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回位密度を考慮した有限変形結晶塑性モデルに基づくMg基LPSO相の変形挙動に関するメッシュフリー解析 (本文)

木村, 祐一 慶應義塾大学

2021.03.23

概要

本論文の構成について述べる.本論文は全 2 部で構成される.第 I 部 (第 2~6 章) では不適合度の成分を回位密度に対応させた高次応力モデルに基づくメッシュフリー解析を実行し,キンク帯を不適合度によって簡便に再現する.また,得られた結果を FEM による解析結果と比較することで,メッシュフリー法が高次ひずみこう配の解析に適することを示す.

第Ⅱ部 (第 7~11 章) では有限変形 Micropolar 結晶塑性モデルを構築し,回位密度を厳密に定義するとともに,メッシュフリー解析を行うことでキンク帯を回位に基づいて再現し,同モデルの妥当性を検証する.

第 2 章では,高次応力モデルに対する運動学を記述する.まず,金属材料の大変形状態を記述するために配置の概念を導入し,各配置におけるひずみを定義する.また,結晶塑性論における諸量の配置変換則を与える.続いて,材料内部に蓄積する転位密度を結晶塑性論に基づいて表現し,孤立転位と転位対の表現にはそれぞれ個別の量の定義が必要であることを示す.さらに,GN 転位密度および SS 転位密度の発展式をそれぞれすべり速度こう配およびすべり速度に対応する量として定義する.

第 3 章では,高次応力を GN 転位密度に共役な力として全自由エネルギーに導入している.また,仮想仕事の原理に基づいて高次理論における各種保存則を導出し,釣合い方程式から高次応力の発散が背応力となることを示している.さらに,仮想仕事の原理を速度形へ拡張するとともに,高次理論において新たに導入される境界条件について言及し,高次応力に対する境界条件が GN 転位密度の境界条件に対応することを述べている.

第 4 章では構成式の導出を行う.Clausius-Duhem の不等式から応力および高次応力の構成式の一般形を導くとともに,それらの構成式に対する熱力学的制限について述べる.熱力学的ポテンシャルとなる準保存エネルギーを具体化することで速度形の弾粘塑性構成式および高次応力の構成式をそれぞれ導出している.その際,高次応力には GN 転位の分布に関連した特性長が含まれることを示す.また,従来の転位-結晶塑性論における硬化則を高次理論に適する形に修正し,本モデルに対する従来の転位-結晶塑性モデルの位置づけを説明している.さらに,数値計算の低コスト化・安定化を実現するために接線係数法を導入する

第 5 章では,メッシュフリー法について概説し,RKPM に基づく形状関数の構築方法について述べる.また,RKPM による数値解析を行うために 3 章で導出した仮想仕事の原理を離散化し,節点積分について言及する.さらに,第 7 章で定義した GN 転位密度,GN 回位密度および GN 不適合度の数値解析上の算出方法を述べる.また,解析に用いる時間増分を制御する 𝑟mi𝚗 法を導入する.

第 6 章では,本モデルを用いた Mg 基 LPSO 相に対するメッシュフリー解析の結果およびそれに対する考察を述べている.まず,キンク変形が生じやすい初期方位を設定した短冊状の単結晶に対する解析を行い,メッシュフリー法においても FEM と同様のキンク帯が形成されることを示す.このとき,キンク帯の幅は積分領域の解像度に依存することを示す.そして,高次応力の導入によるキンク帯形成過程の変化を議論し,メッシュフリー法においても FEM と同様の結果がより少ない節点数で得られることを示す.また,不適合度による回位四重極の表現方法について述べ,異なる寸法を持つ試験片に対する解析を行うことで本モデルが寸法効果を表現可能なことを明らかにし,回位四重極によってキンクが特徴づけられることを示している.

第 7 章では,有限変形 micropolar 理論における運動学について論じる.まず,Shizawa&Zbib(2) の 4 配置構成を micropolar 理論における微視回転に導入し,Cosserat ひずみおよび微視曲率を弾塑性分解する.また,結晶塑性論に基づいて塑性回転変形を幾何学的に表現する.最後に,転位密度の定義を微視回転依存形に拡張し,速度形の回位密度,転位密度そして不適合度を定義する.回位密度および転位密度には非線形項が表れることを示し,結晶塑性論における扱いについて述べている.

第 8 章では,偶応力を考慮した釣合い法則について述べる.まず,本研究で用いる釣合い法則およびエントロピー不等式を示し,偶応力が考慮されることで応力の対称性が損なわれることを示す.続いて,釣合い法則と力学的境界条件を組み合わせて釣合い法則の弱形式である仮想仕事率の原理を導出する.さらに,大変形に対する解析を扱えるようにするために仮想仕事率の原理を速度形の仮想仕事の原理である updated Lagrange 形式の仮想仕事の原理に書き換える.最後に,本研究で追加される偶応力および微視回転自由度に関する境界条件の物理的な意味について述べる.

第 9 章では,弾性構成式の導出および材料硬化則の引数の検討を行う.まず,第 8 章で得られた釣合い法則から Clausius-Planck の不等式を導出する.続いて,Clausius-Planck の不等式の保存部分から応力とひずみの弾性構成式および偶応力とよじれの弾性構成式を導出し,散逸部分に基づいて硬化則に用いる引数の検討を行う.次に,得られた弾性構成式と第 7 章で得られた運動学を組み合わせて弾・粘塑性構成式を導出する.また,本研究で採用するすべりおよび塑性微視曲率の硬化則について述べる.さらに,接線係数法により弾・粘塑性構成式を数値解析に適した形式のものに変形する.

第 10 章では,メッシュフリー解析のための定式化を行う.第 8 章において導出した updated Lagrange 形式の仮想仕事の原理をマトリクス表示したのち,RKPM に基づいて離散化する.さらに,キンク変形などの不安定現象を解析する際に有用な手法である 𝑟mi𝚗 法 (103) について説明する.

第 11 章では,数値解析結果の考察を行う.まず,一般化連続体の結晶塑性モデルを比較する際に広く用いられる単純せん断問題に結晶塑性 Cosserat モデルを適用して解析を行い,従来の転位–結晶塑性モデルとの結果を比較することで本モデルが寸法効果を表現可能であることを示す.続いて,第 6章と同様の材料モデルに対するメッシュフリー解析を実施し,キンク変形を転位密度および回位密度を介して確認する.続いて,本モデルの有する寸法効果やメッシュ依存性に対する検討を行う.次に, LPSO 相単結晶平板に対する負荷–除荷解析を実施し,キンク帯の形成による加工硬化を再現する.さらに,硬化則に導入する転位密度を回位依存のものとし,非依存の場合と比較する.最後に,多結晶に対する解析を実施し,Ortho 形キンクに加えて Ridge 形キンクを結晶欠陥を介して再現する.

第 12 章は結言であり,本研究で提案した計算手法の性質およびメッシュフリー解析から得た知見を要約している.

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