JP2004037397A - Plain weave membrane material analytical system - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a highly-accurate plain weave membrane analytical system for removing restriction on a finite element or a boundary shape, subdividing the element only in a part requiring accuracy, and minimizing a calculation cost, concerning a thread not by a discrete model but by a pseudo continuum model. <P>SOLUTION: This plain weave membrane material analytical system has a cloth structure operation part 1 for determining a strain from deformation of a membrane material by a strain-deformation relation defined in the pseudo continuum model for showing the woven shape between the warp and the weft of a cloth by a waviness coefficient and showing the deformation by a continuous function, and operating a stress by a stress-strain relation from the strain, a coating material structure operation part 2 for determining the strain from the deformation of the membrane material by the strain-deformation relation of a coating material by the continuum and a finite deformation model, and determining the stress by the stress-strain relation from the strain, and a tangent rigidity matrix operation part 3 for performing finite element formulation of the cloth and the coating material by using each stress-strain relation and strain-deformation relation, and determining a rigidity matrix of the plain membrane material. <P>COPYRIGHT: (C)2004,JPO

Description

【００３３】
ここで、うねり高さをｈｉの変化は、有限変位の範囲であるのに対して、中心厚さｄｉの変化は微小ひずみの範囲なので、中心厚さｄｉを定数として扱う。
このとき、無次元パラメータμ（０≦μ≦１）を導入して、緯糸及び経糸の各うねり高さを、以下の（２）式で表す。
【数２】

以下では、上記μを空間に対する連続関数として構成されるうねり係数と定義し、うねり状態を代表するパラメータとして、織布を連続体と見なす（疑似連続体とする）変位ベクトルの１成分として扱う。
すなわち、上記擬似連続体モデルとは、織布を経糸及び緯糸の交差の周期と厚さとうねり高さとそのうねり状態を代表するパラメータであるうねり係数とで表現される擬似的な連続体モデルであり、うねり係数を空間に対する連続関数として定義して構成されている。
【００３４】
そして、織布構造演算部１は、このうねり係数μを用いて、擬似連続体モデルとして、以下のように、平織布のひずみ−変位関係を求める。
単位織構造の変形例を図５に示すが、糸相互の摩擦を無視しているので、糸はせん断変形を生じず、伸縮変形による軸方向の引張変形、及びずれ変形と伸展変形による面内横方向の圧縮変形のみを生じる。
【００３５】
このとき、軸引張ひずみをεＴｉとし、横圧縮ひずみをεＣｉとして表す。
ここで、ずれ変形（面内変形）による糸の面内回転を考慮すると、図５で示したように、緯糸及び経糸の回転角を各々ζ１，ζ２とし、ξ１−ξ２座標系で示した変位ベクトルの成分を、ｖ１，ｖ２とすれば、緯糸と経糸との相対的回転角ζは以下の（３）式により近似することができる。
【数３】

【００３６】
平織布の織り目が稠密であることから、糸の間隔変化が、糸の横圧縮変形に等しいとする。
このとき、伸展変形による糸のうねり波長の変化と、ずれ変形による糸の回転の影響とに基づき、緯糸の横圧縮ひずみεＣ１，及び経糸の横圧縮ひずみεＣ２とは、各々（４）式，（５）式で表される。
【数４】

【数５】

【００３７】
変形の前後において、単位織構造内でのうねりは直線で近似できるとし、その中心線長さをｓｉで表す。
この中心線長さｓｉは、図４の１点鎖線の長さであり、本モデルでは直線で近似されている。
このとき、ずれ変形及び伸展変形による糸の回転を考慮すると、緯糸の軸引張りひずみεＴ１，及び経糸の軸引張りひずみεＴ２とは、各々（６）式，（７）式で表される。
【数６】

【数７】

ただし、φｉ及びψｉは、各々以下の（８）式で定められる定数である。
【数８】

【００３８】
（４）式から（７）式までで定義された糸のひずみは、微小ひずみ・有限回転の仮定の下でのＧｒｅｅｎ−Ｌａｇｒａｎｇｅひずみに相当する。
また、織布構造演算部１において、（４）式から（７）式で求められるひずみから、糸の構成則を用いて、経糸及び緯糸の軸引張応力σＴｉと横圧縮応力σＣｉを求める。
ここで、後述する増分解析のため、応力−ひずみ関係も、以下の（９）式で表される速度形構成則を用いる。
【数９】

ただし、軸引張応力σＴｉと横圧縮応力σＣｉは、微小ひずみ・有限回転の仮定の下での第２Ｐｉｏｌａ−Ｋｉｒｃｈｈｏｆｆ応力に相当する。
【００３９】
また、糸が線形弾性体である場合には、上記（９）式は速度形ではない応力−ひずみ関係と一致する。
ここで、糸の速度形応力−ひずみマトリクス［Ｄ_ｉ］は、入力データに応じて、糸の材料に適したものを、データベース４より読み出してくる。
さらに、コーティング材構造演算部２において、ひずみ−変位関係として定義されている連続体・有限変形モデルは、サンドウィッチモデルにおけるコーティングフィルムの厚さをどちらもｂ_Ｍ／２とする。
【００４０】
このとき、対称積層であることから面外変形は考慮する必要がないため、両面のコーティングフィルムを厚さｂ_Ｍの１枚のフィルムとして面内変形に限定した定式化ができる。
そして、コーティング材のひずみ−変位関係には、大変形まで評価できるように（有限ひずみを考慮できる）、以下に示す（１０）式で表されるＧｒｅｅｎ−Ｌａｇｒａｎｇｅひずみを用いる。
【数１０】

ここで，ｘｉは直交デカルト座標であり、ｕｉは変位ベクトルのｘｉ方向成分を表す。
ただし、上記（１０）式は総和規約を用いている。
また、総和規約とは、数式中において、同じ項に同一の添字が現れた場合、その添字について１から２まで（ｘ１及びｘ２の２つ，または適宜定義されたベクトルの成分全て）の総和を取ることにするという、計算上の取り決めである。
【００４１】
また、（１０）式のＧｒｅｅｎ−Ｌａｇｒａｎｇｅひずみは、テンソルひずみであるので、せん断ひずみに関しては、以下に示す（１１）式の工学せん断ひずみに変換する。
【数１１】

そして、コーティング材構造演算部２において、（１０）式および（１１）式で定義されるひずみから、コーティング材の構成則を用いて、コーティング材の応力σ_ｉｊ ^Ｍを求める。
【００４２】
ここで、後述の増分解析のため、応力−ひずみ関係も、以下の（１２）式で表される速度形構成則を用いる。
【数１２】

上記（１２）式において、コーティング材の応力速度ベクトル（左辺）及びひずみ速度ベクトル（右辺）は、各々、以下の（１３）式及び（１４）式で各々定義されている。
【数１３】

【数１４】

ただし、コーティング材の応力は第２Ｐｉｏｌａ−Ｋｉｒｃｈｈｏｆｆ応力である。
【００４３】
また、コーティング材が線形弾性体である場合には、上記（１２）式は速度形ではない応力−ひずみ関係と一致する。
ここで、コーティング材の速度形応力−ひずみマトリクス［Ｄ_Ｍ］は、入力データに応じて、コーティング材の材料に適したものを、データベース４より読み出してくる。
ここで、接線剛性マトリクス演算部３において行われる、仮想仕事の原理に基づく有限要素定式化について、以下に述べる。
【００４４】
平織膜材料に対する仮想仕事の原理は、有限変形を考慮するため、接線剛性マトリクス演算部３において、以下に示す（１５）式で定義されている。
【数１５】

この上記（１５）式において、Ω_ｉはτｉの作用する経糸もしくは緯糸の初期体積であり、Ω_Ｍはコーティング材の初期体積であり、Ｌは外力のなす仕事である。
そして、有限変形を考慮する場合、幾何学的非線形性（ひずみ−変位関係の非線形性）を考慮するため、時刻ｔから時刻ｔ’までの増分解析を行わなければならない。
【００４５】
つまり、時刻ｔにおいては、応力やひずみ等の、全ての変数が求められているものとし、Δｔ経過後の時刻ｔ’における応力やひずみを計算する。
また、有限要素定式化のため、織布構造演算部１においては、織布に関して、応力ベクトル｛τ｝，ひずみベクトル｛γ｝，変位勾配ベクトル｛κ｝をそれぞれ、以下に示す（１６）式、（１７）式、（１８）式により定義されている。
【数１６】

【数１７】

【数１８】

【００４６】
以下、肩符Ｔでマトリクスの転置を表すこととする。
ここで、応力ベクトル｛τ｝とひずみベクトル｛γ｝の関係は、織布構造演算部１における上記（９）式の速度形構成則により、以下に示す（１９）式で関係付けられる。
【数１９】

上記（１９）式における、右辺でひずみ速度ベクトルに乗ぜられる応力−ひずみマトリクス［Ｄｘ］が以下に示す（２０）式で定められている。
【数２０】

【００４７】
変位勾配ベクトル｛κ｝が平織布の織り目に沿った座標系ξ１−ξ２で定義されているが、実際の解析は織り目に依存しない全体座標系ｘ１−ｘ２で行うため、以下に示す（２１）〜（２３）式に示す座標変換により変位勾配ベクトル｛κ｝を求める。
【数２１】

【数２２】

【数２３】

ここで、（ｕ１，ｕ２）は、ｘ１−ｘ２座標系での変位ベクトルであり、以下の有限要素定式化においては総和規約を用いる。
【００４８】
また、上記増分解析を行うため、（１６）式から（１８）式の表記を用い、（４）式から（７）式の平織布のひずみ−変位関係を、時刻ｔの近傍においてＴａｙｌｏｒ級数展開し、以下に示す（２４）式により二次近似する。
【数２４】

上記（２４）式において、左肩符ｔはその値の生じる時刻であり、Δ（　）は時刻ｔから時刻ｔ’までの増分を表し、^ｔＲ_ｉｊ及び^ｔＳ_ｉｊｋはそれぞれγｉのΔκｊに関する一次微分係数及び二次微分係数である。
【００４９】
一方、以下の（２５）式に示すように、糸の応力についても、増分分解を行う。
【数２５】

また、上記増分解析のため、コーティング材構造演算部３において、コーティング材のひずみ−変位関係式を、以下に示す（２６）式により増分表示する。（Ｇｒｅｅｎ−Ｌａｇｒａｎｇｅひずみは、もともと２次関数であるので、（２６）式は近似式ではない。）
【数２６】

ここで、上記（２６）式の右辺第２項と第３項の和がコーティング材のひずみの増分である。
【００５０】
さらに、以下の（２７）式に示すように、コーティング材の応力についても、増分分解を行う。
【数２７】

ここで、接線剛性マトリックス演算部３において、（２４）式から（２７）式までを、（１５）式に代入し、その線形化を行うと、以下に示す（２８）式の速度形仮想仕事式が得られる。
【数２８】

【００５１】
また、全体座標系ｘ１−ｘ２で表された変位（ｕ１，ｕ２）を、形状関数を導入して節点変位から内挿することにより、（２３）式の変位勾配ベクトルの速度を、節点変位速度ベクトルの関数として、以下の（２９）式により表す。
【数２９】

ここで、（２１）式および（２９）式を、（２８）式に代入すると、最終的に解くべき連立一次方程式が、以下に示す（３０）式として得られる。
【数３０】

ここで、［^ｔＫ_Ｆ］及び｛^ｔＱ_Ｆ｝はそれぞれ平織布の接線剛性マトリクス，節点内力ベクトルであり、［^ｔＫ_Ｍ］及び｛^ｔＱ_Ｍ｝はそれぞれコーティング材の接線剛性マトリクス，節点内力ベクトルであり、｛^ｔ’Ｆ｝は節点外力ベクトルであり、それぞれ以下に示す（３１）式から（３５）式で求められる。
【数３１】

