JP2008298429A - Material status estimating method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To reflect strain-speed dependence actually occurring in stress-strain relation. <P>SOLUTION: In stress operational processing constituted by employing the dynamic solution method of a finite-element method, a D matrix is formed by taking in the relation of stress σ, strain ε and a strain-speed ε' represented by the equation: (σ-σ<SB>0</SB>)[σ-(blog(ε')+c)]=d (wherein ε' is a strain-speed, σ<SB>0</SB>is a value having strain ε as a variable and b, c and d are constants set corresponding to a material). <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a material state estimation method suitable for analysis of steel materials and metal materials such as automobiles.

2. Description of the Related Art Conventionally, in the field of automobile crash analysis, various methods have been proposed that take strain rate dependence into consideration for stress-strain relationships, and such methods have been incorporated into CAE analysis. However, the accuracy of the analysis result of such CAE analysis is not sufficient.
For example, the Cooper-Symmonds equation shown below is known as a stress-strain relationship in consideration of strain rate dependency.
σ _y = σ _y (ε) (1+ (ε ′ / C)) ^{1 / p}
Here, C and p are constants, ε is strain (equivalent strain or equivalent plastic strain), ε ′ is strain rate (equivalent strain rate or equivalent plastic strain rate), and σ _y is yield stress. .
Further, there is a method in which a table is provided for each strain rate with respect to the stress-strain relationship, and the table data is obtained by linearly complementing the table with no data (undefined region).

However, in the Cooper-Symmonds formula, as can be seen from the written formula, when the strain rate is changed, the stress changes with respect to the strain, but the ratio of the stresses obtained at different strain rates was taken. In this case, the ratio becomes a constant value with respect to the strain.
Here, FIG. 12 shows the relationship between the equivalent plastic strain ε (ε _a ^p ) and the stress ratio σ _0.002 / σ ₂₀ obtained by testing using JSC270D (indicated by the Japan Iron and Steel Federation standard). Indicates. Here, sigma _0.002 is strain rate Ipushiron' (equivalent plastic strain rate ε _a ^p ') is a stress at the 0.002 (1 / sec), σ 20 is strain rate Ipushiron' (equivalent plastic This is the stress when the strain rate ε _a ^p ') is 20 (1 / sec).

  As shown in FIG. 12, the stress ratio changes according to the equivalent plastic strain ε. That is, this result says that when an actual material is used, the relationship between the equivalent plastic strain ε and the stress σ is exceptional for each equivalent plastic strain rate ε ′. However, as shown in FIG. 12, in the Cowper-Symmonds equation, the stress ratio is a constant value regardless of the equivalent plastic strain. That is, for each equivalent plastic strain rate ε ′, the relationship between the equivalent plastic strain ε and the stress σ is not an exceptional relationship. For this reason, even though the Cowper-Symmonds formula takes into account the strain rate dependence of the stress-strain relationship, in reality, the characteristics exhibited by the actual material cannot be expressed. Therefore, when such a Cowper-Symmonds formula is used for numerical calculations such as CAE analysis, there arises a problem that the calculation result includes an error.

  On the other hand, as a method for solving such a problem, there is a method of having a table with a stress-strain relationship for each strain rate as described above. For example, FIG. 13 shows the stress-strain relationship for each strain rate A, B,..., Y, Z, and the stress for each strain rate A, B,. A strain relationship is used as a table. Here, when the stress-strain relationship is used as a table for each strain rate, the table data is obtained by a test. However, when a test is performed, it is practically difficult to obtain a stress-strain relationship by keeping the strain rate constant, and the strain rate often varies greatly during the test.

FIG. 14 shows the relationship between the equivalent plastic strain and strain rate during the test to obtain the stress strain relationship. FIG. 14 shows a true strain rate in a JIS No. 5 tensile test and a nominal strain rate obtained at a relatively small strain rate (for example, 0.002 (1 / sec)).
As shown in FIG. 14, even at the nominal strain rate, there is a swing with respect to the change in the equivalent plastic strain, and the true strain rate in the JIS No. 5 tensile test changes greatly with respect to the change in the equivalent plastic strain.

