JP3855075B2 - Plain weave membrane material analysis system - Google Patents

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]
BACKGROUND OF THE INVENTION
The present invention is an analysis technique for plain weave membrane materials and structures, and is a plain weave membrane material analysis system that can be applied to highly accurate evaluation of deformation characteristics and strength reliability with respect to application of plain weave membrane materials.
[0002]
[Prior art]
In recent years, this plain woven membrane material has been applied to various fields by taking advantage of its flexibility in addition to the characteristics of light weight and high strength.
In addition, it has already been used as a deployable air membrane structure such as space structure and large space structure, and as an anti-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 methods for analyzing plain weave membrane materials have been proposed, but those with relatively high practicality as structural analysis methods include the analysis method by Minami et al. (Minami Hirokazu, Nakahara Yoshio, “Coating Textile analysis method, Materials, Vol. 29, No. 324 (1980), pp.916-921) and Nishikawa et al. (Akira Nishikawa, Kazuo Ishii, Tatsuya Kotake, “Stress of membrane structure considering fabric characteristics” "Deformation analysis method", Membrane Structure Research Papers '89 (1989), pp. 41-55).
[0004]
The basic concepts of these analysis methods are almost the same, but the formulation regarding microscopic yarn deformation is different.
However, the difference in consideration of this microscopic deformation is not very essential.
The difference between these analysis methods is the difference in the shape that approximates the geometrical shape of the warp and weft waviness.
Minami et al. Have approximated the yarn undulation with a zigzag piecewise straight line, while Nishikawa et al. Proposed a model that approximated the yarn undulation 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).
[0005]
In other words, the coating material is an isotropic / homogeneous ordinary material, while the woven fabric has special material properties, and therefore modeling that is appropriate for each property is required.
The models of the above analysis methods are formulated by a truss structure that handles each warp and weft as one bar element.
Then, by adding a plane element representing the coating material to the truss model, a plain weave film material model is obtained.
[0006]
[Problems to be solved by the invention]
However, the above-described conventional analysis method has a drawback 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 expressed as an analysis model. ing.
As a result, the conventional analysis method cannot perform a structural analysis of an actual plain weave film material having a complicated boundary shape.
In addition, the conventional analysis method, even if several actual yarns are combined and modeled with one bar element, the number of bar elements required for the structural analysis of the plain woven membrane material becomes enormous, and the calculation load by a computer etc. Becomes very large and is not realistic in terms of actual calculation processing.
[0007]
Furthermore, when analyzing the structure of a plain woven membrane material, it is necessary to assume that the material characteristics are uniform in the thickness direction in order to analyze as a planar member.
However, the plain woven membrane has a three-dimensional microscopic structure in which yarns are intertwined, that is, warps and wefts are intertwined, and a coating that is a polymer-based flexible material coated on the plain weave fabric. It is a composite material composed of materials.
For this reason, for plain woven membrane materials, if there is a methodology for homogenizing (averaging) a three-dimensional microscopic structure into a planar member, the calculation load can be greatly reduced.
[0008]
In the composite material analysis, homogenization is performed in the stress-strain relationship (constitutive law).
That is, in a plain woven fabric, a homogenized macroscopic material constant is obtained from various analysis results for the microscopic structure, 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 an arbitrary 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 basically cannot be obtained as a nonlinear function.
In the case of plain weave membrane materials, this constitutive law has a complex non-linearity, that is, a geometric non-linearity in which this macro constitutive law depends on the microscopic deformation state of the membrane material.
For this reason, the conventional composite material analysis method based on a linear elastic body is difficult to apply to plain woven membrane materials.
Minami et al. Introduced the 3D truss model to overcome the above-mentioned homogenization problem due to the stress-strain relationship. However, the coating material is analyzed using a continuum / small deformation model. In general, a woven fabric causes a finite (large) deformation due to displacement deformation or extension deformation, which is inconsistent as a model, and causes a large decrease in accuracy in the analysis of the stress and strain of the woven fabric and the coating material. It will be.
[0010]
The present invention has been made under such a background. Rather than considering the yarn discretely as in the truss model, the woven shape of the warp and the weft is represented by a swell coefficient, and the deformation of the woven fabric is represented. 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 the elements are refined and calculated only for the parts that require accuracy. The object is to provide a highly accurate plain weave membrane material analysis system that can minimize costs.
[0011]
[Means for Solving the Problems]
The plain weave membrane material analysis system of the present invention is a plain weave membrane material analysis system that performs stress and deformation analysis of a plain weave membrane material which is a composite material obtained by finishing a woven fabric with a coating material, and weaves the warp and weft of the woven fabric. The strain-displacement relationship of the fabric is defined by a pseudo-continuum model that expresses the shape as a swell coefficient and the deformation of the fabric as a continuous function with respect to space. The woven fabric structure calculation unit that calculates the strain from the obtained strain by the yarn stress-strain relationship, and the continuum and the finite deformation model, using the strain-displacement relationship of the coating material, Coating material structure calculation that calculates the coating material stress using the stress-strain relationship of the coating material from the strain obtained from the deformation. The woven fabric and the coating material are respectively finite element formulated based on the principle of virtual work using the respective stress-strain relationship and strain-displacement relationship, and the tangent of the plain woven membrane material is obtained by each obtained tangential stiffness matrix. And a stiffness matrix calculation unit for obtaining a stiffness matrix.
[0012]
In the plain weave membrane material analysis system according to the present invention, the quasi-continuum model is expressed as a continuous coefficient in which the woven fabric is expressed by an undulation coefficient that is a parameter representative of the period, thickness, undulation height, and undulation state of warp and weft yarns. The swell coefficient is configured as a continuous function for space.
In the plain weave membrane material analysis system of 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 the material of the woven fabric, and the coating material The structure calculation unit calculates each stress based on the material of the coating material using the stress-strain relationship of either a linear elastic body, a super elastic body, or an elastic-plastic body model.
[0013]
The plain weave membrane material analysis system of the present invention is characterized in that, in the coating material structure calculation unit, the strain-displacement relationship of the coating material uses a Green-Language strain capable of considering a finite strain.
The plain weave membrane material analysis system of 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 an elastic-plastic body.
[0014]
The plain weave membrane material analysis system of the present invention is characterized in that the stress-strain relationship of the coating material is modeled as a super-elastic body (Money-Rivlin body) in the coating material structure calculation unit.
The plain weave membrane material analysis system of the present invention is characterized in that 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.
[0015]
As described above, the conventional analysis method cannot basically perform the stress / deformation analysis of the plain weave membrane structure, and it has a limited structure and the limited load condition by Minami et al. Or Nishikawa et al. An analysis 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 shown as a problem.
[0016]
However, the quasi-continuum model in the present invention faithfully considers the three deformation modes of plain weave fabrics in plain weave membranes: displacement deformation, extension deformation, and stretch deformation, and the strain of the yarn with respect to space. As required as a continuous function, an arbitrary structure and load condition can be analyzed by defining the strain of a yarn using a continuous swell coefficient and displacement with respect to space.
[0017]
In particular, the plain weave membrane material analysis system of the present invention uses a quasi-continuum model and does not perform discrete operations, and represents the waviness coefficient of the warp and weft of the plain weave fabric, which is a continuous function with respect to space. By using it, the displacement analysis and the extension deformation can be evaluated simultaneously and quantitatively, and this method greatly exceeds the conventional analysis method.
Further, the plain weave membrane material analysis system of the present invention is a new analysis method that employs a pseudo-continuum model as a model of plain weave fabric and also includes analysis of a coating material necessary for actual plain weave membrane material analysis.
