Eshelby Mori Tanaka Approach Research Articles

Vibrational behavior of non‐uniform piezoelectric sandwich beams made of CNT‐reinforced polymer nanocomposite by considering the agglomeration effect of CNTs

The present work deals with natural frequency analysis of non‐uniform nanocomposite beams with surface‐bonded piezoelectric layers by considering the agglomeration effect of Carbon Nanotubes (CNTs). The material properties of the nanocomposite beam are predicted employing Eshelby–Mori–Tanaka approach based on an equivalent fiber and compared with available experimental data in the literature. The equation of motion is derived employing Hamilton's principle and Generalized Differential Quadrature technique as an efficient and accurate numerical tool is employed for free vibration analysis of the structures. Detailed parametric studies are carried out to examine influences of CNTs volume fraction, CNTs agglomeration, thickness of piezoelectric layers, piezoelectric constant, dielectric constant, boundary conditions etc., on the behavior of freely vibrating smart structure. Results represent that for some cases, consideration of agglomeration effect of CNTs is not required while for some other cases it has a significant role on the response. The numerical results also reveal that using thick piezoelectric layer does not necessarily improve the natural frequency of the structure. POLYM. COMPOS., 38:E553–E562, 2017. © 2016 Society of Plastics Engineers

Buckling and Vibration Analysis of Functionally Graded Carbon Nanotube-Reinforced Beam Under Axial Load

Buckling and vibration analysis of cantilever functionally graded (FG) beam that reinforced with carbon nanotube (CNT) is the purpose of this paper. The beam is graded in the thickness direction, and compressive axial force impressed the beam. The volume fractions of randomly oriented agglomerated single-walled CNTs (SWCNTs) are assumed to be graded in the thickness direction. To determine the effect of CNT agglomeration on the elastic properties of CNT-reinforced FG-beam, a two-parameter micromechanics model of agglomeration is employed. In this paper, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is obtained. The stability and motion equations are based on the two-dimensional elasticity theory and Hamilton’s principle. The generalized differential quadrature method (GDQM) that has high accuracy is used to discretize the equations of stability and motion and to implement the boundary conditions. To study the accuracy of the present analysis, a compression is carried out with a known data. Convergence rate, the influence of graded agglomerated CNTs, and the effect of axial forces exerted on the beam, on the natural frequencies of reinforced beam by randomly oriented agglomerated CNTs are investigated.

Vibration and damping analysis of functionally graded carbon nanotubes reinforced hybrid composite shell structures

The present article deals with the vibration and damping analysis of functionally graded carbon nanotubes reinforced hybrid composite (FG-CNTRHC) shell structure which consists of conventional carbon fiber as reinforcing phase and single-walled carbon nanotubes based polymer as matrix phase. The Eshelby–Mori–Tanaka approach in conjunction with strength of material approach is implemented to obtain the material properties of FG-CNTRHC structures. The material properties of FG-CNTRHCs are assumed to be graded through the thickness direction according to power law distributions of the volume fraction of carbon fibers and fiber orientations. An eight node shell element considering transverse shear effect according to Mindlin’s hypothesis has been formulated for finite element modelling and analysis of such functionally graded hybrid composite shell structures. The formulation of shell midsurface in an arbitrary curvilinear coordinate system based on the tensorial notation also presented. The Rayleigh damping model has been implemented in order to study the effects of carbon nanotubes (CNTs) on damping capacity of such hybrid shell structures. Different types of spherical shell panels have been analyzed in order to study the time and frequency responses. The influences of CNTs, carbon fiber, geometry of the shell, power law index and material distributions on the vibration damping characteristics of FG-CNTRHC shell structures have also been presented and discussed. Various types of FG-CNTRHC shell structures (such as spherical, ellipsoidal, doubly curved and cylindrical) have been analyzed and discussed in order to comparative studies in terms of settling time, first resonant frequency and absolute amplitude corresponding to first resonant frequency and the effects of CNTs on vibrations responses of such shell structures are also presented.

