探討缺陷石墨板之局部性質

Abstract

本研究目的在探討具有自由面和裂紋之石墨板受到單軸拉伸載重下之局部性質。利用分子動力模擬，可以得到石墨板受到單軸拉伸外力下，各個碳原子的平衡位置。接著利用Hardy，Lutsko以及Tsai所提出的局部應力公式，可以計算出石墨板之局部應力場。對於具有自由面的石墨板，研究結果顯示在無應力(stress free)的條件下且凡德瓦爾(van der Waals)力存在時，石墨板之自由面存在著壓縮應力，其他部分則受到些微的拉伸應力;而在無應力(stress free)且凡德瓦爾力不存在的條件下，石墨板的每個位置均沒有受力。 對於具有裂紋之石墨板，同樣先利用分子動力模擬得到拉伸載重下碳原子的平衡位置。接著利用局部應力公式以及線彈性破壞力學(linear elastic fracture mechanics)，有限元素法(finite element method)和非局部彈性理論(non-local elasticity theory)來探討裂紋端附近的應力場。研究結果顯示，線彈性破壞力學以及有限元素法的裂紋應力場皆顯示出應力奇異性(stress singularity)，且在小裂紋情況下，線彈性破壞力學與有限元素法的應力場有明顯的差異；而Hardy的應力場在靠近裂紋附近顯示出非局部(non-local)性質，所得到的最大應力與非局部理論解非常近似。另外由Hardy的應力場，也能直接從最大應力推導出石墨板之破壞性質。分析結果顯示，由Hardy應力場所得到的應力強度因子(stress intensity factor)與有限元素和線彈性破壞力學一致；而破裂韌性(fracture toughness)在裂紋長度小於40個晶格時，會隨著裂紋長度變小而降低；在裂紋長度大於40個晶格的時候，破裂韌性才會趨於一個常數定值，此結果與非局部理論所推導出的結果一致。因此，應力強度因子可有效地描述裂紋附近的應力場，而破裂韌性並不適合用來描述具有小裂紋的破壞性質。

This paper aims to investigate the local properties of graphene sheet with free surface or central cracks subjected to uniaxial loading. The equilibrium configuration of the graphene sheet subjected to uniaxial loading was determined through molecular dynamic (MD) simulation. For the graphene with free surfaces, three local stress formulations, i.e., Hardy, Lutsko and Tsai stress, were employed to calculate the local stress distribution near the free surfaces. It was found that when van der Waals force was present, only Hardy stress expression can describe the stress field effectively. Results indicated that the graphene sustained compressive stress on the edge and tensile stress in the interior at stress free state. On the other hand, when van der Waals force was absent, both Hardy and Tsai stress can describe the stress distribution accurately. Results showed that the graphene sustained zero stress at every position at stress free state such that the bond length did not alter. Regarding the graphene with central cracks subjected to remote tensile loading, both the atomistic and continuum stress were employed to investigate the local stress distribution near the crack tip. For the discrete graphene sheet, Hardy and Tsai stress were adopted to calculate the near-tip stress field of the graphene in the absence of van der Waals interaction. For the continuum models, finite element method was used to calculate the stress distribution. In order to describe the numerical results, two analytical solutions were incorporated, such as linear elastic fracture mechanics (LEFM) and the non-local elasticity solution. Results showed that for both LEFM and FEM solutions, the stress fields demonstrated the stress singularity near the crack tip. In addition, it was found LEFM solution cannot describe the stress accurately when the crack length is small. On the other hand, atomistic stress such as Hardy and Tsai stress yielded a more reasonable finite stress near the tip. It was found that the maximum stress obtained from Hardy’s formulation was in agreement with the non-local elasticity solution, whereas Tsai’s maximum stress is larger than the analytical solution; therefore only Hardy stress field exhibited non-local attribute near the crack tip. Based on the maximum stress hypotheses, the fracture properties such as stress intensity factor and fracture toughness were deduced directly from local stress field. Results indicated that stress intensity factor derived from Hardy stress field was in agreement with the FEM and the actual solution of LEFM. On the other hand, the fracture toughness defined in LEFM is found to be cracksize dependent when the crack length is small for discrete models. For crack lengths below 40 lattices, the fracture toughness would decrease with the decrease of the crack lengths; the result was in agreement with the non-local elasticity solution. Therefore, the fracture toughness defined as a material property may not be suitable for describing fracture with small cracks.