由於要在實驗室對於奈米材料進行其材料性質之量測是非常困難的事，所以有很多針對奈米材料的理論或是數值研究方法相應而生。在此篇文章中我們也提出了兩種方法去獲得奈米材料的勁度、強度以及韌度，其一為分子連體模型，其二是非線性有限元素法分析。
整個分子連體模型主要的關鍵是將在分子力學裡面的勢能跟連體力學中的應變能做等效，針對欲取得的性質去施加合適的位移場，即可求得位移前後的整體能量差，利用這個能量差去計算想要得到的性質。如果所選用的勢能函數為調和函數，更可以直接推導出奈米材料彈性性質的明式解。文中我們使用了脆性材料的石墨烯片以及奈米碳管，還有延展性材料的單晶銅當作例子，也比較了不同的大小，結構，還有不同勢能函數的選取對於結果的影響。
在非線性有限元素法分析中，利用樑元素去模擬原子之間的共價鍵，使得樑元素在施加拉伸或是彎矩力後產生的應變能變化等於分子力學裡的拉伸和彎矩的能量，以此樑結構的非線性有限元素模型來取得奈米材料的勁度和強度。欲取得奈米材料的韌度，必須先以此樑元素結構計算出整體的應力應變曲線，將此曲線輸入連體模型，例如平面或是殼元素，再由此連體模型中的能量釋放率計算出破壞韌度。而此方法對於建構金屬鍵結的延展性材料較為困難，所以我們僅使用石墨烯片和奈米碳管為例子。
總結來說，我們提出了一套全新的方法來預測奈米材料的材料性質，以及改善了前人所用來預測的有限元素法，用此兩種方法與文獻中裡用其他方法求得的材料性質做比較，都得到了合理的結果。

Due to the difficulties of the experimental works on the nanomaterials, many efforts have been recently put on the estimation of their mechanical properties through theoretical and numerical simulations. A semi-analytical method called molecular-continuum (MC) model and a molecular dynamics based (MD-based) nonlinear finite element simulation method are proposed to estimate the stiffness, strength, and fracture toughness of nanomaterials in this thesis.
The MC model is developed by combining the concept of molecular dynamics and continuum mechanics, in which the potential energy describing the interactions of atoms is not restricted to the harmonic potential function, and hence its deriving stress-strain relation is not restricted to be linear. Unlike the usual test performed by applying forces, in this model a displacement field is employed in the representative volume element of a specimen. For predicting the stiffness and strength, the uniform strain field is applied. To estimate fracture toughness, a parameter called the strain intensity factor is introduced, and the near tip solution of linear elastic fracture mechanics rewritten in terms of strain intensity factor is used to locate the atoms of the cracked specimen. Through this model, the Young’s moduli, Poisson’s ratios, and shear modulus of graphene and carbon nanotubes (CNTs) for armchair, zigzag, and chiral types can all be written as simple rational functions in which the dependence of radius, chiral angle, and thickness can be observed clearly from the explicit closed-form expressions by using the harmonic potential functions. Moreover, according to the proposed molecular-continuum model, an integrated symbolic and numerical computational scheme (ISNC) is established to deal with the general nanomaterials. The stiffness defined based upon the initial linear region, and the ultimate strength, yield strength and, mode I/mode II toughness occurring at the later period of the materials can all be predicted. Identical results of the closed-form solutions and ISNC verify the correctness of our derivation. Comparison of the results obtained by other methods or different potential energy functions further justifies the simplicity, validity, and efficiency of the proposed model.
The MD-based nonlinear finite element simulation method to estimate the mechanical properties of nanomaterials is developed by using a frame-like structure to construct the molecular model. The bond stretching energy and bond angle bending energy in molecular dynamics can be simulated by the analogous concept in finite element approach, that is, these strain energies of a beam element caused by tensile stress and bending moment, respectively. The nonlinear modified Morse potential energy is selected and used to calculate the nonlinear stress-strain relation and sectional area of the beam element in our simulation. As the prediction by the MC model, a prescribed-displacement condition is applied in this method. The energy release rate is used to predict the fracture toughness of nanomaterials. Since the value of crack increment must be extremely smaller than the value of crack length for calculating the energy release rate, the continuum model of nanomaterials with nonlinear elastic property is constructed based upon the properties estimated by the beam element model. The mechanical properties for both graphene and CNTs in different types and sizes are presented to illustrate the feasibility of this method.
To verify the correctness of two methods, the existing results provided by the other experimental and numerical methods are compared and discussed in this study. The comparison shows that the results estimated by these models fall in the reasonable range.

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