Study on variational inequality model of fracture

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K03356
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Musashino University
• Principal Investigator: Takaishi Takshi 武蔵野大学, 工学部, 教授 (00268666)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: 変分不等式 / フェーズフィールド / き裂進展 / 数理モデル / 数値シミュレーション

Outline of Final Research Achievements

Using a phase-field crack propagation model, it was confirmed that numerical simulations of two-dimensional opening crack propagation can be performed at reasonable computational costs by using the IPOPT package in FreeFEM as a variational inequality.
In addition, the external contact conditions were also modeled as variational inequalities, and for cracks that occur due to material failure due to compression, a numerical simulation was performed using a model in which the unilateral contact condition was applied to a gradient flow as the internal contact condition.It was found that when material failure occurs during a fracture in a cutting machine, the crack cross-section shape deforms significantly when the blade spacing is greater than a certain value.

Free Research Field

応用数学

Academic Significance and Societal Importance of the Research Achievements

本研究では、開口モードと圧縮モードの両方に対応した変分不等式によるフェーズフィールドき裂進展/破壊数理モデルを構築し、数値シミュレーションによって現実の材料破壊現象が再現できるか検証した。材料の圧縮による破壊に伴う非現実的な変形を避けるためにモデルを修正し、変位や変位勾配に基づく拘束条件を用いたエネルギーの変分不等式を導出することで、モデルの有効性を確認した。
計算用のモデルではなく、数学的に解析可能なモデルを構築することで、その解の性質を明らかにする道筋をつけるのみならず、シンプルなモデル化を行うことで、さらなる拡張の可能性を見出すことが期待される。