Mechanical characterization of single-walled carbon nanotubes: Numerical simulation study

The mechanical behaviour of non-chiral and chiral single-walled carbon nanotubes under tensile and bending loading conditions is investigated. For this purpose, three-dimensional finite element modelling is used in order to evaluate the tensile and bending rigidities and, subsequently, the Young's moduli. It is shown that the evolution of rigidity, tensile and bending, as a function of diameter can be described by a unique function for non-chiral and chiral single-walled nanotubes, i.e. regardless of the index or angles of chirality. A comprehensive study of the influence of the nanotube wall thickness and diameter on the Young's modulus values is also carried out. It is established that the evolution of the Young's modulus as a function of the inverse of the wall thickness follows a quasi-linear trend for nanotubes with diameters larger than 1.085 nm. The current numerical simulation results are compared with data reported in the literature. This work provides a benchmark in relation to ascertaining the mechanical properties of chiral and non-chiral single-walled carbon nanotubes by nanoscale continuum models.     

Effect of straight and wavy carbon nanotube on the reinforcement modulus in nonlinear elastic matrix nanocomposites

Elastomers, particularly rubbers, are viscoelastic polymers with low Young’s modulus. In this research, carbon nanotubes were used in the rubber and a rubber–carbon nanotube composite was modeled by ABAQUS™ software. Due to hyperelastic behavior of the rubber, strain function energy was used for the modeling. A sample of rubber was tested and uniaxial, biaxial, as well as planar test data obtained in these tests were used to get an energy function. Polynomial and reduced polynomial form are two common methods to achieve strain energy function. In this paper, elasticity modulus and Poisson ratio were measured for a representative volume element (RVE) of composite. Rubber was also considered as an elastic material and its composite properties in this state compared by hyperelastic rubber matrix assumption. ABAQUS was used to create a three dimensional finite element model of a single long wavy nanotube with diameter of D which perfectly bonded to matrix material. Nanotube waviness was modeled by sinusoidal carbon nanotube shape. Results showed that mechanical properties of the rubber will extremely change by adding carbon nanotube. Furthermore, several volume fractions of carbon nanotube in rubber were modeled and it was shown that stiffness of nanocomposite increases by more volume fraction of carbon nanotubes.     

A Representative volume model for a CNT composite material

The concept of a ‘Representative Volume Model’ is used in combination with ‘Equivalent Mechanical Strain’ or Aboudi's ‘Average Strain’ theorem to illustrate how a carbon nanotube reinforced composite material constitutive law for a nano‐composite material can be implemented into a finite element program for modeling structural applications. Current methods of modeling each individual composite layer to build up an element composed of carbon nanotube reinforced composite material may not be the best approach for modeling structural applications of this composite. The approach presented here is based upon presentations given at the National Science Foundation‐Civil and Mechanical Systems division workshop at John Hopkins University in 2004, which is referred to in this paper as the Williams‐Baxter approach. This approach is also used to demonstrate that damage modeling can be included as was suggested in this workshop. Copyright © 2015 John Wiley & Sons, Ltd.     

Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures

This article deals with the vibration analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shell structures. The material properties of an FG-CNTRC shell are graded smoothly through the thickness direction of the shell according to uniform distribution and some other functionally graded (FG) distributions (such as FG-X, FG-V, FG-O and FG-\({\Lambda}\)) of the volume fraction of the carbon nanotube (CNT), and the effective material properties are estimated by employing the extended rule of mixture. An eight-noded shell element considering transverse shear effect according to Mindlin’s hypothesis has been employed for the finite element modelling and analysis of the composite shell structures. The formulation of the shell midsurface in an arbitrary curvilinear coordinate system based on the tensorial notation is also presented. The Rayleigh damping model has been implemented in order to study the effects of carbon nanotubes (CNTs) on the damping capacity of such shell structures. Different types of shell panels have been analyzed in order to study the impulse and frequency responses. The influences of CNT volume fraction, CNT distribution, geometry of the shell and material distributions on the dynamic behavior of FG-CNTRC shell structures have also been presented and discussed. Various types of FG-CNTRC shell structures (such as spherical, ellipsoidal, doubly curved and cylindrical) have been analyzed and discussed in order to compare studies in terms of settling time, first resonant frequency and absolute amplitude corresponding to first resonant frequency based on the impulse and frequency responses, and the effects of CNTs on vibration responses of such shell structures are also presented. The results show that the CNT distribution and volume fraction of CNT have a significant effect on vibration and damping characteristics of the structure.     