[0018]
Here, the concept of performing material analysis of the plain weave film considering the plain weave fabric and the coating material at the same time is the same as the analysis method of Minami et al.
However, in the analysis model of Minami et al., It is assumed that the plain weave yarn (warp and weft) and the coating material are linear elastic bodies, but in fact, both materials are polymer materials. In this case, the assumption of the 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 microdeformation, but the actual plain weave film material undergoes large deformation, so the analysis accuracy is extremely poor.
[0019]
On the other hand, in the present invention, in the pseudo continuum model, a new analysis model is extended so that the yarn material can be considered as a nonlinear elastic body, and a continuum / finite model is used as a strain-displacement model of the coating material. Deformation (Green-Larrange strain) is used, which corresponds to the analysis corresponding to the deformation of the woven fabric.
[0020]
Furthermore, in the present invention, as an extension of the analysis model, not only a linear elastic body model but also a super elastic body model (Money-Rivlin body) or an elastoplastic body model is adopted in the constitutive law for the coating material. .
The plain weave membrane material analysis system according to the present invention has a lower calculation cost than the conventional analysis method because the conventional analysis method cannot be analyzed due to the combination of the above-described new analysis model of the plain weave fabric and the coating material. In addition, it is possible to perform with higher accuracy.
[0021]
DETAILED DESCRIPTION OF THE INVENTION
First, in the present invention, a plain woven membrane material is separated into a plain woven fabric and a coating material, and an analysis model that takes into account the material nonlinearity and geometric nonlinearity of this plain woven fabric is newly described in detail later. This is an analysis method using a constitutive law of any model of not only a linear elastic body but also a non-linear elastic body in accordance with a homogenized strain-displacement relationship by a pseudo continuum model and corresponding to material and analysis accuracy.
This quasi-continuum model does not use the strain-displacement relationship of the continuum as in the conventional analysis, but instead of obtaining the average strain from the displacement of the membrane material, the deformation of the plain weave structure in which warp and weft are intertwined By formulating the form in detail, the strain of each yarn (warp and weft) is obtained directly from the displacement of the membrane material.
[0022]
Regarding the yarn, it is easy to obtain a constitutive law, and in many cases, the assumption of a linear elastic body can be used.
The geometrical nonlinearity unique to plain weave fabrics is directly taken into account in the strain-displacement relationship, so it is easy to analyze plain weave fabrics using conventional methods for other analysis procedures. It becomes.
Hereinafter, this analysis model is referred to as a quasi-continuum model. This quasi-continuum model can easily evaluate warp and weft strains, and stress concentration occurring in plain woven fabrics can be calculated in the same way as in conventional finite element analysis. If the analysis is possible 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, since most of the plain weave film materials that are actually used industrially are coated with a coating material on the plain weave fabric, an analysis method for considering this coating material is required.
And since the coating material undergoes large deformation together with the woven fabric, the strain-displacement relationship of micro deformation that is often 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]
As a constitutive law for the coating material, any one of a linear elastic body, a super elastic body, and an elastic-plastic body is selected and used corresponding to the material and analysis accuracy in the same manner as the woven fabric.
Furthermore, in the above description, the constitutive law of the yarn is assumed to be a linear elastic body. However, the actual plain woven membrane material often uses polymer fibers, and the model of the linear elastic body may not be sufficient. is there.
Therefore, the analysis method of the present invention described below is an extension of a model of a non-linear elastic body to the analysis of a plain woven fabric.
[0025]
Hereinafter, embodiments of the present invention will be described with reference to the drawings. FIG. 1 is a block diagram showing the configuration of a plain weave membrane material analysis system according to an embodiment of the present invention.
In this figure, the plain weave membrane material analysis system includes a woven fabric structure calculation unit 1 that calculates strain and stress according to a strain-displacement relationship and a constitutive law (stress-strain relationship) of the woven fabric portion, and a strain of the coating material portion. The coating material structure calculation unit 2 that calculates strain and stress by the displacement relationship and constitutive law (stress-strain relationship), and the respective tangential stiffness matrix by the strain-displacement relationship / constitutive law of each woven fabric and coating material And a tangential stiffness matrix calculation unit 3 for obtaining a tangential stiffness matrix of a plain woven membrane material as a composite material.
[0026]
Here, the plain weave film material to be analyzed of the present invention has a microscopic structure in which the coating material is equally coated (applied) on both sides of the plain weave cloth as shown in FIG. This precondition is for ignoring the coupled effects of out-of-plane deformation and in-plane deformation under various stresses).
In the analysis method of the present invention, the microscopic structure is replaced with a sandwich structure in which a plain woven cloth and a coating material having a uniform thickness are separated as shown in FIG. The structure calculation unit 1 and the coating material structure calculation unit 2 obtain the strain-displacement relationship and the constitutive law in an independent state.
[0027]
Returning to FIG. 1, the woven fabric structure calculation unit 1 calculates warp and weft strains and stresses based on the strain-displacement relationship and constitutive rules of the plain woven fabric using a quasi-continuum model of the plain woven fabric.
At this time, the woven fabric structure calculation unit 1 reads the strain-displacement relationship and the constitutive law stored in the database 4 in correspondence with the material and analysis accuracy of the plain woven fabric, and calculates strain and stress. Use.
[0028]
The database 4 stores a linear elastic body model and a nonlinear elastic body model as constitutive models corresponding to the material and analysis accuracy of the plain woven fabric, and corresponds to the material and analysis accuracy of the coating material, A linear elastic body model, a superelastic body model, and an elastoplastic body model are stored as constitutive law models.
Here, for the plain woven fabric part, the constitutive law of warp and weft is extended to a non-linear elastic body, but the presence of a coating material prevents the lateral compression deformation (lateral compression strain) of these yarns from being applied to the coating material. Compared to the case where there is no, it is significantly smaller. .
[0029]
Thereby, in the analysis of the plain weave membrane material, it is often unnecessary to consider the lateral compression strain, and in this analysis method, in order to simplify the constitutive law of the yarn when using the constitutive law of the nonlinear elastic body, Although the accuracy is somewhat lowered, the lateral compression strain can be ignored in the strain-displacement relationship in the woven fabric calculation unit 1, so that the constitutive law of the yarn needs to be considered only for the tensile deformation in the axial direction.
Here, the pseudo continuum model of the present invention defined in the woven fabric structure calculation unit 1 is a plain woven fabric in which warps and wefts are woven orthogonally, and the weave is uniform and dense. It is targeted.
The warp and weft are assumed to be orthotropic linear elastic bodies of the same material each having a uniform cross section.
In the following explanation, the bending rigidity of the yarn (warp and weft) and the friction between the yarns are ignored only for the plane problem, but there is no slip at the center of the intersection (intersection) of the yarn.
[0030]
At this time, the deformation pattern of the plain woven fabric can be classified into three types: displacement deformation, extension deformation, and expansion / contraction deformation of the yarn itself.
Displacement deformation (in-plane deformation of yarn) and extension deformation (or crimp exchange, out-of-plane deformation of yarn) are geometric deformations of the yarn, meaning a finite rigid rotation, and stretching deformation is the axial direction of the yarn The elastic elongation of
However, it is assumed that the strain of the yarn is very small.
[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 ξ1-ξ2.
Here, the thin line in the figure indicates the direction of the yarn, and the ξ1 axis is the axis along the weft and the ξ2 axis is the axis along the warp.
In the description of the coefficients such as the following equations, the lower right subscript is represented as “1” for the value related to the weft and “2” for the value related to the warp.
[0032]
And the conceptual diagram of the unit woven structure enclosed with the wavy line in FIG. 3 is shown in FIG.
In FIG. 4, the cross-sectional area of the yarn is Ai, the center thickness is di, the waviness wavelength is 2pi, and the waviness height (waviness height) is hi.
At this time, there is a relationship of the following expression (1) between the center thickness di and the undulation height hi.
[Expression 1]

