This paper presents an analytical solution procedure for the nonlinear postbuckling analysis of piezoelectric functionally graded carbon nanotubes reinforced composite (FG-CNTRC) cylindrical shells ...subjected to combined electro-thermal loadings, axial compression and lateral loads. The carbon nanotubes are assumed to be aligned and straight with uniform and functionally graded distributions in the thickness direction. The kinematics and constitutive relations are written on the basis of the classic theory and the von Kármán nonlinear strain–displacement relations of large deformation. Applying the Ritz energy approach, analytical solutions are proposed for the nonlinear critical axial load, lateral pressure as well as the load-shortening ratio of the piezoelectric FG-CNTRC shell. Numerical results are presented to study the effects of dimensional parameters, CNT volume fraction, distribution type of the reinforcement and piezoelectric thickness on the nonlinear buckling behavior of the piezoelectric nanocomposite shell. It is revealed that the carrying capacity of the structure increases as the shell is integrated by the piezoelectric layers and reinforced by higher CNT volume fraction. Furthermore, FGX- and FGO- CNTRC piezoelectric shells are indicated to have higher and lower carrying capacities compared to UD-CNTRC piezoelectric shells, respectively.
In this paper, the three dimensional (3D) nonlocal bending and vibration analyses of functionally graded (FG) nanoplates are presented using a novel numerical solution method which is called ...variational differential quadrature (VDQ) due to its numerical essence and the framework of implementation. Through this approach, a quadratic weak formulation of 3D nonlocal elasticity for the considered phenomena is presented. Two types of the distribution of functionally graded materials (FGMs) namely power law distribution and exponentially varied along the thickness of the plate are considered. The energy quadratic representation of the problems is first obtained based on the 3D theory of elasticity. A weak form of local governing equations is then derived from this representation by a variational approach. To incorporate the effects of small size into the local model, a size-dependent energy functional based on the nonlocal elasticity theory is developed. By introducing this functional into Hamilton’s principle, the discretized equations of motion including size effects are derived. By the VDQ method, the need for derivation of strong statement of the problems through minimizing the energy functional in the differential quadrature formulation is bypassed. In several numerical examples, the obtained results are compared with the available solutions in the literature, and the validity and high accuracy as well as fast convergence rate of the VDQ are indicated. It is also found that the small scale has a decreasing effect on the stiffness of nanoplates.
Superlative properties of nanocomposites have motivated considerable research efforts in recent years. Nanocomposite plates of quadrilateral shapes are important structural components used in a ...variety of engineering structures. This article aims to develop a variational formulation to describe the vibrational behavior of functionally graded (FG) nanocomposite straight-sided quadrilateral plates reinforced by carbon nanotubes (CNTs) in thermal environments. Various profiles of single-walled carbon nanotubes (SWCNTs) distribution along the thickness are taken into consideration. The mathematical formulation is developed in the variational form based on the first order shear defamation plate theory (FSDPT) with consideration of thermal effects. Discretization process of the energy functional is done on a computational domain using a mapping-differential quadrature (DQ) methodology. Discrete form of the governing equations is directly derived from a weak formulation which does not involve any transformation and discretization of the high order derivatives appeared in the equations of the strong form. Numerical results are given and compared with the ones reported in the literature to evaluate the convergence behavior and accuracy of the proposed solution. Subsequently, the influences of temperature on natural frequencies of the nanocomposite quadrilateral plates with different geometric parameters, CNT distributions in thickness direction and boundary conditions are investigated.
Based on the higher-order Cauchy–Born (HCB) rule, an atomistic-continuum multiscale model is proposed to address the large-amplitude vibration problem of graphene sheets (GSs) embedded in an elastic ...medium under various kinds of boundary conditions. By HCB, a linkage is established between the deformation of the atomic structure and macroscopical deformation gradients without any parameter fitting. The elastic foundation is formulated according to the Winkler–Pasternak model which considers both normal pressure and transverse shear stress effects. The weak form of nonlinear governing equations is derived via a variational approach, namely based on the variational differential quadrature (VDQ) method and Hamilton’s principle. In order to solve the obtained equations, a numerical scheme is adopted in which the generalized differential quadrature (GDQ) method together with a numerical Galerkin technique is utilized for discretization in the space domain, and the time-periodic discretization method is used to discretize in the time domain. The effects of the arrangement of atoms, the Winkler and Pasternak coefficients of the elastic foundation, and boundary conditions on the frequency–response curves of GSs are illustrated. It is revealed that the nonlinear effects on the response of GSs with larger size in armchair direction are less important.
This article presents analytical explicit frequency expressions for investigating the vibrations of single-layer graphene sheets (SLGSs). The interatomic potential is incorporated into a nonlocal ...continuum plate model through establishing a linkage between the strain energy density induced in the continuum and nonlocal plate constitutive relations. The model which is independent of scattered value of Young's modulus is then applied and explicit frequency formulas for the SLGSs with different edge conditions are derived using static deflection function of the nanoplate under uniformly distributed load. The reliability of the present formulation is verified by the results obtained by the molecular dynamics (MD) simulations and other research workers. The formulas are of a simple short form enabling quick and accurate evaluation of the frequency of the SLGSs and also simple calibration of scale coefficient by the use of MD simulations results.
