Improvement in dynamic properties of laminated MWCNT-polystyrene composite beams via an integrated numerical–experimental approach

A new concept for the optimization of dynamic behavior of laminated nanocomposites is introduced where fiber orientation factor in continuous fiber-reinforced composites is replaced by different wt.% of carbon nanotubes (CNTs) in each layer. First, at a design concept level, an optimum distribution of multi-walled CNTs (MWCNTs) through the thickness of a typical cantilever beam is sought to achieve its highest fundamental natural frequency for a given weight percent of MWCNTs. This is done using a finite element (FE) model in ABAQUS along with a user-defined Python code. Next, based on the obtained optimum distribution, actual laminated MWCNT/polystyrene (PS) composite beams were fabricated and their effective stiffness, fundamental natural frequencies and damping ratios were measured through static deflection and free vibration tests. It was found that the optimum distribution of MWCNTs resulted in an increase of 21.9% and 10.4% in the effective Young’s modulus and the fundamental damped natural frequency values, respectively, which were almost two-fold higher than those of a beam with a uniform MWCNT distribution. In addition, compared to a pure polymer beam, 38.9% and 27.8% improvements in the damping ratio of the uniformly and optimally distributed MWCNT polymer composite beams were achieved.     

Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load

This paper studies the dynamic response of functionally graded beams with an open edge crack resting on an elastic foundation subjected to a transverse load moving at a constant speed. It is assumed that the material properties follow an exponential variation through the thickness direction. Theoretical formulations are based on Timoshenko beam theory to account for the transverse shear deformation. The cracked beam is modeled as an assembly of two sub-beams connected through a linear rotational spring. The governing equations of motion are derived by using Hamilton’s principle and transformed into a set of dynamic equations through Galerkin’s procedure. The natural frequencies and dynamic response with different end supports are obtained. Numerical results are presented to investigate the influences of crack location, crack depth, material property gradient, slenderness ratio, foundation stiffness parameters, velocity of the moving load and boundary conditions on both free vibration and dynamic response of cracked functionally graded beams.     

Dynamics of elastically connected double-functionally graded beam systems with different boundary conditions under action of a moving harmonic load

This paper studies the dynamic responses of an elastically connected double-functionally graded beam system (DFGBS) carrying a moving harmonic load at a constant speed by using Euler–Bernoulli beam theory. The two functionally graded (FG) beams are parallel and connected with each other continuously by elastic springs. Six elastically connected double-functionally graded beam systems (DFGBSs) having different boundary conditions are considered. The point constraints in the form of supports are assumed to be linear springs of large stiffness. It is assumed that the material properties follow a power-law variation through the thickness direction of the beams. The equations of motion are derived with the aid of Lagrange’s equations. The unknown functions denoting the transverse deflections of DFGBS are expressed in polynomial form. Newmark method is employed to find the dynamic responses of DFGBS subjected to a concentrated moving harmonic load. The influences of the different material distribution, velocity of the moving harmonic load, forcing frequency, the rigidity of the elastic layer between the FG beams and the boundary conditions on the dynamic responses are discussed.     

Frequency analysis of sandwich beam with FG carbon nanotubes face sheets and flexible core using high-order element

In the present paper, the vibrational behavior of sandwich beam with a flexible core and anti-symmetric functionally graded carbon nanotubes face sheet is investigated. Carbon nanotubes are considered as functional graded materials in the thickness of the faces and their properties change along the thickness of the face sheets. For the modeling of sandwich beam behavior, the Euler–Bernoulli theory is used for face sheets and the semi-3D elasticity is used for the core, which allowed us to investigate the flexibility of the core. Differential equations of motions are derived using the virtual displacement principle. In this research, a high-order element is presented for solving equations of motion, and then by using this element, the finite element formulation has been extracted and solved. Numerical results are obtained for various boundary conditions, which include simply support, clamped, free-clamped, and simply support-clamped. Also, different volumes of carbon nanotubes have been investigated for different distributions. The results showed that the distribution of the FG-X pattern carbon nanotube leads to the highest natural frequency of the beam. The main conclusion of this research is that, in most cases the FG-O pattern has the lowest natural frequencies and in some cases the FG-Λ pattern has the lowest natural frequencies. In other words, generally, it can be say that the lowest natural frequencies of the sandwich beam with functionally graded carbon nanotubes faces depend on the boundary conditions, thickness ratio, and also the volume fraction of carbon nanotubes. In this paper also the effect of geometric angles of the beam, such as the thickness of the core and face sheet thickness on natural frequency of the system is also investigated.     

Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method

In this paper, dynamic analysis of nanocomposite cylinders reinforced by single-walled carbon nanotubes (SWCNTs) subjected to an impact load was carried out by a mesh-free method. Free vibration and stress wave propagation analysis of carbon nanotube reinforced composite (CNTRC) cylinders are presented. In this simulation, an axisymmetric model is used. Four types of distributions of the aligned carbon nanotubes (CNTs) are considered; uniform and three kinds of functionally graded (FG) distributions along the radial direction of cylinder. Material properties are estimated by a micro mechanical model. In the mesh-free analysis, moving least squares (MLSs) shape functions are used for approximation of displacement field in the weak form of motion equation and the transformation method was used for the imposition of essential boundary conditions. Effects of the kind of distribution and volume fractions of carbon nanotubes and cylinder thickness on the natural frequencies and stress wave propagation of CNTRC cylinders are investigated. Results obtained for this analysis were compared with FEM and previous published work and good agreement was seen between them.     

Dynamic stability analysis of multi-walled carbon nanotubes with arbitrary boundary conditions based on the nonlocal elasticity theory

Dynamic stability of embedded multi-walled carbon nanotubes (MWCNTs) in an elastic medium and thermal environment and subjected to an axial compressive force is studied based on the nonlocal elasticity and Timoshenko beam theory. The developed nonlocal beam model has the capability to consider the small scale effects. The generalized differential quadrature method is employed to discretize the dynamic governing differential equations of MWCNTs with various end supports. A parametric study is conducted to investigate the influences of static load factor, temperature change, nonlocal parameter, slenderness ratio, and spring constant of elastic medium on the dynamic stability characteristics of MWCNTs.     

Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory

This article proposes a higher-order shear deformation beam theory for free vibration analysis of functionally graded carbon nanotube-reinforced composite sandwich beams in a thermal environment. The temperature-dependent material properties of functionally graded carbon nanotube-reinforced composite beams are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture. The governing equations and boundary conditions are derived by using Hamilton's principle, and the Navier solution procedure is used to achieve the natural frequencies of the sandwich beam in a thermal environment. A parametric study is led to carry out the effects of carbon nanotube volume fractions, slenderness ratio, and core-to-face sheet thickness ratio on free vibration behavior of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets. Numerical results are also presented in order to compare the behavior of sandwich beams including uniformly distributed carbon nanotube-reinforced composite face sheets to those including functionally graded carbon nanotube-reinforced composite face sheets.     

Nanocomposite based on modified multiwalled carbon nanotubes: Fabrication by an oriented spinning process and electrical conductivity

We have produced nanocomposite films consisting of modified multiwalled carbon nanotubes (MWCNTs) and a polymer (MWCNT/polymer weight ratio of 95/5). The nanocomposite has been applied to a paper substrate by an oriented spinning process from a liquid phase. The resistivity of the films has been measured as a function of temperature in the range 20–140°C along and across the preferential orientation direction of the nanotubes in the nanocomposite. The results point to an irreversible transition from semiconducting to metallic behavior of the conductivity of the films.     

Dynamic response of a cracked beam subject to a moving load

Summary The dynamic response of a beam with a single-sided crack subject to a moving load on the opposite side is analyzed using Euler beam theory and the assumed mode method. The beam is modeled as two separate beams divided by the crack. Two different sets of admissible functions which satisfy the respective geometric boundary conditions are assumed for these two fictitious sub-beams. The rotational discontinuity at the crack is modeled by a torsional spring with an equivalent spring constant for the crack. The transverse deflection at the crack is matched by a linear spring of very large stiffness. Results of numerical simulations are presented for various combinations of constant axial velocity of the moving load and the crack size.     

Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Kármán geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, vibration amplitude, slenderness ratio and end supports on the nonlinear free vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear free vibration characteristics of FG-CNTRC beams.     

Analysis of postbuckling of graded porous GPL-reinforced beams with geometrical imperfection

This research is concerned with the analysis of post-buckling of a nano-composite beam reinforced by graphene plateletes (GPLs) having geometrical imperfection. GPLs are uniformly and nonuniformly distributed thorough the thickness direction. Different porosity distributions are considered. The elastic properties of the nanocomposite are obtained by employing Halpin-Tsai micromechanics model. The postbuckling load-deflection relation is obtained by solving the governing equations having cubic nonlinearity applying Galerkin's method needless of any iteration process. New results show the importance of porosity coefficient, porosity distribution, GPL distribution, GPL weight fraction, geometrical imperfection, and foundation parameters on nonlinear buckling behavior of porous nanoscale beams. Specially, porosities and GPL reinforcement have a great impact on postbuckling configuration of both ideal and imperfect nanocomposite beams.     