【数３２】

【数３３】

【数３４】

【数３５】

【００５２】
この（３１）式において、［^ｔＫ_Ｌ ^Ｆ］は織布の初期変位マトリクスであり、以下に示す（３６）式により求められ、同様に、［^ｔＫ_ＮＬ ^Ｆ］は織布の初期応力マトリクスであり、（３７）式により求められる。
【数３６】

【数３７】

また、（３３）式において、［^ｔＫ_Ｌ ^Ｍ］はコーティング材の初期変位マトリクスであり、（３８）式により求められ、［^ｔＫ_ＮＬ ^Ｍ］はコーティング材の初期応力マトリクスであり、（３９）式により求められる。
【数３８】

【数３９】

【００５３】
また、（３５）式において、［Ｎ］は形状関数マトリクスであり、｛^ｔ’ｆ｝は入力データとして与えられる平織膜の境界Γσに作用する単位長さあたりの力である。
ここで、［^ｔＲ］は一次微分係数^ｔＲ_ｉｊを配列したマトリクス、［^ｔＳ’］は以下に示す（４０）式で求められるマトリクスであり、［Ｄ’ｘ］は以下に示す（４１）式で求められるマトリクスであり、｛^ｔτ’｝は以下に示す（４２）式で求められるベクトルであり、［^ｔＲ^Ｍ］は一次微分係数^ｔＲ_ｉｊ ^Ｍを配列したマトリクス、［^ｔＳ’^Ｍ］は以下に示す（４３）式で求められるマトリクスである。
【００５４】
【数４０】

【数４１】

【数４２】

【数４３】

上記（４０）式において、［^ｔＳ_ｉ］は二次微分係数^ｔＳ_ｉｊｋをｊ行ｋ列に配置したマトリクスであり、上記（４３）式において、［^ｔＳ_ｉ ^Ｍ］は二次微分係数^ｔＳ_ｉｊｋ ^Ｍをｊ行ｋ列に配置したマトリクスである。
【００５５】
以下、図６及び図７のフローチャート（Ａで互いに接続される）に従い、本発明の平織膜材料解析システムにおけるコーティング材がコーティングされた平織布、すなわち平織膜材の解析動作の説明を行う。
織布構造演算部１，コーティング材構造演算部２及び接線剛性マトリクス演算部３における演算過程における各数値及び式は、データベース４に一時的に格納されるものとする。
【００５６】
＜平織布及びコーティング材の双方の構成則に線形弾性体モデルを用いる場合＞ステップＳ１において、入力部５には、平織布及びコーティング材の解析に必要な材質及び解析精度、解析対象（すなわち平織膜材）の形状と織目の向き、有限要素メッシュデータ、境界条件（荷重を負荷する位置とその大きさ、固定する位置、強制的な変位を与える位置とその大きさ）などの条件が入力される。
次に、ステップＳ２において、織布構造演算部１は、繰り返し計算の各ステップにおける各有限要素の変形状態および応力状態に応じて、織布のひずみ−変位関係を代表するマトリクス［^ｔＲ］及び［^ｔＳ’］の算出を行う。
【００５７】
そして、ステップＳ３において、織布構造演算部１は、織布を構成する糸の材質に応じてた構成則を選択し、上記（９）式における応力−ひずみマトリクスをデータベース４から読み出す。
ここでは、糸は線形弾性体であるとしているが、糸は繊維の集合体としての不連続体であり，そのモデル化においても均質化の問題が生じる。
ただし、糸は基本的には向きの揃った繊維の束であるので、複雑な変形様式はなく、通常の連続体力学に基づく解析理論が適用できる。
【００５８】
本モデルでは、糸が直交異方性の線形弾性体としてモデル化できたものとする。
このとき、上記（９）式で示される糸の速度形構成則における、応力−ひずみマトリクスは、データベース４において、以下に示す（４４）式により定められている。
【数４４】

ただし、糸同士の不連続性から、横圧縮応力は負の値を持ち得ないため、（４４）式の応力−ひずみマトリクスを用いて評価した横圧縮応力が負である場合、織布構造演算部１は、上記（４４）式を、以下に示す単軸引張状態の（４５）式に置き換えて、次の繰り返し計算を行うこととする。
【数４５】

【００５９】
次に、ステップＳ４において、コーティング材構造演算部２は、繰り返し計算の各ステップにおける各有限要素の変形状態および応力状態に応じて、コーティング材のひずみ−変位関係を代表するマトリクス［_ｔＲ^Ｍ］及び［^ｔＳ^’Ｍ］の算出を行う。
次に、ステップＳ５において、コーティング材構造演算部２は、コーティング材の応力−ひずみ関係（構成則）を線形弾性体として解析する場合、上記（１２）式で示されるコーティング材の速度形構成則において、以下に示す（４６）式の応力−ひずみマトリクスを、データベース４から読み出す。
【数４６】

【００６０】
ここで、すでに述べたように、コーティング材は等方性線形弾性体であるとしており、ゴム系材料の場合、線形弾性体モデルは解析精度が悪いが、プラスチック系あるいはセラミックス系材料の場合には、線形弾性体モデルが適している。ただし、プラスチック系あるいはセラミックス系材料の場合でも、最終的な破断を評価したい場合には、弾塑性体モデルを使うことが望ましい。
しかし、線形弾性体は理論的取扱いが簡単であるので、実用上、ある程度の精度を犠牲にして線形弾性体を用いることが多く、その利用価値は高い。
【００６１】
次に、ステップＳ６において、接線剛性マトリクス演算部３は、接線剛性マトリクスの算出に必要な各種のマトリクスを、有限要素メッシュデータに応じて作成し、それらを織布構造演算部１およびコーティング材構造演算部２より読みこまれた［^ｔＲ］、［^ｔＳ’］、［Ｄ_ｉ］、［^ｔＲ^Ｍ］、［^ｔＳ^’Ｍ］、［Ｄ^Ｍ］と統合し、（３１）式及び（３３）式により接線剛性マトリックス［^ｔＫ_Ｆ］，［^ｔＫ_Ｍ］を作成する。
また、織布構造演算部１およびコーティング材構造演算部より読みこまれた現在の応力状態を用いて、（３２）式及び（３４）式により、各々節点内力ベクトル｛^ｔＱ_Ｆ｝，｛^ｔＱ_Ｍ｝を作成する。
【００６２】
次に、ステップＳ７において、外力、変位固定及び強制変位を考慮するための境界条件処理を行う。
ここで、接線剛性マトリクス演算部３は、入力データに基づき、付加された外力をいくつかの増分ステップに分割し、（３５）式により各増分に対する節点外力ベクトル｛^ｔ’Ｆ｝を作成する。
【００６３】
また、入力データにより変位固定の条件が与えられた節点変位については、ステップＳ６で求められた平織膜の接線剛性マトリクスにおいて、その節点変位に関する成分を削除する。
また、入力データにより強制変位が与えられた節点変位については、強制変位を外力と同様に増分ステップに分割し、平織膜の接線剛性マトリクス内の、その節点変位に関する成分と強制変位増分との積を既知量として、節点外力ベクトル｛^ｔ’Ｆ｝に加える。
【００６４】
そして、ステップＳ８において、接線剛性マトリクス演算部３は、ステップＳ６及びステップＳ７により作成された連立一次方程式を解くことにより、Ｎｅｗｔｏｎ−Ｒａｐｈｓｏｎ法に基づく反復計算を行い、各増分に対する解を求める。
次に、ステップＳ９において、接線剛性マトリクス演算部３は、ステップＳ８の結果から、増分解を現在値に足し合わせることにより、次のステップにおける節点変位及び節点うねり係数を算出する。
【００６５】
そして、ステップＳ１０において、織布構造演算部１は、上記節点変位及び節点うねり係数に基づき、（４）〜（７）式により演算を行い、平織布における経糸及び緯糸のひずみ｛^ｔ’γ｝を算出する。
次に、ステップＳ１１において、織布構造演算部１は、ステップＳ１０で求められたひずみ｛^ｔ’γ｝（後述の非線形材料モデルであれば、ひずみ速度）に基づき、（９）式により演算を行い、糸の応力｛^ｔ’τ｝（後述の非線形材料モデルであれば、応力速度）を算出する。
【００６６】
そして、ステップＳ１２において、コーティング材構造演算部２は、上記節点変位及び節点うねり係数に基づき、（１０）式および（１１）式により演算を行い、コーティング材のひずみを算出する。
次に、ステップＳ１３において、コーティング材構造演算部２は、ステップＳ１２で求められたコーティング材のひずみ（後述の非線形材料モデルであれば、ひずみ速度）に基づき、（１２）式により演算を行い、コーティング材の応力（後述の非線形材料モデルであれば、応力速度）を算出する。
そして、ステップＳ１４において、表示部６は、上述したように、織布構造演算部１及びコーティング材構造演算部２それぞれにおいて求められた、平織布及びコーティング材各々の応力をＣＲＴ等の表示装置に、各応力の数値を色の強度に変換した応力の分布図として表示させる。
【００６７】
＜平織布及びコーティング材の構成則として、それぞれ非線形弾性体モデル，超弾性体モデルを用いる場合＞
すでに述べた、図６及び図７のフローチャートにおいて、平織布及びコーティング材の双方を線形弾性体モデルとして説明した場合と同様の動作を行うステップの説明を省略し、異なった処理（演算式等の使用）のステップのみの説明を行う。
ステップＳ３において、織布構造演算部１は、高分子繊維への拡張のため、軸引張応力σＴｉと軸引張ひずみεＴｉとの関係、すなわち非線形弾性体モデルの構成則として、以下に示す（４７）式の非線形関数を、データベース４から読み出し、内部の記憶部に定義する。
【００６８】
すなわち、繊維の材料非線形性を簡単に考慮できるよう、横圧縮応力および横圧縮ひずみを無視する。
このとき、構成則は糸の軸方向の単軸応力状態の応力−ひずみ関係を表しており、軸引張応力σＴｉを次のような軸引張ひずみεＴｉに関する非線形関数（多項式近似で表される非線形弾性体としてモデル化）として表す。
【数４７】

ここで、すでに述べたように、上記軸引張応力σＴｉを示す多項式の次数Ｎとこの次数に対応して用いる係数ａｎは、材料定数であり、応力−ひずみ曲線がステップＳ１５において得られている実験結果と合うように、その数値を適時選定して、その値がステップＳ１６において、データベース４に格納されているとする。
【００６９】
そして、織布構造演算部１は、ステップＳ１の入力部において、入力データにより指定された材料に対応する構成則としての定数ａｎの各数値をデータベース４から読み出し、上記記憶部へ記憶させる。
そして、織布構造演算部１は、（４７）式を時刻ｔで線形化し、（４４）式の速度形応力−ひずみマトリクス［Ｄ_ｉ］として用いることにより、非線形弾性体の解析が行える。
【００７０】
次に、ステップＳ５において、コーティング材構造演算部２は、コーティング材（図２参照）の材料モデルとして超弾性体モデルを使用した場合の構成則の式、すなわち、以下に示す（４８）式を、データベース４から読み出し、内部の記憶部に定義する。
【数４８】

ただし、上記（４８）式において、Ｗはコーティング材のひずみエネルギーである。
ここで、コーティング材はＭｏｏｎｙ−Ｒｉｖｌｉｎ体（超弾性体の一種で最も一般的なモデル）としてモデル化できると仮定する。
【００７１】
このとき、ひずみエネルギＷは、以下の（４９）式で表される。
【数４９】

