JP2003345780A - Purony series approximating method for relaxation elastic modulus - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method for calculating a relaxation elastic modulus being a prony coefficient as input data when a linear viscoelastic material is simulated in a versatile finite element analyzing program. <P>SOLUTION: The method includes: a part 1 for reading relaxation elastic modulus data in time reliability, and an initial relaxation time τi for the portion of N terms in prony series; an initial prony coefficient calculating part 2; a part 3 for re-calculating the prony coefficient with higher approximate precision by a non-linear least square by using an initial value, and calculating the evaluation function of the approximate precision; a part 4 for deciding the increase/ decrease of the evaluation function calculated in the part 3, and determining whether calculation is repeated or not; and a part 5 for outputting the final prony coefficient. Calculation is performed from the part 1 in ascending order. When the evaluation function is being decreased in the part 4, calculation is continued after returning to the part 3. When the decrease is stopped, the calculation is advanced to the part 5 to output a result. <P>COPYRIGHT: (C)2004,JPO

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION Linear viscoelastic simulation of plastic, rubber, glass, etc.

[0002]

2. Description of the Related Art General purpose finite element analysis programs such as MARC and ABAQUS can simulate linear viscoelastic materials. At this time, it is necessary to develop the time-dependent relaxation modulus into a Prony series and to provide a Prony coefficient as input data.

[0003] However, means for calculating the Prony series from the relaxation modulus is not provided, and it is necessary to calculate the coefficient by some method. As a calculation method, a graphical solution method or the like has been proposed.

[0004]

[0005] Conventional techniques include:
Although a graphical solution method has been proposed, there are problems that it takes a very long time, that the method differs depending on the person, and that the obtained value varies. The present invention is intended to solve the above-mentioned problem by calculating a Plonie coefficient using a nonlinear least squares method. When the non-linear least squares method is used, the setting of the initial value greatly affects whether or not a correct answer can be obtained.

[0005] In addition, the method of obtaining approximate parameters when using the nonlinear least squares method greatly affects whether or not a correct answer can be obtained. Therefore, an attempt is also made to provide such a method.

[0006]

SUMMARY OF THE INVENTION The relaxation modulus Gt (t) is expressed by expanding into a Prony series as shown in the following equation (1). The method of calculating the knee coefficients Gi and τi will be described with reference to FIG.

Reference numeral 1 denotes input data, which represents a time-dependent relaxation modulus in a table format. At the same time, an initial relaxation time τi corresponding to the number N of terms of the approximating Prony series is given as input data.

Reference numeral 2 denotes an initial Prony coefficient calculation unit. Here, an initial value for calculating the Plonie coefficient of the relaxation modulus Gt (t) is calculated using the nonlinear least squares method. FIG. 2 is a flowchart showing a method of calculating the initial value. Hereinafter, the flowchart of FIG. 2 will be described.

(1) Data of a set of relaxation time τ _k and corresponding relaxation elastic modulus G _k prepared as input data is read for the number of terms of the Plonie series.

(2) Next, the input data is rearranged as τ ₁ , τ ₂ ,... In descending order of the relaxation time.

(3) Next, k = 1.

(4) Next, G_kIs calculated.
Prony coefficient G of the obtained relaxation modulus_kIs less than 0
The elastic modulus may be negative physically
Therefore, assuming Equation 4, G_k-1Recalculate
I do. At the same time, G _kIs also recalculated.

(5) If k is smaller than the number N of the Prony series, the process returns to (4) with k = k + 1 and the calculation is continued. If k becomes the number N of terms in the Prony series, all the relaxation coefficients are obtained, so that all the relaxation times τ _k and the relaxation coefficients G _k are output.

Reference numeral 3 denotes an initial value of the Proney coefficient (all relaxation times τ _k and relaxation coefficients G _k ) obtained in 2.

Reference numeral 4 denotes a portion for recalculating a Pony coefficient with higher approximation accuracy by the nonlinear least square method using the initial value obtained in 3. The evaluation function when the nonlinear least squares method is applied is as shown in Expression 5.

Reference numeral 5 denotes a portion for determining whether the evaluation function obtained in step 3 has increased or decreased. When the previous evaluation function is compared with the current evaluation function, if it has decreased, the process returns to 3. If it has increased beyond the allowable limit, the previously obtained prony coefficient is used as the final prony coefficient.

Reference numeral 6 denotes a portion for outputting the final Proney coefficient. Also, the relaxation modulus G approximated by the Proney coefficient
Relaxation modulus Gt at time t _i input with t (t _i )
(T _i ) is also displayed as a graph and a table.

[0018]

DESCRIPTION OF THE PREFERRED EMBODIMENTS The following is an example in which the actual relaxation modulus of a resin is approximated by a Plonie series according to the present invention.