Dynamic analysis of partial-interaction composite beams

The state-space method is used to investigate the dynamic behavior of partial-interaction composite beams. The state-space formula is properly established via selecting the appropriate state variables. The characteristic equations of frequency and the corresponding modal shapes of free vibration under generalized boundary conditions are then obtained. Based on the state-space formula and the concept of symplectic inner product, a new route is proposed to prove the orthogonality of vibration modes of composite beams under generalized boundary conditions. By virtue of mode superposition method, we obtain the solution of the inhomogeneous equation for forced vibration and the dynamic response of the partial-interaction composite beams subjected to a uniformly distributed pulse load, a uniformly distributed step load or a moving constant force. Finally, some numerical examples are presented and compared with those available in the literature to validate the present method. The effect of the rigidity of shear connector on dynamic responses of a composite beam under moving constant force is discussed.     

Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory

This paper mainly presents bending and free vibration analyses of thin-to-moderately thick composite plates reinforced by single-walled carbon nanotubes using the finite element method based on the first order shear deformation plate theory. Four types of distributions of the uniaxially aligned reinforcement material are considered, that is, uniform and three kinds of functionally graded distributions of carbon nanotubes along the thickness direction of plates. The effective material properties of the nanocomposite plates are estimated according to the rule of mixture. Detailed parametric studies have been carried out to reveal the influences of the volume fractions of carbon nanotubes and the edge-to-thickness ratios on the bending responses, natural frequencies and mode shapes of the plates. In addition, the effects of different boundary conditions are also examined. Numerical examples are computed by an in-house finite element code and the results show good agreement with the solutions obtained by the FE commercial package ANSYS.     

The effects of carbon nanotube orientation and aggregation on vibrational behavior of functionally graded nanocomposite cylinders by a mesh-free method

In this paper, axisymmetric natural frequencies of nanocomposite cylinders reinforced by straight single-walled carbon nanotubes are presented based on a mesh-free method. The straight carbon nanotubes (CNTs) are oriented, aligned or randomly or locally aggregated into some clusters. Volume fractions of the CNTs and clusters are assumed to be functionally graded along the thickness, so material properties of the carbon nanotube reinforced composite cylinders are variable and are estimated based on the Eshelby–Mori–Tanaka approach. In the mesh-free analysis, moving least squares shape functions are used for an approximation of the displacement field in the weak form of motion equation, and the transformation method is used for the imposition of essential boundary conditions. The effects of orientation and aggregation of the functionally graded CNT are studied. It is observed that kind of distributions, aggregation or even randomly orientations of CNTs has significant effect on the effective stiffness and frequency parameter.     

Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load

In this paper, free vibration characteristics and the dynamic behavior of a functionally graded simply-supported beam under a concentrated moving harmonic load are investigated. The system of equations of motion is derived by using Lagrange’s equations under the assumptions of the Euler–Bernoulli beam theory. Trial functions denoting the transverse and the axial deflections of the beam are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. It is assumed that material properties of the beam vary continuously in the thickness direction according to the exponential law and the power-law form. In this study, the effects of the different material distribution, velocity of the moving harmonic load, the excitation frequency on the dynamic responses of the beam are discussed. Numerical results show that the above-mentioned effects play very important role on the dynamic deflections of the beam.     

Two-dimensional elasticity solution for transient response of simply supported beams under moving loads

A semi-analytical analysis for the transient elastodynamic response of an arbitrarily thick simply supported beam due to the action of an arbitrary moving transverse load is presented, based on the linear theory of elasticity. The solution of the problem is derived by means of the powerful state space technique in conjunction with the Laplace transformation with respect to the time coordinate. The inversion of Laplace transform has been carried out numerically using Durbin??s approach based on Fourier series expansion. Special convergence enhancement techniques are invoked to completely eradicate spurious oscillations and obtain uniformly convergent solutions. Detailed numerical results for the transient vibratory responses of concrete beams of selected thickness parameters are obtained and compared for three types of harmonic moving concentrated loads: accelerated, decelerated and uniform. The effects of the load velocity, pulsation frequency and beam aspect ratio on the dynamic response are examined. Also, comparisons are made against solutions based on Euler?CBernoulli and Timoshenko beam models. Limiting cases are considered, and the validity of the model is established by comparison with the solutions available in the existing literature as well as with the aid of a commercial finite element package.     