Rate dependent modelling of the forming behaviour of viscous textile composites

A predictive approach to modelling the forming of viscous textile composites has been implemented in two finite element codes; Abaqus Standard™ and Abaqus Explicit™. A multi-scale energy model is used to predict the shear force–shear angle–shear rate behaviour of viscous textile composites, at specified temperatures, using parameters supplied readily by material manufacturers, such as fibre volume fraction, weave architecture and matrix rheology. The predictions of the energy model are fed into finite element simulations to provide the in-plane shear properties of two different macro-scale constitutive models implemented in the finite element codes. The manner of coupling predictions of the multi-scale energy model with the macro-scale models is shown to affect the rate-dependent material response in the simulations. These coupling methods are evaluated using picture frame test simulations.     

Studying the Tensile Behaviour of GLARE Laminates: A Finite Element Modelling Approach

Numerical simulations based on finite element modelling are increasingly being developed to accurately evaluate the tensile properties of GLARE (GLAss fibre REinforced aluminium laminates). In this study, nonlinear tensile behaviour of GLARE Fibre Metal Laminates (FML) under in-plane loading conditions has been investigated. An appropriate finite element modelling approach has been developed to predict the stress–strain response and deformation behaviour of GLARE laminates using the ANSYS finite element package. The finite element model supports orthotropic material properties for glass/epoxy layer(s) and isotropic properties with the elastic–plastic behaviour for the aluminium layers. The adhesion between adjacent layers has been also properly simulated using cohesive zone modelling. An acceptable agreement was observed between the model predictions and experimental results available in the literature. The proposed model can be used to analyse GLARE laminates in structural applications such as mechanically fastened joints under different mechanical loading conditions.     

A numerical approach for determining the effective elastic symmetries of particulate-polymer composites

In this paper we deal with the problem of determining on the one hand the effective elastic properties of particulate-polymer composite materials and on the other hand the actual degree of symmetry of the resulting homogenised material. This twofold purpose has been accomplished by building a 2D as well as a 3D finite element model of the heterogeneous material and by using the strain-energy based numerical homogenisation technique. Both finite element models are able to reproduce with a good level of accuracy the real microstructure of the composite material by considering a random distribution of both particles and air bubbles (that are generated by the fabrication process). To assess the effectiveness of the proposed models, we present a numerical study to determine the effective elastic properties of the composite along with a comparison with the existing analytical and experimental results taken from literature and a sensitivity analysis in terms of the spatial distribution of the particles of the unit cell. Numerical results show that both models are able to provide the equivalent elastic properties with a very good level of accuracy when compared to experimental results and that the particulate-reinforced polymer composite could show, depending on the particles volume fraction and arrangement, an isotropic or a cubic elastic symmetry.     

Numerical characterization of CNT-based polymer composites considering interface effects

Carbon nanotubes (CNTs) possess exceptional mechanical properties and are therefore suitable candidates for use as reinforcements in composite materials. Load transfer in nanocomposite materials is achieved through the CNT/matrix interface. Thus, to determine nanocomposite mechanical properties, the interface behavior must be determined. In this investigation, finite element method is used to investigate the effects of interface strength on effective CNT-based composite mechanical properties. Nanocomposite mechanical properties are evaluated using a 3D nanoscale representative volume element (RVE). A single nanotube and the surrounding polymer matrix are modeled. Two cases of perfect bonding and an elastic interface are considered. For the perfect bonding interface, the no slip conditions are applied. To better investigate the elastic interface behavior, two models are proposed for this type of interface. The first elastic interface model consists of a thin layer of an elastic material surrounding the CNT. In the second elastic interface model, a series of spring elements are used as the nanotube/matrix interface. The results of numerical models indicate the importance of adequate interface bonding for a more effective strengthening of polymer matrix by CNT’s.     

Investigation of chirality and diameter effects on the Young’s modulus of carbon nanotubes using non-linear potentials

The main goal of this research is to predict Young’s modulus of carbon nanotubes using a full non-linear finite element model. Spring elements are used to simulate molecular interactions in atomic structure of carbon nanotube. All interactions are simulated non-linearly. A parametric study is performed to investigate effects of chirality and diameter on the Young’s modulus of single walled carbon nanotubes. Unlike the results of presented linear finite element models, the results of current model imply on independency of Young’s modulus from chirality and diameter. Obtained results from this study are in a good agreement with experimental observations and published data.     