ここで、αとは、体積弾性率を意味するペナルティ係数で、実験とは別に適宜定める必要がある。
ただし、αの値には、ステップＳ１における入力データにより指定された値を用いる。
【００７２】
また、非圧縮性を仮定すれば（コーティング材が非圧縮性超弾性体とすると）、ひずみエネルギは以下の（５０）式により求められる。
【数５０】

これら（４９），（５０）式において、ｃ１及びｃ２は実験から得られる材料定数であり、Ｉｃ，ＩＩｃ及びＩＩＩｃはそれぞれ右Ｃａｕｃｈｙ−Ｇｒｅｅｎ変形テンソルの第１，第２及び第３主不変量である。
（５０）式の第２式は非圧縮性を表す制約条件である。
【００７３】
また、上記ｃ１及びｃ２の値は、ステップＳ１における入力データにより指定された材料に対応した値を、データベース４より読み出す。
（４８）式を更にε_ｋｌ ^Ｍで偏微分することにより、時刻ｔにおける線形化を行い、それを（３８）式の［Ｄ_Ｍ］に用いれば、超弾性体の解析を行うことができる。
ただし、非圧縮性を制約条件として入れる場合には、ステップＳ６において、接線剛性マトリックス演算部は、（１５）式の代わりに、Ｌａｇｒａｎｇｅ乗数法を用いて、以下の（５１）式に基づき有限要素定式化を行うことにより、非圧縮の制約条件を組み込む。
【数５１】

ここで、λはＬａｇｒａｎｇｅ乗数である。
【００７４】
そして、接線剛性マトリクス演算部３は、（５１）式を増分分解した後に線形化し、さらに、有限要素離散化を施すことにより、最終的に、以下に示す（５２）式の速度形釣合方程式を導く。
【数５２】

ここで、は節点Ｌａｇｒａｎｇｅ乗数の速度であり、｛^ｔＰ｝は非圧縮制約条件の誤差ベクトルであり、［^ｔＧ］は（５１）式の第１式左辺の第３項に対応する制約条件による接線剛性である。
【００７５】
ただし、ステップＳ６において、接線剛性マトリクス演算部３は、コーティング材が超弾性体であるとして解析する場合には、［^ｔＧ］及び｛^ｔＰ｝も求めて、解くべき連立一次方程式を決定する。
【００７６】
次に、ステップＳ８において、接線剛性マトリクス演算部３は、（５２）式に基づき、ステップＳ６及びステップＳ７により作成された連立一次方程式を解くことにより、Ｎｅｗｔｏｎ−Ｒａｐｈｓｏｎ法に基づく反復計算を行い、各増分に対する解を求める。
次に、ステップＳ９において、接線剛性マトリクス演算部３は、ステップＳ８の結果から、増分解（速度解）を現在値に足し合わせることにより、次のステップにおける節点変位及び節点うねり係数及び節点Ｌａｇｒａｎｇｅ乗数を算出する。
【００７７】
また、ステップＳ１１において、織布構造演算部１は、上記非線形弾性体モデルの線形化された速度形構成則を用いて、（１９）式によりひずみ速度から応力速度を算出し、それを現在の応力｛^ｔτ｝に足すことにより、次のステップの応力｛^ｔ’τ｝を求める。
また、ステップＳ１３において、コーティング材構造演算部２は、上記超弾性体モデルの線形化された速度形構成則を用いて、（１２）式によりひずみ速度ベクトルから応力速度ベクトルを算出し、（２７）式によりそれを現在の応力に足すことにより、次のステップの応力を求める。
以降は、平織布及びコーティング材の双方を線形弾性体モデルとして説明した場合と同様であるため、説明を省略する。
【００７８】
＜平織布及びコーティング材の構成則として、それぞれ非線形弾性体モデル，弾塑性体モデルを用いる場合＞
ＦＲＰ（繊維強化プラスチック）のように，コーティング材（母材）が硬質の樹脂である場合にも、本発明の平織膜材解析システムは適用でき、特に最終的な破断に至る過程での大変形の評価に適している。
このコーティング材がＦＲＰの場合、樹脂の微視的破壊過程を考慮して弾塑性体（等方性）としてのモデル化が有効であり、弾性域において、構成則は（４６）式と同様であり、これを［Ｄ_ｅ ^Ｍ］で表す。
【００７９】
しかし、Ｍｉｓｅｓ相当応力が降伏応力を超えた時点で塑性域に入り、構成則は、以下に示す（５３）式の速度形応力−ひずみマトリックスで表される。
【数５３】

