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.     

Characterizing mechanical properties of graphite using molecular dynamics simulation

The mechanical properties of graphite in the forms of single graphene layer and graphite flakes (containing several graphene layers) were investigated using molecular dynamics (MD) simulation. The in-plane properties, Young’s modulus, Poisson’s ratio, and shear modulus, were measured, respectively, by applying axial tensile stress and in-plane shear stress on the simulation box through the modified NPT ensemble. In order to validate the results, the conventional NVT ensemble with the applied uniform strain filed in the simulation box was adopted in the MD simulation. Results indicated that the modified NPT ensemble is capable of characterizing the material properties of atomistic structures with accuracy. In addition, it was found the graphene layers exhibit higher moduli than the graphite flakes; thus, it was suggested that the graphite flakes have to be expanded and exfoliated into numbers of single graphene layers in order to provide better reinforcement effect in nanocomposites.     

Numerical algorithms for prediction of mechanical properties of single-walled carbon nanotubes based on molecular mechanics model

The objective of this paper is to develop the numerical algorithms for the prediction of mechanical properties of single-walled carbon nanotubes (SWCNTs). By using the energy method, the analytical expressions are obtained and the five independent variables algorithm is developed for the prediction of the elastic properties of SWCNTs via a molecular mechanics model in which the geometrical relationship of carbon nanotube is introduced. It can be found that due to the introduction of the geometrical approximate conditions some errors may exist in the calculation of mechanical properties of SWCNTs in terms of the five independent variables algorithm. Therefore, two improved algorithms, i.e., eigenvalues modified method (EMM) and eigenvalues and eigenvectors modified method (EEMM) are proposed to analyze the possible errors in the numerical results. It is found that the results obtained by the three kinds of algorithms are almost consistent with one another, but EMM and EEMM are preferred to be used because they have properties similar to those of the finite element method, where the consistent equation works just as the constraint condition to void the singularity of the element stiffness matrix. The computational results also reveal that both the surface Young’s modulus and Poisson’s ratio depend on the diameter of carbon nanotubes, and finally converge to the values of the graphite sheet with an increase in the tube diameter in the inverse trends. For SWCNTs with approximately the same diameters, the surface Young’s modulus is in direct and Poisson’s ratio is in inverse proportion to chiral angles, respectively.     

Vibrational analysis of carbon nanotubes and graphene sheets using molecular structural mechanics approach

In this article, the vibrational properties of two kinds of single-layered graphene sheets and single-wall carbon nanotubes (SWCNT) are studied. The simulations are carried out for two types of zigzag carbon nanotubes (6,0), (12,0), armchair carbon nanotubes (4,4), (6,6) and zigzag and armchair graphene sheets with free-fixed and fixed–fixed end conditions.Fundamental frequency is determined by means of molecular structural mechanics approach. In this approach, carbon nanotubes (CNTs) and grapheme sheets are considered as space frames. By constructing equality between strain energies of each element in structural mechanics and potential energies of each bond, equivalent space frames can be achieved. Carbon atoms are considered as concentrated masses placed in beam joints (bond junctions).Results are presented as diagrams stating fundamental frequencies of nanotubes and graphene sheets with respect to aspect ratios. The results indicate that fundamental frequency decreases as aspect ratio increases. So it is preferred to use nanotubes and graphene sheets with lower aspect ratios for dynamic applications in order to prevent resonance and dynamic damage. Fundamental frequency of nanotubes is larger than that of graphene sheets. The results are in good agreement with results of previous researches.     

Investigation of the mechanical properties of the Ni–P–CNTs coated copper composite materials: Experiments and modeling

Recent studies on the mechanical properties of nickel–phosphorous–carbon nanotubes (Ni–P–CNTs) coated copper composite materials have shown surprising results. Their Young’s modulus and tensile strength cannot reach the theoretical values, even falling below those of copper, and the Young’s modulus decreases with the increment of CNT concentrations. Materials used in those studies were prepared through electroless composite plating process, with the Ni–P–CNTs composite electrolessly deposited on the copper substrate. In the present study, however, it is shown that the Young’s modulus and the tensile strength do increase significantly with the increment of the CNT concentrations without activating the CNTs. A composite method of Voigt model and a random distributed discontinuous fiber model is applied to obtain the equivalent Young’s modulus of the composite, which agrees very well with the experimental data.     