Thus, it is difficult to obtain a stress-strain relationship by a test with a constant strain rate. For this reason, even if supplemented with table data composed of data obtained by testing, it is difficult to say that the stress-strain relationship obtained by the complementation shows an appropriate relationship.
An object of the present invention is to reflect the strain rate dependency actually generated in the stress-strain relationship.

In order to solve the above-mentioned problem, the material state estimation method according to claim 1 according to the present invention estimates a material state by numerical analysis using a finite element method based on the relationship between stress σ and strain ε. In the material state estimation method, when ε ′ is a strain rate, σ ₀ is a value having a strain ε as a variable, and b, c, d are constants set according to the material, the following formula (σ− σ ₀ ) (σ− (b · log (ε ′) + c)) = d
The state of the material is estimated based on the relationship between the stress σ, the strain ε, and the strain rate ε ′ shown in FIG.
The material state estimation method according to claim 2 of the present invention is the material state estimation method according to claim 1, wherein the following Swift equation σ ₀ = k obtained by using k, ε ₀ , and n as constants, respectively: (Ε ₀ + ε) ⁿ
The value σ ₀ is determined by the following.

  According to the present invention, since the equation reflects the strain rate dependence actually generated in the stress-strain relationship, numerical analysis using a finite element method to which this equation is applied, The state of the material can be estimated with high accuracy.

The best mode for carrying out the present invention (hereinafter referred to as an embodiment) will be described in detail with reference to the drawings.
(Principle of the present invention)
The principle of the present invention will be described.
First, an actual material was tested to obtain a relationship between strain rate ε ′ and stress σ. FIG. 1 shows the relationship between strain rate ε ′ and stress σ obtained in the test. Further, as shown in FIG. 1, the relationship between the strain rate ε ′ and the stress σ is obtained using the strain ε as a parameter.
As shown in FIG. 1, when the strain rate ε ′ decreases, that is, when log (ε ′) = − ∞, the stress σ corresponding to each strain ε can be obtained asymptotically to a certain value. did it. On the other hand, when the strain rate ε ′ increases, that is, when log (ε ′) = ∞, the stress σ corresponding to each strain ε converges to a certain value while increasing.

  In addition, similar tests were performed using various materials. As a result, the same results were obtained qualitatively, although each material was quantitatively different. That is, for any material, when the strain rate ε ′ decreases, that is, when log (ε ′) = − ∞, the stress σ corresponding to each strain ε is set to a certain value (although this value itself is different). Asymptotic characteristics became. On the other hand, in any material, when the strain rate ε ′ increases, that is, when log (ε ′) = ∞, the stress σ corresponding to each strain ε increases to a value that is increasing (although this value is different). It became the characteristic to converge.

And the following (1) Formula can be obtained by making into a numerical expression the relationship between such stress (sigma), distortion (epsilon), and strain rate (epsilon) '.
(Σ−σ ₀ ) (σ− (b · log (ε ′) + c)) = d (1)
Here, σ ₀ is a value having the strain ε as a variable, and specifically, is a static stress when the strain rate ε ′ is infinitely small. Further, b, c, and d are constants determined by the material. That is, b, c, and d are parameters specific to the material, and are parameters that appropriately indicate the strain rate dependency for each material.
Also, σ ₀ can be expressed as, for example, the following formula (2).
σ ₀ = σ (ε) (2)
For example, the equation (2) can be given by the Swift equation shown as the following equation (3).
σ ₀ = k (ε ₀ + ε) ⁿ (3)
Here, k, ε ₀ , n are constants (material parameters) expressing strain dependence.

Here, FIG. 2 shows a characteristic diagram obtained by the equation (1) for a certain material.
As shown in FIG. 2, when the strain rate ε ′ decreases, that is, when log (ε ′) = − ∞, the stress σ corresponding to each strain ε (= ε ₀ , ε ₁ ,..., Ε _n ). Each show a characteristic that asymptotically approaches a certain value. Further, when the strain rate ε ′ increases, that is, when log (ε ′) = ∞, the stress σ corresponding to each strain ε (= ε ₀ , ε ₁ ,..., Ε _n ) increases. Shows a characteristic that converges to a certain value.

