Modeling the tensile stress–strain response of carbon nanotube/polypropylene nanocomposites using nonlinear representative volume element

This paper presents a finite element model for predicting the mechanical behavior of polypropylene (PP) composites reinforced with carbon nanotubes (CNTs) at large deformation scale. Existing numerical models cannot predict composite behavior at large strains due to using simplified material properties and inefficient interfaces between CNT and polymer. In this work, nonlinear representative volume elements (RVE) of composite are prepared. These RVEs consist of CNT, PP matrix and non-bonded interface. The nonlinear material properties for CNT and polymer are adopted to solid elements. For the first time, the interface between CNT and matrix is simulated using contact elements. This interfacial model is capable enough to simulate wide range of interactions between CNT and polymer in large strains. The influence of adding CNT with different aspect ratio into PP is studied. The mechanical behavior of composites with different interfacial shear strength (ISS) is discussed. The success of this new model was verified by comparing the simulation results for RVEs with conducted experimental results. The results shows that the length of CNT and ISS values significantly affect the reinforcement phenomenon.     

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.     

On the contribution of carbon nanotube deformation to piezoresistivity of carbon nanotube/polymer composites

The change in electrical resistance due to mechanical deformation of carbon nanotube (CNT)/polymer composites can be rationalized in terms of two effects: (i) changes in the composite electrical resistivity due to changes in the CNT network configuration and (ii) deformation of the CNTs themselves. The contribution of CNT dimensional changes (ii) to the piezoresistivity of CNT/polymer composites is investigated here. An analytical model based exclusively on dimensional changes which describes the CNT change of electrical resistance in terms of its mechanical deformation is proposed. A micromechanics approach and finite element analysis are performed to correlate the macroscale composite strain to the individual CNT strain. The CNT change of electrical resistance is quantified for different matrix elastic moduli and CNT weight fractions. The CNT/polymer composite is also modeled as an effective continuum material in terms of both its electrical and mechanical responses so that the effect of dimensional changes on the global piezoresistivity can be investigated. Based on the modeling predictions and previous experimental results, it is estimated that the CNT change of resistance due to the macroscale composite strain is marginal (∼5%) compared to the total composite change of resistance commonly measured in the laboratory, suggesting that the dominant effect in the piezoresistivity of CNT/polymer composites is the change in the CNT network configuration.     

Post-buckling behaviour of carbon-nanotube-reinforced nanocomposite plate

The aim of the present paper is to investigate the buckling and post-buckling behaviour of nanocomposite plate having randomly oriented carbon nanotubes (CNTs) reinforced in magnesium (Mg) under uni-axial compression. The effect of non-bonded interaction at the interface between CNT and matrix is considered through a cohesive zone model, used to predict the elastic property of the interphase, while evaluating the elastic properties of the nanocomposite using a representative volume element. A special purpose program based on finite-element formulation is developed to study the buckling and post-buckling behaviour of nanocomposite plate. The formulation is based on first-order shear deformation theory in conjunction with geometrical non-linearity as per von Karman’s assumptions. A parametric study is conducted to investigate the effects of interphase between CNT and matrix, short-CNT and long-CNT reinforcements and boundary conditions on buckling and post-buckling response of nanocomposite plate. It is found that imperfect bonding between CNT and Mg results in the loss of buckling and post-buckling strength, as compared with perfect bonding, of CNT–Mg nanocomposite plate. It is also concluded that buckling and post-buckling strength is higher for long-CNT-reinforced nanocomposite plate than that of short-CNT reinforcement, irrespective of bonding between CNT and matrix material.     

Mechanical properties of carbon nanotube reinforced polymer nanocomposites: A coarse-grained model

In this work, a coarse-grained (CG) model of carbon nanotube (CNT) reinforced polymer matrix composites is developed. A distinguishing feature of the CG model is the ability to capture interactions between polymer chains and nanotubes. The CG potentials for nanotubes and polymer chains are calibrated using the strain energy conservation between CG models and full atomistic systems. The applicability and efficiency of the CG model in predicting the elastic properties of CNT/polymer composites are evaluated through verification processes with molecular simulations. The simulation results reveal that the CG model is able to estimate the mechanical properties of the nanocomposites with high accuracy and low computational cost. The effect of the volume fraction of CNT reinforcements on the Young's modulus of the nanocomposites is investigated. The application of the method in the modeling of large unit cells with randomly distributed CNT reinforcements is examined. The established CG model will enable the simulation of reinforced polymer matrix composites across a wide range of length scales from nano to mesoscale.     