ここで、塑性構成則は一般的なものなので、詳細な説明は省略する。
弾性構成則［Ｄ_ｅ ^Ｍ］及び塑性構成則［Ｄ_ｐ ^Ｍ］を降伏条件及び除荷判定に基づき適時選択して、選択された弾性構成則［Ｄ_ｅ ^Ｍ］及び塑性構成則［Ｄ_ｐ ^Ｍ］を、（３８）式の［Ｄ^Ｍ］に用いれば、弾塑性体の解析を行うことができる。
以降は、平織布に非線形弾性体モデル、コーティング材に超弾性体モデルを使用した場合と同様であるので、図６及び図７におけるフローチャートによる動作の説明は省略する。
【００８０】
本発明の平織膜材料解析システムは、平織布の糸の構成則を替えることにより、ガラス繊維や炭素繊維などの線形弾性体の解析だけでなく、それらの繊維の逐次破断による見かけの非弾性的挙動や、高分子系繊維の非線形挙動も解析可能となる。
このように、平織膜材料解析システムは、解析対象の糸の材料特性に対応させて、構成則の精度を上げることにより、より正確な変形挙動の把握および破断の予測、すなわち、強度評価が可能となる。
【００８１】
また、上述した本発明の平織膜材料解析システムは、解析対象の母材（コーティング材）の構成則を替えることにより、母材内の破壊を考慮した弾塑性体モデルや、高分子系の柔軟な材料に適した超弾性体モデルを適用でき、より正確な非線形挙動を解析できる。
このように、本発明の平織膜材料解析システムは、母材と糸の材料特性を正確に評価することができれば、最終的な破断（複合材料の終局強度）がより正確に評価できる。
【００８２】
次に、図を参照し、一実施形態の応用例を説明する。
例えば、平織布及びコーティング材の双方の構成則に線形弾性体モデルを用いる場合の計算例を以下に説明する。
ここで、織布にはガラス繊維平織物、一方、コーティング材には四フッ化エチレン（ＰＴＦＥ）を想定し、構造諸元を図８のテーブルに示す値とする。
そして、本発明の平織膜材解析システムにより、この平織膜材の単軸引張試験の解析を行う。
【００８３】
試験片形状は、図９（ａ）に示すものとし、図９（ｂ）に示すように有限要素分割する。
後に述べるように、試験片の四隅には応力が集中することが予想されるため、応力が集中する部分の解析精度を高めるため、四隅だけ細かく分割している。
有限要素には９節点アイソパラメトリック要素を用い、変位とうねり係数のどちらも２次で内挿することとする。
平織布における織目の向きはθ＝０（経糸方向引張）、π／２及びπ／４（緯糸方向引張）の３通りを解析する。
【００８４】
解析より得られた荷重と変位との関係を図１０に示す。横軸が変位の数値（単位ｍｍ）を示し、縦軸が試験片に負荷された引張荷重を試験片幅で除した値、つまり平均膜応力（単位Ｎ／ｍｍ）を示している。
緯糸または経糸方向の引張に関しては、定性的に文献に見られる実験結果と良好な一致を示している。
この荷重−変位曲線を実際の単軸引張試験の結果と比較することにより、糸のＹｏｕｎｇ率とＰｏｉｓｓｏｎ比とを修正すれば、以後、その平織膜材料を利用した構造の解析が非常に高精度に行える。
【００８５】
また、従来の解析法では、バイアス方向の引張を定量的に解析できなかったが、本モデルでは解析パラメータを変えるだけで容易に解析でき、膜材料実験との比較から定量的精度も保証することが可能である。
このように、本発明の平織膜材料解析システムは、任意の形状，任意の負荷条件に対して、平織膜材料の解析が高精度に行うシュミレーションを実行することができる。
【００８６】
以下では、従来の解析において最も解析が困難とされてきた、バイアス方向引張の結果について詳細に説明する。
図１１は荷重１００Ｎ／ｍｍのときのうねり係数の分布である。右横に、階調度とうねり係数の数値との対応関係を示す標識がある。
黒い部分は緯糸が伸展しているところを示し、白い部分は経糸が伸展しているところを示している。
試験片の四隅において、一方の糸が大きく伸展する不均一な変形状態となることが容易に見て確認できる。
このように、うねり係数を導入することにより、糸（緯糸及び経糸）のうねり状態を簡単かつ定量的に評価可能であることも、従来法に対する本システムの大きなメリットである。
【００８７】
図１２は荷重１００Ｎ／ｍｍのときの経糸軸引張応力の分布である。右横に、経糸軸引張応力σＴ２の数値の階調度を示す標識がある。
図１１のうねり係数分布から確認できたように、試験片の四隅では一方の糸がほぼ真直ぐになるまで伸展変形を生じており、その原因となる強い軸引張応力が、図１２の試験片の四隅に現れている。
このような、強い応力集中により、試験片は固定部の端から破断すると予測でき、糸の破断強度があらかじめわかっていれば、本解析の軸引張応力がその値を超えた時点で破断が開始すると予測できる。
このように、本発明の平織膜材料解析システムは、不均一な変形場においても膜材料の強度を定量的に評価できる、従来手法に対して高い解析精度を有していると言える。
【００８８】
図１３および図１４はそれぞれ荷重１００Ｎ／ｍｍのときのコーティング材のｘ方向引張応力σ^Ｍ _１１及びＭｉｓｅｓ相当応力σ^Ｍ _ｅｑの分布をそれぞれ示したグラフである。左横に、それぞれｘ方向引張応力及びＭｉｓｅｓ相当応力の数値の階調度を示す標識がある。
図１３より、試験片中央に大きな圧縮応力が作用していることが確認され、これがしわ発生の主な原因であると考えられる。
従来、織布に関して、しわ発生現象を定量的に評価できる解析モデルは無く、その発生メカニズムもあまりよくわかっていない。
しかし、本発明の平織膜材料解析システムは、単純な引張試験でも膜材に強い圧縮応力を生じることがわかり、この圧縮応力により平織膜材料は容易にしわを発生し得るということが理論的に説明できた。
【００８９】
図１４においては、コーティング材においても、糸の軸引張応力の分布と同様に試験片四隅に強い応力集中が現れる様子がわかる。
以上の結果からも、バイアス方向引張では非常に大きな変形を生じることが明らかであり、このときのコーティング材の応力・ひずみを評価するには、有限変形ひずみを用いなければならず、本発明の平織膜材料解析システムは、その点においても、従来法と異なり高精度で定量的な評価ができる。
【００９０】
上述した応用例では、平織布及びコーティング材双方を線形弾性体としてモデル化しているが、織布の擬似連続体モデルとコーティングの有限変形モデルの組合せを用いることにより、平織膜材料の定量的応力・変形評価が初めて可能となることが確認でき、その有効性が示されたと考える。
コーティング材の構成則を超弾性体などに変えることは、本発明の平織膜材料システムにおいて、材料に応じて解析モデルを高精度化することを意味する。
そのような高精度な構成則を用いれば、糸の破断を厳密に評価することができるため、平織膜材料の強度評価に対しても、高精度な解析精度を得ることができる。
【００９１】
また、上記した本発明の実施形態においては、平織膜材料解析システムの動作において実行される手順をプログラムとしてコンピュータ読取り可能な記録媒体に記録し、この記録媒体に記録されたプログラムをコンピュータシステムに読み込ませ、実行することにより本発明の利用者属性登録情報自動更新装置が実現されるものとする。ここでいうコンピュータシステムとは、ＯＳや周辺機器等のハードウエアを含むものである。
【００９２】
更に、「コンピュータシステム」は、ＷＷＷシステムを利用している場合であれば、ホームページ提供環境（あるいは表示環境）も含むものとする。
また、「コンピュータ読取り可能な記録媒体」とは、フレキシブルディスク、光磁気ディスク、ＲＯＭ、ＣＤ−ＲＯＭ等の可搬媒体、コンピュータシステムに内蔵されるハードディスク等の記憶装置のことをいう。さらに「コンピュータ読取り可能な記録媒体」とは、インターネット等のネットワークや電話回線等の通信回線を介してプログラムが送信された場合のシステムやクライアントとなるコンピュータシステム内部の揮発性メモリ（ＲＡＭ）のように、一定時間プログラムを保持しているものも含むものとする。
【００９３】
また、上記プログラムは、このプログラムを記憶装置等に格納したコンピュータシステムから、伝送媒体を介して、あるいは、伝送媒体中の伝送波により他のコンピュータシステムに伝送されてもよい。ここで、プログラムを伝送する「伝送媒体」は、インターネット等のネットワーク（通信網）や電話回線等の通信回線（通信線）のように情報を伝送する機能を有する媒体のことをいう。
また、上記プログラムは、前述した機能の一部を実現するためのものであっても良い。さらに、前述した機能をコンピュータシステムにすでに記録されているプログラムとの組み合わせで実現できるもの、いわゆる差分ファイル（差分プログラム）であっても良い。
【００９４】
以上、本発明の一実施形態を図面を参照して詳述してきたが、具体的な構成はこの実施形態に限られるものではなく、本発明の要旨を逸脱しない範囲の設計変更等があっても本発明に含まれる。
【００９５】
【発明の効果】
本発明の平織膜材料解析システムは、上述した平織布及びコーティング材の新しい解析モデル、すなわち平織布においては、ひずみ−変位関係に疑似連続体モデルを用い、構成則には線形弾性体及び非線形弾性体のモデルを用い、また、コーティング材においては、ひずみ−変位関係に連続体・有限変形のモデルを用い、構成則には線形弾性体，超弾性体及び弾塑性体モデルを用い、各々評価する材料及び解析の精度に対応して組合せて解析を行うことにより、従来の有限要素解析法が線形弾性体モデルのみしか使用しないため、解析できなかった平織膜材料の高い精度の解析を、従来の有限要素解析に比較してより低い計算コストで、かつ、より高精度に行うことが可能となる。
【図面の簡単な説明】
【図１】本発明の一実施形態による平織膜材料解析システムの構成を示すブロック図である。
【図２】図１に示す平織膜材料解析システムの解析対称である平織膜材料の構成を示す概念図である。
【図３】織布の織り目を示す座標系の概念図である。
【図４】図３における波線で囲まれた単位織構造を示す概念図である。
【図５】外力の印加による、図４に示す単位織構造の変形状態を示す概念図である。
【図６】図１の平織膜材料解析システムの動作例を説明するフローチャートである。
【図７】図１の平織膜材料解析システムの動作例を説明するフローチャートである。
【図８】織布にガラス繊維平織物を用い、コーティング材に四フッ化エチレンを用いた場合の各構造諸元の数値を示すテーブルである。
【図９】有限要素分割する試験片形状を示す概念図である。
【図１０】解析より得られた荷重と変位との関係を示すグラフである。
【図１１】荷重１００Ｎ／ｍｍのときのうねり係数の分布を示す分布図である。
【図１２】荷重１００Ｎ／ｍｍのときの経糸軸引張応力の分布を示す分布図である。
【図１３】荷重１００Ｎ／ｍｍのときのコーティング材のｘ方向引張応力の分布を示す分布図である。
【図１４】荷重１００Ｎ／ｍｍのときのコーティング材のＭｉｓｅｓ相当応力の分布を示す分布図である。
【符号の説明】
１　織布構造演算部
２　コーティング材構造演算部
３　接線剛性マトリクス演算部
４　データベース
５　入力部
６　表示部[0001]
TECHNICAL FIELD OF THE INVENTION
The present invention is an analysis technique for a plain-woven membrane material / structure, and is a plain-woven membrane material analysis system that can be applied to highly accurate evaluation of deformation characteristics and strength reliability of the application of the plain-woven membrane material.
[0002]
[Prior art]
In recent years, this plain woven membrane material has been applied to various fields by utilizing its flexibility in addition to the features of light weight and high strength.
In addition, it has already been used as a deployable air film structure for space structures and large space structures, and as a seismic reinforcement film for reinforced concrete columns supporting bridges, and its use in medical fields such as artificial skin and artificial blood vessels is being studied. is there.
[0003]
A number of analysis methods for this plain woven membrane material have been proposed, but the one with relatively high practicality as a structural analysis method is the analysis method by Minami et al. (Hirokazu Minami, Yoshio Nakahara, “Coating flatness applying finite element method”). Textile Analysis Method, Materials, Vol. 29, No. 324 (1980), pp. 916-921), and Nishikawa et al.'S analysis method (Kaoru Nishikawa, Kazuo Ishii, Tatsuya Kotake, "Stress of membrane structure considering woven fabric characteristics" -Deformation analysis method ", Membrane Structure Research Transactions '89 (1989), pp. 41-55).
[0004]
Although the basic concept of these analysis methods is almost the same, the formulation regarding microscopic yarn deformation is different.
However, the difference in consideration of the microscopic deformation is not so essential.
The difference between these analysis methods is the difference in the shape approximating the waviness of the warp and the weft.
Minami et al. Approximate the waviness of the yarn with a zigzag piecewise straight line, whereas Nishikawa et al. Propose a model that approximates the waviness of the yarn with a piecewise circular arc in addition to the zigzag shape.
Nishikawa et al. Also proposed a method in which the gap at the intersection of the warp and the weft is variable (compressible deformation possible).
[0005]
In other words, the coating material is a normal material that is isotropic and homogeneous, while the woven fabric has special material characteristics, and therefore, it is necessary to model the material according to each characteristic.
In the models of the above analysis methods, the warp and the weft are formulated by a truss structure that treats each of the warp and the weft as one rod element.
Then, a plane element representing a coating material is added to this truss model to make a model of a plain woven membrane material.
[0006]
[Problems to be solved by the invention]
However, the conventional analysis method described above has a disadvantage that only the rectangular elements along the warp and the weft can be used for the finite element method because the yarn itself must be geometrically represented as an analysis model. ing.
As a result, the conventional analysis method cannot perform a structural analysis of an actual plain woven membrane material having a complicated boundary shape.
Further, according to the conventional analysis method, even if several actual yarns are grouped and modeled with one rod element, the number of rod elements required for structural analysis of the plain woven membrane material becomes enormous, and the load of calculation by a computer or the like becomes large. Becomes very large, which is not practical in actual calculation processing.
[0007]
Furthermore, when analyzing the structure of a plain woven membrane material, it is usually necessary to assume that the material properties are uniform in the thickness direction in order to analyze as a planar member.
However, the plain woven film is a plain woven fabric having a three-dimensional microscopic structure in which yarns are intertwined, that is, a warp and a weft are intertwined, and a coating, which is a polymer-based flexible material coated on the plain woven fabric. And a composite material.
For this reason, for a plain weave film material, the calculation load can be significantly reduced if there is a methodology for homogenizing (averaging) a three-dimensional microscopic structure into a planar member.
[0008]
In the composite material analysis, homogenization is performed in the stress-strain relationship (constitutive law).
That is, in the plain woven cloth, a macroscopic material constant homogenized from various analysis results on the microscopic structure is obtained, and the macroscopic structure is analyzed using the material constant.
Therefore, the above-mentioned constitutive law of the macroscopic structure means the relationship between the average stress and the average strain at a certain point in the plain woven fabric.
[0009]
At this time, the constitutive law of the macroscopic structure is generally obtained under the assumption of a linear elastic body, and cannot be basically obtained as a nonlinear function.
In the case of plain woven membrane materials, this constitutive law has complex non-linearities, that is, geometrical non-linearities whose macroscopic constitutive laws depend on the state of microscopic deformation of the membrane material.
For this reason, it is difficult to apply the conventional composite material analysis method based on the linear elastic body to the plain woven membrane material.
Also, Minami et al. Introduced the three-dimensional truss model in order to overcome the problem of homogenization due to the stress-strain relationship, but analyzed the coating material using a continuum / small deformation model. Usually, the woven fabric causes a finite (large) deformation due to shear deformation or extension deformation, which is inconsistent as a model, and causes a large decrease in accuracy in analyzing the stress-strain of the woven fabric and the coating material. It will be.
[0010]
The present invention has been made under such a background, and instead of considering the yarn discretely as in the truss model, the woven shape of the warp and the weft is represented by a waviness coefficient, and the deformation of this woven fabric is described. By using a quasi-continuum model expressed as a continuous function for space, there are no restrictions on finite elements and boundary shapes, the size and shape of finite elements are arbitrary, and elements are refined only for those parts that require precision, and calculations are performed. An object of the present invention is to provide a high-precision plain woven membrane material analysis system capable of minimizing costs.
[0011]
[Means for Solving the Problems]
The plain woven membrane material analysis system of the present invention is a plain woven membrane material analysis system for analyzing stress and deformation of a plain woven membrane material, which is a composite material obtained by finishing a woven fabric with a coating material, and weaving a warp and a weft of the woven fabric. A quasi-continuum model expressing the shape by a waviness coefficient and expressing the deformation of the woven fabric as a continuous function with respect to space defines the strain-displacement relationship of the woven fabric. The woven fabric structure calculation unit that calculates the strain and calculates the yarn stress from the obtained strain based on the stress-strain relationship of the yarn, and the continuum and finite deformation model, uses the strain-displacement relationship of the coating material to calculate the film material. Calculate the coating material strain from the deformation and calculate the coating material stress from the obtained strain using the coating material stress-strain relationship. The woven fabric and the coating material are each formulated into a finite element based on the principle of virtual work using the respective stress-strain relationships and strain-displacement relationships. A stiffness matrix calculation unit for obtaining a stiffness matrix.
[0012]
In the plain woven membrane material analysis system according to the present invention, the quasi-continuous model is a model in which the woven fabric is expressed by a swell coefficient which is a parameter representative of a period of intersection of warp and weft, a thickness, a swell height, and a swell state. And a swell coefficient is configured as a continuous function with respect to space.
In the plain woven membrane material analysis system according to the present invention, the woven fabric structure calculation unit uses the stress-strain relationship of either a linear elastic body or a nonlinear elastic body model based on a material of the woven fabric, and the coating material The structure calculation unit calculates a stress based on the material of the coating material, using the stress-strain relationship of one of a linear elastic body, a superelastic body, and an elasto-plastic body model.
[0013]
The plain woven membrane material analysis system according to the present invention is characterized in that the coating material structure calculating section uses a Green-Lagrange strain in which the strain-displacement relationship of the coating material can consider a finite strain.
In the plain woven membrane material analysis system according to the present invention, the stress-strain relationship of the coating material is modeled as an elastoplastic body in the coating material structure calculation unit.
[0014]
The plain woven membrane material analysis system according to the present invention is characterized in that, in the coating material structure calculation unit, the stress-strain relationship of the coating material is modeled as a superelastic body (Moony-Rivlin body).
The plain woven membrane material analysis system according to the present invention is characterized in that, in the woven fabric structure calculation unit, the stress-strain relationship between the warp and the weft is modeled as a nonlinear elastic body by polynomial approximation.
[0015]
As described above, the conventional analysis method cannot basically analyze the stress / deformation of the plain weave membrane structure, and with a limited structure and under limited load conditions, Minami et al. Or Nishikawa et al. An analytical model has been proposed.
However, the analysis model has a drawback that it cannot be applied to an arbitrary structure and an arbitrary load condition, as described above.
[0016]
However, the quasi-continuum model in the present invention faithfully considers three typical modes of deformation of a plain woven fabric in a plain woven membrane, namely, shear deformation, extension deformation, and expansion / contraction deformation, and reduces the distortion of the yarn with respect to the space. By defining the strain of the yarn using a waviness coefficient and displacement that are continuous in space so that it can be obtained as a continuous function, it is possible to analyze any structure and load conditions.
[0017]
In particular, the plain woven membrane material analysis system of the present invention uses a quasi-continuum model to calculate the swell coefficient, which is a continuous function with respect to the space, representing the swell state of the warp and the weft of the plain woven cloth without performing a discrete operation. By using this method, the shear deformation and the extension deformation can be simultaneously and quantitatively evaluated, which greatly exceeds the conventional analysis method.
Further, the plain-woven membrane material analysis system of the present invention is a new analysis method that employs a quasi-continuous model as a model of a plain-woven fabric and also includes an analysis of a coating material necessary for actual plain-woven membrane material analysis.
[0018]
Here, the concept of performing the material analysis of the plain woven membrane by simultaneously considering the plain woven fabric and the coating material is the same as the analysis method of Minami et al.
However, in the analysis model of Minami et al., The plain woven fabric yarn (warp and weft) and the coating material are assumed to be linear elastic bodies, but in practice, polymer materials are used for both materials. In this case, the assumption of a linear elastic body does not hold.
Furthermore, in the analysis model of Minami et al., The strain of the coating material is obtained based on the assumption of minute deformation. However, since the actual plain woven membrane material undergoes large deformation, the analysis accuracy is extremely poor.
[0019]
On the other hand, in the present invention, in the quasi-continuum model, a new analysis model is extended so that the yarn material can be considered as a non-linear elastic body, and a continuum / finite Deformation (Green-Lagrange distortion) is used, and analysis corresponding to the deformation of the woven fabric is performed.
[0020]
Furthermore, in the present invention, as an extension of the analysis model, for the coating material, not only the linear elastic body model but also a superelastic body model (Moony-Rivlin body) or an elasto-plastic body model is adopted in the constitutive law. .
The plain woven membrane material analysis system of the present invention, by the combination of the new analysis model of the plain woven fabric and the coating material described above, the problem that could not be analyzed by the conventional analysis method, lower calculation cost compared to the conventional analysis method , And can be performed with higher accuracy.
[0021]
BEST MODE FOR CARRYING OUT THE INVENTION
First, the present invention separates a plain woven membrane material into a plain woven fabric and a coating material, and is newly described in detail later as an analytical model that takes into account the material nonlinearity and geometric nonlinearity of the plain woven fabric. This is an analysis method using a strain-displacement relationship homogenized by a quasi-continuum model and a constitutive rule of any model of not only a linear elastic body but also a nonlinear elastic body in correspondence with a material and an analysis accuracy.
This quasi-continuum model uses the strain-displacement relationship of the continuum as in the conventional analysis, and does not calculate the average strain from the displacement of the membrane material, but rather deforms the plain-woven structure in which the warp and weft are entangled. By formulating the style in detail, the strain of each yarn (warp and weft) is obtained directly from the displacement of the membrane material.
[0022]
With respect to the yarn, it is easy to find a constitutive law, and in many cases, the assumption of a linear elastic body can be used.
Because the geometric nonlinearity inherent in plain woven fabric is directly considered in the strain-displacement relationship, analysis of plain woven fabric can be easily performed by using conventional methods for other analysis procedures. It becomes.
In the following, this analytical model is called a quasi-continuum model, and this quasi-continuum model can easily evaluate the warp and weft strains, and calculates the stress concentration that occurs in plain woven fabric in the same way as the conventional finite element analysis. If the analysis can be performed at a cost and the strength of the yarn (that is, the strength of the fiber) is known, the strength of the plain woven fabric structure can be easily evaluated.
[0023]
In addition, most of the plain woven membrane materials actually used industrially have a coating material coated on a plain woven fabric, and therefore an analysis method is required to take this coating material into consideration.
Since the coating material undergoes a large deformation together with the woven fabric, the strain-displacement relationship of micro-deformation usually used is insufficient. In the analysis method of the present invention described below, the strain-displacement relationship of finite deformation ( Green-Lagrange strain) is employed.
[0024]
Further, as a constitutive law of the coating material, any one of a linear elastic body, a superelastic body, and an elasto-plastic body is selected and used according to the material and the analysis accuracy as in the case of the woven fabric.
Furthermore, in the above description, the constitution rule of the yarn is described as a linear elastic body. However, in many cases, a polymer fiber is used as an actual plain woven membrane material, and the linear elastic body model is sometimes insufficient. is there.
Therefore, the analysis method of the present invention described below extends the analysis of a plain woven fabric to a model of a nonlinear elastic body.
[0025]
Hereinafter, embodiments of the present invention will be described with reference to the drawings. FIG. 1 is a block diagram showing a configuration of a plain woven membrane material analysis system according to an embodiment of the present invention.
In this figure, a plain woven membrane material analysis system includes a woven fabric structure calculation unit 1 for calculating strain and stress based on a strain-displacement relationship and a constitutive law (stress-strain relationship) of a woven fabric portion, The coating material structure calculation unit 2 that calculates strain and stress based on the displacement relation and the constitutive law (stress-strain relation), and calculates the respective tangential stiffness matrices based on the strain-displacement relation / constitutive rule of the woven fabric and the coating material. And a tangential stiffness matrix calculation unit 3 for obtaining a tangential stiffness matrix of the plain woven membrane material as the composite material.
[0026]
Here, the plain woven membrane material to be analyzed in the present invention has a microscopic structure in which a coating material is equally coated (applied) on both sides of a plain woven cloth as shown in FIG. 2A. This precondition is to ignore the coupling effect of out-of-plane deformation and in-plane deformation under various stresses).
In the analysis method of the present invention, as shown in FIG. 2B, the microscopic structure is replaced with a sandwich structure separated into a plain woven cloth and a coating material having a uniform thickness. The structure calculation unit 1 and the coating material structure calculation unit 2 determine the strain-displacement relation and the constituent rule in an independent state.
[0027]
Returning to FIG. 1, the woven fabric structure calculation unit 1 calculates the strain and stress of the warp and the weft based on the strain-displacement relationship and the constitutive law of the plain woven fabric using a quasi-continuous model of the plain woven fabric.
At this time, the woven fabric structure calculation unit 1 reads out the strain-displacement relationship and the constituent rule stored in the database 4 in accordance with the material and the analysis accuracy of the plain woven fabric, and performs the calculation of strain and stress. Used.
[0028]
In the database 4, a linear elastic body model and a nonlinear elastic body model are stored as constitutive models in accordance with the material and the analysis accuracy of the plain woven fabric, and in correspondence with the material and the analysis accuracy of the coating material, A linear elastic body model, a superelastic body model, and an elasto-plastic body model are stored as models of the constitutive law.
Here, for the plain woven fabric, the constitutive law of the warp and the weft is extended to a nonlinear elastic body. However, the presence of the coating material allows the lateral compression deformation (lateral compression strain) of these yarns to be reduced by the coating material. It is much smaller than the case without. .
[0029]
As a result, in the analysis of the plain woven membrane material, it is often not necessary to consider the lateral compression strain, and in this analysis method, when using the constitutive law of the nonlinear elastic body, to simplify the constitutive law of the yarn, Although the accuracy is slightly lowered, the lateral compression strain can be ignored in the strain-displacement relationship in the woven fabric structure operation unit 1, and therefore, the constitution rule of the yarn may be considered only for the tensile deformation in the axial direction.
Here, the quasi-continuous model of the present invention defined in the woven fabric structure calculation unit 1 is a plain woven fabric in which the warp and the weft are orthogonally woven and the weave is uniform and dense. It is targeted.
The warp and the weft are assumed to be orthogonally anisotropic linear elastic bodies of the same material, each having a uniform cross section.
In the following description, the bending stiffness of the yarns (warp and weft) and the friction between the yarns are ignored only for the plane problem, but it is assumed that there is no slip at the intersection of the yarns (intersection).
[0030]
At this time, the deformation mode of the plain woven fabric can be classified into three types: shear deformation, extension deformation, and elastic deformation of the yarn itself.
Shear deformation (in-plane deformation of the yarn) and extension deformation (or crimp exchange, out-of-plane deformation of the yarn) are geometrical deformations of the yarn, meaning a finite rigid rotation, and elastic deformation is the axial direction of the yarn. Means the elastic elongation of
However, it is assumed that the strain of the yarn is minute.
[0031]
As shown in FIG. 3, the texture is inclined by θ from the entire coordinate system x1-x2, and the coordinate system along the texture is defined as ξ1-ξ2.
Here, the thin line in the figure indicates the direction of the yarn, and the # 1 axis is the axis along the weft, and the # 2 axis is the axis along the warp.
In the following description of the coefficients of the equations and the like, the lower right suffix is expressed as “1” for the value relating to the weft and “2” for the value relating to the warp.
[0032]
FIG. 4 shows a conceptual diagram of the unit woven structure surrounded by wavy lines in FIG.
In FIG. 4, the sectional area of the yarn is Ai, the center thickness is di, the swell wavelength is 2 pi, and the swell height (swell height) is hi.
At this time, there is a relationship of the following equation (1) between the center thickness di and the swell height hi.
(Equation 1)