As can be seen from the comparison of the characteristic diagram of FIG. 2 obtained by the equation (1) and the test result of FIG. 1, the equation (1) appropriately shows the test result, and Equation (1) properly shows the relationship between stress σ, strain ε, and strain rate ε ′. Therefore, according to the equation (1), the stress-strain relationship can be shown with an appropriate strain rate dependency.
For this reason, if the stress-strain relationship represented by equation (1) is applied to a stress-strain relationship required for stress calculation (for example, CAE analysis) by the finite element method, a favorable calculation result can be obtained. . In the embodiment described below, an example of such stress calculation using the equation (1) is shown.

(Embodiment)
The embodiment is a stress calculation device to which the present invention is applied, that is, to perform a stress simulation adopting the principle of the present invention.
FIG. 3 shows an example of the configuration of the stress calculation apparatus.
As shown in FIG. 3, the stress calculation device includes an input unit 1 to which data is input by an external operation of an operator, an arithmetic processing unit 2 that performs calculations using various data, an output unit that outputs calculation results 3 and a storage unit 4 for storing data. The stress calculation device is constituted by, for example, a personal computer.

FIG. 4 shows a processing procedure of stress calculation using the stress calculation device. In the present embodiment, the stress calculation process is constructed by adopting a dynamic explicit method of a finite element method.
First, when the process is started, nodes and element data necessary for the calculation of the finite element method are input to the calculation processing unit 2 in step S1. For example, the data is input from the storage unit 4.
Subsequently, in step S <b> 2, material characteristic data is input to the arithmetic processing unit 2. For example, the data is input from the input unit 1. Here, the material property data is data specific to the material to be subjected to the stress calculation processing, that is, data including b, c, d, etc. shown in the above equation (1).

Subsequently, in step S <b> 3, boundary conditions necessary for the calculation of the finite element method are input to the calculation processing unit 2. For example, the data is input from the storage unit 4.
Subsequently, in step S4, the arithmetic processing unit 2 creates a mass and attenuation matrix necessary for the calculation of the finite element method.
Subsequently, in step S5, the arithmetic processing unit 2 creates a shape function expressed by the following equation (4).
{u} = [N _m ] {u _m } (4)
Subsequently, in step S6, the arithmetic processing unit 2 creates a B matrix expressed by the following equation (5).
{ε} = [B] {u} (5)
Subsequently, in step S7, the arithmetic processing unit 2 creates a D matrix of a material constitutive equation expressed by the following equation (6).
{σ ′} = [D] {ε ′} (6)
When this equation (6) is specifically described as an isotropic hardening constitutive equation based on the Prandtl-Royce equation, it is expressed as the following equation (7).

Here, the D value in the formula (6) or the formula (7) can be expressed as the following formula (8).

Here, E is a longitudinal elastic modulus (Young's modulus), G is a transverse elastic modulus, and ν is a Poisson's ratio.
Further, the calculation is premised on satisfying the conditions of the following expressions (9) and (10).

For example, in operation, as a equivalent plastic strain increment between the (10) the value d? _A ^p obtained by the formula time step, to obtain the equivalent plastic strain epsilon _a ^p at each time step.
FIG. 5 shows an image of a change in yield surface during isotropic hardening. FIG. 5 shows changes in the yield surface in the direction of the principal stresses σ ₁ and σ ₂ (major axis direction) so that the change in the shear stress τ can be ignored. The yield surface in isotropic hardening is enlarged as shown in FIG.
And the said (1) Formula which shows a stress-strain relationship is applied to (7) Formula which is a structural formula of isotropic hardening shown as mentioned above.
For example, in the equation (8), H '(specifically, the equivalent stress) stress is a value obtained by differentiating the sigma _a In the equivalent plastic strain epsilon _a ^p, represented as the following equation (11).
H ′ = dσ _a / dε _a ^p (11)

The stress obtained by the equation (1) is applied to the equivalent stress σ _a in the equation (11). Specifically, the application is performed by changing the expression (1) to the expression shown in the following expression (12).
_{(Σ a -σ a0) (σ} a - (b · log (ε a p ') + c)) = d ··· (12)
At this time, σ ₀ shown in the equation (2) is expressed by the following equation (13), and the Swift equation shown in the equation (3) is expressed by the following equation (14).
σ _a0 = σ _a0 (ε _a ^p ) (13)
_{σ a0 = k (ε 0 +} ε a p) n ··· (14)
Here, sigma _a is in equivalent strain, epsilon _a ^p is the strain equivalent plastic, ε _{a p} ^'is a equivalent plastic strain rate, sigma _a0 is the equivalent plastic strain rate epsilon _{a ^p'} in the case of 0 Stress And is a function of equivalent plastic strain only.
In addition, the said (12) Formula can also be shown as following (15) Formula.

