Patent Publication Number: US-2013245964-A1

Title: Method for estimating energy loss of high polymer material

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to a computerized method for accurately estimating an energy loss of a high polymer material. 
     The energy loss of a high polymer material such as plastic and rubber is a very important physical quantity which affects various performances of products from the high polymer material. For example, a tire is a typical rubber product, and the energy loss of rubber used therein closely relates to grip performance, fuel consumption performance and the like of the tire. Therefore, it is important to exactly know and control the energy loss. 
     In general, in order to obtain, for example, a rubber compound having desired physical quantities, a development cycle—namely, designing of a rubber composition, trial production of the rubber compound, and measurement of the energy loss—is repeated until the desired physical quantities can be obtained. 
     In recent years, on the other hand, it has been proposed to estimate the energy loss of a high polymer material from results of a computerized simulation of the high polymer material, for example as disclosed in the following Patent documents 1 and 2.
     [Patent document 1] Japanese Patent Application Publication No. 2007-107968   [Patent document 2] Japanese Patent Application Publication No. 2009-271719   

     In the methods disclosed in the Patent documents 1 and 2, roughly, the following steps (a)-(e) are implemented in this sequence: (a) a model of a molecular structure of the high polymer material is defined, (b) potentials are defined on the molecular structure model, (c) a structure relaxation calculation is made using a high polymer material model in which the molecular structure models are randomly disposed in a cell, (d) after the structure relaxation calculation has been done, by applying a cyclic deformation to the high polymer material model, the stress and strain are computed, and (e) a hysteresis loop is obtained from the computed results, and the energy loss is computed from the area of the hysteresis loop. 
     In such methods, however, there are problems as follows: in the step (d), sometimes, the calculation becomes unstable and abends during the calculation; and in the step (e), sometimes, the calculated value of the energy loss differs greatly from the actual measurement value. 
     The present inventor made a study and found that the calculation becomes unstable and abends when the oscillation cycle of the molecular structure model determined by the characteristic frequency of the potential, becomes substantially equal to the cycle of the cyclic deformation of the high polymer material model. 
     SUMMARY OF THE INVENTION 
     It is therefor, an object of the present invention to provide a computerized method for estimating an energy loss of a high polymer material in which a deformation calculation is made only when the oscillation cycle of the molecular structure model does not coincide with the cycle of the cyclic deformation of the high polymer material model, and thereby the energy loss of the high polymer material can be accurately stably estimated. 
     According to the present invention, a computerized method for estimating an energy loss of a high polymer material, comprises: 
     a step in which a molecular structure model of a molecular structure of the high polymer material is defined, wherein the molecular structure model includes particle models each being of an atom or atoms and joining chains specifying relative positions between the particle models, 
     a step in which potentials between the particle models of the molecular structure model are defined, 
     a step in which a high polymer material model in which the molecular structure models are disposed in its cell is defined, and, based on the potentials and predetermined conditions, a structure relaxation calculation is made on the high polymer material model, 
     a step in which, after the structure relaxation calculation has been done, by applying a cyclic deformation to the high polymer material model, a deformation calculation to obtain stress and strain is made, 
     a step in which, based on results of the deformation calculation about the stress and strain, an energy loss is computed, wherein 