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

In the following, μ is defined as a swell coefficient configured as a continuous function with respect to space, and is treated as a component of a displacement vector that regards the woven fabric as a continuum (set as a pseudo continuum) as a parameter representative of the swell state.
In other words, the pseudo continuum model is a pseudo continuum model in which a woven fabric is expressed by the intersecting period, thickness, undulation height, and undulation coefficient representing the undulation state of warp and weft yarns. The swell coefficient is defined as a continuous function for space.
[0034]
And the woven fabric structure calculating part 1 calculates | requires the distortion-displacement relationship of a plain woven fabric as follows as a pseudo continuum model using this waviness coefficient (micro | micron | mu).
FIG. 5 shows a modification of the unit woven structure. Since the friction between the yarns is ignored, the yarn does not undergo shear deformation, and the axial tensile deformation due to expansion / contraction deformation, and in-plane due to displacement deformation and extension deformation. Only lateral compression deformation occurs.
[0035]
At this time, the axial tensile strain is expressed as εTi, and the lateral compressive strain is expressed as εCi.
Here, considering the in-plane rotation of the yarn due to displacement deformation (in-plane deformation), as shown in FIG. 5, the rotation angles of the weft and the warp are ζ1 and ζ2, respectively, and the displacement shown in the ξ1-ξ2 coordinate system If the vector components 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 yarn interval is equal to the lateral 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 displacement deformation, the lateral compression strain εC1 of the weft and the lateral compression strain εC2 of the warp are expressed by equations (4), ( 5) It is expressed by the formula.
[Expression 4]

