孔隙材料的微觀力學建模與崩塌面

Abstract

先前的研究已經對孔隙材料的靜態行為進行了分析，但尚不全面。本論文旨在補足孔隙材料各項力學分析，涵蓋其動態、黏彈性及彈塑性行為。在動態分析部分，本研究從材料層次引入力偶應力，根據力學層次概念，並透過若干適當假設，推導出孔隙材料單元格（unit cell）層次的動態控制方程式，並考量簡諧波求出解析解，進而探討不同形狀及柏松比下的尺寸效應意義。通過兩種不同的輸入形式分析其響應，並引入 Bloch 定理，求出孔隙材料在特定頻率下的帶隙。結果顯示，不同形狀和柏松比對動態行為有顯著影響，且在特定頻率下存在明顯的帶隙現象。在黏彈性分析中，基於先前的研究，探討了不同微結構幾何對孔隙材料能量吸收及耗散的影響。結果表明，特定的幾何結構在能量吸收和耗散方面具有明顯優勢，為設計提供了一種新的想法。最後，在彈塑性分析部分，本研究採用片段線性模式來描述孔隙材料在多維應力下的行為，並進行了與路徑相關的增量分析。在給定已知路徑下，分析孔隙材料的行為直至崩塌點。此外，在孔隙材料極限分析上，本研究打破了既有的路徑依賴分析的概念，提出了一種與路徑無關並且可以一步求得所有崩塌點形成崩塌面的極限分析理論，並透過增量分析驗證其正確性。結果顯示，新的理論能有效預測材料在多維應力下的崩塌行為。

Previous research has already conducted an analysis of the static behavior of cellular materials, but it remains incomplete. This thesis aims to complement the mechanical analysis of cellular materials, covering their dynamic, viscoelastic, and elastoplastic behaviors. In the dynamic analysis section, this study introduces couple stress at the material point. Based on the concept of mechanical hierarchy and through several appropriate assumptions, dynamic governing equations of the unit cell of cellular materials are derived. By considering harmonic waves, analytical solutions are obtained to explore the significance of size effects under different shapes and Poisson's ratios. The response under two different kinds of input is analyzed, and Bloch's theorem is adopted to determine the band gaps of cellular materials at specific frequencies. The results show that different shapes and Poisson's ratios significantly affect dynamic behavior, and there are notable band gap phenomena at specific frequencies. In the viscoelastic analysis, based on our previous research, the impact of different microstructural geometries on the energy absorption and dissipation of cellular materials is investigated. The results indicate that specific geometries have significant advantages in energy absorption and dissipation, providing new ideas for design. Finally, in the elastoplastic analysis section, this study uses piecewise linear multi-yield surface model to describe the behavior of cellular materials under multidimensional stress and conducts path-dependent incremental analysis. Given a known path, the behavior of cellular materials is analyzed until the collapse point. Additionally, in the limit analysis of cellular materials, this study breaks away from the existing concept of path-dependent analysis and proposes an ultimate analysis theory that is path-independent and can obtain all collapse points forming the collapse surface in one step. Incremental analysis is used to verify its correctness. The results show that the new theory can effectively predict the collapse behavior of materials under multidimensional stress.