Mechanical analysis of bi-component-fibre nonwovens: Finite-element strategy

In thermally bonded bi-component fibre nonwovens, a significant contribution is made by bond points in defining their mechanical behaviour formed as a result of their manufacture. Bond points are composite regions with a sheath material reinforced by a network of fibres’ cores. These composite regions are connected by bi-component fibres — a discontinuous domain of the material. Microstructural and mechanical characterization of this material was carried out with experimental and numerical modelling techniques. Two numerical modelling strategies were implemented: (i) traditional finite element (FE) and (ii) a new parametric discrete phase FE model to elucidate the mechanical behaviour and underlying mechanisms involved in deformation of these materials. In FE models the studied nonwoven material was treated as an assembly of two regions having distinct microstructure and mechanical properties: fibre matrix and bond points. The former is composed of randomly oriented core/sheath fibres acting as load-transfer link between composite bond points. Randomness of material’s microstructure was introduced in terms of orientation distribution function (ODF). The ODF was obtained by analysing the data acquired with scanning electron microscopy (SEM) and X-ray micro computed tomography (CT). Bond points were treated as a deformable two-phase composite. An in-house algorithm was used to calculate anisotropic material properties of composite bond points based on properties of constituent fibres and manufacturing parameters such as the planar density, core/sheath ratio and fibre diameter. Individual fibres connecting the composite bond points were modelled in the discrete phase model directly according to their orientation distribution. The developed models were validated by comparing numerical results with experimental tensile test data, demonstrating that the proposed approach is highly suitable for prediction of complex deformation mechanisms, mechanical performance and structure-properties relationships of composites.     

Carbon nanotube–polymer interactions in nanocomposites: A review

The interaction between carbon nanotubes and polymers is critically reviewed. The interfacial characteristics directly influence the efficiency of nanotube reinforcements in improving mechanical, thermal, and electrical properties of the polymer nanocomposite. In this review, various techniques of interaction measurements, including experimental and modelling studies, are described. From the experimental approaches, wetting, spectroscopy and probe microscopy techniques are discussed in detail. Molecular dynamics, coarse grain simulation and density functional theory are also explained as the main modelling approaches in interaction measurement studies. Different methods of interaction improvement, mainly categorized under covalent and noncovalent interactions, are described afterwards. Modelling predictions of nanocomposite properties, such as Young’s modulus, are compared with the experimental results in the literature and the challenges are discussed. Finally, it is concluded that an optimum carbon nanotube–polymer interaction is a key factor towards reaching the full potential of carbon nanotubes in nanocomposites.     

A semi-continuum finite element approach to evaluate the Young’s modulus of single-walled carbon nanotube reinforced composites

This paper describes a micromechanical finite element approach for the estimation of the effective Young’s modulus of single-walled carbon nanotube reinforced composites. These composite materials consist of aligned carbon nanotubes that are uniformly distributed within the matrix. Based on micromechanical theory, the Young’s modulus of the nanocomposite is estimated by considering a representative cylindrical volume element. Within the representative volume element, the reinforcement is modeled according to its atomistic microstructure while the matrix is modeled as a continuum medium. Spring-based finite elements are employed to simulate the discrete geometric structure and behavior of each single-walled carbon nanotube. The load transfer conditions between the carbon nanotubes and the matrix are modeled using joint elements of changeable stiffness that connect the two materials, simulating the interfacial region. The proposed model has been tested numerically and yields reasonable results for variable stiffness values of the joint elements. The effect of the interface on the performance of the composite is investigated for various volume fractions. The numerical results are compared with experimental and analytical predictions.     

Finite element crash simulations of the human body: Passive and active muscle modelling

Conventional dummy based testing procedures suffer from known limitations. This report addresses issues in finite element human body models in evaluating pedestrian and occupant crash safety measures. A review of material properties of soft tissues and characterization methods show a scarcity of material properties for characterizing soft tissues in dynamic loading. Experiments imparting impacts to tissues and subsequent inverse finite element mapping to extract material properties are described. The effect of muscle activation due to voluntary and non-voluntary reflexes on injuries has been investigated through finite element modelling.     