[Equation 5]

[0037]
Before and after deformation, the waviness in the unit woven structure can be approximated by a straight line, and the center line length is represented by si.
This center line length si is the length of the one-dot chain 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 displacement 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 equations (6) and (7), respectively.
[Formula 6]

[Expression 7]

However, 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-Langrange strain under the assumption of minute strain and finite rotation.
Further, in the woven fabric calculation unit 1, the axial tensile stress σTi and the lateral compressive stress σCi of the warp and the weft are obtained from the strain obtained from the equations (4) to (7) using the constitutive law of the yarn.
Here, for the incremental analysis described later, the velocity form constitutive law expressed by the following equation (9) is also used for the stress-strain relationship.
[Equation 9]

However, the axial tensile stress σTi and the lateral compressive stress σCi correspond to the second Piola-Kirchoff stress under the assumption of minute strain and finite rotation.
[0039]
Further, when the yarn is a linear elastic body, the above equation (9) coincides with a stress-strain relationship that is not a velocity type.
Here, the velocity type stress-strain matrix [D_i] Is read out from the database 4 in accordance with the input data, which is suitable for the yarn material.
Furthermore, in the coating material structure calculation unit 2, the continuum / finite deformation model defined as the strain-displacement relationship is the thickness of the coating film in the sandwich model._M/ 2.
[0040]
At this time, since it is a symmetrical lamination, it is not necessary to consider out-of-plane deformation._MFormulation limited to in-plane deformation can be made as a single film.
Then, for the strain-displacement relationship of the coating material, Green-Larrange strain represented by the following formula (10) is used so that large deformation can be evaluated (finite strain can be considered).
[Expression 10]

Here, xi represents Cartesian Cartesian coordinates, and ui represents the xi direction component of the displacement vector.
However, the above equation (10) uses the sum rules.
Also, the sum rule is the sum of 1 to 2 (two of x1 and x2 or all the components of an appropriately defined vector) for the subscript when the same subscript appears in the same term in the formula. It is a computational convention to decide to take.
[0041]
Further, 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).
## EQU11 ##

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

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

[Expression 14]

However, the stress of the coating material is the second Piola-Kirchoff stress.
[0043]
Further, 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] Is read out from the database 4 in accordance with the input data, which is suitable for the material of the coating material.
Here, the finite element formulation based on the principle of virtual work performed in the tangential stiffness matrix calculation unit 3 will be described below.
[0044]
The principle of virtual work for the plain weave film material is defined by the following equation (15) in the tangential stiffness matrix calculation unit 3 in order to consider finite deformation.
[Expression 15]

In this equation (15), Ω_iIs the initial volume of warp or weft where τi acts, Ω_MIs the initial volume of the coating material, and L is the work performed by the external force.
When considering finite deformation, an incremental analysis from time t to time t ′ must be performed in order to consider geometric nonlinearity (nonlinearity of strain-displacement relationship).
[0045]
That is, at time t, all variables such as stress and strain are obtained, and the stress and strain at time t ′ after the lapse of Δt are calculated.
In addition, for the finite element formulation, in the woven fabric structure calculation unit 1, the stress vector {τ}, the strain vector {γ}, and the displacement gradient vector {κ} are respectively expressed by the following equations (16) for the woven fabric. , (17) and (18).
[Expression 16]

[Expression 17]

[Formula 18]

[0046]
Hereinafter, the transposition of the matrix is represented by the shoulder 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 law of the 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 defined by the following equation (20).
[Expression 20]

[0047]
Although the displacement gradient vector {κ} is defined by the coordinate system ξ1−ξ2 along the weave of the plain woven fabric, the actual analysis is performed by the global coordinate system x1−x2 which does not depend on the weave. ) To (23) to obtain the displacement gradient vector {κ} by coordinate transformation.
[Expression 21]

[Expression 22]

[Expression 23]

Here, (u1, u2) is a displacement vector in the x1-x2 coordinate system, and sum rules are used in the following finite element formulation.
[0048]
In addition, in order to perform the incremental analysis, the notation of the equations (16) to (18) is used, and the strain-displacement relationship of the plain woven fabrics of the equations (4) to (7) is expressed in the Taylor series near the time t. This is expanded and quadratic approximated by the following equation (24).
[Expression 24]

In the above equation (24), the left shoulder 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 Δi for γi, respectively.
[0049]
On the other hand, as shown in the following equation (25), the yarn stress is also incrementally decomposed.
[Expression 25]

Further, for the above-described incremental analysis, the coating material structure calculation unit 3 displays the strain-displacement relational expression of the coating material incrementally according to the following equation (26). (Because the Green-Lagrange distortion is originally a quadratic function, equation (26) is not an approximate equation.)
[Equation 26]

Here, the sum of the second term and the third term on the right side of the equation (26) is the increase in strain of the coating material.
[0050]
Furthermore, as shown in the following equation (27), incremental decomposition is also performed on the stress of the coating material.
[Expression 27]