Here, the constant (material parameter) is given as follows, for example.
When the material to be calculated is JSC270D (indicated by the Japan Iron and Steel Federation standard), k = 602.9, ε ₀ = 0.00575, n = 0.360, b = 59.75, c = 13. 24, d = 34195.5. If the material to be calculated is JSC980Y (indicated by the Japan Iron and Steel Federation standard), k = 1548.6, ε ₀ = 0.000392, n = 0.172, b = 82.43, c = −319.6, d = 317604.

The D matrix is created based on the above relationship.
Note that the above equations (12) and (15) form a flow stress curve equation (flow stress equation) in the calculation using the finite element method, and when generalized, the equation of the Cower-Symmonds is also expressed. In addition, it can be expressed as the following equation (16).
σ _a = H (σ _a ^p , σ _a ^p ′) (16)
Subsequently, in step S8, the arithmetic processing unit 2 gives a nodal internal force F and a nodal external force P for deforming the material, and in subsequent step S9, performs a difference calculation of the equation of motion. Then, in the subsequent step S10, the arithmetic processing unit 2 obtains a displacement increment, a strain increment, and a stress increment.

Subsequently, in step S11, the calculation processing unit 2 updates the coordinates and stress for the calculation target material based on the displacement increment, strain increment, and stress increment calculated in step S10.
Subsequently, in step S12, the arithmetic processing unit 2 determines whether or not it is a final time step. Here, if it is the final time step, the arithmetic processing unit 2 ends the process, and if it is not the final time step, the arithmetic processing unit 2 proceeds to step S13.

In step S13, the arithmetic processing unit 2 increases the time step of the arithmetic processing and restarts the processing from step S5. That is, the arithmetic processing unit 2 executes the processes of Step S5 to Step S11 while increasing the time step until reaching the final time step.
Then, the calculation result of the calculation processing unit 2 is stored in the storage unit 4 or output to the output unit 3.

  As described above, based on the relationship between the stress σ, the strain ε, and the strain rate ε ′ expressed in mathematical formulas (the formula (1), the formula (12), or the formula (15)), the stress calculation of the finite element method is performed. Is going. For example, such stress calculation can be used for analysis of collision performance evaluation of steel and metal materials such as automobiles, and for example, can be used for simulation for evaluating collision safety of automobiles.

(Function and effect)
The action and effect are as follows.
As described above, a mathematical expression ((1), (12) or (15)) showing a stress-strain relationship appropriately reflecting the strain rate dependency can be obtained, and the stress calculation is performed using the mathematical expression. Is going. That is, the stress calculation is performed using a mathematical formula that reproduces the true material characteristics. Thereby, the calculation precision of such stress calculation can be made high. Moreover, data input can be simplified when compared with the case of using table data.

FIG. 6 shows the relationship between the stress (regression equation estimated stress) obtained by the computation process shown in FIG. 4 and the stress obtained by actually performing the test under the same condition as the computation condition, assuming that the material to be computed is JSC270D. Indicates. As shown in FIG. 6, the stress obtained by the arithmetic processing and the stress obtained by the experiment substantially coincide.
FIG. 7 shows the relationship between the equivalent plastic strain and stress (equivalent stress) obtained by the calculation process shown in FIG. 4, assuming that the material to be calculated is JSC270D. As shown in FIG. 7, the relationship between the equivalent plastic strain and the stress can be obtained as having a particular characteristic for each equivalent plastic strain rate ε _a ^p ′.

FIG. 8 shows the relationship between the equivalent plastic strain and the stress by the calculation process shown in FIG. 4, assuming that the material to be calculated is JSC980Y. As shown in FIG. 8, even when the material is JSC980Y, the relationship between the equivalent plastic strain and the stress can be obtained as having a particular characteristic for each equivalent plastic strain rate ε _a ^p ′.
In addition, the said embodiment can also be implement | achieved by the following structures.
That is, in the above-described embodiment, the case where the finite element method used for the stress calculation is the dynamic explicit method is described, but the present invention is not limited to using the dynamic explicit method.
Moreover, in the said embodiment, although an example of the process sequence of the stress calculation process which applied (1) Formula was shown using FIG. 4, the calculation process to which (1) Formula is applied is limited to this. is not.

