JP2014010047A - Program for outputting stress/strain curve equation and its device and method for evaluating physical property of elastic material - Google Patents

Program for outputting stress/strain curve equation and its device and method for evaluating physical property of elastic material Download PDF

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JP2014010047A
JP2014010047A JP2012146552A JP2012146552A JP2014010047A JP 2014010047 A JP2014010047 A JP 2014010047A JP 2012146552 A JP2012146552 A JP 2012146552A JP 2012146552 A JP2012146552 A JP 2012146552A JP 2014010047 A JP2014010047 A JP 2014010047A
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stress
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Koji Nishiguchi
浩司 西口
Kazuhisa Maeda
和久 前田
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Nitto Denko Corp
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N3/00—Investigating strength properties of solid materials by application of mechanical stress
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N2203/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N2203/0058—Kind of property studied
    • G01N2203/0092—Visco-elasticity, solidification, curing, cross-linking degree, vulcanisation or strength properties of semi-solid materials
    • G01N2203/0094—Visco-elasticity
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N2203/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N2203/02—Details not specific for a particular testing method
    • G01N2203/0202—Control of the test
    • G01N2203/0212—Theories, calculations
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/10—Numerical modelling

Abstract

PROBLEM TO BE SOLVED: To provide a model for evaluating the physical property of an elastic material by using both visco-elasticity and super-elasticity.SOLUTION: The correlation equation of a stress and a strain to be searched on the basis of a Maxwell model in which elastic elements and viscous elements are serially arranged, that is, the correlation equation of a stress and a strain including a strain energy density function introducing the high order item of the invariant of a strain as an item related to elasticity is used to evaluate the physical property of an elastic material.

Description

本発明は、粘弾性とゴム弾性（超弾性）とを併せ持つ物性を示す弾性材料の大変形挙動をシミュレーション上で定量性よく表現するために必要となる応力−ひずみ曲線式を出力するためのプログラム及びその装置並びに弾性材料の物性評価方法に関する。   The present invention is a program for outputting a stress-strain curve equation necessary for expressing a large deformation behavior of an elastic material having physical properties having both viscoelasticity and rubber elasticity (superelasticity) with good quantitativeness on a simulation. The present invention also relates to a device for evaluating the physical properties of the elastic material.

従来より、弾性材料の物性を評価する指標として粘弾性があり、その粘弾性を表現するモデルとしては、弾性要素及び粘性要素を直列に配置した一般化Ｍａｘｗｅｌｌモデルが用いられる。一般化Ｍａｘｗｅｌｌモデルでは、応力及びひずみの相関式、すなわち、応力−ひずみ曲線式を求めることができる。   Conventionally, there is viscoelasticity as an index for evaluating the physical properties of elastic materials, and a generalized Maxwell model in which elastic elements and viscous elements are arranged in series is used as a model for expressing the viscoelasticity. In the generalized Maxwell model, a correlation equation between stress and strain, that is, a stress-strain curve equation can be obtained.

しかしながら、近年は、弾性材料（例えば粘着剤）の高品質化乃至高付加価値化に伴い、粘弾性だけでは弾性材料の物性を十分に評価できなくなってきている。具体的に言えば、非線形で且つ大変形を起こすような弾性材料であれば、その物性を評価するためには、粘弾性だけでなく、超弾性を指標に加える必要がある。   However, in recent years, with the improvement in quality and added value of elastic materials (for example, pressure-sensitive adhesives), it has become impossible to sufficiently evaluate the physical properties of elastic materials only with viscoelasticity. Specifically, if the elastic material is nonlinear and causes large deformation, it is necessary to add not only viscoelasticity but also superelasticity to the index in order to evaluate the physical properties.

そこで、本願出願人は、粘弾性モデルである一般化Ｍａｘｗｅｌｌモデルを発展させ、粘弾性−超弾性両方の物性を評価できるモデル（一般化Ｍａｘｗｅｌｌ発展モデル）を作成し、その定式化を行った上で、それを活用できるツールを提案した（特許文献１）。   Therefore, the applicant of the present application developed a generalized Maxwell model that is a viscoelastic model, created a model (generalized Maxwell development model) that can evaluate both physical properties of viscoelasticity and superelasticity, and formulated it. Then, the tool which can utilize it was proposed (patent document 1).

特許第４８５２６２６号公報Japanese Patent No. 4852626

本願出願人は、上記特許文献１に記載されたものとは別のアプローチで、新たに粘弾性−超弾性両方の物性を評価できるモデルを作成し、その定式化を行った上で、それを活用できるツールを提案する。   The applicant of the present application creates a new model that can evaluate the physical properties of both viscoelasticity and superelasticity by an approach different from that described in Patent Document 1, and formulates the model. Suggest tools that can be used.

ツールとしての本発明に係るプログラムは、
弾性要素及び粘性要素を直列に配置したＭａｘｗｅｌｌモデルに基づいて求められる応力及びひずみの相関式であって、弾性に関する項として、ひずみの不変量の高次項を導入したひずみエネルギー密度関数を含む、応力及びひずみの相関式を、応力−ひずみ曲線式として出力するためのプログラムであって、
コンピュータを、
前記応力及びひずみの相関式におけるパラメータを入力する入力手段、
該入力されたパラメータによって前記応力及びひずみの相関式を特定する特定手段、
及び、該特定された応力及びひずみの相関式を、応力−ひずみ曲線式として出力する出力手段、
として機能させることを特徴とする。 The program according to the present invention as a tool is:
A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And a program for outputting a correlation equation of a strain as a stress-strain curve equation,
Computer
Input means for inputting parameters in the correlation equation of stress and strain;
A specifying means for specifying a correlation equation of the stress and strain according to the input parameters;
And an output means for outputting the identified stress-strain correlation equation as a stress-strain curve equation,
It is made to function as.

