Full length articleEffect of CNT volume fractions on nonlinear vibrations of PMMA/CNT composite plates: A multiscale simulation
Graphical abstract
Introduction
Carbon nanotube (CNT) possesses the superior mechanical and physical properties, which is demonstrated as one of the most promising reinforcements for the polymer composites [1], [2], [3]. The previous studies have been indicated that the volume fractions of single-walled carbon nanotubes (SWCNT) affected the mechanical properties and dynamic behaviors of the composites significantly [4], [5], [6]. Hence, the investigation of the various CNT volume fractions is meaningful to explore the sensitivity of the composite to reinforcement and to provide the useful references for the subsequent theoretical and experimental investigations. The functionally graded CNT reinforced composite (FG-CNTRC) is a new generation of the advanced composite materials, in which the CNTs are applied as the reinforcements in a linear gradient arrangement [7], [8]. There are four linear gradient arrangement types of the functionally graded distribution for the CNT, which are defined as the FG-O, FG-X and FG-V based on the shape of the CNT distribution, as shown in Fig. 1(d). The polymeric FG-CNTRC is considered as one of the most promising materials in the aerospace field.
In the practical engineering applications, the long-term vibration behaviors raised from the environmental disturbance usually cause the failure of the composite structure [9], [10], [11], [12], [13], [14], [15]. The linear vibration theory has been widely and effectively applied in investigating the small amplitude vibration of the long and slender objects like the bridges, airplane wings, and helicopter blades. Comparatively, some phenomena of the nonlinearities in the mechanical systems including the gears, bearings and friction elements occur frequently. This cannot be explained by the linear theory [16], [17], [18], [19]. The nonlinear vibration theory provides an effective approach to study these complex phenomena. Considering the PMMA/CNT composite plate has been increasingly adopted as the cabin window and windshield in the airplane, which is usually subjected to the unexpected vibration due to the external environment. In this condition, the deflection leads to the nonlinear deformation. It is noticed that in such large deformation, the linear model would result in the significant errors without considering the nonlinear terms. Hence, the nonlinear vibrations of the composite plate is investigated in this study.
Due to the limitation of the test conditions, it is difficult to accurately measure the nonlinear vibrations using the experiments. Numerical analysis has been increasingly adopted to investigate the nonlinear vibration responses of the macroscopic composite structure, as shown in Fig. 1. For the nanoscale arrangements of the CNT in the molecular unit cell, the CNT are distributed uniformly, as shown in Fig. 1(b). The molecular model is then converted into a homogenized continuous representative volume element, as shown in Fig. 1(c). The representative volume elements are then considered as the micro-inclusions in the macro-structure, as shown in Fig. 1(d), where the CNT regions surrounded by the PMMA matrix have the same diameter with the single CNT arrangement at the nanoscale. In the aforementioned process, the CNT volume fractions always keep a constant value from the nanoscale to macroscale for each content case, and the macroscale dimension is the centimeter-level. Notably, the mesh-based method are applied to solve the governing equation of the CNTRC at the macroscale [20], [21], [22]. As the interpolations of the unknown variables are related to the geometry of the elements and generation of the meshes, the mesh-based method requires the meshing and remeshing of the solution domains in the case of the large deformation. Comparatively, the meshless methods are adopted to investigate the nonlinear vibrations [23], in which the shape functions require only the nodes rather than the meshes to avoid the mesh distortion from the large deformation in the mesh-based method [24]. Specifically, the free vibration of the FG-CNTRC plates was analyzed by using the meshless method [14], [25], which indicates that the dimensionless fundamental frequency of the plates has a higher value with a larger CNT volume fraction, and decreased as the aspect ratio increase for the plates. Meanwhile, an equivalent continuum meshless approach was adopted for the nonlinear analysis of the CNTRC, in which the SWCNT is considered as a hollow circular cylinder, and the longitudinal modulus increases significantly as the SWCNT length or SWCNT volume fraction increase [26]. With the aid of the meshless method, the nonlinear vibration of the carbon fiber reinforced composite rectangular plate with the random material properties was studied and found that the dimensionless nonlinear frequencies increase slowly with increasing the vibration amplitude [27]. Furthermore, the nonlinear behavior of the CNTRC with the uniform transverse mechanical loading was carried out using meshless method, and the increase in the CNT volume fraction from 11% to 17% led to the decrease in the deflection [28].
It is noticed that most of these relevant studies obtain the constitutive relation of the composite using the extended rule of mixtures (EROM). In the EROM, the efficiency parameters are introduced to account for the size-dependence of the composite, which are derived based on the fundamental inputs obtained from the molecular simulations. Specifically, the assumption of the conventional rule of mixtures is that the whole system is continuum and that the interfaces between the matrix and reinforcement remain fully intact. In some situations, the results of the nanocomposite are not consistent with MD simulation due to the nanoscale, surface and other factor effects. The efficiency parameters are then introduced to the EROM to capture the nanoscale effects including the strain rate, size of reinforcement, polymer chain length, and intermolecular-coupled stress. However, there are few derivations of the efficiency parameters relevant to the FG-based composite. Therefore, it still requires the derivation of the efficiency parameters via the molecular simulation.