A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities

This article proposes a four-variable shear deformation refined beam theory for thermo-mechanical vibration characteristics of porous, functionally graded (FG) beams exposed to various kinds of thermal loadings by using an analytical method. Thermo-mechanical properties of functionally graded material (FGM) beams are supposed to vary through the thickness direction, and are estimated through the modified power-law rule in which the porosities with even and uneven types are approximated. The material properties of FGM beams are supposed to be temperature dependent. Porosities possibly occur inside FGMs during fabrication because of technical problems that lead to the creation of microvoids in these materials. The variation of pores along the thickness direction influences the mechanical properties. Thus, it is incumbent to predict the effect of porosities on the thermo-mechanical vibration behavior of FG beam in the present study. Four types of thermal loading, namely, uniform, linear, nonlinear, and sinusoidal temperature rises through the z-axis direction are discussed. The governing differential equations and boundary conditions of FG porous beams subjected to thermal loadings are formulated through Hamilton's principle, based on a four-variable refined theory that considers a constant transverse displacement and higher order variation of axial displacement through the depth of the beam without the need of any shear correction factors. An analytical solution procedure is used to achieve the natural frequencies of porous FG beams subjected to various temperature fields. The impact of several specific parameters such as power-law exponent, porosity volume fraction, different porosity distribution, and thermal effect on the vibration of the porous FG beams is perused and discussed in detail. It is deduced that these parameters play a notable role on the thermo-dynamic behavior of porous FG beams. Presented numerical results can serve as benchmarks for the future analyses of FG beams with porosity phases.     

Dispersion of multi-walled carbon nanotubes in biodegradable poly(butylene succinate) matrix

This communication describes the preparation, characterization and properties of biodegradable poly(butylene succinate) (PBS)/multi-walled carbon nanotubes (MWCNTs) nanocomposite. Nanocomposite was prepared by melt-blending in a batch mixer and the amount of MWCNTs loading was 3 wt%. State of dispersion-distribution of the MWCNTs in the PBS matrix was examined by scanning and transmission electron microscopic observations that revealed homogeneous distribution of stacked MWCNTs in PBS matrix. The investigation of the thermomechanical behavior was performed by dynamic mechanical thermal analysis. Results demonstrated substantial enhancement in the mechanical properties of PBS, for example, at room temperature, storage flexural modulus increased from 0.64 GPa for pure PBS to 1.2 GPa for the nanocomposite, an increase of about 88% in the value of the elastic modulus. The tensile modulus and thermal stability of PBS were moderately improved after nanocomposite preparation with 3 wt% of MWCNTs, while electrical conductivity of neat PBS dramatically increased after nanocomposite formation. For example, the in plane conductivity increased from 5.8 x 10(-9) S/cm for neat PBS to 4.4 x 10(-3) for nanocomposite, an increase of 10(6) fold in value of the electrical conductivity.     

Nonlinear forced vibration analysis of clamped functionally graded beams

Multiple time scale solutions are presented to study the nonlinear forced vibration of a beam made of symmetric functionally graded (FG) materials based on Euler?CBernoulli beam theory and von Kármán geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through the thickness direction. A Galerkin procedure is used to obtain a second-order nonlinear ordinary equation with cubic nonlinear term. The natural frequencies are obtained for the nonlinear problem. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FG beams are discussed. Also, forced vibrations of the system in primary and secondary resonances have been studied, and the effects of different parameters on the frequency-response have been investigated.     

Effect of multiwalled carbon nanotubes on the crystallization and dielectric properties of BP-PEN nanocomposites

In this work, polymer-based nanocomposite films formed from biphenol poly(arylene ether nitrile) (BP-PEN) and multiwalled carbon nanotubes (MWCNTs) were successfully prepared by the solution casting method combined with continuous ultrasonic dispersion technology. The micromorphology, thermal, mechanical and dielectric properties of the nanocomposite films were investigated in detail. Non-isothermal crystallization behavior studies indicate that the presence of MWCNTs enhances the crystallization of BP-PEN in the nanocomposites, which is consistent with the XRD analysis. Most importantly, it could be observed that the film containing 0.8 wt% MWCNTs reached the maximum crystallinity. Although, incorporation of MWCNTs did not obviously increase the mechanical of the films, all the nanocomposite films still exhibited excellent mechanical strength. The SEM micrographs of the nanocomposite films showed that the MWCNTs were uniformly coated by BP-PEN crystals, and indicating significantly improved nucleation ability of MWCNTs for polymer crystallization.