Mathematical Analysis on dynamic rupture problems and its seismological application

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03380
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tokyo University of Science
• Principal Investigator: Itou Hiromichi 東京理科大学, 理学部第二部数学科, 教授 (30400790)
• Co-Investigator(Kenkyū-buntansha): 田中 良巳 横浜国立大学, 大学院環境情報研究院, 教授 (10315830)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 動弾性体 / き裂 / 摩擦 / 接触問題 / 逆問題 / 非線形弾性体 / 粘弾性体 / 多孔性媒質

Outline of Final Research Achievements

In this study, we consider a mathematical model that incorporates the unique properties of the earthquake progression process, such as scale invariance and energy dissipation due to friction. As a result, a qualitative theory for an initial value boundary value problem with a given friction force and a generalized non-penetration condition (named SCD condition) on a fixed crack surface in a linear dynamic elastic body is obtained. As an application to seismology, an indication formula for the slip velocity was derived for a problem in which several friction conditions are imposed on a self-similarly expanding crack. Furthermore, the method for the inverse crack problems in elastic bodies was developed, and a mathematical theory for the crack problem was constructed for a new elastic model describing porous media such as concrete and rock.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

線形動弾性体におけるき裂問題に摩擦条件と接触条件を課した問題については未解決なことが多かったが、現象に見合ったき裂面における摩擦条件や接触条件を考察し、数学理論を発展させた。これらは今後、地震学で用いられている摩擦モデルの理論的解析に貢献できると考えられる。また、地震の特徴を組み込んだ数理モデルの解析解の導出は地震学における破壊伝播速度の決定機構の解明につながるものである。さらに、非破壊検査に関連するき裂の逆問題および多孔質弾性体を記述する新しい数理モデルにおけるき裂問題の数学解析で得られた成果は工学分野への波及効果が期待できる。