Modelagem de nanomateriais e nanoestruturas pelo método não linear de elementos finitos em escala atômica

Orientadores: Euclides de Mesquita Neto, Nimalsiri Dharmakeerthi RajapakseTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecânicaResumo: Este trabalho apresenta o comportamento mecânico de materiais em nanoescala aplicando o método de elementos finitos de escala atômica (AFEM), proposto por Liu et al., (2004), utilizando diferentes campos de potencial atômico. O método AFEM é formulado com base no conceito de campos de potenciais que descrevem a interação entre átomos. Os potenciais considerados neste estudo são o potencial de Lennard-Jones (Jones, 1924), o potencial de Tersoff (Tersoff, 1987), e o potencial REBO de segunda geração (Second-Generation Reactive Empirical Bond Order) (Brenner et al., 2002). O objetivo de considerar e implementar o potencial de Lennard-Jones é introduzir e discutir questões fundamentais sobre a aplicação do método AFEM, como a convergência, e o tratamento consistente dos átomos nas bordas (contornos). Propomos uma abordagem para os elementos atômicos uni- e bidimensionais, para a inclusão das condições de contorno e também para explorar a ideia de relacionar o tamanho do elemento atômico ao conceito de raio de corte aplicado na Dinâmica Molecular (MD). Em oposição ao caráter local dos elementos dentro do quadro do método de elementos finitos (FEM) clássico, os potenciais utilizados no método AFEM têm caráter não-local. Depois de examinar as questões fundamentais da formulação do método AFEM considerando o potencial de Leonard-Jones, a análise é estendida a potenciais interatômicos mais complexos, como o potencial de Tersoff e o potencial REBO de segunda geração, os quais são capazes de descrever o comportamento mecânico de folhas de grafeno. Serão consideradas folhas de grafeno com e sem defeitos, nas direções armchair e zigzag. Será analisada pelo método AFEM a influência da presença de defeitos no comportamento mecânico da folha de grafeno quando submetida a um carregamento uniaxial. Os resultados numéricos obtidos pelo método AFEM, tais como, as relações força-deformação, são comparados com a simulação MD obtida a partir do software LAMMPS, e com os resultados apresentados na literatura. Precisão, convergência e estabilidade do AFEM serão comparadas com MD. Como a formulação básica do método AFEM é não-linear, sendo assim, o método de Newton-Raphson é usado para executar as iterações. As relações força-deformação e tensão-deformação são obtidas numericamente considerando malhas de diferentes tamanhos, defeitos e direçõesAbstract: This work presents the mechanical behavior of materials at the nanoscale by applying the atomic-scale finite element method (AFEM), proposed by Liu et al., (2004), using different atomic potential fields. AFEM is formulated based on the concept of potentials describing the interaction among atoms. The potentials considered in this study are the Lennard-Jones potential (Jones, 1924), Tersoff potential (Tersoff, 1987) and second-generation Reactive Empirical Bond Order (REBO) potential (Brenner et al., 2002). The purpose of considering and implementing the Lennard-Jones potential is to introduce and discuss fundamental issues about the application of the AFEM such as convergence and consistent treatment of boundary atoms. We propose a new arrangement for the atomic finite elements in one and two dimensions, for the inclusion of the boundary conditions and also to exploit the idea of linking the size of the atomic-scale element to the concept of cut-off radius applied in Molecular Dynamics (MD) simulations. Opposite to the local character of elements within the framework of classical FEM, the potentials used to generate the AFEM have non-local character. After examining the key issues of the formulation of AFEM using Leonard-Jones potential, the analysis is extended to more complex interatomic potentials such as Tersoff potential and second-generation REBO potential, which can describe the mechanical behavior of graphene sheets. Pristine and graphene sheets with cracks with armchair and zigzag edges are considered. The influence of vacancy defects on mechanical behavior under uniaxial tensile loading is analyzed by the AFEM. The numerical results obtained from AFEM such as the force-strain relations are compared with the MD simulation obtained from LAMMPS software, and with the results presented in the literature. Accuracy, convergence and stability of the AFEM compared to MD are examined. As the basic formulation of the AFEM is non-linear, the Newton-Raphson method is used to perform the iterations. Force-strain and stress-strain relations are obtained numerically for meshes of different sizes, defects and orientationsDoutoradoMecanica de Solidos e Projeto MecanicoDoutora em Engenharia Mecânica141173/2013-0CNP