[0033]
Here, the change in the swell height hi is in the range of finite displacement, whereas the change in the center thickness di is in the range of minute strain. Therefore, the center thickness di is treated as a constant.
At this time, the dimensionless parameter μ (0 ≦ μ ≦ 1) is introduced, and each swell height of the weft and the warp is expressed by the following equation (2).
(Equation 2)

In the following, μ is defined as a swell coefficient configured as a continuous function with respect to space, and is treated as one component of a displacement vector that regards a woven fabric as a continuum (a pseudo continuum) as a parameter representing the swell state.
That is, the pseudo continuum model is a pseudo continuum model in which the woven fabric is expressed by the period of intersection of the warp and the weft, the thickness, the swell height, and the swell coefficient which is a parameter representing the swell state. , The swell coefficient is defined as a continuous function with respect to space.
[0034]
Then, the woven fabric structure calculation unit 1 obtains the strain-displacement relationship of the plain woven fabric as a pseudo continuum model using the undulation coefficient μ as follows.
FIG. 5 shows a modified example of the unit woven structure. Since the friction between the yarns is ignored, the yarn does not undergo shear deformation, and is subjected to axial tensile deformation due to elastic deformation and in-plane due to shear deformation and extension deformation. Only lateral compressive deformation occurs.
[0035]
At this time, the axial tensile strain is represented by εTi, and the lateral compressive strain is represented by εCi.
Here, considering the in-plane rotation of the yarn due to the displacement deformation (in-plane deformation), as shown in FIG. 5, the rotation angles of the weft and the warp are respectively ζ1, ζ2, and the displacement shown in the ξ1-ξ2 coordinate system. Assuming that the components of the vector are v1 and v2, the relative rotation angle ζ between the weft and the warp can be approximated by the following equation (3).
(Equation 3)

[0036]
Since the weave of the plain woven fabric is dense, it is assumed that the change in the distance between the yarns is equal to the transverse compression deformation of the yarn.
At this time, based on the change in the waviness wavelength of the yarn due to the extension deformation and the influence of the rotation of the yarn due to the shear deformation, the lateral compressive strain εC1 of the weft and the lateral compressive strain εC2 of the warp are expressed by the equations (4) and (4), respectively. 5) It is expressed by the equation.
(Equation 4)

(Equation 5)