かかる構成によれば、物性に応じて異なるパラメータ（必要に応じてその他のパラメータも含む）を入力することで、弾性材料の物性に応じた応力−ひずみ曲線式が出力される。例えば、適宜、パラメータを変更して入力することで、所望の応力−ひずみ曲線式を追求し、また、それに合うような材料設計を行うといったアプリケーションが期待できる。   According to this configuration, by inputting different parameters according to physical properties (including other parameters as necessary), a stress-strain curve formula corresponding to the physical properties of the elastic material is output. For example, it is possible to expect an application in which a desired stress-strain curve equation is pursued by appropriately changing and inputting parameters, and a material design is performed so as to match it.

また、本発明に係るプログラムにおいては、一例として、
前記応力及びひずみの相関式は、Cauchy応力を太字σ、固体の圧力をps、２階の単位テンソルを太字Ｉ、体積変化率をＪ、左Cauchy-Green変形テンソルの第一低減不変量（ひずみの第一不変量）を上線付きIB、左Cauchy-Green変形テンソルの第二低減不変量（ひずみの第二不変量）を上線付きIIB、左Cauchy-Green変形テンソルの低減不変量を上線付きＢ、実験により定められる定数をｃ1，ｃ2，ｃ3、実験により定められる定数をｐ，ｑ（ｐ＞０，ｑ＞０）、時間増分をΔｔ、実験により定められる緩和時間をτα（τα＞０）、実験により定められる定数をβα∞（βα∞＞０）で表した場合、以下の式で規定される。
In the program according to the present invention, as an example,
The correlation between stress and strain is as follows: Cauchy stress is bold σ, solid pressure is p s , second-order unit tensor is bold I, volume change rate is J, first Cauchy-Green deformation tensor first invariant ( The first invariant of strain) is overlined I B , the second reduced invariant of the left Cauchy-Green deformation tensor (the second invariant of strain) is overlined II B , and the reduced invariant of the left Cauchy-Green deformation tensor B with an overline, c 1 , c 2 , c 3 constants determined by experiment, p, q (p> 0, q> 0) determined by experiment, Δt time increment, relaxation time determined by experiment When τα (τα> 0) and a constant determined by experiment are expressed by βα∞ (βα∞> 0), they are defined by the following equations.

また、本発明に係るプログラムにおいては、温度依存性を含めた例として、
前記応力及びひずみの相関式は、Cauchy応力を太字σ、固体の圧力をps、２階の単位テンソルを太字Ｉ、体積変化率をＪ、左Cauchy-Green変形テンソルの第一低減不変量（ひずみの第一不変量）を上線付きIB、左Cauchy-Green変形テンソルの第二低減不変量（ひずみの第二不変量）を上線付きIIB、左Cauchy-Green変形テンソルの低減不変量を上線付きＢ、実験により定められる定数をｃ1，ｃ2，ｃ3、実験により定められる定数をｐ，ｑ（ｐ＞０，ｑ＞０）、時間増分をΔｔ、実験により定められる緩和時間をτα（τα＞０）、実験により定められる定数をβα∞（βα∞＞０）、弾性材料の温度をθ、基準温度（２０℃）をθS、ガラス転移温度であって、材料固有の温度をθg、実験により定められる定数をｄ1，ｄ2で表した場合、以下の式で規定される。
In the program according to the present invention, as an example including temperature dependence,
The correlation between stress and strain is as follows: Cauchy stress is bold σ, solid pressure is p s , second-order unit tensor is bold I, volume change rate is J, first Cauchy-Green deformation tensor first invariant ( The first invariant of strain) is overlined I B , the second reduced invariant of the left Cauchy-Green deformation tensor (the second invariant of strain) is overlined II B , and the reduced invariant of the left Cauchy-Green deformation tensor B with an overline, c 1 , c 2 , c 3 constants determined by experiment, p, q (p> 0, q> 0) determined by experiment, Δt time increment, relaxation time determined by experiment τα (τα> 0), the constant determined by experiment is βα∞ (βα∞> 0), the temperature of the elastic material is θ, the reference temperature (20 ° C.) is θ S , the glass transition temperature, and the temperature inherent to the material the theta g, when expressed a constant determined by experiment d 1, d 2, defined by the following formula .

また、他のツールとしての本発明に係る装置は、
弾性要素及び粘性要素を直列に配置したＭａｘｗｅｌｌモデルに基づいて求められる応力及びひずみの相関式であって、弾性に関する項として、ひずみの不変量の高次項を導入したひずみエネルギー密度関数を含む、応力及びひずみの相関式を、応力−ひずみ曲線式として出力する装置であって、
前記応力及びひずみの相関式におけるパラメータを入力する入力手段と、
該入力されたパラメータによって前記応力及びひずみの相関式を特定する特定手段と、
及び、該特定された応力及びひずみの相関式を、応力−ひずみ曲線式として出力する出力手段と
を含むことを特徴とする。 The device according to the present invention as another tool is
A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And a strain correlation equation as a stress-strain curve equation,
Input means for inputting parameters in the correlation equation of stress and strain;
A specifying means for specifying a correlation equation of the stress and strain according to the input parameters;
And an output means for outputting the specified correlation equation of stress and strain as a stress-strain curve equation.