Quasi-continuum Non-local Plate and Shell Models of Carbon-Based 2D Nanomaterials

Non-local plate and shell models have attracted much interest in the area of grap-hene and carbon nanotube simulations. This work explores further into this area and aims to provide more accurate and reliable non-local modeling methods to graphene and carbon nanotubes. At first, a semi-analytical model for determining the equilibrium configuration of single wall carbon nanotubes is presented. By taking advantage of the symmetry charac-teristics, a carbon nanotube structure is represented by five independent variables. A line search optimization procedure is employed to determine the equilibrium values of these variables by minimizing the potential energy. With the equilibrium configuration obtai-ned, the semi-analytical model enables a straightforward calculation of the radial breathing mode frequency of carbon nanotubes. The radius and radial breathing mode frequency results obtained from the semi-analytical approach are compared with those from molecu-lar dynamics and ab initio calculations. The results demonstrate that the semi-analytical approach offers an efficient and accurate way to determine those properties. Next, we investigate several issues in the local and non-local plate models of sin-gle layer graphene sheets. The issues include the ambiguity of the plate thickness in the moment-curvature relation, the definition of clamped boundary condition at graphene ed-ges, and the value of the non-local parameter. For error analysis, the results obtained from a REBO potential based atomic lattice mechanics model are used as reference results. Errors of the plate models are analyzed and remedies are proposed within the framework of the non-local plate model. Numerical results of static and modal analysis of graphene are presented to demonstrate the effectiveness of the remedies. In the last part of this work, a non-local finite element shell model is established for single-walled carbon nanotubes. Based on the accurately relaxed radius, bond lengths and angles obtained from the semi-analytical model, it is possible to calculate more accurate elastic constants directly from the interatomic potentials. Then through the combination of the classical first order shell theory, the non-local elasticity, and the potential-based elastic properties, a more accurate shell representation of single wall carbon nanotubes is establis-hed. The improvement in accuracy is demonstrated by comparing the spectral frequency analysis and dispersion relation results with those obtained from lattice mechanics and mo-lecular dynamics simulations, respectively

A new shell formulation for graphene structures based on existing ab-initio data

An existing hyperelastic membrane model for graphene calibrated from ab-initio data (Kumar and Parks, 2014) is adapted to curvilinear coordinates and extended to a rotation-free shell formulation based on isogeometric finite elements. Therefore, the membrane model is extended by a hyperelastic bending model that reflects the ab-inito data of Kudin et al. (2001). The proposed formulation can be implemented straight-forwardly into an existing finite element package, since it does not require the description of molecular interactions. It thus circumvents the use of interatomic potentials that tend to be less accurate than ab-initio data. The proposed shell formulation is verified and analyzed by a set of simple test cases. The results are in agreement to analytical solutions and satisfy the FE patch test. The performance of the shell formulation for graphene structures is illustrated by several numerical examples. The considered examples are indentation and peeling of graphene and torsion, bending and axial stretch of carbon nanotubes. Adhesive substrates are modeled by the Lennard-Jones potential and a coarse grained contact model. In principle, the proposed formulation can be extended to other 2D materials.Comment: New examples are added and some typos are removed. The previous results are unchanged, International Journal of Solids and Structures (2017

A Review on Mechanics and Mechanical Properties of 2D Materials - Graphene and Beyond

Since the first successful synthesis of graphene just over a decade ago, a variety of two-dimensional (2D) materials (e.g., transition metal-dichalcogenides, hexagonal boron-nitride, etc.) have been discovered. Among the many unique and attractive properties of 2D materials, mechanical properties play important roles in manufacturing, integration and performance for their potential applications. Mechanics is indispensable in the study of mechanical properties, both experimentally and theoretically. The coupling between the mechanical and other physical properties (thermal, electronic, optical) is also of great interest in exploring novel applications, where mechanics has to be combined with condensed matter physics to establish a scalable theoretical framework. Moreover, mechanical interactions between 2D materials and various substrate materials are essential for integrated device applications of 2D materials, for which the mechanics of interfaces (adhesion and friction) has to be developed for the 2D materials. Here we review recent theoretical and experimental works related to mechanics and mechanical properties of 2D materials. While graphene is the most studied 2D material to date, we expect continual growth of interest in the mechanics of other 2D materials beyond graphene