Effect of an interphase region on debonding of a CNT reinforced polymer composite

The effect on stiffness and debonding of an interphase zone of altered polymer properties surrounding each carbon nanotube (CNT) in a CNT reinforced polymer composite is investigated. The interphase zone has position dependent material properties that merge with those of the polymer at a sufficiently large distance from the inclusion. There is evidence that such an interphase zone must be included in models in order to represent the overall composite properties. The analyses are based on an axisymmetric unit cell model of the composite. An elastic–viscoplastic conventional continuum constitutive relation (a size-independent relation between stress, strain and strain rate) is taken to characterize the bulk polymer material and the interphase, with the material properties being position dependent in the interphase. The interface between the polymer and the CNT is modeled by a phenomenological cohesive relation that allows for complete separation and the creation of new free surface. The effect of varying interface strength on the composite stress–strain response and on debonding is analyzed both with and without an interphase. The presence of an interphase increases the composite stiffness but promotes debonding which ultimately reduces composite stress carrying capacity. The compliance of the interface also affects the stress–strain response prior to debonding and leads to stress redistributions within both the fiber and the matrix (and/or interphase) which can affect the fracture mode that occurs.     

Modulus–density negative correlation for CNT-reinforced polymer nanocomposites: Modeling and experiments

Mechanical and weight properties of polymer nanocomposites (PNCs) are measured and modeled at the interlaminar region, predicting the density and elastic modulus of individual carbon nanotubes (CNTs). A simple model of the CNTs density and elastic modulus within the PNC, accounting for fundamental material properties, geometry, and interactions, is developed, capable of predicting CNT contributions in the PNCs. Furthermore, the model is validated with experimental results that demonstrate enhancement of the elastic modulus, while reducing density in the presence of aligned CNTs. By establishing an inverse relation of density and elastic modulus (negative correlation), it is demonstrated the potential of increasing mechanical properties while reducing weight. Therefore, by introducing controlled nanoporosity through suitable CNT distributions within the interlayer of multi-lamina structures, it is possible to simultaneously control effective weight reduction and enhanced modulus, toward bio-inspired carbon fiber reinforced polymer composites.     

Micromechanical analysis of fuzzy fiber reinforced composites

A novel fuzzy fiber reinforced composite (FFRC) reinforced with zig-zag single-walled carbon nanotubes (CNTs) and carbon fibers is proposed. The distinct constructional feature of this composite is that the uniformly aligned CNTs are radially grown on the surface of carbon fibers. Analytical models based on the mechanics of materials approach and the Mori–Tanaka method are derived to estimate the effective elastic constants of this proposed FFRC. The values of the effective elastic properties of this composite are estimated with and without considering an interphase between the CNT and the polymer matrix. It has been found that the transverse effective properties of this composite are significantly improved due to the radial growing of CNTs on the surface of carbon fiber. The effective properties are also found to be sensitive to the CNT diameter.     

Identification of effective properties of particle reinforced composite materials

For the determination of effective elastic properties an energy averaging procedure has been used for particle reinforced composite materials. This procedure is based on finite element calculations of the deformation energy of a characteristic volume element. The proposed approach allows the determination of effective properties of particle reinforced composite with acceptable precision. The calculated effective properties of the composite are found in range between upper and lower Hashin-Shtrikman bounds. The averaging elastic properties of the composite depend on the properties of the particles, matrix volume fraction of the particles and some parameters taking into account the influence of the interphase between matrix and particles. These dependencies can be presented by simple analytical functions approximatically. An identification procedure basing on numerical experiments allows the estimation of the unknown approximation parameters. The obtained functions describe precisely the numerical data for any relationship between material constituents.     

Predicting mechanical properties of fuzzy fiber reinforced composites: radially grown carbon nanotubes on the carbon fiber

It is intended to predict mechanical properties of fuzzy fiber reinforced polymer. An appropriate computational modeling is developed on the basis of bottom-up modeling covering all involved scales of nano, micro, meso and macro. The effective parameters of each scale are identified using top-down scanning approach and defining a representative volume element for each scale of analysis. At nano-scale, mechanical properties of isolated CNT are estimated. Then, at the upper scale of micro, the interaction between CNT and surrounding polymer is investigated considering non-bonded van der Waals interactions. Mechanical properties of the CNT/polymer nanocomposite with radial arrangement of CNT are derived at meso scale. Subsequently, mechanical properties of a single fuzzy fiber encompassing core carbon fiber and surrounding CNT/polymer are calculated. Finally, mechanical properties of the uni-directional and short fuzzy fiber reinforced composites are evaluated at the scale of macro. Treating CNT volume fraction and its arbitrary non-straight shapes as random parameters, developed modeling is conducted stochastically. The results imply on the importance of stochastic modeling, since deterministic modeling is led to a noticeable overestimation in predicted results. A very good agreement is reported between predicted results by stochastic modeling and published experimental data in literature.     