また、別のツールとしての弾性材料の物性評価方法は、
弾性要素及び粘性要素を直列に配置したＭａｘｗｅｌｌモデルに基づいて求められる応力及びひずみの相関式であって、弾性に関する項として、ひずみの不変量の高次項を導入したひずみエネルギー密度関数を含む、応力及びひずみの相関式を用い、弾性材料の物性を評価することを特徴とする。 In addition, the physical property evaluation method of the elastic material as another tool is
A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And the physical property of the elastic material is evaluated using a correlation equation of strain and strain.

以上の如く、本発明によれば、粘弾性−超弾性両方の物性を評価できるモデルを作成し、その定式化を行った上で、それを活用できるツールを提供することができる。また、そのツールを用いて、粘弾性と超弾性とを併せ持つ物性を示す弾性材料の大変形挙動をシミュレーション上で定量性よく表現することができるようになる。   As described above, according to the present invention, it is possible to provide a tool capable of creating a model that can evaluate both physical properties of viscoelasticity and superelasticity, formulating the model, and using the model. In addition, by using the tool, large deformation behavior of an elastic material exhibiting physical properties having both viscoelasticity and superelasticity can be expressed on a simulation with good quantitativeness.

一般化Ｍａｘｗｅｌｌモデルの概念図を示す。The conceptual diagram of a generalized Maxwell model is shown. Operator Split法の概念図を示す。A conceptual diagram of the Operator Split method is shown. （ａ）３種類の引張り速度にて実測した実際の応力−ひずみ曲線図と、各引張り速度に対し、比較形態に係るモデルに基づく計算解によって表現される応力−ひずみ曲線図とを示し、（ｂ）同じ３種類の引張り速度にて実測した実際の応力−ひずみ曲線図と、各引張り速度に対し、本実施形態に係るモデルに基づく計算解によって表現される応力−ひずみ曲線図とを示す。(A) An actual stress-strain curve diagram actually measured at three types of tension speeds, and a stress-strain curve diagram expressed by a calculation solution based on a model according to a comparative form for each tension speed, b) An actual stress-strain curve diagram actually measured at the same three types of tension speeds and a stress-strain curve diagram expressed by a calculation solution based on the model according to this embodiment for each tension speed are shown. （ａ）３種類の温度にて実測した実際の応力−ひずみ曲線図と、各温度に対し、比較形態に係るモデルに基づく計算解によって表現される応力−ひずみ曲線図とを示し、（ｂ）同じ３種類の温度にて実測した実際の応力−ひずみ曲線図と、各温度に対し、本実施形態に係るモデルに基づく計算解によって表現される応力−ひずみ曲線図とを示す。(A) An actual stress-strain curve diagram actually measured at three types of temperatures, and a stress-strain curve diagram expressed by a calculation solution based on a model according to a comparative form for each temperature are shown, (b) An actual stress-strain curve diagram actually measured at the same three types of temperatures and a stress-strain curve diagram expressed by a calculation solution based on the model according to the present embodiment for each temperature are shown.

以下、本発明の一実施形態について説明する。まず、応力及びひずみの相関式を導出するまでの流れを説明する。   Hereinafter, an embodiment of the present invention will be described. First, the flow until the correlation formula between stress and strain is derived will be described.

応力（Cauchy応力）は、以下の式（１）で表される。
The stress (Cauchy stress) is represented by the following formula (1).

また、式（１）の右辺の第１項は、Cauchy応力の体積変化項であり、以下の式（２）で表される。
ここで、psは固体の圧力、太字Ｉは２階の単位テンソル、を表す。 The first term on the right side of the equation (1) is a volume change term of the Cauchy stress and is represented by the following equation (2).
Here, p s represents the pressure of the solid, and bold I represents the unit tensor on the second floor.

また、式（１）の右辺の第２項は、Cauchy応力の等積変化項であり、図１に示す一般化Ｍａｘｗｅｌｌモデルを有限変形理論に拡張し、弾性部分が超弾性体として表わされるＳｉｍｏの粘弾性モデルに基づき、定式化され、以下の式（３）で表される。
Further, the second term on the right side of the equation (1) is an equivalent product change term of the Cauchy stress, and the generalized Maxwell model shown in FIG. 1 is expanded to a finite deformation theory, and the elastic part is represented as a superelastic body. Based on this viscoelastic model, it is formulated and expressed by the following equation (3).

また、式（３）の右辺の第１項は、弾性ユニットの平衡応力であり（図１参照）、以下の式（４）で表される。
ここで、Ｊは体積変化率、Ψiso∞はひずみエネルギー密度関数、上線付きIBは左Cauchy-Green変形テンソルの第一低減不変量（ひずみの第一不変量）、上線付きIIBは左Cauchy-Green変形テンソルの第二低減不変量（ひずみの第二不変量）、上線付きＢは左Cauchy-Green変形テンソルの低減不変量、太字Ｉは２階の単位テンソル、を表す。 The first term on the right side of the equation (3) is the equilibrium stress of the elastic unit (see FIG. 1), and is represented by the following equation (4).
Where J is the volume change rate, Ψ iso ∞ is the strain energy density function, I B with the upper line is the first reduced invariant (first invariant of strain) of the left Cauchy-Green deformation tensor, and II B with the upper line is left The second reduction invariant of the Cauchy-Green deformation tensor (second invariant of strain), B with an overline represents the reduction invariant of the left Cauchy-Green deformation tensor, and bold I represents the second-order unit tensor.