     the deformation calculation is made only when the cycle Ts of the cyclic deformation is substantially not equal to the oscillation cycle Tp determined based on the characteristic frequency of the potential. 
     Preferably, the structure relaxation calculation is made under such a condition that a density lower than the actual density of the high polymer material is defined on the high polymer material model. Preferably, the structure relaxation calculation for at least 10 picoseconds is made under a constant pressure and a constant temperature. Preferably, the deformation calculation is made for only the molecular structure models whose radii of inertia are less that 0.5 times a size of a cell. It is preferable that, when the ratio Ts/Tp of the cycle Ts to the oscillation cycle Tp is out of a range from 0.8 to 1.2, the deformation calculation is made. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is the structural formula of polybutadiene. 
         FIG. 2  is a flow chart of a simulation method as an embodiment of the present invention. 
         FIG. 3  shows a molecular structure model used in this embodiment. 
         FIG. 4  is a perspective view of a molecular structure model whose degree of polymerization is three. 
         FIG. 5  is a schematic perspective view for explaining a high polymer material model. 
         FIG. 6  is a graph showing a relationship between the density of the molecular structure model and the computing time of the structure relaxation calculation. 
         FIG. 7  is a flow chart of a deformation calculation in this embodiment. 
         FIG. 8(   a ) is a diagram of a cell for explaining the deformation calculation. 
         FIG. 8(   b ) is a graph for explaining a variation of the strain. 
         FIG. 9  is a graph showing a relationship between the strain and stress obtained by the deformation calculation. 
         FIGS. 10(   a ) and  10 ( b ) are diagrams for explaining relationship between the radius of inertia of the molecular structure model and the size of the cell. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     An embodiment of present invention will now be described in detail in conjunction with accompanying drawings. 
     According to the present invention, the method for estimating an energy loss of a high polymer material is implemented by a computer. 
     The high polymer material encompasses at least rubber, resin and elastomer. 
     The high polymer material used in this embodiment is, as shown in  FIG. 1 , cis-1,4 polybutadiene (hereinafter, simply referred to as polybutadiene). 
     The polybutadiene is a polymer formed from the polymerization process of the monomer {—[CH2—CH═CH—CH2]—} made from a methylene group (—CH2—) and a methine group (—CH—). 
       FIG. 2  shows a flow chart of the simulation method as an embodiment of the present invention. 
     In this embodiment, firstly, a molecular structure model of the high polymer material is generated by the computer and stored. (step S 1 ). 
       FIG. 3  and  FIG. 4  show a molecular structure model  2  which represents a unit of a macromolecular chain of polybutadiene. 
     Of course, the reality of the molecular structure model is numerical data capable of being dealt with in a molecular dynamics calculation by the computer. 
     The molecular structure model  2  is formed by linking a number (n) of monomer models  3  of a butadiene monomer. In the example shown in  FIG. 4 , the polymerization degree is three. 
     The monomer model  3  in this example is so called “full atom model” in which all of the atoms (C and H) are dealt with as particle models  4  in a computer simulation. 
     However, it is also possible that the monomer model  3  is so called “united atom model” in which a plurality of atoms are incorporated into one particle represented by one particle model. 
     In a computer simulation based on a molecular dynamics calculation, each particle model  4  incorporated in the molecular structure model  2  is dealt with as a material point in a motion equation. In other words, on each particle model  4 , numerical data about its parameters—mass, volume, diameter, electric charge and/or initial coordinates—are defined and stored in the computer. 
     Between the particle models  4 , joining chains  5  are defined to bind each other. The joining chains  5  are dealt with as springs, and for that purpose, for each of the joining chains  5 , its equilibrium length and spring constant are defined and stored in the computer. 
     Each of the monomer models  3  is three-dimensional. 
