| 摘要: | 長久以來，有限元素法（Finite Element Method, FEM）在固體力學問題的分析中得到了廣泛應用與驗證。然而，當面臨爆炸、高速撞擊、複雜幾何結構、裂紋動態擴展等大變形問題時，有限元素法常會遇到模型建構困難和網格需要重新分配等挑戰，而物質點法具有易於建構模型和處理大變形與運動問題的特性，近年來已成為處理前述問題的熱門數值方法。
本論文主要利用廣義插值物質點法( Generalized Interpolation Material Point Method, GIMP) 探討物質點法在極限狀態下和結構穩定性兩類問題中的適用性和可靠性。極限狀態問題包含爆炸和高速撞擊，這些情況涉及劇烈的大變形和高應變率，由於廣義插值物質點法能夠很好地處理這些複雜的現象，因此在這些極端條件下顯示出其獨特的優勢。此外，結構不穩定問題，如由微小偏心引發的挫屈現象，也是一個難以模擬的挑戰。
在研究方法上，針對極限狀態下的高應變率問題，本研究進行了不同速度和網格劃分的收斂性分析；對於結構穩定性問題，則採用柱挫屈問題進行數值模擬，研究了不同偏心率對試體荷載反應的影響，並比較了三種應力截取方法之準確性。
結果顯示，在高應變率條件下，確保變形速度低於材料應力波之波速，物質點法能夠準確模擬材料的應力-應變行為。隨著網格細密程度的提高，模擬結果更接近理論解，但計算複雜性和時間成本也隨之增加。此外，研究表明在不同偏心率下，試體固接處的反力曲線更接近理論解，證實了物質點法在結構挫屈現象分析中的可靠性。
本研究證實了物質點法在高應變率和穩定性問題中的應用潛力，並為未來的數值模擬提供了參考建議，有助於提高工程設計和分析的準確性和可靠性。
;The Finite Element Method (FEM) has long been widely applied and validated in the analysis of solid mechanics problems. However, when faced with large deformation problems such as explosions, high-speed impacts, complex geometries, and dynamic crack propagation, FEM often encounters challenges such as model construction difficulties and the need for remeshing. In recent years, the Material Point Method (MPM) has become a popular numerical method for addressing these issues due to its ease of model construction and its ability to handle large deformation and motion problems.
This thesis primarily utilizes the Generalized Interpolation Material Point Method (GIMP) to explore the applicability and reliability of MPM in two types of problems: extreme conditions and structural stability. Extreme condition problems include explosions and high-speed impacts, which involve severe large deformations and high strain rates. Due to GIMP′s capability to effectively handle these complex phenomena, it demonstrates unique advantages under such extreme conditions. Additionally, structural instability problems, such as buckling induced by slight eccentricity, also pose a significant challenge for simulation.
In terms of research methods, this study conducts convergence analysis with different speeds and mesh divisions for high strain rate problems under extreme conditions. For structural stability problems, the buckling of columns is numerically simulated to investigate the impact of different eccentricities on the load response of specimens, and the accuracy of three stress extraction methods is compared.
The results show that under high strain rate conditions, ensuring the deformation speed is lower than the material′s stress wave velocity allows MPM to accurately simulate the material′s stress-strain behavior. As the mesh refinement increases, the simulation results become closer to theoretical solutions, although computational complexity and time costs also increase. Furthermore, the study indicates that under different eccentricities, the reaction force curves at the fixed ends of the specimens closely match theoretical solutions, confirming the reliability of MPM in analyzing structural buckling phenomena.
This research confirms the application potential of MPM in high strain rate and stability problems and provides reference suggestions for future numerical simulations, contributing to improving the accuracy and reliability of engineering design and analysis. |