ひずみエネルギー密度関数Ψiso∞として、本発明者は、「山下義裕, 川端季雄: 補強ゴムのひずみエネルギー密度関数の近似式, 日本ゴム協会誌, 65(9), 517-528, 1992.」に開示されたひずみエネルギー密度関数（以下の式（５）’）を検討した。
ここで、上線付きIBは左Cauchy-Green変形テンソルの第一低減不変量（ひずみの第一不変量）、上線付きIIBは左Cauchy-Green変形テンソルの第二低減不変量（ひずみの第二不変量）、ｃ1，ｃ2，ｃ3は実験により定められる定数、ｐは実験により定められる定数（ｐ＞０）、を表す。 As the strain energy density function Ψ iso ∞, the present inventor wrote in “Yoshihiro Yamashita, Tokio Kawabata: Approximate Formula of Strain Energy Density Function of Reinforced Rubber, Journal of Japan Rubber Association, 65 (9), 517-528, 1992.” The disclosed strain energy density function (the following equation (5) ′) was studied.
Here, the overlined I B is the first reduced invariant of the left Cauchy-Green deformation tensor (first invariant of strain), and the overlined II B is the second reduced invariant of the left Cauchy-Green deformation tensor (first strain invariant). Bivariate), c 1 , c 2 , and c 3 are constants determined by experiment, and p is a constant determined by experiment (p> 0).

しかし、このひずみエネルギー密度関数では、後述する図３（ａ）や図４（ａ）に示されるように、実測の値との相関性は必ずしも良くない。というのも、超弾性材料（例えば粘着剤）の応力−ひずみ曲線は、上記非特許文献が対象とするカーボンブラック補強ゴムの応力−ひずみ曲線に比べ、非線形性が強いからである。
この非線形性は、過去の運動履歴に依存する等積変化項で生じる。従って、等積変化を表す左Cauchy-Green変形テンソルの第二不変量の高次項を追加し、独立な係数を設定することで、超弾性材料の応力−ひずみ曲線をより高精度に記述することができる。そこで、ひずみエネルギー密度関数Ψiso∞として、以下の式（５）を考え出した。
ここで、上線付きIBは左Cauchy-Green変形テンソルの第一低減不変量（ひずみの第一不変量）、上線付きIIBは左Cauchy-Green変形テンソルの第二低減不変量（ひずみの第二不変量）、ｃ1，ｃ2，ｃ3は実験により定められる定数、ｐ，ｑは実験により定められる定数（ｐ＞０，ｑ＞０）、を表す。式（５）’と比べるとわかるように、式（５）の右辺の第４項が、追加されたひずみの第二不変量の高次項である。 However, in this strain energy density function, as shown in FIG. 3A and FIG. 4A described later, the correlation with the actually measured value is not necessarily good. This is because the stress-strain curve of a superelastic material (for example, an adhesive) is more nonlinear than the stress-strain curve of carbon black reinforced rubber, which is the subject of the non-patent literature.
This non-linearity is caused by an equal product change term that depends on the past motion history. Therefore, the stress-strain curve of the superelastic material can be described with higher accuracy by adding the second invariant high-order term of the left Cauchy-Green deformation tensor that represents the change in equal volume and setting an independent coefficient. Can do. Therefore, the following equation (5) was devised as the strain energy density function Ψ iso ∞.
Here, the overlined I B is the first reduced invariant of the left Cauchy-Green deformation tensor (first invariant of strain), and the overlined II B is the second reduced invariant of the left Cauchy-Green deformation tensor (first strain invariant). Bivariate), c 1 , c 2 , c 3 are constants determined by experiment, and p, q are constants determined by experiment (p> 0, q> 0). As can be seen from comparison with equation (5) ′, the fourth term on the right side of equation (5) is the higher-order term of the second invariant of the added strain.

また、式（３）の右辺の第２項は、α番目の粘弾性ユニットの非平衡応力であり（図１参照）、以下の式（６）で表される。
ここで、Δｔは時間増分、ａ（θ）は時間−温度換算因子（移動因子）、ταは実験により定められる緩和時間（τα＞０）、βα∞は実験により定められる定数（βα∞＞０）、を表す。 The second term on the right side of the equation (3) is the non-equilibrium stress of the α-th viscoelastic unit (see FIG. 1) and is represented by the following equation (6).
Here, Δt is a time increment, a (θ) is a time-temperature conversion factor (transfer factor), τα is a relaxation time determined by experiment (τα> 0), βα∞ is a constant determined by experiment (βα∞> 0) ).

