Microcontinuum Analysis of Granular Materials

材料之微觀結構對其宏觀行為有著顯著之影響者，一般統稱此類材料為複雜材料。近年來，此類材料在各個領域中均有相當廣泛之應用。顆粒材料屬於複雜材料的一種，為日常生活中隨處可見的材料類型之一，然而，顆粒材料多變的宏觀力學特性，卻也增加分析之難度。眾多文獻提及局部膨脹與旋轉這兩個微結構效應，於顆粒材料之運動變形過程中，均扮演重要之角色。而在顆粒材料分析之連續體理論中，得以處理微結構效應且較為常見的有：針對局部膨脹效應之 Goodman-Cowin(GC) 理論與分析局部旋轉之 Cosserat 形式連續體理論。本論文將以微連續體力學理論為工具，分析顆粒材料之力學行為，以此論述微連續體力學理論於分析複雜材料時之優勢，並提供此理論一可能之應用範例。 本文一共可分為三個部分，首先陳述微連續體力學之架構與相關場量。將微觀運動相關之場量分解為體膨脹、剪切與旋轉三部分後，以此定義出微連體力學之七種不同分支。此些分支可分別處理不同微觀運動機制，並用以分析大部分複雜材料之力學行為。其次採用微連續體分支中，微觀場量限制為僅具有體膨脹部分之微膨脹連續體理論，將其應用於分析乾性顆粒材料之上，獲得一可描述顆粒材料微觀膨脹效應之分析模型。將其與GC理論取得連結後，不僅證明兩理論幾乎完全一致，亦可賦予 GC 理論中各變數與方程式完整之物理詮釋。最後以微連續體力學之另一分支：微拉伸連續體的概念出發，將 GC 與 Cosserat 形式連續體理論所分別考量之微結構效應，合併於同一連續體理論中。論文中建構一雙軸試驗離散元素模擬，並計算微拉伸連續體之微迴旋張量中，對應於微觀膨脹效應之體膨脹部分與微觀旋轉效應之旋轉部分，以此二場量討論微拉伸連續體作為顆粒材料基本分析架構之可行性與必要性。Materials with microstructure characteristics are called complex materials. Nowadays, these kinds of materials have been applied widely in several technologies. Although granular materials are everywhere in our daily life, they are kind of complex materials, which are difficult to be analyzed for their multiple macroscopic mechanical behaviors. Recently, many studies mentioned that two microstructure effects, local dilatant and rotational behavior, play significant roles in the process of deformation for granular materials. There are two analytical approaches widely used in the continuum theory of granular materials to deal with the behavior of microstructure: one of which is the Goodman-Cowin (GC) theory for dealing with the local dilatant effect, the other is the Cosserat type continua on the analysis of the local rotational behavior of granular assemblies. The key issue of this study is to analyze the mechanical behavior of granular materials by using the microcontinuum theory. Based on this work, we regard the microcontinuum theory to have a considerably predominance to analyze complex materials and propose one of the possible application for this theory. This thesis is divided into three parts. Firstly, we clearly describe the framework and the related quantities of microcontinuum theory. By simply decomposing the various field quantities of the microcontinuum into the dilatancy, shearing, and rotational parts, a microcontinuum can be classified into seven classes. Each class has its micro-deformation and can be used to characterize the behavior of different materials. Secondly, we used one of the microcontinuum classes, called a microdilatation continuum, in which only the dilatant motion is taken into account, to analyze dry granular materials. Furthermore, we presented the derivation of GC theory by using this model and render a newly physical meaning for equations and quantities of GC theory from the aspect of microcontinuum. Moreover, we model the granular materials by another microcontinuum class, called microstretch continuum, which include the two microstructure effects concerned by the GC theory and the Cosserat type continua. A DEM simulation of a granular system under a biaxial compression is constructed. By calculating the bulk part and the rotational part of the gyration tensor, which respectively measure the local dilatancy and the local rotation, we verify the feasibility and necessity of the microstretch continuum as a basic framework of analysis granular assembly.謝誌................................................i文摘要..........................................iii文摘要...........................................iv、緒論................................................1.1 前言................................................1.2 文獻回顧............................................2.3 研究動機與論文架構.................................26、數學符號與張量演算規則.............................29.1 數學符號通則.......................................29.2 張量演算符號.......................................30、微連續體力學：微形連續體...........................35.1 微形連續體之運動與變形描述.........................35.2 場量之時間導數與應變率.............................41.3 微觀場量與宏觀場量之關係...........................44.4 微形連續體之平衡方程式.............................53.5 討論：微形連續體平衡方程式.........................65、連續體力學體系之分支...............................67.1 定義連續體力學體系之分支...........................68.2 場量分解法則.......................................71.3 微連續體力學分支之微迴旋、微應變與微應變率張量.....75.4 微連續體力學分支之平衡方程式.......................83.5 討論...............................................91、顆粒材料之Goodman-Cowin理論與微膨脹連續體分析模型..95.1 Goodman-Cowin 理論介紹.............................95.2 Modified Goodman-Cowin 理論.......................102.3 顆粒材料之微膨脹連續體近似模型....................107.4 討論..............................................115、顆粒材料之剪切帶-雙軸試驗模擬.....................121.1 雙軸試驗模擬配置..................................122.2 雙軸壓縮模擬試驗之定量計算........................133.3 雙軸試驗模擬結果與討論............................142.4 結論與建議........................................171、結論與展望........................................173.1 結論..............................................173.2 研究展望..........................................174考文獻..............................................176錄A：符號表.........................................184錄B：微連體力學之熱力學分析.........................189錄C：微形連續體之第二組應變張量.....................198錄D：Eringen 場量分解對照表.........................200錄E：離散元素法軟體:PFC-2D..........................20