Here, in the tangential stiffness matrix calculation unit 3, when the equations (24) to (27) are substituted into the equation (15) and linearized, the velocity type virtual work of the following equation (28) is obtained. The formula is obtained.
[Expression 28]

[0051]
Also, the displacement (u1, u2) represented by the global coordinate system x1-x2 is interpolated from the nodal displacement by introducing a shape function, so that the velocity of the displacement gradient vector of the equation (23) is obtained as the nodal displacement velocity. As a vector function, it is expressed by the following equation (29).
[Expression 29]

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

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

[Expression 32]

[Expression 33]

[Expression 34]

[Expression 35]

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

[Expression 37]

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

[39]

[0053]
In the 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_ijA matrix with an array of [^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), and [^tR^M] Is the first derivative^tR_ij ^MA matrix with an array of [^tS '^M] Is a matrix obtained by the following equation (43).
[0054]
[Formula 40]

[Expression 41]

[Expression 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, according to the flowcharts of FIG. 6 and FIG. 7 (connected to each other in A), the analysis operation of the plain woven fabric coated with the coating material in the plain woven membrane material analysis system of the present invention, that is, the plain woven membrane material will be described.
Each numerical value and expression in the calculation process in the woven structure calculation unit 1, the coating material structure calculation unit 2, and the tangential stiffness matrix calculation unit 3 are temporarily stored in the database 4.
[0056]
<When a linear elastic body model is used for the constitutive law of both plain woven fabric and coating material> In step S1, the input unit 5 includes a material, analysis accuracy, and analysis target necessary for analysis of the plain woven fabric and coating material ( That is, conditions such as the shape and direction of the weave, finite element mesh data, boundary conditions (positions and sizes where the load is applied, positions where they are fixed, positions and sizes where the forced displacement is applied), etc. Is entered.
Next, in step S2, the woven fabric structure calculation unit 1 is 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]
In step S <b> 3, the woven fabric structure calculation unit 1 selects a constitutive law corresponding to the material of the yarn constituting the woven fabric, and reads the stress-strain matrix in the above equation (9) from the database 4.
Here, the yarn is assumed to be a linear elastic body, but the yarn is a discontinuous body as an aggregate of fibers, and the problem of homogenization also arises in its modeling.
However, since the yarn is basically a bundle of fibers having the same orientation, there is no complicated deformation mode, and an analysis theory based on normal 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 velocity form constitutive law of the yarn expressed by the above equation (9) is defined in the database 4 by the following equation (44).
(44)

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

[0059]
Next, in step S4, the coating material structure calculation unit 2 determines 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 the coating material structure calculation unit 2 analyzes the stress-strain relationship (constitutive law) of the coating material as a linear elastic body, the velocity form constitutive law of the coating material represented by the above equation (12). Then, the stress-strain matrix of the following equation (46) is read from the database 4.
[Equation 46]

[0060]
Here, as already mentioned, the coating material is assumed to be an isotropic linear elastic body. In the case of rubber-based materials, the linear elastic body model has poor analysis accuracy, but in the case of plastic-based or ceramic-based materials. A linear elastic model is suitable. However, even in the case of plastic or ceramic materials, it is desirable to use an elastoplastic model when it is desired to evaluate the final fracture.
However, since the linear elastic body is easy to handle theoretically, in practice, the linear elastic body is often used at the expense of a certain degree of accuracy, and its utility value is high.
[0061]
Next, in step S6, the tangential stiffness matrix calculation unit 3 creates various matrices necessary for the calculation of the tangential stiffness matrix according to the finite element mesh data, and creates them in the woven fabric structure calculation unit 1 and the coating material structure. Read from the calculation unit 2 [^tR], [^tS '], [D_i], [^tR^M], [^tS^'M], [D^M] And the tangential stiffness matrix [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, each of the nodal internal force vectors {^tQ_F}, {^tQ_M}.
[0062]
Next, in step S7, boundary condition processing for considering external force, displacement fixation, and forced displacement is performed.
Here, the tangential stiffness matrix calculation unit 3 divides the added external force into several incremental steps based on the input data, and the nodal external force vector {^{t '}F} is created.
[0063]
For the nodal displacement for which the condition for fixing the displacement is given by the input data, the component related to the nodal displacement is deleted from the tangential stiffness matrix of the plain weave film obtained in step S6.
For the nodal displacement for which the forced displacement is given by the input data, the forced displacement is divided into incremental steps in the same way as the external force, and the product of the nodal displacement component and the forced displacement increment in the tangential stiffness matrix of the plain weave membrane. Is a known quantity and the nodal external force vector {^{t '}F}.
[0064]
In step S8, the tangential stiffness matrix computing unit 3 performs 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 tangential stiffness matrix calculation unit 3 calculates the nodal displacement and the nodal swell coefficient in the next step by adding the augmentation to the current value from the result of step S8.
[0065]
In step S10, the woven fabric structure calculation unit 1 performs calculations according to the equations (4) to (7) based on the above-mentioned nodal displacement and nodal swell coefficient, and warp and weft distortions {^{t '}γ} is calculated.
Next, in step S11, the woven fabric structure calculation unit 1 calculates the strain {^{t '}Based on γ} (the strain rate in the case of a nonlinear material model described later), the calculation is performed by equation (9), and the stress of the yarn {^{t '}τ} (stress rate in the case of a nonlinear material model described later) is calculated.
[0066]
In step S12, the coating material structure calculation unit 2 calculates the coating material strain based on the above-described nodal displacement and nodal swell coefficient by the equations (10) and (11).
Next, in step S13, the coating material structure calculation unit 2 performs calculation according to equation (12) based on the strain of the coating material obtained in step S12 (in the case of a nonlinear material model described later, strain rate), The stress of the coating material (stress rate in the case of a nonlinear material model described later) is calculated.
In step S14, as described above, the display unit 6 displays the stresses 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, as a display device such as a CRT. Then, the numerical value of each stress is displayed as a stress distribution map converted into color intensity.
[0067]
<When nonlinear elastic body model and superelastic body model are used as constitutive rules for plain woven fabric and coating material, respectively>
In the flowcharts of FIG. 6 and FIG. 7 described above, the description of the steps for performing the same operation as that 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 (calculation formula etc.) is omitted. Only the steps of (Use) are explained.
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 the expansion to the polymer fiber (47) The nonlinear function of the equation is read from the database 4 and defined in the 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 expresses the stress-strain relationship in the uniaxial stress state in the axial direction of the yarn. The axial tensile stress σTi is expressed by the following nonlinear function related to the axial tensile strain εTi (nonlinear elasticity expressed by polynomial approximation). Expressed as a model).
[Equation 47]