一般に超弾性材料（例えば粘着剤）は、公称ひずみで数千パーセント以上の極めて大きな変形を生じる。その際の応力−ひずみ曲線は超弾性によりＳ字型に立ち上がり、粘性により速度依存性を示す。また、これらの粘性-超弾性挙動には温度依存性がある。なお、時間−温度換算則は粘弾性材料において一般に成立し、時間と温度を等価に扱えることが知られている。すなわち、粘弾性材料の温度変化を時間変化に変換することにより、温度依存性を表現することができる。本実施形態では、代表的な時間−温度換算則であるＷＬＦ（Williams-Landel-Ferry）式（以下の式（７））を用いた。
ここで、θは弾性材料の温度、θSは基準温度（２０℃）、θgはガラス転移温度であって、材料固有の温度、ｄ1，ｄ2は実験により定められる定数、を表す。 In general, superelastic materials (such as adhesives) cause very large deformations of more than a few thousand percent at nominal strain. The stress-strain curve at that time rises into an S-shape due to superelasticity, and shows speed dependence due to viscosity. These viscous-superelastic behaviors are temperature dependent. The time-temperature conversion rule is generally established in viscoelastic materials, and it is known that time and temperature can be handled equivalently. That is, the temperature dependence can be expressed by converting the temperature change of the viscoelastic material into a time change. In this embodiment, the WLF (Williams-Landel-Ferry) formula (the following formula (7)) which is a typical time-temperature conversion rule is used.
Here, θ is the temperature of the elastic material, θ S is the reference temperature (20 ° C.), θ g is the glass transition temperature, and is a temperature specific to the material, and d 1 and d 2 are constants determined by experiments.

ところで、弾性材料の数値解析について、従来はLagrange型有限要素法が用いられてきた。しかし、Lagrange型有限要素法では超弾性材料（例えば粘着剤）の大変形挙動や剥離現象・破壊現象を取り扱うのは難しい。一方、Euler型有限要素法では、空間に固定されたメッシュを超えて物質が変形するため、任意の大変形解析が可能である。また、新たな物質境界面を容易に生成できるため、破壊解析に有利である。そこで、本実施形態では、粘着剤の粘性−超弾性に着目し、その挙動をEuler型有限要素法で解析した。   By the way, the Lagrange type finite element method has been used for numerical analysis of elastic materials. However, the Lagrange finite element method is difficult to handle the large deformation behavior, exfoliation phenomenon and fracture phenomenon of superelastic materials (for example, adhesive). On the other hand, in the Euler type finite element method, since the material deforms beyond the mesh fixed in the space, arbitrary large deformation analysis is possible. In addition, since a new material interface can be easily generated, it is advantageous for fracture analysis. Therefore, in this embodiment, paying attention to the viscosity-superelasticity of the adhesive, its behavior was analyzed by the Euler type finite element method.

分子レベルの微視的スケールは考慮せず、連続体として超弾性材料をモデル化すると、連続体の運動方程式は、Euler記述では以下の式（８）で表される。
ここで、ρは質量密度、v は速度ベクトル、σはCauchy応力テンソル、bは物体力ベクトル、を表す。 When a hyperelastic material is modeled as a continuum without considering the microscopic scale at the molecular level, the equation of motion of the continuum is expressed by the following formula (8) in the Euler description.
Here, ρ represents a mass density, v represents a velocity vector, σ represents a Cauchy stress tensor, and b represents an object force vector.

次に、Euler表示における固体変形の記述法を説明する。一般に固体力学において用いられるLagrange表示の場合、物質点の変位ベクトルから各種のひずみテンソルを求め、体変形を記述する。一方、Euler表示の場合、物質点を追跡しないため、変位ベクトルから固体変形を記述できない。そこで、速度ベクトル場から固体変形を記述できるように、左Cauchy-Green変形テンソルの定義式、すなわち、以下の式（９）
の両辺を物質時間微分することにより得られる以下の式（10）を導入する。
ここで、Ｌは速度勾配テンソル、を表す。すなわち、式（10）を用いることにより、速度ベクトル場から左Cauchy-Green変形テンソルを求めることができる。 Next, the description method of the solid deformation in the Euler display will be described. In the case of Lagrange display, which is generally used in solid mechanics, various strain tensors are obtained from the displacement vector of a material point, and body deformation is described. On the other hand, in the case of Euler display, the solid point cannot be described from the displacement vector because the substance point is not tracked. Therefore, in order to describe the solid deformation from the velocity vector field, the definition formula of the left Cauchy-Green deformation tensor, that is, the following equation (9)
The following formula (10), which is obtained by differentiating the two sides of the substance with respect to material time, is introduced.
Here, L represents a velocity gradient tensor. That is, by using Equation (10), the left Cauchy-Green deformation tensor can be obtained from the velocity vector field.

式（８）（運動方程式）と、式（10）（左Cauchy-Green変形テンソルの時間発展式）は、以下の式（11）のように一般化して表される。
ここで、φとｆは任意関数、ハット付きｖはメッシュ速度、を表す。すなわち、Euler表示の基礎方程式をALE(arbitrary Lagrangian-Eulerian) 表示に拡張しておく。本手法のEuler型解法では物質境界面に境界条件を付与できないため、物質境界面に境界条件を与える問題ではALE型解法を用いる。 Equation (8) (equation of motion) and equation (10) (time evolution equation of the left Cauchy-Green deformation tensor) are generalized as the following equation (11).
Here, φ and f represent arbitrary functions, and v with a hat represents a mesh speed. That is, the basic equation of Euler display is expanded to ALE (arbitrary Lagrangian-Eulerian) display. Since the Euler type solution of this method cannot give boundary conditions to the material boundary surface, the ALE type solution method is used for the problem that gives boundary conditions to the material boundary surface.

式（11）を時間方向に前進差分近似すれば、以下の式（12）となる。
If the equation (11) is approximated to the forward difference in the time direction, the following equation (12) is obtained.

