本研究主要以分子力學方法及能量等效模型來探討石墨烯的彈性性質及預測破壞應變，考慮一代表性單位晶格在施加平面負載下的變形，在分子力學法中僅考慮碳原子間鍵長與鍵角變化造成能量變化，分別以最小能量假設與均勻應變假設可求得穩態原子分佈，在小變形下由應變能變化可推得彈性性質，結果發現由最小能量假設與均勻應變假設求得的彈性性質並不相同，且在最小能量時，因代表性單位晶格內的原子為不均勻離散排列的關係，因此石墨烯內部的變形並不均勻，大加載下結合以臨界鍵長為判斷準則可用以預測石墨烯的破壞應變，並有效的發現石墨烯薄膜的彈性性質具平面等向性，但破壞則具方向性。
另外分別以彈簧及樑等有限元素進行碳原子間共價鍵作用的能量等效，嘗試結合代表性單位晶格的概念建立等效的修正樑元素，並與分子力學計算的結果相比較驗證，可發現修正樑元素較文獻中常採用的單一鍵等效方式更為接近分子力學的計算結果，並與分子力學法在最小能量假設下的結果一致。此外，分子力學方法與原子等效模型皆得到石墨烯的彈性性質呈現平面等向性，但在加載下內部鍵長變化卻有方向性，以臨界鍵長為破壞準則預測石墨烯之破壞應變，可發現扶手椅型石墨烯的破壞應變比鋸齒型的高，與文獻中以分子動力學模擬結果一致。
本研究結合了理論推導及有限元素法計算成功的探討石墨烯的力學性質，更發現文獻中常用的能量等效樑元素需加以修正才能提供正確的結果，等效元素的建立結合有限元素法計算能改善分子動力學受限於電腦計算能力的限制，更快速的探討石墨烯奈米帶或奈米碳管的機械性質受尺寸及螺旋性等的影響，將可幫助石墨烯或碳管元件的設計。

This research developed molecular mechanics method and energy equivalent model to investigate the elastic properties of graphene as well as to predict its fracture. In molecular mechanics method, a representative unit cell of graphene was considered and only the bond stretching and bending energies were considered assuming that the infinite graphene sheet was under uniform in-plane loading. The deformation of the graphene could be calculated under either the minimum energy or uniform strain assumptions. It was found that the deformation inside the representative unit cell is not uniform at minimum energy condition due to the non-uniform discrete distribution to the atoms. The fracture criteria based on critical bond length was used to predict the fracture strain of graphene. Moreover, the elastic properties were observed to be in-plane isotropic while the fracture strains were directional dependent.
The spring and beam elements were employed as replacements for covalent bond between atoms using the energy equivalent concept. Other than single bond equivalence as commonly suggested in the literature, an equivalence based on the representative unit cell was proposed and a modified beam element was suggested. The finite element calculations of the graphene under loading were performed and the results were compared with the one based on molecular mechanics method. It would found that the mechanical properties calculated using modified beam element were in good agreement with molecular mechanics method. The predicted fracture strains of armchair graphene were higher than the zigzag one, which was consistent with the molecular dynamics simulation results reported in the literature.
In this research, we successfully implemented the analytical calculation and finite element method to study the mechanical properties of graphene. It was concluded that the beam element proposed in the literature needs to be modified in order to provide correct results. With the modified equivalent element integrating with finite element analysis, we could easily extend the model size to overcome the computational obstacle usually encountered in molecular dynamics. We could effectively and efficiently investigate the size and chirality effects of graphene nanoribbons and carbon nanotubes to benefit the design of nano-devices.

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