Elastic constants and high pressure structural transitions in yttrium pnictides

We present first-principles calculations on the structural, elastic and high pressure properties of yttrium pnictide compounds, using the pseudo-potential plane-waves approach based on density functional theory, within the generalized gradient approximation. Results are given for lattice constant, bulk modulus and its pressure derivative. The pressure transition at which these compounds undergo structural phase transition from NaCl-type (B1) to CsCl-type (B2) structure is calculated and compared with previous calculations and available experimental data. The elastic constants and their pressure dependence are calculated using the static finite strain technique. We derived the bulk and shear moduli, Young’s modulus and Poisson’s ratio of the B1 phase for YN, YP, YAs and YSb compounds. We estimated the Debye temperature of these compounds from the average sound velocity. This is the first quantitative theoretical prediction of the elastic properties of YN, YP, YAs and YSb compounds, and it still awaits experimental confirmation.     

Influence of external stress and plastic strain on morphological evolution of precipitates in Ni-based superalloys

An analytical method is proposed to investigate the morphological evolution of γ^′ precipitates in Ni-based superalloys. Two types of superalloys are considered with different ratio between the Young’s moduli of the precipitates and the matrix. Based on Eshelby’s equivalent inclusion theory and Mori–Tanaka’s mean field method the elastic energy is calculated as a function of the particle shape. The plastic strain induced by the formation of dislocation networks at the phase interface is taken into account. The results show that the external stress and the plastic matrix strain play a significant role for shape stability and the rafting mechanism of the precipitates. The elastic–plastic analysis provides a satisfactory explanation for all available experimental observations superalloys undergoing plastic creep.     

Young’s moduli of functionalized single-wall carbon nanotubes under tensile loading

This paper quantitatively investigates the effect of chemical functionalization on the axial Young’s moduli of single-walled carbon nanotubes (SWCNTs) based on molecular mechanics (MM) simulation, in which the COMPASS force field is used to model the interatomic interactions in a nonfunctionalized nanotube or a functionalized nanotube grafted with vinyl groups. We obtain the axial Young’s moduli of both functionalized and nonfunctionalized SWCNTs. The influences of the number and distribution density of the sp³-hybridized carbon atoms and the radius and chirality of the SWCNTs on Young’s moduli are studied. The results indicate that Young’s moduli depend strongly on the chirality of the SWCNTs and the distribution density of the sp³-hybridized carbon atoms. A 37.50% content of sp³-hybridized carbon atoms may degrade Young’s modulus by up to 33.36%. In addition, MM simulations show that the functionalization of SWCNTs results in a decrease of Young’s moduli of the corresponding SWCNT/polyethylene composites.     

Prediction study of the structural, elastic and electronic properties of ANSr3 (A = As, Sb and Bi)

Based on first-principles total energy calculations, we predict the elastic and electronic properties of the anti-perovskites AsNSr₃, SbNSr₃ and BiNSr₃ compounds. The calculated lattice constants are in good agreement with the available results. The independent elastic constants (C₁₁, C₁₂ and C₄₄) and their pressure dependence are calculated using the static finite strain technique. The isotropic elastic moduli, namely, bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (σ) and Lame’s constants (λ and μ) are calculated in framework of the Voigt–Reuss–Hill approximation for ideal polycrystalline ANSr₃ aggregates. By analysing the ratio between the bulk and shear moduli, we conclude that ANSr₃ compounds are brittle in nature. We estimated the Debye temperature of ANSr₃ from the average sound velocity. The band structures show that all studied materials are semiconductors. The analysis of the site and momentum projected densities, charge transfer and charge densities show that bonding is of covalent–ionic nature. This is the first quantitative theoretical prediction of the elastic and electronic properties of AsNSr₃, SbNSr₃ and BiNSr₃ compounds that requires experimental confirmation.     

Different mechanical properties of the pristine and hydrogen passivated ZnO nanowires

We comparatively calculate the Young’s moduli of the pristine and the hydrogen passivated ZnO nanowires using the first-principles approaches. It is found that the pristine nanowire has the higher Young’s modulus, but the corresponding hydrogen passivated nanowire has the lower Young’s modulus than that of bulk ZnO. The physical origin of the opposite tendency can be attributed to both the different surface relaxations of the two kinds of ZnO nanowires and the core nonlinear effect.     