     For each of the monomer models  3 , bond lengths between particle models  4 , bond angles each formed by three particle models  4  linked by joining chains  5 , and dihedrals each formed by three adjacent particle models  4  of four particle models  4  linked by joining chains  5  are defined and stored in the computer. 
     when the monomer model  3  is subjected to an external force or an internal force, the bond lengths, bond angles and dihedrals may be changed, and accordingly, the three-dimensional structure of the molecular structure model  2  may be changed. 
     Such modeling is possible by the use of a software “J-OCTA” developed by JSOL corporation, for example. 
     Next, between the particle models  4 , potentials are defined and stored in the computer. (step S 2 ) 
     The potential is a function of an angle or a distance between the particles and used to calculate a force acting between two particle models  4 . 
     Major potentials are as follows: Potential of bond stretching between particle models  4  linked by joining chain  5 ; Potential of bending formed by three continuous particle models  4 ; Potential of torsion formed by three or four continuous particle models  4 ; and Potential of Van der Waals&#39; force between particle models  4  not linked with each other. 
     Next, the molecular structure models  2  initialized as above are arranged in a predetermined cell (space) S as shown in  FIG. 5 , and thereby a high polymer material model P is defined. Here, the molecular structure models  2  are randomly arranged to form a stable or metastable state. 
     Then, a structure relaxation calculation is carried out according to the potentials, predetermined conditions, and a molecular dynamics calculation. (step S 3  and S 4 ) 
     The cell S corresponds to a microstructural part of the high polymer material. In this example, the cell S is regular hexahedron. In the cell S, multiple molecular structure models  2  whose polymerization degree is three are randomly arranged. 
     Conditions for the structure relaxation calculation may be variously defined. 
     In this embodiment, the density of the high polymer material model P is set to a value less than the actual density of the polybutadiene. Here, the model&#39;s density is a quotient of the total mass of the molecular structure models  2  disposed in the cell S divided by the volume of the cell S. 
     In this embodiment, the density of the molecular structure model is set to 0.01 g/cm̂3 by default. 
     If a large number of molecular structure models  2  are disposed in the cell S from the start, there is a possibility that abnormal forces occur in the molecular dynamics calculation, and the calculation by the computer becomes unstable, when the molecular structure models  2  are overlapped each other. In this embodiment, however, since the structure relaxation calculation starts with the lower density, the probability of overlapping the molecular structure models  2  with each other is decreased and it is possible to stabilize the calculation. 
     It is desirable that the structure relaxation calculation for at least 10 picoseconds is made under a constant pressure and a constant temperature and under a periodic boundary condition. 
       FIG. 6  shows a relationship between the density of the molecular structure model  2  and the calculation time as an example of the result of the structure relaxation calculation. As shown, by making the structure relaxation calculation for at least 10 picoseconds, the density of the high polymer material model is restored to its actual value. 
     As a result, a deformation calculation made on the high polymer material model P in the next step becomes stable. 
     Next, a deformation calculation is made by applying a cyclic deformation to the high polymer material model P stabilized through the structure relaxation calculation in order to calculate stress and strain. (step S 5 ) 
       FIG. 7  shows a flow chart of an example of the deformation calculation step S 5 . 
     In this example, firstly, based on the characteristic frequency of each potential defined on the molecular structure models  2 , an oscillation cycle Tp is calculated by the computer. (step S 51 ) 
     In the case of the potential of bond stretching, for example, if the potential u of a C—C bond stretching is given by the following expression (1): 
         U= 0.5* k *( x−x 0)̂2   (1)
 
