Mathematics for wrinkle and crack -- from a viewpoint of energy variation --

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K18733
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Analysis, Applied mathematics, and related fields
• Research Institution: Kanazawa University
• Principal Investigator: Kimura Masato 金沢大学, 数物科学系, 教授 (70263358)
• Co-Investigator(Kenkyū-buntansha): 田中 良巳 横浜国立大学, 大学院環境情報研究院, 准教授 (10315830) ; VANMEURS PATRICK 金沢大学, 数物科学系, 助教 (20815378)
• Project Period (FY): 2017-06-30 – 2020-03-31
• Keywords: 亀裂 / しわのパターン

Outline of Final Research Achievements

As a first step toward understanding of the interaction between periodic wrinkle pattern and crack shape from energy variation, we tried mathematical and numerical analysis of mathematical models together with some physical experiments of pattern generation. For viscoelastic materials like gel with which wrinkles and cracks are both observed, we studied variational structures of the Maxwell and Zener type viscoelastic models. As an extension of the classical Cahn-Hilliard equation which can produce periodic pattens, we also considered a generalized Cahn-Hilliard model by adding a new term which corresponds to an external dynamic stimulation or a crack dynamics. Moreover, as an approach from physical experiments, we performed a fracture experiment of an elastic sheet with wrinkle pattern and succeeded to obtain some scaling laws.

Free Research Field

応用数学

Academic Significance and Societal Importance of the Research Achievements

しわと裂け目のなすパターン形成の問題は、ナノテクノロジーや生物の発生、産業デザイン、氷河や断層・しゅう曲といった、非常に幅広い分野と関連して将来的に大きな意味を持つ可能性を秘めている。今回の研究では、しわなどのパターンと亀裂を同時に考えるために必要な簡略化された数理モデルや物理実験系を確立できたことで、将来的に様々な分野の問題と関連が生まれてくるものと期待される。