[0037]
Before and after the deformation, it is assumed that the undulation in the unit weave structure can be approximated by a straight line, and the center line length is represented by si.
The center line length si is the length of the dashed line in FIG. 4, and is approximated by a straight line in this model.
At this time, considering the rotation of the yarn due to the shear deformation and the extension deformation, the axial tensile strain εT1 of the weft and the axial tensile strain εT2 of the warp are expressed by the equations (6) and (7), respectively.
(Equation 6)

(Equation 7)

Here, φi and ψi are constants defined by the following equation (8).
(Equation 8)

[0038]
The strain of the yarn defined by the equations (4) to (7) corresponds to the Green-Lagrange strain under the assumption of small strain and finite rotation.
Further, in the woven fabric structure calculation unit 1, the axial tensile stress σTi and the transverse compressive stress σCi of the warp and the weft are obtained from the strains obtained from the expressions (4) to (7) using the yarn constitution rule.
Here, for the incremental analysis described later, the stress-strain relationship also uses the velocity form constitutive law expressed by the following equation (9).
(Equation 9)

However, the axial tensile stress σTi and the lateral compressive stress σCi correspond to the second Piola-Kirchhoff stress under the assumption of small strain and finite rotation.
[0039]
When the yarn is a linear elastic body, the above equation (9) is consistent with a stress-strain relationship that is not a velocity type.
Here, the velocity type stress-strain matrix [D_i] Reads out from the database 4 a material suitable for the material of the yarn according to the input data.
Further, in the coating material structure calculation unit 2, the continuum / finite deformation model defined as the strain-displacement relationship indicates that the thickness of the coating film in the sandwich model is b_M/ 2.
[0040]
At this time, it is not necessary to consider out-of-plane deformation due to the symmetrical lamination, so that the coating film on both sides is_MCan be formulated as a single film limited to in-plane deformation.
Then, for the strain-displacement relationship of the coating material, Green-Lagrange strain expressed by the following equation (10) is used so that large deformation can be evaluated (finite strain can be considered).
(Equation 10)

Here, xi is Cartesian Cartesian coordinates, and ui represents the xi component of the displacement vector.
However, the above equation (10) uses the sum rule.
Also, the sum rule is that, when the same subscript appears in the same term in a mathematical expression, the sum of 1 to 2 (two of x1 and x2 or all components of a vector defined as appropriate) for the subscript is defined as It's a computational agreement to take it.
[0041]
Since the Green-Lagrange strain of the equation (10) is a tensor strain, the shear strain is converted into an engineering shear strain of the following equation (11).
[Equation 11]

Then, the coating material structure calculation unit 2 calculates the stress σ of the coating material from the strain defined by the equations (10) and (11) using the constitutive rule of the coating material._ij ^MAsk for.
[0042]
Here, for the incremental analysis described later, the stress-strain relationship also uses the velocity form constitutive law expressed by the following equation (12).
(Equation 12)

In the above equation (12), the stress rate vector (left side) and strain rate vector (right side) of the coating material are defined by the following equations (13) and (14), respectively.
(Equation 13)

[Equation 14]

However, the stress of the coating material is the second Piola-Kirchhoff stress.
[0043]
When the coating material is a linear elastic body, the above equation (12) agrees with a stress-strain relationship that is not a velocity type.
Here, the velocity type stress-strain matrix [D_M] Reads out from the database 4 a material suitable for the material of the coating material according to the input data.
Here, the finite element formulation based on the principle of virtual work, which is performed in the tangent stiffness matrix calculation unit 3, will be described below.
[0044]
The principle of the virtual work for the plain woven film material is defined by the following equation (15) in the tangential stiffness matrix calculation unit 3 in order to consider the finite deformation.
(Equation 15)

In the above equation (15), Ω_iIs the initial volume of the warp or weft on which τi acts, and Ω_MIs the initial volume of the coating material, and L is the work performed by the external force.
When considering finite deformation, incremental analysis from time t to time t 'must be performed in order to take into account geometric non-linearity (non-linearity of the strain-displacement relationship).
[0045]
That is, at time t, all variables such as stress and strain are determined, and stress and strain at time t 'after elapse of At are calculated.
Further, in order to formulate the finite element, the woven fabric structure calculation unit 1 calculates a stress vector {τ}, a strain vector {γ}, and a displacement gradient vector {κ} with respect to the woven fabric using the following equation (16). , (17), and (18).
(Equation 16)

[Equation 17]

(Equation 18)

[0046]
Hereinafter, the transposition of the matrix is represented by a superscript T.
Here, the relationship between the stress vector {τ} and the strain vector {γ} is related by the following equation (19) according to the velocity form constitutive rule of the above equation (9) in the woven fabric structure calculation unit 1.
[Equation 19]

In the above equation (19), the stress-strain matrix [Dx] multiplied by the strain rate vector on the right side is determined by the following equation (20).
(Equation 20)

[0047]
Although the displacement gradient vector {κ} is defined in the coordinate system {1- {2} along the weave of the plain woven cloth, the actual analysis is performed in the overall coordinate system x1-x2 independent of the weave, so that (21) ) To (23), a displacement gradient vector {κ} is obtained.
(Equation 21)

(Equation 22)

[Equation 23]

Here, (u1, u2) is a displacement vector in the x1-x2 coordinate system, and a sum rule is used in the following finite element formulation.
[0048]
Further, in order to perform the above-described incremental analysis, the notation of the equations (16) to (18) is used, and the strain-displacement relationship of the plain woven cloth of the equations (4) to (7) is calculated by using the Taylor series near the time t. It is developed and secondarily approximated by the following equation (24).
(Equation 24)

In the above equation (24), the left t is the time at which the value occurs, Δ () represents the increment from time t to time t ′,^tR_ijas well as^tS_ijkAre the first and second derivatives of γi with respect to Δκj, respectively.
[0049]
On the other hand, as shown in the following expression (25), incremental decomposition is also performed on the yarn stress.
(Equation 25)

For the above-mentioned incremental analysis, the coating material structure calculating section 3 incrementally displays the strain-displacement relational expression of the coating material according to the following expression (26). (Since the Green-Lagrange distortion is originally a quadratic function, equation (26) is not an approximate equation.)
(Equation 26)

Here, the sum of the second and third terms on the right side of the above equation (26) is the increment of the strain of the coating material.
[0050]
Further, as shown in the following equation (27), the stress of the coating material is also subjected to incremental decomposition.
[Equation 27]

Here, in the tangent stiffness matrix calculation unit 3, by substituting the expressions (24) to (27) into the expression (15) and linearizing the expression, the speed type virtual work of the following expression (28) is obtained. An expression is obtained.
[Equation 28]

[0051]
Further, by interpolating the displacement (u1, u2) represented by the global coordinate system x1-x2 from the nodal displacement by introducing a shape function, the speed of the displacement gradient vector of the equation (23) is calculated as the nodal displacement speed. It is represented by the following equation (29) as a function of the vector.
(Equation 29)

Here, when the equations (21) and (29) are substituted into the equation (28), a simultaneous linear equation to be finally solved is obtained as the following equation (30).
[Equation 30]

here,[^tK_F]as well as{^tQ_F｝ Is the tangential stiffness matrix of the plain woven cloth and the nodal force vector, respectively.^tK_M]as well as{^tQ_M｝ Is the tangential stiffness matrix of the coating material and the nodal force vector, respectively.^{t '}F｝ is a nodal external force vector, which is obtained by the following equations (31) to (35), respectively.
(Equation 31)

(Equation 32)

[Equation 33]

(Equation 34)

(Equation 35)

[0052]
In this equation (31), [^tK_L ^F] Is an initial displacement matrix of the woven cloth, which is obtained by the following equation (36).^tK_NL ^F] Is an initial stress matrix of the woven fabric, which is obtained by Expression (37).
[Equation 36]

(37)

Also, in equation (33), [^tK_L ^M] Is an initial displacement matrix of the coating material, which is obtained by Expression (38).^tK_NL ^M] Is an initial stress matrix of the coating material, which is obtained by Expression (39).
[Equation 38]

[Equation 39]

[0053]
In equation (35), [N] is a shape function matrix, and ｛^{t '}f｝ is a force per unit length acting on the boundary Γσ of the plain weave film given as input data.
here,[^tR] is the first derivative^tR_ijMatrix,^tS ′] is a matrix obtained by the following equation (40), [D′ x] is a matrix obtained by the following equation (41), and ｛^tτ ′｝ is a vector obtained by the following equation (42),^tR^M] Is the first derivative^tR_ij ^MMatrix,^tS '^M] Is a matrix obtained by the following equation (43).
[0054]
(Equation 40)

(Equation 41)

(Equation 42)

[Equation 43]

In the above equation (40), [^tS_i] Is the second derivative^tS_ijkAre arranged in j rows and k columns, and in the above equation (43), [^tS_i ^M] Is the second derivative^tS_ijk ^MAre arranged in j rows and k columns.
[0055]
Hereinafter, the analysis operation of the plain woven fabric coated with the coating material, that is, the plain woven membrane material in the plain woven membrane material analysis system of the present invention will be described with reference to the flowcharts of FIGS.
Each numerical value and formula in the calculation process in the woven fabric structure calculation unit 1, the coating material structure calculation unit 2, and the tangential rigidity matrix calculation unit 3 are temporarily stored in the database 4.
[0056]
<In the case where a linear elastic body model is used for both the constitutive law of the plain woven fabric and the coating material> In step S1, the input unit 5 includes the material and analysis accuracy required for the analysis of the plain woven fabric and the coating material, and the analysis target ( In other words, conditions such as the shape and weave direction of the plain woven membrane material, the finite element mesh data, and the boundary conditions (the position where the load is applied and its size, the position where it is fixed, the position where the forced displacement is applied and its size) Is entered.
Next, in step S2, the woven fabric structure calculation unit 1 determines a matrix representing the strain-displacement relationship of the woven fabric according to the deformation state and stress state of each finite element in each step of the iterative calculation.^tR] and [^tS ′] is calculated.
[0057]
Then, in step S3, the woven fabric structure calculation unit 1 selects a constitutive rule according to the material of the yarns constituting the woven fabric, and reads the stress-strain matrix in the above equation (9) from the database 4.
Here, it is assumed that the yarn is a linear elastic body, but the yarn is a discontinuous body as an aggregate of fibers, and there is a problem of homogenization in modeling the yarn.
However, since the yarn is basically a bundle of oriented fibers, there is no complicated deformation mode, and an analysis theory based on ordinary continuum mechanics can be applied.
[0058]
In this model, it is assumed that the yarn can be modeled as an orthotropic linear elastic body.
At this time, the stress-strain matrix in the yarn velocity form constitutive law expressed by the above equation (9) is determined in the database 4 by the following equation (44).
[Equation 44]

However, since the transverse compressive stress cannot have a negative value due to the discontinuity between the yarns, when the transverse compressive stress evaluated using the stress-strain matrix of the equation (44) is negative, the woven fabric structure calculation is performed. The part 1 replaces the above equation (44) with the following equation (45) in a uniaxial tension state, and performs the following repeated calculation.
[Equation 45]

[0059]
Next, in step S4, the coating material structure calculation unit 2 calculates the matrix [representing the strain-displacement relationship of the coating material according to the deformation state and stress state of each finite element in each step of the iterative calculation._tR^M]as well as[^tS^'M] Is calculated.
Next, in step S5, when analyzing the stress-strain relationship (constitutive law) of the coating material as a linear elastic body, the coating material structure calculating unit 2 determines the velocity shape constitutive law of the coating material represented by the above equation (12). , The stress-strain matrix of the following equation (46) is read from the database 4.
[Equation 46]

