.49) is used as it is for the Poisson's ratio. Therefore, an element with small deformation is treated as exhibiting incompressibility without substantial volume change. On the other hand, the Poisson's ratio of an element that has a large deformation with a maximum principal strain exceeding 3.0 is changed to a smaller value as the maximum principal strain increases.

  The determination in step S7 will be described. For example, for an arbitrary element, the maximum principal strain calculated in the previous step (step before the time increment) is 1.0, and the maximum principal strain calculated in the current step. Is 2.0, according to FIG. 4, Table 1, the element does not need to change the first material parameter. That is, according to Table 1, the Poisson's ratio of the element may remain at 0.49. Therefore, such an element does not correspond to “an element for which the first material parameter must be changed” in step S7. On the other hand, when the maximum principal strain calculated in the previous step is 1.0 and the maximum principal strain calculated in the current step is 4.0 for any one element, according to Table 1, the element The Poisson's ratio needs to be changed from 0.49 to 0.45. Therefore, such an element corresponds to “an element whose first material parameter must be changed” in step S7. Note that the correspondence relationship of the maximum principal distortion-Poisson ratio not described in this table is calculated as appropriate by linear interpolation. Such a determination is made for all elements.

  Next, if there is no element (hereinafter referred to as “target element”) for which the first material parameter needs to be changed (N in step S7), the process continues until the analysis end condition is satisfied. S4 to 6 are repeated. Thereby, the shape of the rubber analysis model 3 that changes every moment from the moment when the tensile load is applied is calculated and output.

  On the other hand, if the target element exists (Y in step S7), the first material parameter (Poisson's ratio) of the target element is changed according to Table 1 (parameter change process), and the entire stiffness matrix is determined using other boundary conditions. Is set again (step S8). Then, the re-calculation of the deformed shape, that is, the second deformation calculation is performed using the newly set overall stiffness matrix (step S9). Here, as shown in Expression (6), the stiffness matrix [D] includes the Poisson's ratio. Therefore, by changing the Poisson's ratio to be small for an element having a large degree of deformation, the volume change can be expressed more greatly in the second deformation calculation than in the first deformation calculation. That is, a shape change accompanied by a volume change as shown when a real rubber material undergoes a large deformation can be reproduced by making the first material parameter that does not change originally variable. Therefore, it is possible to accurately reproduce the deformation behavior of the real object.

In the simulation method of the present embodiment, after the second deformation calculation, a difference between the second deformation calculation result and the first deformation calculation result is calculated, and the difference is within a predetermined allowable range. Is determined (step S10). This criterion can be appropriately set according to the contents of the simulation. As an example, the total number of nodes of the rubber analysis model 3 is N, the displacement of the i-th node (i is an integer from 1 to N) in the first deformation calculation is d1 _i , and the displacement in the second deformation calculation is d2 _{where i} is within the allowable range when the maximum value of the displacement difference | d1 _i −d2 _i | is equal to or less than a predetermined percentage of the average displacement (Σd1 _i / N) of all elements in the first deformation calculation. It is desirable. In addition, although a numerical range etc. can be changed according to material, about 5% is suitable as an example.

  If the difference between the second deformation calculation result and the first deformation calculation result is within an allowable range (Y in step S10), steps S4 to S6 are repeated until a sufficient analysis end condition is satisfied, and rubber analysis is performed. The deformed shape of model 3 is calculated again. On the other hand, if the difference between the calculation results is outside the allowable range (N in step S10), the time increment is decreased (step S11), and the first material parameter (Poisson's ratio) of the target element is returned to the value before the change, The overall stiffness matrix is set again using other boundary conditions (step S12). Then, step S4 and subsequent steps are executed.

  When the calculation result changes rapidly due to the change in the first material parameter, there is a possibility that an accurate calculation result (correct answer) cannot be reached unless the process of this change is closely tracked. Therefore, as in the present embodiment, when it is determined in step S10 that the difference between the first deformation calculation result and the second deformation calculation result is not within the allowable range, the update increment amount of the boundary condition is set. The calculation accuracy can be increased by reducing the rigidity matrix and resetting the rigidity matrix. Note that, as an aspect of reducing the boundary condition update increment, for example, the calculation time increment can be reduced. Further, in the case of static balance calculation, for example, the original 1 mm deformation can be set to 0.5 mm, or the load amount can be reduced. According to the present embodiment, in the case of N in step S10, both the first deformation calculation result and the second deformation calculation result calculated based on the previous boundary condition are discarded.

