Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach

Analytical formulations are presented to predict the elastic moduli of graphene sheets and carbon nanotubes using a linkage between lattice molecular structure and equivalent discrete frame structure. The obtained results for a graphene sheet show an isotropic behavior, in contrast to limited molecular dynamic simulations. Young’s modulus of CNT represents a high dependency of stiffness on tube thickness, while dependency on tube diameter is more tangible for smaller tube diameters. The presented closed-form solution provides an insight to evaluate finite element models constructed by beam elements. The results are in a good agreement with published data and experimental results.     

Chirality-dependent anisotropic elastic properties of a monolayer graphene nanosheet

An analytical approach is presented to predict the elastic properties of a monolayer graphene nanosheet based on interatomic potential energy and continuum mechanics. The elastic extension and torsional springs are utilized to simulate the stretching and angle variation of carbon-carbon bond, respectively. The constitutive equation of the graphene nanosheet is derived by using the strain energy density, and the analytical formulations for nonzero elastic constants are obtained. The in-plane elastic properties of the monolayer graphene nanosheet are proved to be anisotropic. In addition, Young's moduli, Poisson's ratios and shear modulus of the monolayer graphene nanosheet are calculated according to the force constants derived from Morse potential and AMBER force field, respectively, and they were proved to be chirality-dependent. The comparison with experimental results shows a very agreement.     

A new REBO potential based atomistic structural model for graphene sheets

A new atomistic structural model is developed here for graphene sheets based on the stiffnesses from the REBO potential. Using this model, the flexural vibration natural frequencies and buckling loads of rectangular single-layer graphene sheets of different sizes, chiralities and boundary conditions are calculated. The newly developed atomistic structural model is verified by comparing the calculated fundamental natural frequencies for small-sized graphene sheets with those obtained from ab initio density functional theory (DFT) frequency analysis. The vibration and buckling analysis results are also compared with those of an earlier atomistic structural model based on the AMBER potential as well as the equivalent continuum model for graphene sheets. Through this study, it is observed that graphene sheets display very slight anisotropic characteristics in flexural vibration and buckling. Also, it is shown that the atomistic structural model cannot be replaced by a classical equivalent continuum model such as a plate model. Most significantly, we verify that the new atomistic structural model based on the REBO potential predicts more accurate natural frequencies and buckling loads for graphene sheets, which are considerably lower than those predicted by the earlier atomistic structural model based on the AMBER potential.     

Models for the elastic deformation of honeycombs

A theoretical model has been developed for predicting the elastic constants of honeycombs based on the deformation of the honeycomb cells by flexure, stretching and hinging. This is an extension of earlier work based on flexure alone. The model has been used to derive expressions for the tensile moduli, shear moduli and Poisson's ratios. Examples are given of structures with a negative Poisson's ratio. It is shown how the properties can be tailored by varying the relative magnitudes of the force constants for the different deformation mechanisms. Off-axis elastic constants are also calculated and it is shown how the moduli and Poisson's ratios vary with applied loading direction. Depending on the geometry of the honeycomb the properties may be isotropie (for regular hexagons) or extremely anisotropic. Again, the degree of anisotropy is also affected by the relative magnitude of the force constants for the three deformation mechanisms.     

Numerical derivation of homogenized dynamic masonry material properties with strain rate effects

Masonry is a composite material composed of bricks and mortar disposed in a regular arrangement. It is commonly used as load bearing or partition walls in building structures. Owing to limitations of computer power, detailed distinctive modelling of brick and mortar of a realistic masonry structure or a structure with masonry infilled walls is usually not possible. Moreover, no dynamic masonry material model can be found in the open literature. Dynamic masonry material properties are important for an accurate prediction of masonry failure and fragmentation under dynamic loads. In this paper, a continuum damage model with strain rate effect is developed for masonry materials based on the homogenization method. The equivalent elastic properties, strength envelope and dynamic increase factors (DIFs) of strength and moduli for the homogenized masonry material are numerically derived from the simulated responses of a representative volume element (RVE). A numerical model of an RVE is analyzed with detailed distinctive modelling of brick and mortar with their respective dynamic material properties obtained from laboratory tests. The homogenized material model can be used to analyse large-scale masonry structures subjected to dynamic loading.     