[0060]
Here, as described above, the coating material is assumed to be an isotropic linear elastic body, and in the case of a rubber-based material, the linear elastic body model has poor analysis accuracy, but in the case of a plastic-based or ceramic-based material, A linear elastic body model is suitable. However, even in the case of plastic-based or ceramic-based materials, it is desirable to use an elasto-plastic body model when evaluating ultimate fracture.
However, a linear elastic body is easy to theoretically handle. Therefore, in practice, a linear elastic body is often used at the expense of some accuracy, and its utility value is high.
[0061]
Next, in step S6, the tangent stiffness matrix computing unit 3 creates various matrices necessary for calculating the tangent stiffness matrix in accordance with the finite element mesh data, and uses them in the woven fabric structure computing unit 1 and the coating material structure. Read from the arithmetic unit 2 [^tR], [^tS '], [D_i], [^tR^M], [^tS^'M], [D^MAnd the tangent stiffness matrix [Equation (31) and (33)]^tK_F], [^tK_M] Is created.
In addition, using the current stress state read from the woven fabric structure calculation unit 1 and the coating material structure calculation unit, the nodal force vector ｛is given by Expressions (32) and (34), respectively.^tQ_F｝ 、 ｛^tQ_MCreate｝.
[0062]
Next, in step S7, boundary condition processing for taking into account external force, fixed displacement and forced displacement is performed.
Here, the tangent stiffness matrix calculation unit 3 divides the added external force into several increment steps based on the input data, and calculates the nodal external force vector ｛for each increment by Expression (35).^{t '}Create F｝.
[0063]
In addition, for the nodal displacement for which the condition of the fixed displacement is given by the input data, the component relating to the nodal displacement is deleted from the tangential rigidity matrix of the plain weave film obtained in step S6.
In addition, for the nodal displacement given the forcible displacement by the input data, the forcible displacement is divided into increment steps in the same manner as the external force, and the product of the component relating to the nodal displacement and the forcible displacement increment in the tangential rigidity matrix of the plain weave membrane. As a known quantity, the nodal external force vector ｛^{t '}Add to F｝.
[0064]
Then, in step S8, the tangent stiffness matrix calculation unit 3 performs an iterative calculation based on the Newton-Raphson method by solving the simultaneous linear equations created in steps S6 and S7, and obtains a solution for each increment.
Next, in step S9, the tangent stiffness matrix calculation unit 3 calculates the nodal displacement and the nodal swell coefficient in the next step by adding the increased resolution to the current value from the result of step S8.
[0065]
Then, in step S10, the woven fabric structure calculation unit 1 performs calculations according to the equations (4) to (7) based on the node displacement and the node waviness coefficient to obtain the warp and weft distortions 平 of the plain woven fabric.^{t '}Calculate γ｝.
Next, in step S11, the woven fabric structure calculation unit 1 determines the strain ｛obtained in step S10.^{t '}Based on γ｝ (strain rate in the case of a non-linear material model described later), a calculation is performed by equation (9), and the yarn stress ｛^{t '}τ｝ (stress rate in the case of a nonlinear material model described later) is calculated.
[0066]
Then, in step S12, the coating material structure calculation unit 2 calculates the strain of the coating material by performing calculations according to the equations (10) and (11) based on the node displacement and the node undulation coefficient.
Next, in step S13, the coating material structure calculation unit 2 performs a calculation based on the equation (12) based on the strain of the coating material obtained in step S12 (for a nonlinear material model described later, the strain rate), The stress of the coating material (stress rate in the case of a nonlinear material model described later) is calculated.
Then, in step S14, as described above, the display unit 6 displays the stress of each of the plain woven fabric and the coating material obtained by the woven fabric structure calculation unit 1 and the coating material structure calculation unit 2, respectively, on a display device such as a CRT. Next, a numerical value of each stress is displayed as a distribution map of the stress converted into a color intensity.
[0067]
<When a nonlinear elastic model and a superelastic body model are used as the constitutive laws of the plain woven fabric and the coating material, respectively>
In the flowcharts of FIGS. 6 and 7 described above, the description of steps for performing the same operation as in the case where both the plain woven fabric and the coating material are described as the linear elastic body model is omitted, and different processing (operational formulas and the like) is performed. ) Will be described.
In step S3, the woven fabric structure calculation unit 1 shows the relationship between the axial tensile stress σTi and the axial tensile strain εTi, that is, the constitutive law of the nonlinear elastic body model, for expansion to the polymer fiber, as follows (47) The nonlinear function of the equation is read from the database 4 and defined in an internal storage unit.
[0068]
That is, the lateral compressive stress and the lateral compressive strain are ignored so that the material nonlinearity of the fiber can be easily considered.
At this time, the constitutive law represents the stress-strain relationship in the uniaxial stress state in the axial direction of the yarn, and the axial tensile stress σTi is converted into a nonlinear function (axial nonlinear approximation expressed by polynomial approximation) related to the axial tensile strain εTi as follows. (Modeled as a body).
[Equation 47]

Here, as described above, the order N of the polynomial representing the axial tensile stress σTi and the coefficient an used in accordance with this order are material constants, and an experiment in which the stress-strain curve was obtained in step S15. It is assumed that the numerical value is appropriately selected so as to match the result, and the value is stored in the database 4 in step S16.
[0069]
Then, the woven fabric structure calculation unit 1 reads each numerical value of the constant an as a constitutive law corresponding to the material specified by the input data from the database 4 and stores the numerical value in the storage unit in the input unit of step S1.
Then, the woven fabric structure calculation unit 1 linearizes Expression (47) at time t, and calculates the velocity-type stress-strain matrix [D_i], It is possible to analyze a nonlinear elastic body.
[0070]
Next, in step S5, the coating material structure calculation unit 2 calculates the expression of the constitutive law when the superelastic body model is used as the material model of the coating material (see FIG. 2), that is, the following expression (48). , And are defined in the internal storage unit.
[Equation 48]

Here, in the above equation (48), W is the strain energy of the coating material.
Here, it is assumed that the coating material can be modeled as a Moony-Rivlin body (one of the most common models of a superelastic body).
[0071]
At this time, the strain energy W is expressed by the following equation (49).
[Equation 49]

Here, α is a penalty coefficient meaning the bulk modulus, and needs to be appropriately determined separately from the experiment.
However, the value specified by the input data in step S1 is used as the value of α.
[0072]
Assuming incompressibility (assuming that the coating material is an incompressible superelastic body), the strain energy can be obtained by the following equation (50).
[Equation 50]

In these equations (49) and (50), c1 and c2 are material constants obtained from experiments, and Ic, IIc and IIIc are the first, second and third principal invariants of the right Cauchy-Green deformation tensor, respectively. is there.
The second expression of the expression (50) is a constraint condition indicating incompressibility.
[0073]
As the values of c1 and c2, values corresponding to the material specified by the input data in step S1 are read from the database 4.
Equation (48) is further converted to ε_kl ^M, A linearization at the time t is performed, and the result is expressed by [D_M], It is possible to analyze a superelastic body.
However, if incompressibility is to be entered as a constraint, in step S6, the tangent stiffness matrix calculation unit uses the Lagrange multiplier method instead of the expression (15) and uses a finite element based on the following expression (51). Formulating incorporates uncompressed constraints.
(Equation 51)

Here, λ is a Lagrange multiplier.
[0074]
Then, the tangent stiffness matrix calculation unit 3 performs linearization after incrementally decomposing the equation (51), and further performs finite element discretization to finally obtain the velocity type balancing equation of the following equation (52). Lead.
(Equation 52)

Here, is the speed of the node Lagrange multiplier, and ｛^tP｝ is the error vector of the uncompressed constraint,^tG] is the tangential stiffness due to the constraint corresponding to the third term on the left side of the first equation of the equation (51).
[0075]
However, in step S6, the tangential stiffness matrix calculation unit 3 analyzes the coating material as a superelastic body by selecting [^tG] and ｛^tP｝ is also determined to determine a simultaneous linear equation to be solved.
[0076]
Next, in step S8, the tangent stiffness matrix calculation unit 3 performs an iterative calculation based on the Newton-Raphson method by solving the simultaneous linear equations created in steps S6 and S7 based on the equation (52). Find the solution for each increment.
Next, in step S9, the tangent stiffness matrix calculation unit 3 adds the increased resolution (velocity solution) to the current value based on the result of step S8, so that the node displacement, the node swell coefficient, and the node Lagrange multiplier in the next step are obtained. Is calculated.
[0077]
Further, in step S11, the woven fabric structure calculation unit 1 calculates the stress rate from the strain rate by the equation (19) using the linearized velocity form constitutive rule of the nonlinear elastic body model, and calculates it as the current stress rate. stress{^tBy adding to τ｝, the stress of the next step ｛^{t '}Find τ｝.
Further, in step S13, the coating material structure calculation unit 2 calculates the stress rate vector from the strain rate vector by the equation (12) using the linearized velocity form constitutive rule of the superelastic body model, and (27) ) Equation is added to the current stress to find the stress for the next step.
The subsequent steps are the same as in the case where both the plain woven fabric and the coating material are described as the linear elastic body model, and a description thereof will be omitted.
[0078]
<When the nonlinear elastic model and the elasto-plastic model are used as the constitutive laws of the plain woven fabric and the coating material, respectively>
Even when the coating material (base material) is a hard resin, such as FRP (fiber reinforced plastic), the plain woven membrane material analysis system of the present invention can be applied, and particularly, large deformation in the process of finally breaking. Suitable for evaluation.
When this coating material is FRP, modeling as an elasto-plastic body (isotropic) is effective in consideration of the microscopic fracture process of the resin, and in the elastic range, the constitutive law is the same as in equation (46). Yes, this is [D_e ^M].
[0079]
However, when the Mises equivalent stress exceeds the yield stress, it enters the plastic region, and the constitutive law is expressed by a velocity type stress-strain matrix of the following equation (53).
[Equation 53]