•Explicit nonlocal frequency formulas based on interatomic potentials are derived for SLGSs with various edge supports.•The formulas not depending on Young's modulus enable simple calibration of scale coefficient using MD results.•The analytical expressions are shown to be capable of predicting vibrational response of SLGSs accurately.•The prominence of size effect in frequency of SLGS is found to be dependent on side length and boundary conditions.
The present study aims to examine the vibrational behavior of buckled Functionally Graded (FG) circular plates under clamped and simply-supported edge conditions. To this end, von Kármán assumptions ...were taken into account to incorporate geometric nonlinearity into Kirchhoff plate theory and derive the nonlinear governing equations of motion using Hamilton's principle. Critical buckling load and linear natural frequencies were first calculated using Generalized Differential Quadrature (GDQ) method. Next, the postbuckling characteristics of the circular plate were identified through direct solving of the nonlinear governing equations. Several comparative studies have confirmed the reliability of the proposed model. Finally, the fundamental natural frequency of the plate was evaluated for pre- and postbuckled configurations. This study also evaluated the effects of material property and boundary conditions on the static bifurcation diagram and natural frequency of the initial undeflected and buckled plate. According to the findings, the trend of the fundamental natural frequency changes with the applied radial load around the pre-buckled configuration differed from the one around the buckled configuration.
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In the present article, an atomistic-continuum multiscale model is developed to study the free-vibration response of single-layered graphene sheets (SLGSs) embedded in an elastic medium based upon ...the higher-order Cauchy-Born (HCB) rule. In order to take both transverse shear stress and normal pressure into account, the elastic foundation is considered to be of Winkler-Pasternak type. The governing equations are derived within a variational formulation using a newly proposed method called Variational Differential Quadrature (VDQ). Using the VDQ approach together with the Generalized Differential Quadrature (GDQ) technique, the variational form of the governing equation is discretized in a computationally efficient manner. Finally, a generalized eigenvalue problem is solved to calculate the frequencies of SLGSs. The convergence and correctness of the presented numerical solutions are examined firstly. Then, a number of numerical examples are given to study the effects of boundary conditions, elastic medium and arrangement of atoms on the vibrational response of SLGSs. The present model does not involve any additional phenomenological input, and it considers size effect and material nonlinearity due to atomic interactions.
Abstract
In this paper, a three-dimensional (3D) size-dependent formulation is developed for the free vibrations of functionally graded quadrilateral nanoplates subjected to thermal environment. The ...plate model is constructed within the frameworks of the Gurtin–Murdoch surface and the 3D elasticity theories. In this way, the effect of surface free energy and all the components of stress and strain tensors are considered without any initial assumption on them as there is no need to assume the variation of transverse normal stress inside the bulk material in advance. The variational differential quadrature approach and the mapping technique are applied to derive a discretized weak form of the governing equations. The present solution method bypasses the transformation and discretization of the higher order derivatives appearing in the equations of the strong form. The effects of surface stress, thermal environment, material gradient index and geometrical properties on the size-dependent vibrational behavior of quadrilateral nanoplates are investigated. It is observed that the thermal load intensifies the effect of surface free energy on the natural frequency of the nanoplates. The present model is exact in the extent of the continuum models and can be employed for structures with any thickness–span ratios.
In this paper, a non-classical plate model capturing the size effect is developed to study the forced vibration of functionally graded (FG) microplates subjected to a harmonic excitation transverse ...force. To this, the modified couple stress theory (MCST) is incorporated into the first-order shear deformation plate theory (FSDPT) to account for the size effect through one length scale parameter, only. Strong form of nonlinear governing equations and associated boundary conditions are obtained using Hamilton's principle. The solution process is implemented on two domains. The generalized differential quadrature (GDQ) method is first employed to discretize the governing equations on the space domain. A Galerkin-based scheme is then applied to extract a reduced set of the nonlinear equations of Duffing-type. On the second domain, through a time differentiation matrix operator, the set of ordinary differential equations are transformed into the discrete form on time domain. Eventually, a system of the parameterized nonlinear equations is acquired and solved via the pseudo-arc length continuation method. The frequency response curve of the microplate is sketched and the effects of various material and geometrical parameters on it are evaluated.
Due to the high surface to volume ratio of the nanoscale domain, the surface stress effect is a major concern in the analysis of mechanical response of the nanomaterials and nanostructures. This ...paper is concerned with the applicability of a continuum model including the surface properties for describing the bending and buckling configuration of the nanoscale plates. The Gurtin–Murdoch surface theory of elasticity is first incorporated into Mindlin’s plate theory. Then, the principle of virtual work is applied to derive the size-dependent governing equations along with various boundary conditions. To solve the governing equations, the generalized differential quadrature (GDQ) method is employed. The critical uniaxial and biaxial buckling loads and the maximum deflection of the nanoplate due to a uniform transverse load are calculated in the presence and absence of the surface effects for various edge conditions. It is found that the significance of the surface effects on the response of the nanoplate relies on its size, type of edge support and selected surface constants.
A Mindlin plate model is developed to describe the bending and buckling characteristics of nanoplates including the surface stress effect. Display omitted
•Development of a non-classical continuum model using Gurtin–Murdoch surface theory.•Study of bending and buckling behaviors of nanoplates with different boundary conditions.•Surface property is found to be effective in the surface effect on the response of nanoplates.•Nanoplates with stiffer edge conditions is observed to be less affected by the surface influences.
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