压电集成碳纳米管CNT增强功能梯度结构非线性建模与仿真

将碳纳米管(carbon nanotube,CNT)以梯度形式分布与基体材料结合,形成功能梯度(functionally graded,FG)结构。为了实现FG-CNT增强复合板在发生大变形时的准确计算,考虑四种典型的CNT分布形式,均匀分布、V型分布、O型分布和X型分布,建立基于Reissner-Mindlin板壳假设的大变形几何非线性有限元模型。该模型不仅包含了几何全非线性应变位移关系,还考虑了薄板结构法向发生大转角的情形。该研究与文献结果进行比较,验证模型的准确性;对FG-CNT增强复合板进行几何大变形非线性计算,分析CNT分布形式、CNT增强角度等因素对FG-CNT增强复合板刚度的影响。研究结果表明,CNT分布形式及增强角度对FG-CNT复合板的力学特征有显著的影响。     

Nonlinear vibration of nanotube-reinforced composite plates in thermal environments

This paper deals with the large amplitude vibration of nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation in thermal environments. The SWCNTs are assumed aligned, straight and a uniform layout. Two kinds of carbon nanotube-reinforced composite (CNTRC) plates, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRC plates are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The motion equations are based on a higher-order shear deformation plate theory that includes plate-foundation interaction. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. The equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies of CNTRC plates. Numerical results reveal that the natural frequencies as well as the nonlinear to linear frequency ratios are increased by increasing the CNT volume fraction. The results also show that the natural frequencies are reduced but the nonlinear to linear frequency ratios are increased by increasing the temperature rise or by decreasing the foundation stiffness. The results confirm that a functionally graded reinforcement has a significant effect on the nonlinear vibration characteristics of CNTRC plates.     

Effect of fibre coating and geometry on the tensile properties of hybrid carbon nanotube coated carbon fibre reinforced composite

Hierarchically structured hybrid composites are ideal engineered materials to carry loads and stresses due to their high in-plane specific mechanical properties. Growing carbon nanotubes (CNTs) on the surface of high performance carbon fibres (CFs) provides a means to tailor the mechanical properties of the fibre–resin interface of a composite. The growth of CNT on CF was conducted via floating catalyst chemical vapor deposition (CVD). The mechanical properties of the resultant fibres, carbon nanotube (CNT) density and alignment morphology were shown to depend on the CNT growth temperature, growth time, carrier gas flow rate, catalyst amount, and atmospheric conditions within the CVD chamber. Carbon nanotube coated carbon fibre reinforced polypropylene (CNT-CF/PP) composites were fabricated and characterized. A combination of Halpin–Tsai equations, Voigt–Reuss model, rule of mixture and Krenchel approach were used in hierarchy to predict the mechanical properties of randomly oriented short fibre reinforced composite. A fractographic analysis was carried out in which the fibre orientation distribution has been analyzed on the composite fracture surfaces with Scanning Electron Microscope (SEM) and image processing software. Finally, the discrepancies between the predicted and experimental values are explained.     

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.     

A study on the tensile response and fracture in carbon nanotube-based composites using molecular mechanics

A study on the mechanical properties of polyethylene and carbon nanotube (CNT) based composites is presented using molecular mechanics simulations. The systems being investigated consist of amorphous as well as crystalline polyethylene (PE) composites with embedded single-walled CNTs. All the systems are subjected to quasi-static tensile loading, with the assumption that no cross-link chemical bonds exist between the CNT and polyethylene matrix in the case of nanocomposites. Based on the numerical simulations, we report Young’s moduli (C₃₃) of 212–215 GPa for crystalline PE, which closely match the experimental measurement. Furthermore, elastic stiffness of 3.19–3.69 GPa and tensile strength of 0.21–0.25 GPa are obtained for amorphous PE. The tensile responses are found to be highly isotropic. In the case of crystalline PE reinforced by long through CNTs, moderate improvements in the tensile strength and elastic stiffness are observed. However, the results differ from the predictions using the rule of mixtures. On the other hand, although significant increase in the overall tensile properties is observed when amorphous PE is reinforced by long through CNTs, the load transfer at the nanotube/polymer interface has negligible effect. Finally, degradations in both tensile strength and elastic stiffness are reported when amorphous PE is reinforced by embedded CNTs. The study presented indicates the importance of specific CNT and polymer configurations on the overall properties of the nanocomposite.     