Here, as already described, the order N of the polynomial representing the axial tensile stress σTi and the coefficient an used corresponding to this order are material constants, and the experiment in which the stress-strain curve is 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 structure calculation unit 1 reads each numerical value of the constant an as a constitutive rule corresponding to the material specified by the input data from the database 4 and stores it in the storage unit in the input unit of step S1.
Then, the woven fabric structure calculation unit 1 linearizes the equation (47) at time t, and calculates the velocity type stress-strain matrix [D_i] Can be used to analyze a nonlinear elastic body.
[0070]
Next, in step S5, the coating material structure calculation unit 2 uses the constitutive law equation when the superelastic body model is used as the material model of the coating material (see FIG. 2), that is, the following equation (48). , Read from the database 4 and defined in the internal storage unit.
[Formula 48]

However, 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 (a kind of superelastic body and the most general model).
[0071]
At this time, the strain energy W is expressed by the following equation (49).
[Formula 49]

Here, α is a penalty coefficient that means a bulk modulus, and should be determined appropriately separately from the experiment.
However, the value designated by the input data in step S1 is used as the value of α.
[0072]
Further, 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 representing incompressibility.
[0073]
Further, 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.
(48)_kl ^MIs linearized at time t, and the result is converted to [D] in equation (38)._M], It is possible to analyze a superelastic body.
However, when incompressibility is included as a constraint, in step S6, the tangential stiffness matrix calculation unit uses the Larange multiplier method instead of the equation (15), and based on the following equation (51), the finite element Incorporate uncompressed constraints by formulating.
[Formula 51]

Here, λ is a Larange multiplier.
[0074]
Then, the tangential stiffness matrix calculation unit 3 performs linear decomposition after incrementally decomposing the equation (51), and further performs finite element discretization, so that the velocity type balance equation of the following equation (52) is finally obtained. Lead.
[Formula 52]

Where is the speed of the node Larange multiplier, {^tP} is the error vector of the uncompressed constraint,^tG] is the tangential rigidity according to the constraint condition corresponding to the third term on the left side of the first expression of expression (51).
[0075]
However, in step S6, when the tangential stiffness matrix calculation unit 3 analyzes that the coating material is a superelastic body,^tG] and {^tP} is also obtained, and the simultaneous linear equations to be solved are determined.
[0076]
Next, in step S8, the tangential stiffness matrix calculation unit 3 performs 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 a solution for each increment.
Next, in step S9, the tangential stiffness matrix calculation unit 3 adds the expansion (velocity solution) to the current value from the result of step S8, so that the nodal displacement, the nodal swell coefficient and the nodal Larange multiplier in the next step. Is calculated.
[0077]
In step S11, the woven fabric structure calculation unit 1 uses the linear velocity form constitutive law of the nonlinear elastic body model to calculate the stress rate from the strain rate according to the equation (19), and calculates the current rate. stress{^tBy adding to τ}, the stress of the next step {^{t '}τ} is obtained.
In step S13, the coating material structure calculation unit 2 calculates the stress rate vector from the strain rate vector using equation (12) using the linear velocity form constitutive law of the superelastic body model, and (27 The stress of the next step is obtained by adding it to the current stress by the equation (1).
The subsequent steps are the same as when both the plain woven fabric and the coating material are described as the linear elastic body model, and thus the description thereof is omitted.
[0078]
<When nonlinear elastic body model and elasto-plastic body model are used as constitutive rules for plain woven fabric and coating material, respectively>
Even when the coating material (base material) is a hard resin, such as FRP (fiber reinforced plastic), the plain weave film material analysis system of the present invention can be applied, and particularly large deformation in the process leading to the final breakage. 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 the constitutive law is the same as the equation (46) in the elastic region. Yes, this is [D_e ^M].
[0079]
However, when the Mises equivalent stress exceeds the yield stress, the plastic zone is entered, and the constitutive law is expressed by a velocity type stress-strain matrix of the following equation (53).
[53]