本手法では，式（12）のOperator Split法により計算する。Operator Split法の概念を図２に示す。Operator Split法では、式（12）を以下の式（13）と式（14）のように二つの式に分離する。
式（13）は時間を進める非移流ステップであり、陽的有限要素法により離散化する。なお、右上付きの＊印は非移流ステップ後の値を意味する。式（14）は時間を止めた移流ステップである。速度の移流は中央差分法により計算し、左Cauchy-Green 変形テンソルは数値安定化のために２次精度風上差分法であるMUSCL法により計算する。界面捕捉法としてはVOF(Volume of Fluid)法を採用し、VOF関数の移流方程式もMUSCL法により計算する。 In this method, calculation is performed by the Operator Split method of Equation (12). The concept of the Operator Split method is shown in FIG. In the Operator Split method, the equation (12) is divided into two equations as in the following equations (13) and (14).
Equation (13) is a non-advection step that advances time and is discretized by an explicit finite element method. In addition, * mark with an upper right means the value after a non-advection step. Equation (14) is an advection step that stops time. The velocity advection is calculated by the central difference method, and the left Cauchy-Green deformation tensor is calculated by the MUSCL method, which is a second-order upwind method for numerical stabilization. The VOF (Volume of Fluid) method is adopted as the interface trapping method, and the advection equation of the VOF function is also calculated by the MUSCL method.

次に、式（１）〜（７）によって求められる応力及びひずみの相関式の妥当性を検証するため、温度及び引張速度の異なる試験条件において、アクリル系粘着剤の一軸引張試験シミュレーションを行った。図３（ｂ）は、温度２０℃の試験体を速度５ｍｍ／ｍｉｎ、５０ｍｍ／ｍｉｎ及び５００ｍｍ／ｍｉｎで一軸伸長させたときの応力−ひずみ曲線をそれぞれ示している。また、図４（ｂ）は、温度０℃、２０℃及び４０℃の試験体を５０ｍｍ／ｍｉｎで一軸伸長させたときの応力−ひずみ曲線をそれぞれ示している。   Next, in order to verify the validity of the correlation equation of stress and strain obtained by the equations (1) to (7), a uniaxial tensile test simulation of an acrylic adhesive was performed under test conditions with different temperatures and tensile speeds. . FIG. 3B shows stress-strain curves obtained when the test specimen at a temperature of 20 ° C. is uniaxially stretched at speeds of 5 mm / min, 50 mm / min, and 500 mm / min, respectively. Moreover, FIG.4 (b) has each shown the stress-strain curve when the test body of temperature 0 degreeC, 20 degreeC, and 40 degreeC is extended uniaxially at 50 mm / min.

式（５）の代わりに式（５）’を用い、温度２０℃の試験体を速度５ｍｍ／ｍｉｎ、５０ｍｍ／ｍｉｎ及び５００ｍｍ／ｍｉｎで一軸伸長させた場合（図３（ａ））の応力−ひずみ曲線と実測の値との相関性は、
であり、式（５）の代わりに式（５）’を用い、温度０℃、２０℃及び４０℃の試験体を５０ｍｍ／ｍｉｎで一軸伸長させた場合（図４（ａ））の応力−ひずみ曲線と実測の値との相関性は、
であり、どちらも相関性は必ずしも良くない。 Using formula (5) ′ instead of formula (5), the stress at the time when the specimen at a temperature of 20 ° C. was uniaxially stretched at speeds of 5 mm / min, 50 mm / min, and 500 mm / min (FIG. 3A) The correlation between the strain curve and the measured value is
Using the formula (5) ′ in place of the formula (5), the stress when the test specimens at temperatures of 0 ° C., 20 ° C. and 40 ° C. were uniaxially stretched at 50 mm / min (FIG. 4 (a)) The correlation between the strain curve and the measured value is
In both cases, the correlation is not always good.

しかし、式（１）〜（７）を用い、温度２０℃の試験体を速度５ｍｍ／ｍｉｎ、５０ｍｍ／ｍｉｎ及び５００ｍｍ／ｍｉｎで一軸伸長させた場合（図３（ｂ））の応力−ひずみ曲線と実測の値との相関性は、
であり、式（１）〜（７）を用い、温度０℃、２０℃及び４０℃の試験体を５０ｍｍ／ｍｉｎで一軸伸長させた場合（図４（ａ））の応力−ひずみ曲線と実測の値との相関性は、
であり、どちらも相関性は良い。これにより、粘性と超弾性とによる速度依存性が再現されており、試験体の温度の低下による応力値の上昇を再現できていることがわかる。 However, using the formulas (1) to (7), the stress-strain curve when the specimen at a temperature of 20 ° C. is uniaxially stretched at a speed of 5 mm / min, 50 mm / min, and 500 mm / min (FIG. 3B). And the correlation between the measured values and
Using the formulas (1) to (7), the stress-strain curve and actual measurement when the specimens at temperatures of 0 ° C., 20 ° C., and 40 ° C. were uniaxially stretched at 50 mm / min (FIG. 4A). The correlation with the value of
Both have good correlation. As a result, the speed dependency due to the viscosity and superelasticity is reproduced, and it can be seen that the increase in the stress value due to the decrease in the temperature of the specimen can be reproduced.