Interlocking hexagons model for auxetic behaviour

A 2D ‘Rough Particle’ model consisting of interlocking hexagons is reported. Analytical expressions for the in-plane Poisson’s ratios and Young’s moduli due to particle translation along the geometrically matched male and female interlocks are derived for the model. The dependency of the mechanical properties on each of the model (geometrical and stiffness) parameters is provided, and it is shown that the assembly of interlocking hexagons deforming by particle translation along the interlocks displays auxetic (negative Poisson’s ratio) behaviour. The model predictions are compared with experimental mechanical properties for auxetic polypropylene (PP) films and fibres. The model predicts the experimental Poisson’s ratio values very well (model: ν_xy = −1.30, ν_yx = −0.77; experiment (PP films): ν_|| = −1.12, ). The model generally overestimates the Young’s moduli of the films, but is in reasonable agreement with the axial Young’s modulus of the fibres.     

A numerical analysis of sheet metal formability for automotive stamping applications

A theoretical failure model is presented for the numerical prediction of the forming limit strains of automotive sheets. The model uses the Swift’s diffuse necking and Hill’s localized necking concepts in describing tearing-type sheet metal failures and a computational scheme is proposed in which the failure conditions are expressed in incremental forms. The Bauschinger effect is included properly in the deformation modeling using an additive backstress form of the nonlinear-kinematic hardening rule. The necking conditions and plasticity model are transformed into a set of algebraic equations that may be applied both for proportional and non-proportional strain-controlled loadings. An iterative approach is employed in the incremental solution of algebraic equations. The formability analyses are conducted using the proposed theoretical model and the forming limit strains of two new generation auto sheets (Trip600 1.4 mm, DP980 1.15 mm) are estimated. The numerically generated FLC are compared with the experimental data and the FLC calculated with the Keeler–Brazier equation. For both steels, the model produced conservative plain–strain intercept values, FLC₀, when compared with the predictions of Keeler–Brazier equation. Also the negative minor strain part of the experimental FLD’s is estimated with sufficient accuracy. For the positive minor strain side, however, the predictions are lower than both the experimental fit and the standard curve.     

Monitoring quasi-static and cyclic fatigue damage in fibre-reinforced plastics by Poisson’s ratio evolution

Even if the extent of fatigue damage in fibre-reinforced plastics is limited, it can already affect the elastic properties. Therefore, the damage initiation and propagation in composite structures is monitored very carefully. Beside the use of nondestructive testing methods (ultrasonic inspection, optical fibre sensing), the follow-up of the degradation of engineering properties such as the stiffness is a common approach.In this paper, it is proved that the Poisson’s ratio can be used as a sensitive indicator of fatigue damage in fibre-reinforced plastics. Static tests, quasi-static cyclic tests and fatigue tests were performed on [0°/90°]_2s glass/epoxy laminates, and longitudinal and transverse strain were measured continuously. The evolution of the Poisson’s ratio ν_xy versus time and longitudinal strain ε_xx is studied. As the transverse strain measurement is crucial to monitor the degradation of the Poisson’s ratio, three techniques were applied to measure the transverse strain (strain gauges, mechanical extensometer and external optical fibre sensor).Finally, the technique has been applied to a totally different material: a carbon fabric thermoplastic composite. The results show a very similar degradation of the Poisson’s ratio, although no stiffness degradation can be observed during fatigue loading of this material.It is concluded that the degradation of the Poisson’s ratio can be a valuable indicator of fatigue damage, in combination with the stiffness degradation.     

Evaluation of the in-plane effective elastic moduli of single-layered graphene sheet

In this paper, an equivalent continuum-structural mechanics approach is used to characterize the mechanical behaviour of nanostructured graphene. The in-plane elastic deformation of armchair graphene sheets is simulated by using finite element modelling. The model is based on the assumption that force interaction among carbon atoms can be modelled by load-carrying beams in a representative two-dimensional honeycomb lattice structure. The elastic properties of beam elements are determined by equating the energies of the molecular structure and the continuum beam model subjected to small strain deformation. Then an equivalent continuum technique is adopted to estimate effective elastic moduli from which elastic constants are extracted. A comparison of elastic constants obtained from current modelling concur with results reported in literature. With the multifunctional properties of graphene sheets as manifested in a broad range of industrial applications, determination of their elastic moduli will facilitate a better design of the corresponding materials at macroscopic level.     