Here, since the plastic constitutive law is general, detailed description is omitted.
Elastic constitutive law [D_e ^M] And plastic constitutive law [D_p ^M] In a timely manner based on the yield condition and unloading judgment, and the selected elastic constitutive law [D_e ^M] And plastic constitutive law [D_p ^M] In the equation (38) [D^M], It is possible to analyze an elastic-plastic body.
The 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 thus the description of the operation according to the flowcharts in FIGS. 6 and 7 is omitted.
[0080]
The plain weave membrane material analysis system of the present invention is not only capable of analyzing linear elastic bodies such as glass fibers and carbon fibers by changing the constitutive rules of plain weave yarns, but also apparent inelasticity due to sequential breakage of those fibers. It is also possible to analyze dynamic behavior and nonlinear behavior of polymer fibers.
In this way, the plain weave membrane material analysis system enables more accurate understanding of deformation behavior and prediction of breakage, that is, strength evaluation, by increasing the accuracy of the constitutive law in accordance with the material properties of the yarn to be analyzed. It becomes.
[0081]
In addition, the plain weave membrane material analysis system of the present invention described above changes the constitutive law of the base material (coating material) to be analyzed, thereby allowing an elastic-plastic model that takes into account the fracture in the base material and a flexible polymer system. A superelastic model suitable for various materials can be applied, and more accurate nonlinear behavior can be analyzed.
Thus, the plain weave membrane material analysis system of the present invention can more accurately evaluate the final fracture (the ultimate strength of the composite material) if the material properties of the base material and the yarn can be accurately evaluated.
[0082]
Next, an application example of one embodiment will be described with reference to the drawings.
For example, a calculation example in the case where a linear elastic body model is used as a constitutive law for both plain woven fabric and coating material will be described below.
Here, a glass fiber plain woven fabric is assumed as the woven fabric, while ethylene tetrafluoride (PTFE) is assumed as the coating material, and the structural specifications are values shown in the table of FIG.
Then, the plain woven membrane material analysis system of the present invention analyzes the uniaxial tensile test of this plain woven membrane material.
[0083]
The test piece shape is as shown in FIG. 9A, and is divided into finite elements as shown in FIG. 9B.
As will be described later, since stress is expected to concentrate at the four corners of the test piece, only the four corners are finely divided in order to improve the analysis accuracy of the portion where the stress is concentrated.
A 9-node isoparametric element is used for the finite element, and both the displacement and the swell coefficient are interpolated in a second order.
The direction of the weave in the plain woven fabric is analyzed in three ways: θ = 0 (warp direction tension), π / 2 and π / 4 (weft direction tension).
[0084]
FIG. 10 shows the relationship between the load and displacement obtained from the analysis. The horizontal axis indicates the numerical value of displacement (unit: mm), and the vertical axis indicates the value obtained by dividing the tensile load applied to the test piece by the test piece width, that is, the average film stress (unit: N / mm).
As for the tension in the weft or warp direction, it is in good agreement with the experimental results found in the literature.
If the Young's modulus and Poisson's ratio of the yarn are corrected by comparing this load-displacement curve with the actual uniaxial tensile test results, the structure analysis using the plain weave membrane material will be very accurate. Can be done.
[0085]
In addition, the conventional analysis method could not quantitatively analyze the tension in the bias direction, but this model can be easily analyzed simply by changing the analysis parameters, and guarantees quantitative accuracy from comparison with membrane material experiments. Is possible.
As described above, the plain weave membrane material analysis system of the present invention can execute a simulation in which analysis of a plain weave membrane material is performed with high accuracy for any shape and any load condition.
[0086]
In the following, the results of bias direction tension, which has been considered to be the most difficult in the conventional analysis, will be described in detail.
FIG. 11 shows the distribution of the swell coefficient when the load is 100 N / mm. On the right side is a sign indicating the correspondence between the gradation and the numerical value of the swell coefficient.
The black part shows where the weft is extended, and the white part shows where the warp is extended.
It can be easily seen and confirmed that the four corners of the test piece are in a non-uniform deformation state in which one of the yarns is greatly extended.
Thus, by introducing the undulation coefficient, it is also a great merit of the present system over the conventional method that the undulation state of the yarn (weft and warp) can be evaluated easily and quantitatively.
[0087]
FIG. 12 shows the distribution of the warp axis tensile stress when the load is 100 N / mm. On the right side, there is a mark indicating the gradation of the numerical value of the warp axis tensile stress σT2.
As can be confirmed from the undulation coefficient distribution in FIG. 11, at one of the four corners of the test piece, one of the threads is stretched until it becomes almost straight. Appears in the four corners.
With such a strong stress concentration, the specimen can be predicted to break from the end of the fixed part, and if the breaking strength of the yarn is known in advance, the fracture starts when the axial tensile stress in this analysis exceeds that value. Then it can be predicted.
Thus, it can be said that the plain weave 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 a non-uniform deformation field.
[0088]
13 and 14 show the tensile stress σ in the x direction of the coating material when the load is 100 N / mm, respectively.^M ₁₁And Mises equivalent stress σ^M _eqIt is the graph which showed 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.
FIG. 13 confirms that a large compressive stress is acting on the center of the test piece, and this is considered to be the main cause of wrinkle generation.
Conventionally, there is no analytical model that can quantitatively evaluate the wrinkle generation phenomenon in woven fabrics, and the generation mechanism is not well understood.
However, the plain weave membrane material analysis system of the present invention shows that even a simple tensile test generates strong compressive stress on the membrane material, and it is theoretically possible that plain weave membrane material can easily wrinkle due to this compressive stress. I was able to explain.
[0089]
In FIG. 14, it can be seen that a strong stress concentration appears 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 a very large deformation is caused by tensile in the bias direction. In order to evaluate the stress / strain of the coating material at this time, a finite deformation strain must be used. In this respect, the plain weave membrane material analysis system is capable of quantitative evaluation with high accuracy, 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. However, by using a combination of the quasi-continuum model of the woven fabric and the finite deformation model of the coating, It was confirmed that the stress / deformation evaluation was possible for the first time, and its effectiveness was shown.
Changing the constitutive law of the coating material to a superelastic body or the like means that the analysis model of the plain weave membrane material system of the present invention is highly accurate according to the material.
If such a high-accuracy constitutive law is used, it is possible to strictly evaluate the yarn breakage, and therefore high-accuracy analysis accuracy can be obtained 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. By executing, the user attribute registration information automatic updating apparatus 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 the 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, and a CD-ROM, and a storage device such as a hard disk built in the computer system. Further, the “computer-readable recording medium” refers to a volatile memory (RAM) in a computer system that becomes a client or a system when a program is transmitted through a network such as the Internet or a communication line such as a telephone line. In addition, those holding programs for a certain period of time are also included.
[0093]
The 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 the 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.
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 mentioned above, although one embodiment of the present invention has been described in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and there are design changes and the like without departing from the gist of the present invention. Are also included in the present invention.
[0095]
【The invention's effect】
The plain weave membrane material analysis system of the present invention uses a new analysis model of the above-described plain weave fabric and coating material, that is, the plain weave fabric uses a pseudo-continuum model for the strain-displacement relationship, and the constitutive law includes a linear elastic body and Nonlinear elastic body models are used. In coating materials, continuum and finite deformation models are used for the strain-displacement relationship, and linear elastic bodies, superelastic bodies, and elastoplastic bodies models are used for the constitutive rules. By performing analysis in combination with the material to be evaluated and the accuracy of the analysis, the conventional finite element analysis method uses only a linear elastic body model, so high-accuracy analysis of plain weave membrane materials that could not be analyzed, Compared with the conventional finite element analysis, the calculation cost can be reduced with higher accuracy.
[Brief description of the drawings]
FIG. 1 is a block diagram showing a configuration of a plain weave membrane material analysis system according to an embodiment of the present invention.
FIG. 2 is a conceptual diagram showing a configuration of a plain weave membrane material that is symmetrical to the plain weave membrane material analysis system shown in FIG.
FIG. 3 is a conceptual diagram of a coordinate system showing a texture of a woven fabric.
4 is a conceptual diagram showing a unit woven structure surrounded by a wavy line in FIG. 3;
5 is a conceptual diagram showing a deformed state of the unit woven structure shown in FIG. 4 by application of an external force.
6 is a flowchart for explaining an operation example of the plain weave membrane material analysis system of FIG. 1. FIG.
FIG. 7 is a flowchart for explaining an operation example of the plain weave membrane material analysis system of FIG. 1;
FIG. 8 is a table showing numerical values of each structural specification when glass fiber plain fabric is used for the woven fabric and ethylene tetrafluoride is used for the coating material.
FIG. 9 is a conceptual diagram showing the shape of a test piece to be divided into finite elements.
FIG. 10 is a graph showing the relationship between load and displacement obtained from analysis.
FIG. 11 is a distribution diagram showing the distribution of the swell coefficient when the load is 100 N / mm.
FIG. 12 is a distribution diagram showing the distribution of warp axis tensile stress when the load is 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 the distribution of Mises equivalent stress of the coating material when the load is 100 N / mm.
[Explanation of symbols]
1 Weaving structure calculation unit
2 Coating material structure calculation unit
3 Tangent stiffness matrix calculator
4 Database
5 Input section
6 Display section