A continuum mechanics framework and a constitutive model for predicting the orthotropic elastic properties of microtubules

This paper presents a theoretical framework for modeling the orthotropic elastic properties of microtubules. We propose a constitutive model to describe the detailed microscale information and continuum properties of the microtubules. The microtubule is viewed as being transformed from an equivalent planar structure, a fictitious-bond is introduced to evaluate the system energy and related with deformation gradients, and a representative unit cell is considered to bridge the microscale energy and the continuum strain energy. In a representative unit cell, the tubulin monomers and guanosine molecules are treated as spheroids, and the fictitious-bond vectors are evaluated through the higher-order Cauchy–Born rule. To deal with this polyatomic bio-composite structure that has large quantities of different types of chemical elements, a homogenization technique is performed to calculate the fictitious-bond energy. The structure of the microtubule is thus determined by minimizing the potential of the representative unit cell. With the established model, the fictitious-bond lengths between adjacent molecules are evaluated and the longitudinal and circumferential moduli are calculated.     

The elastic properties, elastic models and elastic perspectives of metallic glasses

Bulk metallic glass (BMG) provides plentiful precise knowledge of fundamental parameters of elastic moduli, which offer a benchmark reference point for understanding and applications of the glassy materials. This paper comprehensively reviews the current state of the art of the study of elastic properties, the establishments of correlations between elastic moduli and properties/features, and the elastic models and elastic perspectives of metallic glasses. The goal is to show the key roles of elastic moduli in study, formation, and understanding of metallic glasses, and to present a comprehensive elastic perspectives on the major fundamental issues from processing to structure to properties in the rapidly moving field.A plentiful of data and results involving in acoustic velocities, elastic constants and their response to aging, relaxation, applied press, pressure and temperature of the metallic glasses have been compiled. The thermodynamic and kinetic parameters, stability, mechanical and physical properties of various available metallic glasses especially BMGs have also been collected. A survey based on the plentiful experimental data reveals that the linear elastic constants have striking systematic correlations with the microstructural features, glass transition temperature, melting temperature, relaxation behavior, boson peak, strength, hardness, plastic yielding of the glass, and even rheological properties of the glass forming liquids. The elastic constants of BMGs also show a correlation with a weighted average of the elastic constants of the constituent elements. We show that the elastic moduli correlations can assist in selecting alloying components with suitable elastic moduli for controlling the elastic properties and glass-forming ability of the metallic glasses, and thus the results would enable the design, control and tuning of the formation and properties of metallic glasses.We demonstrate that the glass transition, the primary and secondary relaxations, plastic deformation and yield can be attributed to the free volume increase induced flow, and the flow can be modeled as the activated hopping between the inherent states in the potential energy landscape. We then propose an extended elastic model to understand flow in metallic glass and glass-forming supercooled liquid, and the model presents a simple and quantitative mathematic expression for flow activation energy of various glasses. The elastic perspectives, which consider all metallic glasses exhibit universal behavior based on a small number of readily measurable parameters of elastic moduli, are presented for understanding the nature and diverse properties of the metallic glasses.     

Functional grading in hierarchical honeycombs: Density specific elastic performance

The introduction of hierarchy into structures has been credited with improving their elastic and other properties. Similarly, functional grading has been demonstrated to increase the damage tolerance of honeycomb structures, although with the penalty of reduced Young’s modulus or increased density. The combination of both hierarchy and functional grading has not been reported for honeycomb structures, although it is known in natural materials. A parametric numerical modelling study has been made of the in-plane elastic properties of honeycombs and how they are affected by functional grading and hierarchy, and importantly to establish whether it is possible to avoid reductions in Young’s modulus. A set of analytical models has been developed to describe functional grading and hierarchy in honeycombs, based upon beam mechanics and the transform section method. The conditions for transition of a hierarchical honeycomb in behaviour from that of a discrete structure to that of a continuum are established. Furthermore, conditions are established for which hierarchical honeycombs, uniform or functionally graded, can surpass in-plane Young’s moduli of conventional honeycombs a by factor of up to 2, on an equal density basis.     