このように、本実施形態に係るモデルは、粘弾性−超弾性の物性を示す粘着剤（弾性材料）の変形を表現するモデルとして、妥当性があるモデルであることが確認できたわけである。そして、以上により、粘弾性については従来と同じようにして評価をし、超弾性については、今回提案する方法で評価をすることで、粘弾性と超弾性とを併せ持つ物性を示す弾性材料の大変形挙動をシミュレーション上で定量性よく表現することができるようになる。   Thus, it has been confirmed that the model according to the present embodiment is a valid model as a model expressing the deformation of an adhesive (elastic material) exhibiting viscoelastic-superelastic properties. As described above, viscoelasticity is evaluated in the same manner as in the past, and superelasticity is evaluated by the method proposed here, so that an elastic material having physical properties having both viscoelasticity and superelasticity is large. Deformation behavior can be expressed with high quantitativeness on simulation.

ここで、本実施形態に係るソフトウェアは、弾性要素及び粘性要素を直列に配置したＭａｘｗｅｌｌモデルに基づいて求められる応力及びひずみの相関式であって、弾性に関する項として、ひずみの不変量の高次項を導入したひずみエネルギー密度関数を含む、応力及びひずみの相関式を、応力−ひずみ曲線式として出力するためのプログラムであって、コンピュータを、応力及びひずみの相関式におけるパラメータを入力する入力手段、該入力されたパラメータによって応力及びひずみの相関式を特定する特定手段、及び、該特定された応力及びひずみの相関式を、応力−ひずみ曲線式として出力する出力手段、として機能させるものである。   Here, the software according to the present embodiment is a correlation equation of stress and strain obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and a high-order term of an invariant of strain as a term related to elasticity. A program for outputting a stress-strain correlation equation including a strain energy density function introduced as a stress-strain curve equation, and a computer for inputting parameters in the stress-strain correlation equation; A specifying means for specifying a correlation equation of stress and strain according to the input parameters, and an output means for outputting the specified correlation equation of stress and strain as a stress-strain curve equation.

本実施形態に係る装置は、コンピュータによって構成され、それぞれバスに接続されたＣＰＵ、ＲＯＭ、ワーキングメモリ、フレームメモリ、データ入出力装置、ハードディスク及びディスプレイを備える。ＲＯＭは、上記プログラムや各種パラメータを記憶し、ワーキングメモリは、ＣＰＵが制御を行うために必要なメモリであり、バッファやレジスタ等を含む。ＣＰＵは、ＲＯＭに記憶されたコンピュータプログラムに従って各種演算や処理を行う。データ入出力装置が上記入力手段、ＣＰＵ及びＲＯＭやメモリからなる制御部が上記特定手段を構成する。出力手段については、応力−ひずみ曲線式によって表現される応力−ひずみ曲線をディスプレイに表示させるならば、ディスプレイが出力手段を構成し、また、応力−ひずみ曲線式を外部に送信するならば、入出力装置が出力手段を構成し、また、応力−ひずみ曲線式を制御部内で保有しておくならば（それを用いるアプリケーションが制御部内にインストールされていて、当該パソコン内でアプリケーションを起動させるケースなど）、制御部自体が出力手段を構成する。   The device according to the present embodiment is configured by a computer, and includes a CPU, a ROM, a working memory, a frame memory, a data input / output device, a hard disk, and a display, each connected to a bus. The ROM stores the program and various parameters, and the working memory is a memory necessary for the CPU to perform control, and includes a buffer, a register, and the like. The CPU performs various calculations and processes according to the computer program stored in the ROM. A data input / output device is the input means, and a control unit comprising a CPU, ROM and memory constitutes the specifying means. For the output means, if the display displays the stress-strain curve expressed by the stress-strain curve formula, the display constitutes the output means, and if the stress-strain curve formula is transmitted to the outside, the input is entered. If the output device constitutes the output means and has the stress-strain curve equation in the control unit (such as a case where an application using the same is installed in the control unit and the application is activated in the personal computer) ), The control unit itself constitutes the output means.

また、粘弾性も一緒に評価する観点から、それに関するものもプログラミングされ、装置に保有させる（コンピュータにインストールする）ようにするのが好ましい。   Also, from the viewpoint of evaluating viscoelasticity together, it is preferable that the related one is programmed and held in the apparatus (installed in a computer).

なお、本発明は、上記実施形態に限定されるものではなく、本発明の要旨を逸脱しない範囲で種々の変更が可能である。   In addition, this invention is not limited to the said embodiment, A various change is possible in the range which does not deviate from the summary of this invention.

例えば、応力−ひずみ曲線式並びにそれの前提となる各種の相関式は、以上に挙げられたものに限定されるものではなく、その他の適切な近似式に基づいて各種の相関式及びそれに基づいて導出される応力−ひずみ曲線式であってもよい。   For example, the stress-strain curve equation and the various correlation equations that serve as the premise thereof are not limited to those listed above, but are based on various correlation equations based on other appropriate approximation equations. It may be a derived stress-strain curve equation.

本発明は、粘弾性と超弾性とを併せ持つ性質を示す高品質乃至高付加価値な弾性材料を設計するために利用することができる。   The present invention can be used to design a high-quality or high-value-added elastic material exhibiting properties having both viscoelasticity and superelasticity.