基于周期性边界条件的炭黑填充橡胶复合材料力学行为的预测

在分析炭黑填充橡胶复合材料的宏观与细观特征之间联系的基础上,提出了具有随机分布形态的代表性体积单元,推导并应用了周期性细观结构的边界约束条件,建立了三维多颗粒夹杂代表性体积单元的数值模型,对炭黑填充橡胶复合材料的宏观力学行为进行了模拟仿真。研究表明,该模型通过周期性边界条件的约束保证了宏观结构变形场和应力场的协调性;计算得到的炭黑填充橡胶复合材料的弹性模量明显高于未填充橡胶材料,并随着炭黑颗粒所占体积分数的增加而增大;该模型对复合材料有效弹性模量的预测结果与实验结果吻合较好,而且比Bergstrom三维模型的预测结果更好,证实了该模型能够用于炭黑颗粒增强橡胶基复合材料有效性能的模拟分析。     

Effects of the carbon nanotube distribution on the macroscopic stiffness of composite materials

Having extremely high stiffness and low specific weight, carbon nanotubes (CNTs) have been known recently as perfect reinforcing fibers in nanotechnology. They can improve the stiffness and strength of nanocomposites by being used as reinforcing elements for example in polymer matrices. The corresponding properties of the fibers and matrix, such as volume fraction and aspect ratio are some of the significant factors in the characterization of mechanical properties of CNT reinforced composites. In recent years, the way in which fibers are distributed inside the matrix, in terms of randomness, has introduced another important factor in characterizing the mechanical properties of such composites. Based on this factor, composites can be classified into two types namely, aligned and randomly distributed. This research has studied the effect of random distribution of fibers in the matrix on the elastic modulus and initial yield stress of the nanocomposite. Therefore, several models of composites, with different distribution of fibers, were considered while holding the volume fractions and aspect ratio constant. As a result, the effect of randomness on the effective modulus of CNT reinforced composites was estimated. The finite element method (FEM), using the MSC.Marc software, was employed to predict the effective modulus of CNT reinforced composites and the results were successfully validated by comparison with the analytical Halpin-Tsai method.     

连续纤维增强树脂复合材料纵向压缩强度预测模型的发展及其影响因素

基于高强、高韧、高模和压拉平衡为特征的第三代先进复合材料的需求,综述了连续纤维增强树脂复合材料纵向压缩强度预测模型的发展历程。基于纤维微屈曲、纤维扭结带、联合预测模型及渐进损伤失效模型,分别讨论了连续纤维增强树脂复合材料压缩失效机制,并在联合预测模型基础上,探究了碳纤维（直径、模量、体积分数、初始偏角）、树脂基体（弹性模量、剪切模量）及纤维/树脂界面三要素对连续纤维增强树脂复合材料纵向压缩强度和压缩失效形式的影响。      

Modeling the elastic properties of carbon nanotube array/polymer composites

In this work, the elastic properties of single-walled carbon nanotube (SWNT) arrays and their composites are investigated. The properties of twisted SWNT nano-arrays or ropes of circular cross-section are predicted through a finite element analysis by applying proper boundary conditions to the model and using the strain energy method. The nano-array properties are then used to describe the properties of twisted SWNT nano-array/polymer composites. The effect of volume fraction and aspect ratio of the nano-array as reinforcement for dilute polymer composite systems are examined for aligned and random reinforcement distribution using conventional micromechanics. Finally, elastic properties of the twisted SWNT nano-array/polymer composites are compared to the results from constitutive model of individual nanotube-reinforced polymer composites.     

Vibration damping characteristics of carbon nanotubes-based thin hybrid composite spherical shell structures

The present work deals with the evaluation of elastic properties and dynamic analyses of thin hybrid composite shell structures, which consist of conventional carbon fiber as the reinforcing phase and multiwalled carbon nanotubes-based polymer as the matrix phase. The Mori-Tanaka and strength of material method has been implemented to determine the elastic properties of such hybrid composite structures without and with considering agglomerations. An eight-noded shell element, which considers stress resultant-type Koiter's shell theory and transverse shear effect as per Mindlin's hypothesis having five degrees of freedom at each node, has been utilized for discretizing and analysis of such hybrid shell structures. The Rayleigh damping model has been implemented in order to study the effect of carbon nanotubes (CNTs) on damping capacity of such hybrid composite shell structures. Different types of spherical shell panels have been analyzed in order to study the time and frequency responses. Results show that the elastic properties as well as damping properties of such hybrid composite structures improved with the addition of CNTs as compared to conventional carbon fiber reinforced composites laminates; effects of some important parameters on the vibration characteristic of such hybrid composite shell structures have also been presented. The effects of agglomeration parameters on the elastic properties and their influences on the dynamic responses considering different layers and stacking sequences have also been investigated.