Here, since the plastic constitutive law is general, detailed description is omitted.
Elastic constitutive law [D_e ^M] And the plastic constitutive law [D_p ^M] Is appropriately selected based on the yield condition and the unloading judgment, and the selected elastic constitutive law [D_e ^M] And the plastic constitutive law [D_p ^M] To [D] in equation (38).^M], An elasto-plastic body can be analyzed.
Subsequent steps are the same as when the non-linear elastic body model is used for the plain woven fabric and the super elastic body model is used for the coating material, and the description of the operations in the flowcharts in FIGS. 6 and 7 will be omitted.
[0080]
The plain woven membrane material analysis system of the present invention not only analyzes linear elastic bodies such as glass fibers and carbon fibers by changing the constitutional rules of the yarns of the plain woven fabric, but also apparent inelasticity due to the sequential breakage of those fibers. Behavior and non-linear behavior of polymer fibers can be analyzed.
In this way, the plain-woven membrane material analysis system enables more accurate grasp of deformation behavior and prediction of breakage, that is, strength evaluation, by improving the accuracy of the constitutive law in accordance with the material properties of the yarn to be analyzed. It becomes.
[0081]
Further, the above-described plain woven membrane material analysis system of the present invention can change the constitutive law of the base material (coating material) to be analyzed to provide an elasto-plastic body model in consideration of fracture in the base material or a polymer-based flexible material. A superelastic model suitable for various materials can be applied, and more accurate nonlinear behavior can be analyzed.
As described above, if the plain woven membrane material analysis system of the present invention can accurately evaluate the material properties of the base material and the yarn, the final break (the ultimate strength of the composite material) can be more accurately evaluated.
[0082]
Next, an application example of one embodiment will be described with reference to the drawings.
For example, a calculation example when a linear elastic body model is used for both the constitutive laws of the plain woven fabric and the coating material will be described below.
Here, a glass fiber plain fabric is used for the woven fabric, and ethylene tetrafluoride (PTFE) is used for the coating material, and the structural specifications are set to the values shown in the table of FIG.
And the analysis of the uniaxial tensile test of this plain woven membrane material is performed by the plain woven membrane material analysis system of the present invention.
[0083]
The test piece shape is as shown in FIG. 9 (a), and is divided into finite elements as shown in FIG. 9 (b).
As will be described later, it is expected that stress will be concentrated at the four corners of the test piece. Therefore, in order to improve the analysis accuracy of the portion where the stress is concentrated, the test piece is divided into only four corners.
A nine-node isoparametric element is used as the finite element, and both the displacement and the waviness coefficient are interpolated quadratic.
The orientation of the weave in the plain woven fabric is analyzed in three ways: θ = 0 (tensile in the warp direction), π / 2 and π / 4 (tensile in the weft direction).
[0084]
FIG. 10 shows the relationship between the load and the displacement obtained from the analysis. The horizontal axis shows the numerical value of displacement (unit: mm), and the vertical axis shows the value obtained by dividing the tensile load applied to the test piece by the width of the test piece, that is, the average film stress (unit: N / mm).
Regarding the tension in the weft or warp direction, it shows a good agreement qualitatively with the experimental results found in the literature.
By comparing this load-displacement curve with the results of the actual uniaxial tensile test, the Young's modulus and Poisson's ratio of the yarn can be corrected. Can be done.
[0085]
In addition, the conventional analysis method could not quantitatively analyze the tension in the bias direction, but this model can easily analyze the tension simply by changing the analysis parameters, and guarantees the quantitative accuracy by comparing with the film material experiment. Is possible.
As described above, the plain woven membrane material analysis system of the present invention can execute a simulation in which the analysis of the plain woven membrane material is performed with high accuracy for an arbitrary shape and an arbitrary load condition.
[0086]
In the following, a result of the bias direction tension, which has been most difficult to analyze in the conventional analysis, will be described in detail.
FIG. 11 shows the distribution of the undulation coefficient when the load is 100 N / mm. On the right side, there is a sign indicating the correspondence between the gradient and the value of the swell coefficient.
The black part indicates where the weft is extending, and the white part indicates where the warp is extending.
At the four corners of the test piece, it can be easily seen and confirmed that one of the yarns is in a non-uniform deformation state in which the yarn is greatly extended.
As described above, the introduction of the waviness coefficient makes it possible to easily and quantitatively evaluate the waviness state of the yarn (weft and warp), which is a great advantage of the present system over the conventional method.
[0087]
FIG. 12 shows the distribution of the warp shaft tensile stress when the load is 100 N / mm. On the right side, there is a sign indicating the gradation of the numerical value of the warp axis tensile stress σT2.
As can be confirmed from the waviness coefficient distribution in FIG. 11, the four corners of the test piece undergo extension deformation until one of the yarns becomes substantially straight, and the strong axial tensile stress that causes the deformation is shown in FIG. Appears in the four corners.
Due to such strong stress concentration, the test piece can be predicted to break from the end of the fixing part, and if the breaking strength of the yarn is known in advance, the break starts when the axial tensile stress of this analysis exceeds that value. Then you can predict.
Thus, it can be said that the plain-woven membrane material analysis system of the present invention has higher analysis accuracy than the conventional method, which can quantitatively evaluate the strength of the membrane material even in an uneven deformation field.
[0088]
13 and 14 show tensile stress σ in the x direction of the coating material at a load of 100 N / mm, respectively.^M ₁₁And Mises equivalent stress σ^M _eqIs a graph showing the distribution of each. On the left side, there are signs indicating the gradation of numerical values of the tensile stress in the x direction and the Mises equivalent stress, respectively.
From FIG. 13, it was confirmed that a large compressive stress was acting at the center of the test piece, and this is considered to be the main cause of wrinkles.
Conventionally, there is no analytical model capable of quantitatively evaluating the wrinkle generation phenomenon of a woven fabric, and its generation mechanism is not well understood.
However, it has been found that the plain-woven membrane material analysis system of the present invention generates a strong compressive stress even in a simple tensile test, and it is theoretically possible that the plain-woven membrane material can easily generate wrinkles due to the compressive stress. I was able to explain.
[0089]
In FIG. 14, it can be seen that strong stress concentrations appear at the four corners of the test piece as well as the distribution of the axial tensile stress of the yarn in the coating material.
From the above results, it is clear that very large deformation occurs in the bias direction tension, and in order to evaluate the stress and strain of the coating material at this time, finite deformation strain must be used. In this respect, the plain woven membrane material analysis system can also perform high-precision and quantitative evaluation unlike the conventional method.
[0090]
In the application example described above, both the plain woven fabric and the coating material are modeled as linear elastic bodies, but by using a combination of a quasi-continuous model of the woven fabric and a finite deformation model of the coating, the quantitative analysis of the plain woven membrane material is performed. It was confirmed that stress / deformation evaluation became possible for the first time, and it is considered that its effectiveness was demonstrated.
Changing the constitutive law of the coating material to a superelastic body or the like means improving the accuracy of the analysis model according to the material in the plain woven membrane material system of the present invention.
If such a high-precision constitutive rule is used, the breakage of the yarn can be strictly evaluated, so that a high-precision analysis accuracy can be obtained even for the strength evaluation of the plain woven membrane material.
[0091]
In the above-described embodiment of the present invention, the procedure executed in the operation of the plain weave membrane material analysis system is recorded as a program on a computer-readable recording medium, and the program recorded on the recording medium is read into the computer system. Then, by executing, the user attribute registration information automatic updating device of the present invention is realized. The computer system here includes an OS and hardware such as peripheral devices.
[0092]
Further, the "computer system" includes a homepage providing environment (or display environment) if a WWW system is used.
The “computer-readable recording medium” refers to a portable medium such as a flexible disk, a magneto-optical disk, a ROM, a CD-ROM, and a storage device such as a hard disk built in a computer system. Further, the “computer-readable recording medium” refers to a volatile memory (RAM) inside a computer system which is a system or a client when a program is transmitted via a network such as the Internet or a communication line such as a telephone line. In addition, programs that hold programs for a certain period of time are also included.
[0093]
Further, the above program may be transmitted from a computer system storing the program in a storage device or the like to another computer system via a transmission medium or by a transmission wave in the transmission medium. Here, the “transmission medium” for transmitting a program refers to a medium having a function of transmitting information, such as a network (communication network) such as the Internet or a communication line (communication line) such as a telephone line.
Further, the program may be for realizing a part of the functions described above. Furthermore, what can implement | achieve the function mentioned above in combination with the program already recorded on the computer system, and what is called a difference file (difference program) may be sufficient.
[0094]
As described above, one embodiment of the present invention has been described in detail with reference to the drawings. However, the specific configuration is not limited to this embodiment, and there are design changes and the like within a range not departing from the gist of the present invention. Are also included in the present invention.
[0095]
【The invention's effect】
The plain woven membrane material analysis system of the present invention uses a new analytic model of the plain woven fabric and the coating material described above, that is, in the plain woven fabric, a pseudo-continuum model is used for the strain-displacement relationship. A model of a non-linear elastic body is used. For a coating material, a continuum / finite deformation model is used for the strain-displacement relation. By performing the analysis in combination in accordance with the material to be evaluated and the accuracy of the analysis, the conventional finite element analysis method uses only the linear elastic body model, so high-precision analysis of the plain weave membrane material that could not be analyzed, Compared to the conventional finite element analysis, it can be performed at lower calculation cost and with higher accuracy.
[Brief description of the drawings]
FIG. 1 is a block diagram showing a configuration of a plain woven membrane material analysis system according to an embodiment of the present invention.
FIG. 2 is a conceptual diagram showing a configuration of a plain woven membrane material which is analysis symmetrical of the plain woven membrane material analysis system shown in FIG.
FIG. 3 is a conceptual diagram of a coordinate system showing a texture of a woven fabric.
FIG. 4 is a conceptual diagram showing a unit woven structure surrounded by wavy lines in FIG.
FIG. 5 is a conceptual diagram showing a deformed state of the unit woven structure shown in FIG. 4 due to application of an external force.
FIG. 6 is a flowchart illustrating an operation example of the plain weave membrane material analysis system in FIG. 1;
FIG. 7 is a flowchart illustrating an operation example of the plain weave membrane material analysis system in FIG. 1;
FIG. 8 is a table showing numerical values of respective structural specifications when a glass fiber plain woven fabric is used for a woven fabric and ethylene tetrafluoride is used for a coating material.
FIG. 9 is a conceptual diagram showing a test piece shape to be divided into finite elements.
FIG. 10 is a graph showing the relationship between load and displacement obtained from the analysis.
FIG. 11 is a distribution diagram showing a distribution of a waviness coefficient at a load of 100 N / mm.
FIG. 12 is a distribution diagram showing a distribution of a warp shaft tensile stress at a load of 100 N / mm.
FIG. 13 is a distribution diagram showing the distribution of tensile stress in the x direction of the coating material when the load is 100 N / mm.
FIG. 14 is a distribution diagram showing a distribution of Mises equivalent stress of a coating material at a load of 100 N / mm.
[Explanation of symbols]
1 Woven structure calculation unit
2 Coating material structure calculation section
3 Tangential stiffness matrix calculator
4 database
5 Input section
6 Display

Claims (7)

It is a plain woven membrane material analysis system that performs stress and deformation analysis of a plain woven membrane material, which is a composite material obtained by finishing a woven fabric with a coating material.
The woven shape of the warp and the weft of the woven fabric is represented by a waviness coefficient, and the strain-displacement relationship of the woven fabric is defined by a quasi-continuous model expressing the deformation of the woven fabric as a continuous function with respect to space. A woven fabric calculation unit that calculates the yarn strain from the deformation of the membrane material based on the displacement relationship and calculates the yarn stress from the obtained strain based on the stress-strain relationship of the yarn;
Using the continuum and the finite deformation model, the strain of the coating material is obtained from the deformation of the film material using the strain-displacement relationship of the coating material, and the stress of the coating material is obtained from the obtained strain using the stress-strain relationship of the coating material. A coating material structure calculating unit for calculating,
The woven fabric and the coating material are each formulated into a finite element based on the principle of virtual work using the respective stress-strain relations and strain-displacement relations. And a stiffness matrix calculation unit for obtaining the following equation. The pseudo continuum model is a continuum expressed by a undulation coefficient, which is a parameter representing the period and thickness of the intersection of the warp and weft, the swell height, and the swell state, and the swell coefficient is a continuous function with respect to space. The plain woven membrane material analysis system according to claim 1, wherein the system is configured as: The woven fabric structure calculation unit uses the stress-strain relationship of either a linear elastic body or a non-linear elastic body model based on the material of the woven fabric, and the coating material structure calculation unit determines the material of the coating material. 3. The plain woven fabric according to claim 1, wherein the stress is calculated using the stress-strain relationship of any of a linear elastic body, a superelastic body, and an elastoplastic body model. Material analysis system. The plain-woven membrane according to any one of claims 1 to 3, wherein in the coating material structure calculation unit, the strain-displacement relationship of the coating material uses a Green-Lagrange strain capable of considering a finite strain. Material analysis system. The stress-strain relationship of the coating material is modeled as a superelastic body (Moony-Rivlin body) in the coating material structure calculation unit, according to any one of claims 1 to 3, wherein Plain weave material analysis system. 4. The plain woven membrane analysis system according to claim 1, wherein the stress-strain relationship of the coating material is modeled as an elasto-plastic body in the coating material structure calculation unit. The plain woven membrane according to any one of claims 1 to 6, wherein the stress-strain relationship between the warp and the weft is modeled as a nonlinear elastic body by polynomial approximation in the woven fabric structure calculation unit. Material analysis system.

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