A specific example of the simulation method described above will be shown. FIG. 5 shows a deformation shape of the rubber analysis model as a simulation result of the tensile test of the rubber analysis model performed according to the procedure of FIG. Further, FIG. 6 shows a side view when a real rubber material is subjected to a tensile test similar to the simulation. The other conditions under which the simulation was stopped by causing the tensile load to increase as slowly and gradually as possible and breaking the model were as follows.
Shape of rubber analysis model: thin hexahedron with height 30 mm x horizontal length 30 mm x thickness 1 mm Slit: depth of cut 10 mm
Elastic modulus of the element: 5 MPa
Poisson's ratio: variable as shown in Table 1. Density: 1.1 g / cm ³

  As is apparent from FIGS. 5 and 6, it can be confirmed in the simulation of the present embodiment that a deformed shape very close to the actual rubber material is obtained. On the other hand, FIG. 7 shows a simulation result of a comparative example at the same time when the Poisson's ratio is constant at 0.49. In this case, since the volume change does not occur, it can be seen that the deformed shape of the slit is greatly different from that of the actual one.

  Although the simulation method of the present invention has been described above, the present invention can be variously modified without departing from the scope of the present invention. For example, in the above embodiment, the Poisson ratio is used as the first material parameter, but it goes without saying that the Poisson number that is the reciprocal thereof may be adopted. In this case, the Poisson number is changed to increase with increasing deformation of the element. Also, the bulk modulus can be employed as the first material parameter. In this case, the initial value of the bulk modulus is set to an extent corresponding to a Poisson's ratio of 0.49, preferably 20 times or more of the modulus of elasticity (longitudinal modulus) defined in the element e of the rubber analysis model 3 and It is desirable to set it to 1000 times or less. In the parameter changing process, the volume modulus of elasticity can be reduced as the strain or stress of the rubber analysis model 3 increases as in the case of the Poisson's ratio.

It is a perspective view which shows an example of a computer apparatus. It is a flowchart which shows an example of the process sequence of this embodiment. (A) is a perspective view of a rubber material to be analyzed, and (b) is a side view of a rubber analysis model for the same. It is a graph which shows the relationship between a Poisson's ratio and the largest principal distortion. It is a side view which visualizes and shows the simulation result of a rubber analysis model. It is a side view which shows the tension test result of a real rubber material. It is a side view which visualizes and shows the simulation result of the rubber analysis model outside this invention.

Explanation of symbols

1 Computer device 2 Rubber material 3 Rubber analysis model

Claims (6)

A simulation method for performing deformation calculation of rubber material in minute time increments,
A process for setting a rubber analysis model in which a rubber material is modeled by a plurality of elements capable of numerical analysis;
A process of setting a value of a first material parameter that defines a volume change of the rubber analysis model so that the rubber analysis model exhibits substantially incompressible characteristics;
A process for performing a first deformation calculation of the rubber analysis model using at least the first material parameter and a predetermined analysis condition;
A process of calculating strain or stress of each element after the first deformation calculation;
And a parameter changing process for changing the value of the first material parameter so that a substantial volume change occurs in the element when the strain or stress exceeds a predetermined value. Rubber material simulation method. The first material parameter is a Poisson's ratio;
The rubber material simulation method according to claim 1, wherein the parameter changing process decreases the value of the Poisson's ratio based on strain or stress of the element.   The method for simulating a rubber material according to claim 2, wherein an initial value of the Poisson's ratio is 0.49 or more and less than 0.5.   The rubber material simulation method according to claim 1, wherein the first material parameter is a bulk modulus. An initial value of the bulk modulus is 20 times or more of a modulus of an element of a rubber analysis model, and the parameter changing process decreases the bulk modulus based on strain or stress of the rubber analysis model. The rubber material simulation method according to claim 4. After the parameter changing process, a process of performing a second deformation calculation using the changed first material parameter;
A process of calculating a difference between the result of the second deformation calculation and the result of the first deformation calculation;
The rubber material simulation method according to claim 1, further comprising a process of reducing the time increment when the difference exceeds a predetermined value.