     wherein 
     “k” is a spring constant, 
     “x” is a c-c bond length, 
     “x0” is the equilibrium length of the joining chain, then the characteristic frequency f of the bond stretching can be obtained by the following expression (2): 
         f= 1/(2pi)*( k/M )̂0.5   (2)
 
     wherein 
     “pi” is the circle ratio, 
     “M” is the mass of a carbon atom. 
     The oscillation cycle Tp is the reciprocal of a characteristic frequency. 
     Therefore, the oscillation cycle Tp of the characteristic frequency f of the potential of the bond stretching is obtained by the following expression (3): 
         Tp= 1/ f    (3)
 
     Next, a cycle Ts of the deformation calculation made on high polymer material model P is set to a certain value. (step S 52 ) 
     In the deformation calculation in this embodiment, as shown in  FIG. 8(   a ), a tensile and compressive deformation is simulated, wherein the high polymer material model P is subjected to a uniaxial tensile strain and compressive strain of the same absolute value, and the course from the original state to original state (zero-strain to zero-strain) is regarded as one cycle. 
       FIG. 8(   b ) shows an example of the cyclic deformation of the high polymer material model P. In the figure, the strain is shown as a function of the time. In this example, the strain is varied like a sine wave. It is also possible to vary like a triangle wave. 
     Also, the magnitude of the strain, Poisson ratio and the like are set in addition to the cycle Ts. 
     Next, the computer judges if the cycle Ts is substantially equal to the oscillation cycle Tp. (step S 53 ) 
     In the case that a plurality of different oscillation cycles Tp exist, the judgment is made with respect to each of the oscillation cycles Tp. 
     In this embodiment, if the ratio Ts/Tp is within a range of from 0.8 to 1.2, the cycle Ts is regarded as being substantially equal to the oscillation cycle Tp. It is of course possible to set the range differently. 
     If equal (Y in step S 53 ), then the cycle Ts of the deformation calculation is set to a different value than the former value. (step S 52 ) 
     If not equal (N in step S 53 ), then the deformation calculation is made on the high polymer material model P with the cycle TS. (step S 54 ) 
     As explained above, if the oscillation cycle Tp becomes substantially equal to the cycle Ts, the calculation becomes unstable and abends. Further, the accuracy of the calculated results is greatly lowered. The reason is assumed to be abnormally large forces acting on the atoms due to large variations of the distances between the atoms occurring during the repetition of deformation. Such problem seems to be solved by making the deformation calculation for only one cycle (namely, not repeating the deformation calculation). But, when cyclic deformation is made using the molecular dynamics, regardless of whether or not the cycle Tp is equal to the characteristic frequency Ts, the influence of the deformation on the stress can not be recognized or it is hard to recognize for a short while after the beginning of the deformation. For this reason, to make the deformation calculation for only one cycle is not a realistic way. 
     In the present invention, therefore, before the deformation calculation is made on the high polymer material model P, the cycle Ts of the deformation of the high polymer material model P is compared with the oscillation cycle Tp determined based on the characteristic frequency of the potential of the molecular structure models  2 , and if the cycle TS is substantially not equal to the oscillation cycle Tp, the deformation calculation is made on the high polymer material model P. Therefore, in the deformation calculation, the cycle Ts is always different from the oscillation cycle Tp. Accordingly, the abend of the calculation and the deterioration of the calculation accuracy can be prevented. 
       FIG. 9  shows a stress-strain curve (hysteresis loop) of the high polymer material model P obtained by the deformation calculation. 
     The energy loss of the high polymer material model P can be obtained by computing the area surrounded by the hysteresis loop. 
     It is preferable that the deformation calculation is made, targeted at only the molecular structure models whose radii of inertia are less than 0.5 times the size of the cell S. 
       FIG. 10(   a ) shows an example where the radii of inertia Rg are more than 0.5 times the size L of the cell S (in this example, the length of a side of a cell S). 
     In the deformation calculation using the molecular structure models, a cycle boundary condition is defined where the cell S is repeated in a back-and-forth direction, a right-and-left direction and an up-and-down direction. 
     In the cycle boundary condition, a boundary of a cell S abuts on a boundary of the next cell S. 
     If a molecular structure model  2  in a first cell S protrudes partially from a boundary Sa of the first cell S, then a part of a molecular structure model  2  in the next second cell S corresponding to the protruded part of the first cell protrudes from the opposite boundary Sb into the first cell S. 
     Therefore, if the radius of inertia Rg is more than 0.5 times the size L of the cell S, there is a possibility that the particle models of the molecular structure models overlap each other, and as a resultant abnormal force is computed due to their potentials. 
     On the other hand, as shown in  FIG. 10(   b ), if the radius of inertia Rg is less than 0.5 times the size L of the cell S, the above-mentioned interference or overlap can be avoided and an accurate calculation is possible. 
     Comparison Tests 
     In order to confirm the effects of the present invention, an energy loss was estimated according to the above described embodiment. The conditions were as follows: 
       the spring constant k of the potential of the bond stretching corresponding to the bond length=918 [ε/σ̂2],
 
       the mass M of the particle model=1.0 [m], 
       the characteristic frequency  f= 4.82[1/τ], and
 
       the oscillation cycle=0.207 [τ],
 
     wherein
     ε: unit of energy,   σ: unit of length,   m: unit of mass, and   τ: unit of time.   

     In this embodiment, the cycle Ts of the deformation was set to 2.07 [τ], namely 10 times the oscillation cycle Tp based on the characteristic frequency of the potential. As a result, a stress and strain curve like a hysteresis loop shown in  FIG. 9  could be obtained. 
     For comparison, the cycle Ts was set to 0.207 [τ], namely, a value equal to the oscillation cycle, and the calculation was made. In this case, the calculation was abnormally ended. 
     Further, it was also confirmed that when the ratios Ts/Tp were in a range of from 0.8 to 1.2, the calculations were abnormally ended with high probability. 
     As described above, in the method according to the present invention, the calculation can be made stably without being abnormally ended, and accurate estimations are possible.