Claims (7)

It is a plain weave membrane material analysis system that performs stress and deformation analysis of plain weave membrane material, which is a composite material finished with woven fabric with coating material,
The strain-displacement relationship of the woven fabric is defined by a quasi-continuum model in which the woven shape of the warp and weft of the woven fabric is expressed by a undulation coefficient, and the deformation of the woven fabric is expressed as a continuous function with respect to space. A woven fabric structure calculation unit that calculates the strain of the yarn from the deformation of the membrane material by the displacement relationship, and calculates the stress of the yarn by the stress-strain relationship of the yarn from the obtained strain;
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 calculated from the obtained strain using the stress-strain relationship of the coating material. A coating material structure calculation unit to calculate,
Each woven fabric and coating material is formulated based on the principle of virtual work using the respective stress-strain relationship and strain-displacement relationship, and the tangential stiffness matrix of the plain woven membrane material is obtained by each obtained tangential stiffness matrix. A plain weave membrane material analysis system comprising: a stiffness matrix calculation unit for obtaining The quasi-continuum model is a continuum represented by a waviness coefficient that is a parameter representative of the period, thickness, waviness height and waviness state of warp and weft crossing, and the waviness coefficient is a continuous function for space. The plain weave membrane material analysis system according to claim 1, wherein: 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. The plain woven fabric according to claim 1 or 2, wherein the stress is calculated using the stress-strain relationship of any one of a linear elastic body, a super elastic body, and an elastic-plastic 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, a strain-displacement relationship of the coating material uses a Green-Langrange strain that can take into account a finite strain. Material analysis system. The said coating material structure calculating part WHEREIN: The stress-strain relationship of a coating material is modeled as a superelastic body (Moony-Rivlin body), The Claim 1 characterized by the above-mentioned. Plain weave membrane material analysis system. The plain weave film material analysis system according to any one of claims 1 to 3, wherein in the coating material structure calculation unit, the stress-strain relationship of the coating material is modeled as an elastic-plastic body. 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 structure calculation unit. Material analysis system.

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