Experimental measurement and predictive modelling of bending behaviour for viscous unidirectional composite materials

It is widely accepted that the key deformation mechanisms during forming of viscous textile composite (prepreg) sheets are in-plane shear and out-of-plane bending. This paper focuses on the bending deformation mechanism, including experimental characterisation and theoretical modelling of bending behaviour during viscous composite forming. Experimental measurements are obtained by means of a large-displacement buckling test at a variety of displacement rates and temperatures. Some important aspects, such as viscoelastic behaviour, are also investigated. A bending model based on elastic theory combined with uniaxial continuum theory for ideal fibre-reinforced fluids for viscous shear deformation has been developed, using material parameters obtained from industrial manufacturers as input data, such as composite geometry, fibre properties, fibre volume fraction and matrix rheology. Model predictions demonstrate that the model can capture the main characteristics of material properties, such as rate dependence. This bending model can be used in formability analysis for viscous unidirectional composite materials, and might be applied in a finite element forming simulation to account for the bending stiffness.     

Atomistic simulations of J-integral in 2D graphene nanosystems

The J-integral is investigated in discrete atomic systems using molecular mechanics simulations. A method of calculating J-integral in specified atomic domains is developed. Two cases, a semiinfinite crack in an infinite domain under the remote K-field deformation and a finite crack length in a finite geometry under the tensile and shear deformation prescribed on the boundary, are studied in the two-dimensional graphene sheets and the values of J-integral are obtained under small-strain deformation. The comparison with energy release rates in Mode I and Mode II based on continuum theory of linear elastic fracture mechanics show good agreements. Meanwhile, the nonlinear strain and stress relation of a 2D graphene sheet is evaluated and is fitted with a power law curve. With necessary modifications on the Tersoff-Brenner potential, the critical values of J-integral of 2D graphene systems, which denoted as Jc, are eventually obtained. The results are then compared with those from the relevant references.     

石墨烯/Cu复合材料力学性能的分子动力学模拟

结合嵌入原子方法（EAM）、反应经验键序（REBO）作用势和Morse势函数,采用分子动力学方法研究了石墨烯/Cu复合材料的弹性性能和变形机制。分子动力学计算得到复合材料的弹性模量随石墨烯体积分数的增加而线性增加,这与Halpin-Tsai模型的预测趋势吻合。此外,石墨烯的加入同时也提供了复合材料的屈服强度。通过比较预制裂纹在单晶铜和石墨烯/Cu复合材料中的动态扩展,发现石墨烯的加入显著抑制了裂纹的扩展,材料的变形主要表现为沿石墨表面的滑移。石墨烯很大程度上提高了复合材料的塑性变形能力。     

Nanomechanical and Charge Transport Properties of Two‐Dimensional Atomic Sheets

The materials properties of graphene and other two‐dimensional atomic sheets are influenced by atomic‐scale defects, mechanical deformation, and microstructures. Thus, for graphene‐based applications, it is essential to uncover the roles of atomic‐scale defects and domain structures of two‐dimensional layers in charge transport properties. This review highlights recent studies of nanomechanical and charge transport properties of two‐dimensional atomic sheets, including graphene, MoS₂, and boron nitrides. Because of intrinsic structural differences, two‐dimensional atomic sheets give rise to unique nanomechanical properties, including a dependence on layer thickness and chemical modification that is in contrast to three‐dimensional continuum media. Mapping of local conductance and nanomechanical properties on a graphene layer can be used to image the domain and microstructures of two‐dimensional atomic layers. This paper also reviews recent experimental and theoretical findings on the role of bending, defects, and microstructures on nanomechanical and transport properties of graphene‐derived materials.     

