本研究以元素釋放法(Element Free Method, EFM)來處理非均質材料之固體力學的彈性問題，以探討元素釋放法分析不同材料界面(interface)之可行性。此法基於「移動式最小平方(Moving Least Squares, MLS)」內插的觀念來處理定義域內節點資料之一種無網格(meshless)近似的數值方法。因為只需節點資料，而不需節點與元素間之關聯限制，在不妨礙元素與元素的關聯性下來建立形狀函數(shape function)。所以在需要利用「適應性方法(adaptive method)」處理問題時較有限元素法具有更大的靈活性。故在處理漸進式裂縫及不同材料界面之固體-固體的相態轉換(solid-solid phase transformations)等問題具有相當好的成效。 由於元素釋放法可以不受關聯條件的限制，以適用區域的節點資料配合MLS內插觀念推導而出的形狀函數與近似位移函數，不具有Kronecker delta的性質。因此，在界面上施加Lagrange multipliers來滿足不同材料界面的位移(displacements)與曳力(tractions)連續條件，以確保界面上的連續性。 對界面附近節點資料的處理，本文引入簡易的節點選取修正方法，並配合無網格的特性，在界面附近區域任意增加或移動節點，來改善求解的精度。因此，當位移梯度(應變場)越過不同材料界面時，元素釋放法能夠解出具高階連續的變化場，使得位移梯度能滿足跳躍(jump)現象，故在處理非均質材料的問題時，元素釋放法可容易履行應力與應變場的組構定律(constitutive laws)。 The purpose of this research was to deal with the static problems of inhomogeneous materials by element free method (EFM), and to discuss the practicable method about analyzing different material interfaces by EFM. The EF method utilized “moving-least-squares” (MLS) interpolants in which the nodes were only required, unencumbered by elements and elemental connectivity, to construct the shape functions. It was more flexible in solving the problems with EF method than finite element method when an “adaptive method” was needed. Excellent results could be obtained with this method when dealing with the crack propagation and the solid-solid phase transformation problems. Since the EF method was not restricted by connectivity, the shape functions and approximate displacement functions, which deduced from the nodal data within the adapted area and the MLS concept, had no characteristics of Kronecker delta. To ensure the continuing conditions on the interfaces, the Lagrangian multipliers were enforced on the interfaces such that the displacements and the tractions between interfaces of materials were fulfilled. A straightforward modification to the EF method was introduced in this research that enabled the EF method to solve problems involving material discontinuities. Therefore, as the displacement gradients crossing the interfaces of materials, the high-order continuous variation field could be obtained with the EF method such that the jump of displacement gradients could be simulated. EF method satisfied the constitutive laws of stress and strain in dealing with the problems of inhomogeneous materials.