A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams

In this paper, an elastic, rectangular, and simply supported, sigmoid functionally graded material (S-FGM) beam of thick thickness subjected to uniformly distributed transverse loading has been investigated. The S-FGM system consists of ceramic (Al₂O₃) and metal (Al) phases varying through the thickness of beam. Major classes of representative theories such as classical laminate beam theory (CLBT), first-order shear deformation theory (FSDT) and high-order theories (HOTs) have been considered and a unified kinematic formulation is then proposed. The Poisson’s ratio of the thick S-FGM beam is assumed to be constant, but their Young’s moduli vary continuously throughout the thickness direction according to the volume fraction of constituents defined by sigmoid function. The numerical illustrations concern bending response of S-FGM rectangular beams. Qualitative and quantitative assessments of displacement and stress fields have been presented and discussed.     

A comparative study of Young’s modulus of single-walled carbon nanotube by CPMD, MD and first principle simulations

Carbon nanotubes (CNTs), due to their exceptional magnetic, electrical and mechanical properties, are promising candidates for several technical applications ranging from nanoelectronic devices to composites. Young’s modulus holds the special status in material properties and micro/nano-electromechanical systems (MEMS/NEMS) design. The excellently regular structures of CNTs facilitate accurate simulation of CNTs’ behavior by applying a variety of theoretical methods. Here, three representative numerical methods, i.e., Car–Parrinello molecular dynamics (CPMD), density functional theory (DFT) and molecular dynamics (MD), were applied to calculate Young’s modulus of single-walled carbon nanotube (SWCNT) with chirality (3,3). The comparative studies showed that the most accurate result is offered by time consuming DFT simulation. MD simulation produced a less accurate result due to neglecting electronic motions. Compared to the two preceding methods the best performance, with a balance between efficiency and precision, was deduced by CPMD.     

Numerical investigation of elastic mechanical properties of graphene structures

The computation of the elastic mechanical properties of graphene sheets, nanoribbons and graphite flakes using spring based finite element models is the aim of this paper. Interatomic bonded interactions as well as van der Waals forces between carbon atoms are simulated via the use of appropriate spring elements expressing corresponding potential energies provided by molecular theory. Each layer is idealized as a spring-like structure with carbon atoms represented by nodes while interatomic forces are simulated by translational and torsional springs with linear behavior. The non-bonded van der Waals interactions among atoms which are responsible for keeping the graphene layers together are simulated with the Lennard-Jones potential using appropriate spring elements. Numerical results concerning the Young’s modulus, shear modulus and Poisson’s ratio for graphene structures are derived in terms of their chilarity, width, length and number of layers. The numerical results from finite element simulations show good agreement with existing numerical values in the open literature.     

Influence of continuum damage mechanics on the Bree’s diagram of a closed end tube

This paper extends the Bree’s cylinder behaviors, which is subjected to the constant internal pressure and cyclic temperature gradient loadings, with considering continuum damage mechanics coupled with nonlinear kinematic hardening model. The Bree’s biaxial stress model is modified using the unified damage and the Armstrong–Frederick nonlinear kinematic hardening models. With the help of the return mapping algorithm, the incremental plastic strain in axial and tangential directions is obtained. Continuum damage mechanics approach can be used to extend the Bree’s diagram to the damaging structures and reduce the plastic shakedown domain. Kinematic hardening behavior was considered in the material model which shifts the ratcheting zone. The role of the material constants in the Bree’s diagram is also discussed.     

Elastic and anelastic behaviour of icosahedral quasicrystals

A theory is developed, based on theoretical-group analysis, to describe the linear, reversible, time-dependent response of an icosahedral quasicrystal, containing point defects, to a stress field and known as anelastic relaxation. We obtain also anelastic relaxation relationships for the practical Young’s, shear and Poisson’s moduli.     

A nanoindentation analysis of the effects of microstructure on elastic properties of Al2O3/SiC composites

Nanoindentation tests were carried out to investigate certain elastic properties of Al₂O₃/SiC_p composites at microscopic scales (nm up to μm) and under ultra-low loads from 3 mN to 250 mN, with special attention paid to effects caused by SiC particles and pores. The measured Young’s modulus depends on the volume fraction of SiC particles and on the composite porosity and it can compare with that of alumina. The Young’s modulus exhibits large scatters at small penetrations, but it tends to be constant with lesser dispersion as the indentation depth increases. Further analysis indicated that the scatter results from specific microstructural heterogeneities. The measured Young’s moduli are in agreement with predictions, provided the actual role of the microstructure is taken into account.