Elastic in-plane properties of 2D linearized models of graphene

Graphene is a monolayer of carbon atoms packed into a two-dimensional honeycomb lattice. This allotrope can be considered as mother of all graphitic forms of carbon. The elastic in-plane properties of graphene are studied and various existing linearized models of its elastic deformations are critically re-examined. Problems related to modelling of graphene by nonlinear multi-body potentials of interaction are also discussed. It is shown that experimental results for small deformations can be well described by both the two-parametric molecular mechanics model developed by Gillis in 1984, while some popular models have serious flaws and often the results obtained using these models do not have physical meaning. It is argued that in order to study elastic constants of linearized models of graphene layers, it is very convenient to use the four parameter molecular mechanics model. The advantages of this approach is demonstrated by its application to the Tersoff and Brenner nonlinear interaction potentials, and by its comparison with the Gillis two-parametric model.     

Elastic properties of chemically derived single graphene sheets

The elastic modulus of freely suspended graphene monolayers, obtained via chemical reduction of graphene oxide, was determined through tip-induced deformation experiments. Despite their defect content, the single sheets exhibit an extraordinary stiffness ( E = 0.25 TPa) approaching that of pristine graphene, as well as a high flexibility which enables them to bend easily in their elastic regime. Built-in tensions are found to be significantly lower compared to mechanically exfoliated graphene. The high resilience of the sheets is demonstrated by their unaltered electrical conductivity after multiple deformations. The electrical conductivity of the sheets scales inversely with the elastic modulus, pointing toward a 2-fold role of the oxygen bridges, that is, to impart a bond reinforcement while at the same time impeding the charge transport.     

Predicting the properties of micro- and nanocomposites: from the microwhiskers to the bristled nano-centipedes

The paper draws on the similarities between the well-known process of whiskerization of microfibres and the recent idea of bristled nanowires. The new method for evaluation of the effective elastic properties of such materials is suggested based on the model of four-component composition. This model assumes the transverse isotropy of continuum and predicts five elastic moduli and density as independent effective constants. An example of calculation of the constants for the particular materials is given. It shows the significant increase in the shear strength of composites with whiskerized or bristled fibres.     

Microstructure continuum modeling of an elastic metamaterial

Elastic metamaterials have unusual microstructures that can make them exhibit unusual dynamic behavior. For instance, if treated as classical elastic solids, these materials may have frequency-dependent effective mass densities which may become negative in certain frequency range. In this study, an approach for developing microstructure continuum models to represent elastic metamaterials was presented. Subsequently, this continuum model was used to study wave propagation and band gaps in elastic metamaterial with resonators. In contrast to the use of the conventional continuum theory with which the effective mass density would become frequency-dependent and negative, the main advantage of the microstructure continuum model is that the local microstructural deformation/motion is accounted for with the introduction of additional kinematic variables. Moreover, the material constants of the microstructure continuum model are explicitly expressed in term of the properties of the host medium and the resonator. The accuracy of the microstructure continuum model was evaluated by comparing dispersion curves of harmonic waves to those obtained by the finite element analysis based on the exact geometry of the elastic metamaterial.     

Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling

Computational simulation for predicting mechanical properties of carbon nanotubes (CNTs) has been adopted as a powerful tool relative to the experimental difficulty. Based on molecular mechanics, an improved 3D finite element (FE) model for armchair, zigzag and chiral single-walled carbon nanotubes (SWNTs) has been developed. The bending stiffness of the graphene layer has been considered. The potentials associated with the atomic interactions within a SWNT were evaluated by the strain energies of beam elements which serve as structural substitutions of covalent bonds. The out-of-plane deformation of the bonds was distinguished from the in-plane deformation by considering an elliptical cross-section for the beam elements. The elastic stiffness of graphene has been studied and the rolling energy per atom has been calculated through the analysis of rolling a graphene sheet into a SWNT to validate the proposed FE model. The effects of diameters and helicity on Young’s modulus and the shear modulus of SWNTs were investigated. The simulation results from this work are comparable to both experimental tests and theoretical studies from the literatures.     