Effect of nanotube geometry on the elastic properties of nanocomposites

Many analytical models replace carbon nanotubes with “effective fibers” to bridge the gap between the nano and micro-scales and allow for the calculation of the elastic properties of nanocomposites using micromechanics. Although curvature of nanotubes can have a direct impact on these properties, it is typically ignored. In this work, the nanotube geometry in 3D is included in the calculation of the elastic properties of a modified effective fiber. The strain energy of the nanotube and the effective fiber are calculated using Castligiano’s theorem and constraints imposed by the matrix on the deformation are taken into consideration. Model results are compared to results from archived literature, and a reasonable agreement is observed. Results show that the effect of nanotube curvature on reducing the modulus of the effective fiber is not limited to in-plane curvature but also to curvature in 3D. The impact of the nanotube curvature on the elastic properties of nanocomposites is studied utilizing the modified fiber model and the approach developed by Mori–Tanaka. Analytical results show that for a low weight fraction of nanotubes the effect of curvature seems to be minor and as the weight fraction increases, the effect of nanotube curvature becomes critical.     

Industrial challenges for finite element and multiscale methods for material modeling

The automotive industry promotes lightweight design to reduce the CO2-emission and enhances the passenger’s safety using high strength steel grades. One limiting factor to the accuracy of modern stamping simulation are the empirical constitutive models. In particular for high strength multiphase steels the modelling techniques like multi-scale methods are becoming more interesting. However they should meet the industrial needs. Not only the accuracy but also features like time, costs and complexity are rapidly increasing. The challenge is the development of finite element technologies and multi-scale methods in an appropriate framework for industrial projects. The crystal plasticity finite element method bridges the gap between the micro level and macroscopic mechanical properties that opens the way for more profound consideration of metal anisotropy in stamping process simulation. Nevertheless new empirical constitutive models are favourable for spring back prediction in forming simulations, even if the number of material parameters and the amount of tests for their identification increases. In this paper the application of crystal plasticity FEM within the concept of virtual material testing with a representative volume element (RVE) is demonstrated.     

Computational studies on the mechanical properties of diamond nanotoroids

Diamond nanotoroids are a new class of carbon nanostructures with interesting theoretical properties and are ideal for studying the elastic and plastic deformation behavior of diamond nanorods. Various sizes of diamond nanotoroids, along with a carbon nanotube, a carbon nanotube toroid, and a diamond nanorod were simulated using molecular dynamics. We tested these compounds for stability and compared our calculated values for the ultimate tensile strength and the Young’s modulus over a range of strain rates to those reported in the literature and attempted to explain any discrepancies found between our results and those reported. The results of these simulations suggest the tensile strength of diamond nanotoroids would be many times stronger than conventional materials and this novel material has potential for use in many demanding applications.     

Modelling of impact forces and pressures in Lagrangian bird strike analyses

The paper aims at evaluating and improving the accuracy of bird impact numerical analyses performed with finite element explicit codes, focusing on the modelling of the spatial and temporal pressure distributions exerted on the target by the impacting body. A Lagrangian approach is adopted, interfacing the ESI/Pam-Crash solver code with an automatic trial-and-error procedure for the elimination of the excessively distorted elements. The theoretical formulation relevant to the impact of a cylindrical soft body against a rigid target is reviewed and this idealised case is adopted to validate the presented approach with increasingly refined finite element schemes. A sensitivity study is then carried out, adopting differently shaped bird models and varying the material hydrodynamic and deviatoric responses. A set of models is selected comparing the results with the experimental average values and the scattering reported in literature for the most significant loading parameters in impacts on rigid targets. The model shape and the calibration parameters of the bird material used in these models are subsequently adopted in the analyses of impacts on a deformable polycarbonate plate. The numerical results obtained with increasingly refined bird models are presented and discussed. A range of modelling parameters is finally suggested to perform reliable numerical analyses on aircraft structures and a criterion is proposed to select the models for a reasonably conservative approach to the design of a bird proof structure.     