(Example)
An example is as follows.
First, using JSC270D as an applicable material, the strain rate was changed by a test to obtain a stress-strain relationship, and material parameters (b, c, d, etc.) were obtained.
And about the three-point bending crushing test, the calculation result and the test result were compared.
Specifically, as shown in FIG. 9, a three-point bending crush test was performed by supporting both ends of the application material 100 having a substantially square cross section and pressing the punch 101 near the center. In addition, the shape of the applicable material 100 of FIG. 9 has shown the state after a crush test. On the other hand, CAE analysis was performed on the three-point bending crush test by arithmetic processing means such as a personal computer. By applying the present invention, CAE analysis was performed using the material parameters and the equation (1) (the equation (12) or the equation (15)). Furthermore, as a comparative example (conventional example), CAE analysis was performed using a Cowper-Symmonds formula and table data (table formula).

  The calculation conditions for the CAE analysis are the same as the conditions for the actual three-point bending crushing test. For example, the supporting distance of the applied material 100 is 500 mm, and the bending crushing speed (punch 101 speed) is 10 mm / sec. The CAE analysis and the actual three-point bending crushing test were evaluated by the absorbed energy when the punch stroke was 500 mm. For example, as shown in FIG. 10, the absorbed energy is a value that can be conceptualized by an integrated value of the reaction force applied to the support point of the applied material 100 and the movement stroke S of the punch 101 during the crush test.

FIG. 11 shows the evaluation results of the absorbed energy.
As shown in FIG. 11, the results of the examples (results of applying the present invention) are experimental values (actual three-point bending crush test) compared with the results obtained by the Cowper-Symmonds formula and the table formula of the comparative example. The value obtained in (1) was closer, and the CAE analysis could be performed with high accuracy by applying the present invention. Specifically, the value obtained by the conventional table method has an error of about 7.1% with respect to the experimental value, but by applying the present invention, the error is reduced to about half. It was reduced to about 3.6%.

It is a characteristic view which shows the relationship between the strain rate (epsilon) 'obtained by the test, and stress (sigma). It is a characteristic view which shows the relationship between the strain rate (epsilon) 'obtained by (1) Formula, stress (sigma), and strain (epsilon). It is a block diagram which shows the structure of the stress calculating device to which this invention is applied. It is a flowchart which shows the process sequence of stress calculation with a stress calculating device. It is the figure used for description of the change of the yield curve in isotropic hardening. It is a characteristic view which shows the relationship between the stress obtained by experiment, and the stress obtained by (1) Formula. It is a characteristic view which shows the relationship between the equivalent plastic distortion and stress (equivalent stress) which obtained material as JSC270D. It is a characteristic view which shows the relationship between the equivalent plastic strain and stress (equivalent stress) obtained by using JSC980Y as a material. It is the figure used for description of an Example. It is the figure used for description of absorbed energy. It is a figure which shows the evaluation result of absorbed energy. It is a characteristic view which shows the relationship between an equivalent plastic strain and a stress ratio. It is a figure used for explanation when there is a table for each strain rate regarding the stress-strain relationship. It is a characteristic view which shows the relationship between the equivalent plastic strain in the test which obtains a stress-strain relationship, and a strain rate.

Explanation of symbols

  1 input unit, 2 arithmetic processing unit, 3 output unit, 4 storage unit

Claims (2)

Based on the relationship between stress σ and strain ε, the material state estimation method for estimating the material state by numerical analysis using the finite element method,
When ε ′ is a strain rate, σ ₀ is a value with strain ε as a variable, and b, c, and d are constants set according to the material, the following formula (σ−σ ₀ ) (σ− ( b · log (ε ′) + c)) = d
A material state estimation method characterized in that the state of the material is estimated on the basis of the relationship between stress σ, strain ε, and strain rate ε ′ shown in FIG. The following Swift equation σ ₀ = k (ε ₀ + ε) ⁿ obtained by using k, ε ₀ and n as constants, respectively.
The material state estimation method according to claim 1, wherein the value σ ₀ is determined by:

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