Free vibration analysis of functionally graded nanocomposite sandwich beams resting on Pasternak foundation by considering the agglomeration effect of CNTs

In the present work, by considering the agglomeration effect of single-walled carbon nanotubes, free vibration characteristics of functionally graded (FG) nanocomposite sandwich beams resting on Pasternak foundation are presented. The carbon nanotubes (CNTs) volume fraction is graded through the thicknesses of face sheets according to a generalized power–law distribution. The material properties of the FG nanocomposite sandwich beam are estimated using the Eshelby–Mori–Tanaka approach based on an equivalent fiber. The equations of motion are derived based on Timoshenko beam theory and employing Hamilton's principle. Generalized differential quadrature technique as an efficient and accurate numerical tool is employed to obtain the natural frequencies of the structure. The verification study represents the accuracy of the solution for free vibration analysis of the nanocomposite sandwich beams resting on elastic foundation. Detailed parametric studies are carried out to investigate the influences of CNTs agglomeration, different profiles of CNT volume fraction such as symmetric, asymmetric, and classic, Winkler foundation modulus, shear elastic foundation modulus, length to span ratio, thicknesses of face sheets, and boundary conditions on the vibrational behavior of the structure. It is shown that the natural frequencies of structure are seriously affected by the influence of CNTs agglomeration. Results also represent the fact that utilizing FG nanocomposite sandwich beams in most agglomeration states improves the fundamental frequencies of the structure, but in some cases has a destructive effect on the vibrational characteristics.

Stress analysis of functionally graded open cylindrical shell reinforced by agglomerated carbon nanotubes

In this paper, stresses due to bending behavior of functionally graded carbon nanotube-reinforced (FGCNTR) open cylindrical shells subjected to mechanical loads is studied. The material properties of FGCNTR shells are assumed to be graded in the thickness direction, and are estimated using a two-parameter micromechanics model in which Eshelby–Mori–Tanaka approach is employed. The primary bending formulation is based on the linear, small-strain, three-dimensional elasticity theory. In addition, the cylindrical shells are analyzed using the third-order shear deformation theory (TSDT). In order to discretize the governing equations, the two-dimensional generalized differential quadrature method (2-D GDQM) in the thickness and longitudinal directions and the trigonometric functions in tangential direction are used. The effects of agglomeration parameters, CNTs volume fraction, and CNTs distribution through the thickness on the bending behavior of FGCNTR open cylindrical shells are studied. In addition, the mechanical stresses obtained from 3-D elasticity are compared with those obtained using TSDT for a different range of geometric and agglomeration parameters.

Dynamic stability analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the element-free kp-Ritz method

In this paper, a first-known dynamic stability analysis of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels under static and periodic axial force by using the mesh-free kp-Ritz method is presented. The cylindrical panels are reinforced by single-walled carbon nanotubes (SWCNTs) with different types of distributions, i.e. uniform and three kinds of functionally graded distributions of carbon nanotubes along thickness direction of the panels. Eshelby–Mori–Tanaka approach is employed to estimate effective material properties of the resulting nanocomposite panels. By applying the Ritz minimization procedure to the energy expressions, a system of Mathieu–Hill equations is formulated. Then the principal instability regions are analyzed through Bolotin’s first approximation. Detailed parametric studies have been carried out to reveal the influences of volume fraction of carbon nanotubes, edge-to-radius ratio and radius-to-thickness ratio. In addition, effects of different boundary conditions and types of distributions of carbon nanotubes are examined in detail.

Prediction of thermal conductivity for irradiated SiC/SiC composites by informing continuum models with molecular dynamics data

This article proposes a new method to estimate the thermal conductivity of SiC/SiC composites subjected to neutron irradiation. The modeling method bridges different scales from the atomic scale to the scale of a 2D SiC/SiC composite. First, it computes the irradiation-induced point defects in perfect crystalline SiC using molecular dynamics (MD) simulations to compute the defect thermal resistance as a function of vacancy concentration and irradiation dose. The concept of defect thermal resistance is explored explicitly in the MD data using vacancy concentrations and thermal conductivity decrements due to phonon scattering. Point defect-induced swelling for chemical vapor deposited (CVD) SiC as a function of irradiation dose is approximated by scaling the corresponding MD results for perfect crystal β-SiC to experimental data for CVD-SiC at various temperatures. The computed thermal defect resistance, thermal conductivity as a function of grain size, and definition of defect thermal resistance are used to compute the thermal conductivities of CVD-SiC, isothermal chemical vapor infiltrated (ICVI) SiC and nearly-stoichiometric SiC fibers. The computed fiber and ICVI-SiC matrix thermal conductivities are then used as input for an Eshelby–Mori–Tanaka approach to compute the thermal conductivities of 2D SiC/SiC composites subjected to neutron irradiation within the same irradiation doses. Predicted thermal conductivities for an irradiated Tyranno-SA/ICVI-SiC composite are found to be comparable to available experimental data for a similar composite ICVI-processed with these fibers.

Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels

A large deflection geometrically nonlinear behavior of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels under uniform point transverse mechanical loading is studied. The analysis is carried out using the kp-Ritz method with kernel particle function is employed to construct the shape functions for the two-dimensional displacement approximations. Based on the first-order shear deformation shell theory, nonlinear governing equations are developed with geometric nonlinearity taking the form of von Kármán strains. It is assumed that carbon nanotubes are uniaxially aligned in the axial direction and are functionally graded in thickness direction of the cylindrical panels. The effective material properties of resulting CNTR-FG panels are estimated by employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach. A stabilized conforming nodal integration scheme is employed to evaluate the system bending stiffness and the membrane as well as shear terms are calculated by the direct nodal integration method to eliminate shear locking, for a very thin cylindrical panel. Several numerical example problems are examined to reveal the influences of volume fraction of carbon nanotubes, span angle, edge-to-radius ratio and thickness on nonlinear responses of the CNTR-FG panels. Moreover, effects of different boundary conditions and distribution type of carbon nanotubes are also investigated.

Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels

The analysis of flexural strength and free vibration of carbon nanotube reinforced composite cylindrical panels is carried out. Four types of distributions of uniaxially aligned reinforcements are considered, i.e. uniform and three kinds of functionally graded distributions of carbon nanotubes along thickness direction of the panels. Material properties of nanocomposite panels are estimated by employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach. The governing equations are developed based on the first-order shear deformation shell theory. Detailed parametric studies have been carried out to reveal the influences of volume fraction of carbon nanotubes, edge-to-radius ratio and thickness on flexural strength and free vibration responses of the panels. In addition, effects of different boundary conditions and types of distributions of carbon nanotubes are examined.

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.

Fiber/matrix interfacial thermal conductance effect on the thermal conductivity of SiC/SiC composites

Abstract SiC/SiC composites used in fusion reactor applications are subjected to high heat fluxes and require knowledge and tailoring of their in-service thermal conductivity. Accurately predicting the thermal conductivity of SiC/SiC composites as a function of temperature will guide the design of these materials for their intended use, which will eventually include the effects of 14-MeV neutron irradiations. This paper applies an Eshelby–Mori–Tanaka approach (EMTA) to compute the thermal conductivity of unirradiated SiC/SiC composites. The homogenization procedure includes three steps. In the first step EMTA computes the homogenized thermal conductivity of the unidirectional (UD) SiC fiber embraced by its coating layer. The second step computes the thermal conductivity of the UD composite formed by the equivalent SiC fibers embedded in a SiC matrix, and finally the thermal conductivity of the as-formed SiC/SiC composite is obtained by averaging the solution for the UD composite over all possible fiber orientations using the second-order fiber orientation tensor. The EMTA predictions for the transverse thermal conductivity of several types of SiC/SiC composites with different fiber types and interfaces are compared to the predicted and experimental results by Youngblood et al. [J. Nucl. Mater. 307–311 (2002) 1120–1125, Fusion Sci. Technol. 45 (2004) 583–591, Compos. Sci. Technol. 62 (2002) 1127–1139.]

Local aggregation effect of CNT on the vibrational behavior of four‐parameter continuous grading nanotube‐reinforced cylindrical panels

AbstractFree vibrations analysis of four‐parameter continuously graded nanocomposite cylindrical panels reinforced by randomly oriented straight and local aggregation single‐walled carbon nanotubes (CNTs) are presented based on three‐dimensional theory of elasticity. The material properties of continuously graded carbon nanotube‐reinforced composites (CG‐CNTRCs) are estimated through the Eshelby–Mori–Tanaka approach based on an equivalent fiber. The generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and impediment the boundary conditions. One of the contributions of this work is to illustrate the influence of the four parameters of power‐law distributions on the vibration behavior of functionally graded orthotropic cylindrical panels reinforced by nanotube. The properties of CG‐CNTRC are affected by its microstructure, especially the degree of CNT aggregation that is described by an aggregation coefficient. It is shown the degree of aggregation canseriously reduce the effective stiffness and frequency parameter. POLYM. COMPOS., 2013. © 2013 Society of Plastics Engineers