FIG. 3 is a graph showing the relaxation elastic modulus of the resin, and FIG. 5 shows the initial value of the Proney coefficient obtained using the relaxation time shown in FIG. FIG. 6 shows an approximated relaxation elastic modulus using the obtained Prony coefficient by the method indicated by 2. As shown in this graph, the relaxation elasticities are quite consistent, but there are some deviations. FIG. 7 shows a result of recalculation of the Proney coefficient using the nonlinear least square method shown in FIG. FIG. 8 shows the results of calculating the relaxation elastic modulus based on the Prony coefficient calculated using the nonlinear least squares method. FIG. 8 can more accurately approximate the relaxation modulus than FIG.

FIG. 9 shows the value of the real part of the measured complex elastic modulus and the value obtained by approximating the value of the real part with the Plonie series. When expanded to a Prony series, it is well approximated over the entire measurement range. Equation 6 is used to calculate the real-time relaxation modulus from the real part of the relaxation modulus. FIG. 3 shows the real-time relaxation modulus obtained by the equation (6).

[0021]

According to the present invention, the real-time relaxation modulus can be obtained from the measured real-time relaxation modulus, and the Plonie coefficient easily and highly accurately approximated from the real-time relaxation modulus can be calculated. can do. According to the present invention, it is possible to easily calculate a highly accurate Plonie coefficient from the measured time-dependent relaxation elastic modulus.

[Brief description of the drawings]

FIG. 1 Control flow diagram

FIG. 2A is a diagram showing a flow of calculation of an initial Prony coefficient (coefficients are determined so as to satisfy a physical condition that a coefficient Gi of a relaxation modulus does not become negative)

FIG. 2B is an explanatory diagram of Expressions 1 to 3

FIG. 2C is an explanatory diagram of Expressions 4 to 6.

FIG. 3 Relaxation modulus of resin

Fig. 4 Initial value of relaxation time (input data)

Fig. 5 Initial value of Proney coefficient

FIG. 6 is a graph in which a relaxation modulus is calculated using an initial Prony coefficient.

FIG. 7 shows a Prony coefficient calculated by a nonlinear least squares method.

FIG. 8 is a graph in which a relaxation modulus is calculated using a Prony coefficient calculated by a non-linear least square method.

FIG. 9 is a graph in which the real part of a complex elastic modulus is approximated by a Prony series.

[Explanation of symbols]

1: Input data Time t_i, And the relaxation modulus Gt (t_i) And one set
Table data and the approximate number of terms of the Prony coefficient
Min relaxation time τ_i 2: Initial Prony coefficient calculation unit Relaxation time τ given as input data_iRelaxation modulus at
Gt (τ_i) To calculate the Prony coefficient. Mitigation bullet
To satisfy the condition that the coefficient Gi of the power factor does not become negative.
To determine the coefficient. Relaxation time τ given as input data
_iAn approximation error occurs at times other than. 3: Initial value of Proney coefficient 2. Initial value of the Prony coefficient obtained by This value
When the relaxation modulus is calculated from the Plonie coefficient using
Relaxation time τ given by_iRelaxation modulus Gt (τ _i) Is positive
Approximately, but relaxed bullets with other relaxation times
An error occurs when calculating the power factor. 4: Prony coefficient calculator by nonlinear least squares method A section that calculates the Plonie coefficient using the nonlinear least squares method
Minutes. First, fix the relaxation time of the
Only the sum coefficient term is calculated. Next, fix the relaxation coefficient
Calculate the sum time. 5: Evaluation function judgment unit Examine the evaluation function used when performing the nonlinear least squares method.
The increase / decrease of the evaluation function and the current evaluation function are determined. 6: Result output section 4. Outputs the Prony coefficient calculated by
Relaxation modulus Gt (t_i)
Time t entered_iRelaxation modulus Gt (t _i)
Rough, table display.

Claims (4)

[Claims] In a method of calculating a Plonie coefficient based on time-dependent data of a relaxation modulus, a relaxation time τi is given as input data, and relaxation is performed so that an approximation error of the relaxation modulus at the relaxation time is eliminated. Prony coefficient Gi of elastic modulus
A technique for approximating. 2. A method for calculating a Plonie coefficient based on time-dependent data of a relaxation elastic modulus, comprising:
A method of calculating the relaxation time τi and the Prony coefficient Gi using the nonlinear least squares method. 3. The method according to claim 2, wherein the initial value is set as the initial value.
Using the relaxation time τi and the Prony coefficient Gi obtained by the above method to minimize the error of the Prony series approximation. 4. The method according to claim 2, wherein the relaxation time τi is fixed, and the Ploney coefficient Gi is calculated by the nonlinear least squares method.
A method of performing an optimal Ploney series approximation by alternately performing a method of calculating i by a nonlinear least squares method.