Three Phase Composite Cylinder Assemblage Model for Analyzing the Elastic Behavior of MWCNT-Reinforced Polymers

Evolution of computational modeling and simulation has given more emphasis on the research activities related to carbon nanotube (CNT) reinforced polymer composites recently. This paper presents the composite cylinder assemblage (CCA) approach based on continuum mechanics for investigating the elastic properties of a polymer resin reinforced by multi-walled carbon nanotubes (MWCNTs). A three-phase cylindrical representative volume element (RVE) model is employed based on CCA technique to elucidate the effects of inter layers, chirality, interspacing, volume fraction of MWCNT, interphase properties and temperature conditions on the elastic modulus of the composite. The interface region between CNT and polymer matrix is modeled as the third phase with varying material properties. The constitutive relations for each material system have been derived based on solid mechanics and proper interfacial traction continuity conditions are imposed. The predicted results from the CCA approach are in well agreement with RVE-based finite element model. The outcomes reveal that temperature softening effect becomes more pronounced at higher volume fractions of CNTs.     

On the effective elastic moduli of carbon nanotubes for nanocomposite structures

A critical review on the validity of different experimental and theoretical approaches to the mechanical properties of carbon nanotubes for advanced composite structures is presented. Most research has been recently conducted to study the properties of single-walled and multi-walled carbon nanotubes. Special attention has been paid to the measurement and modeling of tensile modulus, tensile strength, and torsional stiffness. Theoretical approaches such as molecular dynamic (MD) simulations, finite element analysis, and classical elastic shell theory were frequently used to analyze and interpret the mechanical features of carbon nanotubes. Due to the use of different fundamental assumptions and boundary conditions, inconsistent results were reported. MD simulation is a well-known technique that simulates accurately the chemical and physical properties of structures at atomic-scale level. However, it is limited by the time step, which is of the order of 10⁻¹⁵ s. The use of finite element modeling combined with MD simulation can further decrease the processing time for calculating the mechanical properties of nanotubes. Since the aspect ratio of nanotubes is very large, the elastic rod or beam models can be adequately used to simulate their overall mechanical deformation. Although many theoretical studies reported that the tensile modulus of multi-walled nanotubes may reach 1 TPa, this value, however, cannot be directly used to estimate the mechanical properties of multi-walled nanotube/polymer composites due to the discontinuous stress transfer inside the nanotubes.     

Buckling of Carbon/Glass Composite Tubes Under Uniform External Hydrostatic Pressure

Abstract: This paper describes an experimental and an analytical investigation into the collapse of 44 circular cylindrical composite tubes under external hydrostatic pressure. The results for 22 of these tubes were from a previous investigation and the results for a further 22 models are reported for the first time in this paper. The investigations concentrated on fibre‐reinforced plastic tube specimens made from a mixture of three carbon and two E‐glass fibre layers. The lay‐up was 0°/90°/0°/90°/0; the carbon fibres were laid lengthwise (0°) and the E‐glass fibres circumferentially (90°). The theoretical investigations were carried out using a simple solution for isotropic materials, namely a well‐known formula by ‘von Mises’. The previous investigation also used a numerical solution based on ANSYS, but this was found to be rather disappointing. The experimental investigations showed that the composite specimens behaved similarly to isotropic materials previously tested, in that the short vessels collapsed through axisymmetric deformation while the longer tubes collapsed through non‐symmetric bifurcation buckling. Furthermore, it was discovered that the specimens failed at changes of the composite lay‐up due to the manufacturing process of these specimens. These changes seem to be the weak points of the specimens. For the theoretical investigations, two different types of material properties were used to analyse the composite. These were calculated properties derived from the properties of the single layers given by the manufacturer and also the experimentally obtained properties. Two different approaches were chosen for the investigation of the theoretical buckling pressures, of the previously analysed models, namely a program called ‘MisesNP’, based on a well‐known formula by von Mises for single‐layer isotropic materials, and two finite element analyses using the famous computer package called ‘ANSYS’. These latter analyses simulated the composite with a single‐layer orthotrophic element (Shell93) and also with a multi‐layer element (Shell99). The results from Shell93 and Shell99 agreed with each other but, in general, their predictions were higher than the analytical solution by von Mises. The von Mises solution agreed better than the finite element solutions for the longer vessels, which collapsed by elastic instability, particularly when the experimentally obtained material properties were used. Thus, it was concluded that the results obtained from the finite element analyses predicted ‘questionable’ buckling pressures. The report provides design charts by all approaches and material types, which allow the possibility of obtaining a ‘plastic knockdown factor’ for these vessels. The theoretical buckling pressures obtained using the computer programs MisesNP or ANSYS can then be divided by the plastic knockdown factor obtained from the design charts, to give the predicted buckling pressures. It is not known whether or not this method can be used for the design of very large vessels.