Claims (5)

弾性要素及び粘性要素を直列に配置したＭａｘｗｅｌｌモデルに基づいて求められる応力及びひずみの相関式であって、弾性に関する項として、ひずみの不変量の高次項を導入したひずみエネルギー密度関数を含む、応力及びひずみの相関式を、応力−ひずみ曲線式として出力するためのプログラムであって、
コンピュータを、
前記応力及びひずみの相関式におけるパラメータを入力する入力手段、
該入力されたパラメータによって前記応力及びひずみの相関式を特定する特定手段、
及び、該特定された応力及びひずみの相関式を、応力−ひずみ曲線式として出力する出力手段、
として機能させることを特徴とするプログラム。 A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And a program for outputting a correlation equation of a strain as a stress-strain curve equation,
Computer
Input means for inputting parameters in the correlation equation of stress and strain;
A specifying means for specifying a correlation equation of the stress and strain according to the input parameters;
And an output means for outputting the identified stress-strain correlation equation as a stress-strain curve equation,
A program characterized by functioning as 前記応力及びひずみの相関式は、Cauchy応力を太字σ、固体の圧力をps、２階の単位テンソルを太字Ｉ、体積変化率をＪ、左Cauchy-Green変形テンソルの第一低減不変量（ひずみの第一不変量）を上線付きIB、左Cauchy-Green変形テンソルの第二低減不変量（ひずみの第二不変量）を上線付きIIB、左Cauchy-Green変形テンソルの低減不変量を上線付きＢ、実験により定められる定数をｃ1，ｃ2，ｃ3、実験により定められる定数をｐ，ｑ（ｐ＞０，ｑ＞０）、時間増分をΔｔ、実験により定められる緩和時間をτα（τα＞０）、実験により定められる定数をβα∞（βα∞＞０）で表した場合、以下の式
で規定される請求項１に記載のプログラム。 The correlation between stress and strain is as follows: Cauchy stress is bold σ, solid pressure is p s , second-order unit tensor is bold I, volume change rate is J, first Cauchy-Green deformation tensor first invariant ( The first invariant of strain) is overlined I B , the second reduced invariant of the left Cauchy-Green deformation tensor (the second invariant of strain) is overlined II B , and the reduced invariant of the left Cauchy-Green deformation tensor B with an overline, c 1 , c 2 , c 3 constants determined by experiment, p, q (p> 0, q> 0) determined by experiment, Δt time increment, relaxation time determined by experiment When τα (τα> 0), a constant determined by experiment is expressed as βα∞ (βα∞> 0),
The program according to claim 1, defined in 前記応力及びひずみの相関式は、Cauchy応力を太字σ、固体の圧力をps、２階の単位テンソルを太字Ｉ、体積変化率をＪ、左Cauchy-Green変形テンソルの第一低減不変量（ひずみの第一不変量）を上線付きIB、左Cauchy-Green変形テンソルの第二低減不変量（ひずみの第二不変量）を上線付きIIB、左Cauchy-Green変形テンソルの低減不変量を上線付きＢ、実験により定められる定数をｃ1，ｃ2，ｃ3、実験により定められる定数をｐ，ｑ（ｐ＞０，ｑ＞０）、時間増分をΔｔ、実験により定められる緩和時間をτα（τα＞０）、実験により定められる定数をβα∞（βα∞＞０）、弾性材料の温度をθ、基準温度（２０℃）をθS、ガラス転移温度であって、材料固有の温度をθg、実験により定められる定数をｄ1，ｄ2で表した場合、以下の式
で規定される請求項１に記載のプログラム。 The correlation between stress and strain is as follows: Cauchy stress is bold σ, solid pressure is p s , second-order unit tensor is bold I, volume change rate is J, first Cauchy-Green deformation tensor first invariant ( The first invariant of strain) is overlined I B , the second reduced invariant of the left Cauchy-Green deformation tensor (the second invariant of strain) is overlined II B , and the reduced invariant of the left Cauchy-Green deformation tensor B with an overline, c 1 , c 2 , c 3 constants determined by experiment, p, q (p> 0, q> 0) determined by experiment, Δt time increment, relaxation time determined by experiment τα (τα> 0), the constant determined by experiment is βα∞ (βα∞> 0), the temperature of the elastic material is θ, the reference temperature (20 ° C.) is θ S , the glass transition temperature, and the temperature inherent to the material Is represented by θ g , and constants determined by experiments are represented by d 1 and d 2 ,
The program according to claim 1, defined in 弾性要素及び粘性要素を直列に配置したＭａｘｗｅｌｌモデルに基づいて求められる応力及びひずみの相関式であって、弾性に関する項として、ひずみの不変量の高次項を導入したひずみエネルギー密度関数を含む、応力及びひずみの相関式を、応力−ひずみ曲線式として出力する装置であって、
前記応力及びひずみの相関式におけるパラメータを入力する入力手段と、
該入力されたパラメータによって前記応力及びひずみの相関式を特定する特定手段と、
及び、該特定された応力及びひずみの相関式を、応力−ひずみ曲線式として出力する出力手段と
を含むことを特徴とする装置。 A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And a strain correlation equation as a stress-strain curve equation,
Input means for inputting parameters in the correlation equation of stress and strain;
A specifying means for specifying a correlation equation of the stress and strain according to the input parameters;
And an output means for outputting the specified correlation equation of stress and strain as a stress-strain curve equation. 弾性要素及び粘性要素を直列に配置したＭａｘｗｅｌｌモデルに基づいて求められる応力及びひずみの相関式であって、弾性に関する項として、ひずみの不変量の高次項を導入したひずみエネルギー密度関数を含む、応力及びひずみの相関式を用い、弾性材料の物性を評価することを特徴とする弾性材料の物性評価方法。   A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And evaluating the physical properties of the elastic material using a correlation equation of the strain and the strain.
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