Characterizing elastic properties of carbon nanotube‐based composites by using an equivalent fiber

AbstractEffective elastic properties for carbon nanotube (CNT)‐reinforced composites are obtained through a variety of micromechanics techniques. An embedded CNT in a polymer matrix and its surrounding interphase is replaced with an equivalent fiber for predicting the mechanical properties of the CNT/polymer composite. Formulas to extract the effective material constants from solutions for the representative volume element under three loading cases are derived based on the elasticity theory. The effects of an interphase layer between the nanotubes and the polymer matrix as result of effective interphase layer are also investigated. Furthermore, this research is aimed at characterizing the elastic properties of CNTs‐reinforced composites using Eshelby–Mori–Tanaka approach based on an equivalent fiber. The variations of mechanical properties with tube radius, interphase thickness, and degree of aggregation are investigated. It is shown that the presence of aggregates has stronger impact than the interphase thickness on the effective modulus of the composite. This is because aggregates have significantly lower modulus than individual CNTs. POLYM. COMPOS., 2013 © 2013 Society of Plastics Engineers

Large deflection analysis of functionally graded carbon nanotube-reinforced composite plates by the element-free kp-Ritz method

A nonlinear analysis is presented for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates using the element-free kp-Ritz method. The nonlinear governing equations are developed to investigate problems related to small strains and moderate rotations, based on the first-order shear deformation plate theory and von Kármán strains. Two-dimensional displacement fields of the plates are approximated by a set of mesh-free kernel particle functions. Single-walled carbon nanotubes (SWCNTs) are selected as reinforcement and effective material properties of FG-CNTRC plates are assumed to be graded through the thickness direction and are estimated through an equivalent continuum model based on the Eshelby–Mori–Tanaka approach. For eliminating shear locking for a very thin plate, a stabilized conforming nodal integration scheme is employed to evaluate the system bending stiffness, and the membrane as well as shear terms are calculated by the direct nodal integration method. Numerical simulations are carried out to investigate effects of various parameters on nonlinear behaviors of FG-CNTRC plates and results for uniformly distributed (UD) CNTRC plates are provided for comparison. Numerical results indicate that carbon nanotube content by volume, plate width-to-thickness ratio, plate aspect ratio and boundary condition have pronounced effects on the nonlinear response of CNTRC plates.

Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method

This paper presents the buckling analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates under various in-plane mechanical loads, using the element-free kp-Ritz method. The first-order shear deformation plate theory is applied and a set of mesh-free kernel particle functions are used to approximate two-dimensional displacement fields. Effective properties of materials of the plates reinforced by single-walled carbon nanotubes (SWCNTs) are estimated through a micromechanical model based on either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. Comparison study and numerical simulations with various parameters are conducted to assess efficacy and accuracy of the present method for analysis of buckling of SWCNT-reinforced composite plates. Results demonstrate that the change of carbon nanotube volume fraction, plate width-to-thickness ratio, plate aspect ratio, loading condition and temperature have pronounced effects on buckling strength of CNTRC plates as well as the boundary condition.

Influence of graded agglomerated CNTs on vibration of CNT-reinforced annular sectorial plates resting on Pasternak foundation

The main objective of this research paper is to present 3-D elasticity solution for free vibration analysis of continuously graded CNT-reinforced (CGCNTR) annular sectorial plates resting on Pasternak elastic foundation. The elastic foundation is considered as a Pasternak model with adding a shear layer to the Winkler model. The volume fractions of randomly oriented agglomerated single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. An embedded CNT in a polymer matrix and its surrounding inter-phase is replaced with an equivalent fiber for predicting the mechanical properties of the carbon nanotube/polymer composite. To determine the effect of CNT agglomeration on the elastic properties of CNT-reinforced composites, a two-parameter micromechanics model of agglomeration is employed. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented straight CNTs. The 2-D generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the equations of motion and to implement the various boundary conditions. The fast rate of convergence of the method is shown and the results are compared against existing results in literature. Then, the influence of graded agglomerated CNTs, elastic foundation stiffness parameters and various geometrical parameters on the vibration characteristics of the annular sectorial plates resting on Pasternak elastic foundation reinforced by randomly oriented agglomerated CNTs is investigated.

Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes

In this paper, the mechanical buckling of a functionally graded nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs) subjected to uniaxial and biaxial in-plane loadings is investigated. The material properties of the nanocomposite plate are assumed to be graded in the thickness direction and vary continuously and smoothly according to two types of the symmetric carbon nanotubes volume fraction profiles. The material properties of SWCNT are determined according to molecular dynamics (MDs), and then the effective material properties at a point are estimated by either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. The equilibrium and stability equations are derived using the Mindlin plate theory considering the first-order shear deformation (FSDT) effect and variational approach. The results for nanocomposite plate with uniformly distributed CNTs, which is a special case in the present study, are compared with those of the symmetric profiles of the CNTs volume fraction. A numerical study is performed to investigate the influences of the different types of compressive in-plane loadings, CNTs volume fractions, various types of CNTs volume fraction profiles, geometrical parameters and different types of estimation of effective material properties on the critical mechanical buckling load of functionally graded nanocomposite plates.

Eshelby–Mori–Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels

In this paper, natural frequencies characteristics of a continuously graded carbon nanotube-reinforced (CGCNTR) cylindrical panels based on the Eshelby–Mori–Tanaka approach is considered. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The CGCNTR shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. The 2-D generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The novelty of the present work is to exploit Eshelby–Mori–Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various mid radius-to-thickness ratio, shell angle, length-to-mean radius ratio and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR cylindrical panels. The interesting and new results show that continuously graded oriented CNTs volume fractions can be utilized for the management of vibrational behavior of structures so that the frequency parameters of structures made of such material can be considerably improved than that of the nanocomposites reinforced with uniformly distributed CNTs.

Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load

This work deals with a study of the vibrational properties of functionally graded nanocomposite beams reinforced by randomly oriented straight single-walled carbon nanotubes (SWCNTs) under the actions of moving load. Timoshenko and Euler–Bernoulli beam theories are used to evaluate dynamic characteristics of the beam. The Eshelby–Mori–Tanaka approach based on an equivalent fiber is used to investigate the material properties of the beam. An embedded carbon nanotube in a polymer matrix and its surrounding inter-phase is replaced with an equivalent fiber for predicting the mechanical properties of the carbon nanotube/polymer composite. The primary contribution of the present work deals with the global elastic properties of nano-structured composite beams. The system of equations of motion is derived by using Hamilton’s principle under the assumptions of the Timoshenko beam theory. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. In order to evaluate time response of the system, Newmark method is also used. Numerical results are presented in both tabular and graphical forms to figure out the effects of various material distributions, carbon nanotube orientations, velocity of the moving load, shear deformation, slenderness ratios and boundary conditions on the dynamic characteristics of the beam. The results show that the above mentioned effects play very important role on the dynamic behavior of the beam and it is believed that new results are presented for dynamics of FG nano-structure beams under moving loads which are of interest to the scientific and engineering community in the area of FGM nano-structures.

Vibrations of carbon nanotube-reinforced composites

This work deals with a study of the vibrational properties of carbon nanotube-reinforced composites by employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach. The theory allows the calculation of the effective constitutive law of the elastic isotropic medium (matrix) with dispersed elastic inhomogeneities (carbon nanotubes). The devised computational approach is shown to yield predictions in good agreement with the experimentally obtained elastic moduli of composites reinforced with uniformly aligned single-walled carbon nanotubes (CNTs). The primary contribution of the present work deals with the global elastic modal properties of nano-structured composite plates. The investigated composite plates are made of a purely isotropic elastic hosting matrix of three different types (epoxy, rubber, and concrete) with embedded single-walled CNTs. The computations are carried out via a finite element (FE) discretization of the composite plates. The effects of the CNT alignment and volume fraction are studied in depth to assess how the modal properties are influenced both globally and locally. As a major outcome, the lowest natural frequencies of CNT-reinforced rubber composites are shown to increase up to 500 percent.