On a possibility of reconstruction of Cosserat moduli in particulate materials using long waves

The paper proposes a method of reconstruction of the Cosserat elastic moduli using the measurements of velocities of the p-wave and the high-frequency twist wave as well as the low-frequency asymptotics of a shear wave dispersion relationship. It is shown that in the case of a general isotropic Cosserat continuum, the information obtained from these wave measurements is insufficient for the complete moduli reconstruction. The reconstruction is shown to be possible in the case of a 3D isotropic Cosserat continuum governed by at most four independent parameters. Such a continuum is suggested for a particulate material consisting of spherical particles connected by normal, shear and rotational links. Another case when the full reconstruction is possible consists of 2D orthotropic Cosserat continuum modelling particulate material with square packing of cylindrical particles and 2D isotropic Cosserat continuum modelling with hexagonal packing of cylindrical particles. In the 2D materials, the measurements of p-wave velocity and the shear wave dispersion relationship are sufficient for complete reconstruction of all moduli. A phase shift method and reconstruction algorithms are presented.     

Nanoscale vibration characterization of multi-layered graphene sheets embedded in an elastic medium

Detailed studies on the nanoscale vibration characteristics of multi-layered graphene sheets (MLGSs) that are embedded in an elastic medium are carried out using continuum-based modelling and Generalized Differential Quadrature (GDQ) method. Natural frequencies and their associated vibration modes of practical interest of single-layered and triple-layered graphene sheets, as well as general MLGSs that are embedded in an elastic medium are established. Numerical simulations are conducted to examine the effects of van der Waals (vdW) interactions, which are present as bonding forces between the layers, on nanoscale vibration natural frequencies and their mode shapes. The results show that for a general MLGSs embedded in an elastic medium, vibration modes can in general be classified into three families - lower classical synchronized modes which are independent of van der Waals forces and are somewhat sensitive to the surrounding elastic medium, middle van der Waals enhanced modes which are largely determined by the presence of van der Waals interactions and are hence less sensitive to the changes of the surrounding elastic medium, and higher mixed modes which are combinations of classical synchronized modes and van der Waals enhanced modes. Detailed characterizations of these modes from their derived mode shapes have been achieved for the typical case of an embedded triple-layered GSs, as well as general embedded MLGSs. Effects of Winkler modulus K_W, the shear layer modulus G_b, different boundary conditions, aspect ratio β and the number L of graphene layers on nanoscale vibration properties have been examined in detail. The results presented in this paper, for the first time, provide accurate and wholesome studies and characterizations on the interesting nanoscale vibration properties of multi-layered graphene sheets embedded in an elastic medium and the results obtained will certainly be useful to those who are concerned with the dynamics of embedded graphene sheets which are increasingly being deployed for various innovative engineering applications such as nano-electro-mechanical systems (NEMS).     

Effective thermoelastic properties of random structure composites reinforced by the clusters of deterministic structure (application to clay nanocomposites)

Summary Polymer/clay nanocomposites consisting of an epoxy matrix reinforced by silicate clay plates have been observed to exhibit enhanced mechanical properties at low volume fraction of clay. The matrix and embedded nanoelements are modeled in the framework of continuum mechanics with known mechanical properties previously evaluated by, e.g., molecular dynamic simulation. Nanoclay composite is modeled by the aligned, uniformly distributed in the matrix stacks of parallel clay sheets separated from one another by interlayer matrix galleries of nanometer scale. Interaction of a finite number of oblate spheroidal inclusions modeling an individual stack inside the infinite matrix is carried by the multipole expansion technique. The obtained accurate numerical solution was incorporated into the multiparticle effective field method [5] for the estimation of effective thermoelastic properties. Detailed parametric analyses demonstrate the influence on the effective elastic moduli and stress concentrator factors of such key factors as the shape of nanoelements, interlayer distance, and the number of nanoelements in the stacks of deterministic structure.