The molecule dynamics (MD) simulation enables the simulation of the molecular interactions of the materials and visualization of the dynamic evolution [29], [30], which provides the fundamental inputs for a bottom-up multiscale modeling of the nonlinear vibration investigation [31]. In recent studies, the longitudinal, transverse and shear moduli of the poly (methyl methacrylate) (PMMA)/SWCNT and poly {(m-phenylene-vinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]}(PmPV)/SWCNT nanocomposites with SWCNT volume fractions over 12% were studied using MD simulation. The efficiency parameters were derived by substituting the simulation results into the EROM to predict the mechanical properties and nonlinear vibrations of the composite at the macroscale [32], [33], [34]. Meanwhile, the longitudinal, transverse and shear moduli of the polyethylene composites with 2.8% SWCNT volume fraction were measured using MD simulation, and the obtained moduli were substituted into the EROM to derive the efficiency parameters [33], [35]. Furthermore, in our recent study, the longitudinal, transverse and shear moduli of the PMMA and PMMA/CNT nanocomposite obtained from MD simulation were used in the calculation of the efficiency parameters, which were applied to investigate the composite structure. The results indicated that vibration responses were affected significantly by the CNT distribution in the composite plate [36]. From these relevant studies, it was found that the CNT volume fraction has a significant influence on the mechanical properties of the nanocomposite. In terms of the nonlinear vibration responses of two-phase composites based on MD simulation, the previous works mainly focused on the vibration responses of the composite system with the high CNT volume fraction, namely, over 10% or a single CNT volume fraction case. In general, the CNT volume fraction in the experimental samples is usually lower than 10% [37], and the nonlinear vibration of the composites with such CNT volume fraction remains unpredictable. Furthermore, the mechanism of the CNT enhancement on the nonlinear vibration of the composite is not clear in the previous investigations.
In this paper, a multiscale homogenization approach is adopted for investigating the nonlinear vibrations of the PMMA/CNT composites plate with various CNT volume fractions, and to reveal the mechanism of the CNT enhancement on the nonlinear vibrations of the composite structure. The present approach combines MD simulation, micro-mechanical method based on the EROM, and macro-mechanical analysis via meshless method in which the mesoscale is not considered [36]. Such multiscale approach has also been widely applied to study the mechanical properties of the CNTRC [4], [38]. The longitudinal, transverse and shear moduli of the CNT, PMMA polymer and PMMA/CNT nanocomposite are integrated into the EROM to derive the efficiency parameters of the PMMA/CNT nanocomposite with various CNT volume fractions. We adopt the FG-based EROM to obtain the constitutive equation of the FG-based PMMA/CNT composite plates. Applying the derived governing equation by the meshless method, the nonlinear vibrations are studied for the FG-based PMMA/CNT composite plate, which provides a theoretical instruction for the design of the composite in the engineering fields.
Section snippets
Simulation procedures
The simulation approach herein starts with the model construction of the PMMA/CNT nanocomposite with the various CNT volume fractions. Carrying out the tensile and shear deformations, the longitudinal, transverse and shear moduli of the CNT, PMMA and PMMA/CNT nanocomposites are calculated respectively, which are used as the inputs in the derivation of the efficiency parameter. The discrete nonlinear vibration equation is established based on the virtual work principle and meshless method,
Results and discussions
The molecular simulation results of the CNT, PMMA polymer, and PMMA/CNT nanocomposites with the various CNT volume fractions are shown herein, and the corresponding efficiency parameters are presented and formulated with analytical expressions. Subsequently, the nonlinear vibration responses of the FG-based composite plates are discussed.
Conclusions
In this study, the nonlinear vibration of PMMA/CNT composite plate with various CNT volume fractions is analyzed by the three-step bottom-up multiscale approach, which includes MD simulation, micro-mechanical approach, and meshless method. MD simulation is initially adopted to obtain the longitudinal, transverse, and shear moduli of CNT, PMMA polymer, and PMMA/CNT nanocomposite at the nanoscale, and then the PMMA/CNT nanocomposites are utilized as micro-inclusions in the micro-mechanical
CRediT authorship contribution statement
J.F. Wang: Conceptualization, Methodology, Formal analysis, Software, Investigation, Validation, Data curation, Writing – original draft. J.P. Yang: Conceptualization, Methodology, Formal analysis, Software, Validation, Data curation, Writing – original draft. L.-h. Tam: Conceptualization, Methodology, Writing – review & editing, Resources, Supervision. W. Zhang: Conceptualization, Methodology, Validation, Writing – review & editing, Resources, Supervision, Funding acquisition, Project
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors gratefully acknowledge the support of National Natural Science Foundation of China through Grants No. 12072003, 11702006, 11832002, and 51808020, the General Program of Science and Technology Development Project of Beijing Municipal Education Commission of China through Grant No. KM201810005003.
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